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Solving Word Problems by Adding and Subtracting Fractions and Mixed Numbers

Learn how to solve fraction word problems with examples and interactive exercises.

Example 1: Rachel rode her bike for one-fifth of a mile on Monday and two-fifths of a mile on Tuesday. How many miles did she ride altogether?

Analysis: To solve this problem, we will add two fractions with like denominators.

Solution: 

Answer: Rachel rode her bike for three-fifths of a mile altogether.

Analysis: To solve this problem, we will subtract two fractions with unlike denominators.

Answer: Stefanie swam one-third of a lap farther in the morning.

Analysis: To solve this problem, we will add three fractions with unlike denominators. Note that the first is an improper fraction.

Answer: It took Nick three and one-fourth hours to complete his homework altogether.

Analysis: To solve this problem, we will add two mixed numbers, with the fractional parts having like denominators.

Answer: Diego and his friends ate six pizzas in all.

Analysis: To solve this problem, we will subtract two mixed numbers, with the fractional parts having like denominators.

Answer: The Cocozzelli family took one-half more days to drive home.

Analysis: To solve this problem, we will add two mixed numbers, with the fractional parts having unlike denominators.

Answer: The warehouse has 21 and one-half meters of tape in all.

Analysis: To solve this problem, we will subtract two mixed numbers, with the fractional parts having unlike denominators.

Answer: The electrician needs to cut 13 sixteenths cm of wire.

Analysis: To solve this problem, we will subtract a mixed number from a whole number.

Answer: The carpenter needs to cut four and seven-twelfths feet of wood.

Summary: In this lesson we learned how to solve word problems involving addition and subtraction of fractions and mixed numbers. We used the following skills to solve these problems: 

• Add fractions with like denominators.
• Subtract fractions with like denominators.
• Find the LCD.
• Add fractions with unlike denominators.
• Subtract fractions with unlike denominators.
• Add mixed numbers with like denominators.
• Subtract mixed numbers with like denominators.
• Add mixed numbers with unlike denominators.
• Subtract mixed numbers with unlike denominators.

Directions: Subtract the mixed numbers in each exercise below.  Be sure to simplify your result, if necessary.  Click once in an ANSWER BOX and type in your answer; then click ENTER. After you click ENTER, a message will appear in the RESULTS BOX to indicate whether your answer is correct or incorrect. To start over, click CLEAR.

Note: To write the fraction three-fourths, enter 3/4 into the form. To write the mixed number four and two-thirds, enter 4, a space, and then 2/3 into the form.

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Fraction word prob.

Fraction word problems

Here you will learn about fraction word problems, including solving math word problems within a real-world context involving adding fractions, subtracting fractions, multiplying fractions, and dividing fractions.

Students will first learn about fraction word problems as part of number and operations—fractions in 4 th grade.

What are fraction word problems?

Fraction word problems are math word problems involving fractions that require students to use problem-solving skills within the context of a real-world situation.

To solve a fraction word problem, you must understand the context of the word problem, what the unknown information is, and what operation is needed to solve it. Fraction word problems may require addition, subtraction, multiplication, or division of fractions.

After determining what operation is needed to solve the problem, you can apply the rules of adding, subtracting, multiplying, or dividing fractions to find the solution.

For example,

Natalie is baking 2 different batches of cookies. One batch needs \cfrac{3}{4} cup of sugar and the other batch needs \cfrac{2}{4} cup of sugar. How much sugar is needed to bake both batches of cookies?

You can follow these steps to solve the problem:

Step-by-step guide: Adding and subtracting fractions

Step-by-step guide: Adding fractions

Step-by-step guide: Subtracting fractions

Step-by-step guide: Multiplying and dividing fractions

Step-by-step guide: Multiplying fractions

Step-by-step guide: Dividing fractions

Common Core State Standards

How does this relate to 4 th grade math to 6 th grade math?

• Grade 4: Number and Operations—Fractions (4.NF.B.3d) Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.
• Grade 4: Number and Operations—Fractions (4.NF.B.4c) Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat \cfrac{3}{8} of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?
• Grade 5: Number and Operations—Fractions (5.NF.A.2) Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result \cfrac{2}{5}+\cfrac{1}{2}=\cfrac{3}{7} by observing that \cfrac{3}{7}<\cfrac{1}{2} .
• Grade 5: Number and Operations—Fractions (5.NF.B.6) Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
• Grade 5: Number and Operations—Fractions (5.NF.B.7c) Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share \cfrac{1}{2} \: lb of chocolate equally? How many \cfrac{1}{3} cup servings are in 2 cups of raisins?
• Grade 6: The Number System (6.NS.A.1) Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for \cfrac{2}{3} \div \cfrac{4}{5} and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that \cfrac{2}{3} \div \cfrac{4}{5}=\cfrac{8}{9} because \cfrac{3}{4} of \cfrac{8}{9} is \cfrac{2}{3}. (In general, \cfrac{a}{b} \div \cfrac{c}{d}=\cfrac{a d}{b c} \, ) How much chocolate will each person get if 3 people share \cfrac{1}{2} \: lb of chocolate equally? How many \cfrac{3}{4} cup servings are in \cfrac{2}{3} of a cup of yogurt? How wide is a rectangular strip of land with length \cfrac{3}{4} \: m and area \cfrac{1}{2} \: m^2?

[FREE] Fraction Operations Worksheet (Grade 4 to 6)

Use this quiz to check your grade 4 to 6 students’ understanding of fraction operations. 10+ questions with answers covering a range of 4th to 6th grade fraction operations topics to identify areas of strength and support!

How to solve fraction word problems

In order to solve fraction word problems:

Determine what operation is needed to solve.

Write an equation.

Solve the equation.

State your answer in a sentence.

Fraction word problem examples

Example 1: adding fractions (like denominators).

Julia ate \cfrac{3}{8} of a pizza and her brother ate \cfrac{2}{8} of the same pizza. How much of the pizza did they eat altogether?

The problem states how much pizza Julia ate and how much her brother ate. You need to find how much pizza Julia and her brother ate altogether , which means you need to add.

2 Write an equation.

3 Solve the equation.

To add fractions with like denominators, add the numerators and keep the denominators the same.

4 State your answer in a sentence.

The last step is to go back to the word problem and write a sentence to clearly say what the solution represents in the context of the problem.

Julia and her brother ate \cfrac{5}{8} of the pizza altogether.

Example 2: adding fractions (unlike denominators)

Tim ran \cfrac{5}{6} of a mile in the morning and \cfrac{1}{3} of a mile in the afternoon. How far did Tim run in total?

The problem states how far Tim ran in the morning and how far he ran in the afternoon. You need to find how far Tim ran in total , which means you need to add.

To add fractions with unlike denominators, first find a common denominator and then change the fractions accordingly before adding.

\cfrac{5}{6}+\cfrac{1}{3}= \, ?

The least common multiple of 6 and 3 is 6, so 6 can be the common denominator.

That means \cfrac{1}{3} will need to be changed so that its denominator is 6. To do this, multiply the numerator and the denominator by 2.

\cfrac{1 \times 2}{3 \times 2}=\cfrac{2}{6}

Now you can add the fractions and simplify the answer.

\cfrac{5}{6}+\cfrac{2}{6}=\cfrac{7}{6}=1 \cfrac{1}{6}

Tim ran a total of 1 \cfrac{1}{6} miles.

Example 3: subtracting fractions (like denominators)

Pia walked \cfrac{4}{7} of a mile to the park and \cfrac{3}{7} of a mile back home. How much farther did she walk to the park than back home?

The problem states how far Pia walked to the park and how far she walked home. Since you need to find the difference ( how much farther ) between the two distances, you need to subtract.

To subtract fractions with like denominators, subtract the numerators and keep the denominators the same.

\cfrac{4}{7}-\cfrac{3}{7}=\cfrac{1}{7}

Pia walked \cfrac{1}{7} of a mile farther to the park than back home.

Example 4: subtracting fractions (unlike denominators)

Henry bought \cfrac{7}{8} pound of beef from the grocery store. He used \cfrac{1}{3} of a pound of beef to make a hamburger. How much of the beef does he have left?

The problem states how much beef Henry started with and how much he used. Since you need to find how much he has left , you need to subtract.

To subtract fractions with unlike denominators, first find a common denominator and then change the fractions accordingly before subtracting.

\cfrac{7}{8}-\cfrac{1}{3}= \, ?

The least common multiple of 8 and 3 is 24, so 24 can be the common denominator.

That means both fractions will need to be changed so that their denominator is 24.

To do this, multiply the numerator and the denominator of each fraction by the same number so that it results in a denominator of 24. This will give you an equivalent fraction for each fraction in the problem.

\begin{aligned}&\cfrac{7 \times 3}{8 \times 3}=\cfrac{21}{24} \\\\ &\cfrac{1 \times 8}{3 \times 8}=\cfrac{8}{24} \end{aligned}

Now you can subtract the fractions.

\cfrac{21}{24}-\cfrac{8}{24}=\cfrac{13}{24}

Henry has \cfrac{13}{24} of a pound of beef left.

Example 5: multiplying fractions

Andre has \cfrac{3}{4} of a candy bar left. He gives \cfrac{1}{2} of the remaining bit of the candy bar to his sister. What fraction of the whole candy bar does Andre have left now?

It could be challenging to determine the operation needed for this problem; many students may automatically assume it is subtraction since you need to find how much of the candy bar is left.

However, since you know Andre started with a fraction of the candy bar and you need to find a fraction OF a fraction, you need to multiply.

The difference here is that Andre did NOT give his sister \cfrac{1}{2} of the candy bar, but he gave her \cfrac{1}{2} of \cfrac{3}{4} of a candy bar.

To solve the word problem, you can ask, “What is \cfrac{1}{2} of \cfrac{3}{4}? ” and set up the equation accordingly. Think of the multiplication sign as meaning “of.”

\cfrac{1}{2} \times \cfrac{3}{4}= \, ?

To multiply fractions, multiply the numerators and multiply the denominators.

\cfrac{1}{2} \times \cfrac{3}{4}=\cfrac{3}{8}

Andre gave \cfrac{1}{2} of \cfrac{3}{4} of a candy bar to his sister, which means he has \cfrac{1}{2} of \cfrac{3}{4} left. Therefore, Andre has \cfrac{3}{8} of the whole candy bar left.

Example 6: dividing fractions

Nia has \cfrac{7}{8} cup of trail mix. How many \cfrac{1}{4} cup servings can she make?

The problem states the total amount of trail mix Nia has and asks how many servings can be made from it.

To solve, you need to divide the total amount of trail mix (which is \cfrac{7}{8} cup) by the amount in each serving ( \cfrac{1}{4} cup) to find out how many servings she can make.

To divide fractions, multiply the dividend by the reciprocal of the divisor.

\begin{aligned}& \cfrac{7}{8} \div \cfrac{1}{4}= \, ? \\\\ & \downarrow \downarrow \downarrow \\\\ &\cfrac{7}{8} \times \cfrac{4}{1}=\cfrac{28}{8} \end{aligned}

You can simplify \cfrac{28}{8} to \cfrac{7}{2} and then 3 \cfrac{1}{2}.

Nia can make 3 \cfrac{1}{2} cup servings.

Teaching tips for fraction word problems

• Encourage students to look for key words to help determine the operation needed to solve the problem. For example, subtracting fractions word problems might ask students to find “how much is left” or “how much more” one fraction is than another.
• Provide students with an answer key to word problem worksheets to allow them to obtain immediate feedback on their solutions. Encourage students to attempt the problems independently first, then check their answers against the key to identify any mistakes and learn from them. This helps reinforce problem-solving skills and confidence.
• Be sure to incorporate real-world situations into your math lessons. Doing so allows students to better understand the relevance of fractions in everyday life.
• As students progress and build a strong foundational understanding of one-step fraction word problems, provide them with multi-step word problems that involve more than one operation to solve.
• Take note that students will not divide a fraction by a fraction as shown above until 6 th grade (middle school), but they will divide a unit fraction by a whole number and a whole number by a fraction in 5 th grade (elementary school), where the same mathematical rules apply to solving.
• There are many alternatives you can use in place of printable math worksheets to make practicing fraction word problems more engaging. Some examples are online math games and digital workbooks.

Easy mistakes to make

• Misinterpreting the problem Misreading or misunderstanding the word problem can lead to solving for the wrong quantity or using the wrong operation.
• Not finding common denominators When adding or subtracting fractions with unlike denominators, students may forget to find a common denominator, leading to an incorrect answer.
• Forgetting to simplify Unless a problem specifically says not to simplify, fractional answers should always be written in simplest form.

Related fractions operations lessons

• Fractions operations
• Multiplicative inverse
• Reciprocal math
• Fractions as divisions

Practice fraction word problem questions

1. Malia spent \cfrac{5}{6} of an hour studying for a math test. Then she spent \cfrac{1}{3} of an hour reading. How much longer did she spend studying for her math test than reading?

Malia spent \cfrac{1}{2} of an hour longer studying for her math test than reading.

Malia spent \cfrac{5}{18} of an hour longer studying for her math test than reading.

Malia spent \cfrac{1}{2} of an hour longer reading than studying for her math test.

Malia spent 1 \cfrac{1}{6} of an hour longer studying for her math test than reading.

To find the difference between the amount of time Malia spent studying for her math test than reading, you need to subtract. Since the fractions have unlike denominators, you need to find a common denominator first.

You can use 6 as the common denominator, so \cfrac{1}{3} becomes \cfrac{3}{6}. Then you can subtract.

\cfrac{3}{6} can then be simplified to \cfrac{1}{2}.

Finally, you need to choose the answer that correctly answers the question within the context of the situation. Therefore, the correct answer is “Malia spent \cfrac{1}{2} of an hour longer studying for her math test than reading.”

2. A square garden is \cfrac{3}{4} of a meter wide and \cfrac{8}{9} of a meter long. What is its area?

The area of the garden is 1\cfrac{23}{36} square meters.

The area of the garden is \cfrac{27}{32} square meters.

The area of the garden is \cfrac{2}{3} square meters.

The perimeter of the garden is \cfrac{2}{3} meters.

To find the area of a square, you multiply the length and width. So to solve, you multiply the fractional lengths by mulitplying the numerators and multiplying the denominators.

\cfrac{24}{36} can be simplified to \cfrac{2}{3}. 

Therefore, the correct answer is “The area of the garden is \cfrac{2}{3} square meters.”

3. Zoe ate \cfrac{3}{8} of a small cake. Liam ate \cfrac{1}{8} of the same cake. How much more of the cake did Zoe eat than Liam?

Zoe ate \cfrac{3}{64} more of the cake than Liam.

Zoe ate \cfrac{1}{4} more of the cake than Liam.

Zoe ate \cfrac{1}{8} more of the cake than Liam.

Liam ate \cfrac{1}{4} more of the cake than Zoe.

To find how much more cake Zoe ate than Liam, you subtract. Since the fractions have the same denominator, you subtract the numerators and keep the denominator the same.

\cfrac{2}{8} can be simplified to \cfrac{1}{4}. 

Therefore, the correct answer is “Zoe ate \cfrac{1}{4} more of the cake than Liam.”

4. Lila poured \cfrac{11}{12} cup of pineapple and \cfrac{2}{3} cup of mango juice in a bottle. How many cups of juice did she pour into the bottle altogether?

Lila poured 1 \cfrac{7}{12} cups of juice in the bottle altogether.

Lila poured \cfrac{1}{4} cups of juice in the bottle altogether.

Lila poured \cfrac{11}{18} cups of juice in the bottle altogether.

Lila poured 1 \cfrac{3}{8} cups of juice in the bottle altogether.

To find the total amount of juice that Lila poured into the bottle, you need to add. Since the fractions have unlike denominators, you need to find a common denominator first.

You can use 12 as the common denominator, so \cfrac{2}{3} becomes \cfrac{8}{12}.  Then you can add.

\cfrac{19}{12} can be simplified to 1 \cfrac{7}{12}. 

Therefore, the correct answer is “Lila poured 1 \cfrac{7}{12} cups of juice in the bottle altogether.”

5. Killian used \cfrac{9}{10} of a gallon of paint to paint his living room and \cfrac{7}{10} of a gallon to paint his bedroom. How much paint did Killian use in all?

Killian used \cfrac{2}{10} gallons of paint in all.

Killian used \cfrac{1}{5} gallons of paint in all.

Killian used \cfrac{63}{100} gallons of paint in all.

Killian used 1 \cfrac{3}{5} gallons of paint in all.

To find the total amount of paint Killian used, you add the amount he used for the living room and the amount he used for the kitchen. Since the fractions have the same denominator, you add the numerators and keep the denominators the same.

\cfrac{16}{10} can be simplified to 1 \cfrac{6}{10} and then further simplified to 1 \cfrac{3}{5}.

Therefore, the correct answer is “Killian used 1 \cfrac{3}{5} gallons of paint in all.”

6. Evan pours \cfrac{4}{5} of a liter of orange juice evenly among some cups.

He put \cfrac{1}{10} of a liter into each cup. How many cups did Evan fill?

Evan filled \cfrac{2}{25} cups.

Evan filled 8 cups.

Evan filled \cfrac{9}{10} cups.

Evan filled 7 cups.

To find the number of cups Evan filled, you need to divide the total amount of orange juice by the amount being poured into each cup. To divide fractions, you mulitply the first fraction (the dividend) by the reciprocal of the second fraction (the divisor).

\cfrac{40}{5} can be simplifed to 8.

Therefore, the correct answer is “Evan filled 8 cups.”

Fraction word problems FAQs

Fraction word problems are math word problems involving fractions that require students to use problem-solving skills within the context of a real-world situation. Fraction word problems may involve addition, subtraction, multiplication, or division of fractions.

To solve fraction word problems, first you need to determine the operation. Then you can write an equation and solve the equation based on the arithmetic rules for that operation.

Fraction word problems and decimal word problems are similar because they both involve solving math problems within real-world contexts. Both types of problems require understanding the problem, determining the operation needed to solve it (addition, subtraction, multiplication, division), and solving it based on the arithmetic rules for that operation.

The next lessons are

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Adding Fractions

Learn about adding fractions., adding fractions lesson, how to add fractions.

To add fractions, we follow three simple steps. They are as follows:

• Make the denominators the same if they aren't already.
• Add the numerators, keeping the denominator the same.
• Simplify the resulting fraction.

The same three steps apply for adding mixed fractions (such as 4 1 / 2 + 1 2 / 3 ) except that we will simply add the whole number and fraction components separately.

In this lesson we will go through how to add fractions and show examples of adding fractions with like and unlike denominators.

Adding Fractions with Like Denominators

Let's go through how to add fractions with like denominators first, since it is most simple type of fraction addition. Here's an example of adding fractions with like denominators, using the three steps from earlier.

Find the sum of 3 / 5 + 1 / 5 .

• The denominators are already the same, so we can skip step 1.
• Let's add the numerators. 3 + 1 = 4, so the sum of our numerators is 4. The denominator is still 5, so our result is 4 / 5 .
• 4 / 5 is already in its simplest form, so there is no simplifying needed here.

The solution is 3 / 5 + 1 / 5 = 4 / 5 .

Adding Fractions with Unlike Denominators

Now let's go through another example but this time with unlike denominators. We will use the same exact three steps.

Find the sum of 1 / 4 + 2 / 3 .

• Let's find the lowest common denominator and convert these fractions to like denominators to make them addable. Multiplying the top and bottom of each fraction by the other fraction's denominator gives us 1 / 4  ·  3 / 3 = 3 / 12 and 2 / 3  ·  4 / 4 = 8 / 12 .
• Now let's add the numerators. 3 + 8 = 11, so the sum of our numerators is 11. The denominator is still 12, so our result is 11 / 12 .
• 11 / 12 is already in its simplest form, so there is no simplifying needed here.

The solution is 1 / 4 + 2 / 3 = 11 / 12 .

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Adding Fractions

A fraction like 3 4 says we have 3 out of the 4 parts the whole is divided into.

To add fractions there are Three Simple Steps:

• Step 1: Make sure the bottom numbers (the denominators ) are the same
• Step 2: Add the top numbers (the numerators ), put that answer over the denominator
• Step 3: Simplify the fraction (if possible)

Step 1 . The bottom numbers (the denominators) are already the same. Go straight to step 2.

Step 2 . Add the top numbers and put the answer over the same denominator:

1 4 + 1 4 = 1 + 1 4 = 2 4

Step 3 . Simplify the fraction:

In picture form it looks like this:

... and do you see how 2 4 is simpler as 1 2 ? (see Equivalent Fractions .)

Step 1 : The bottom numbers are different. See how the slices are different sizes?

We need to make them the same before we can continue, because we can't add them like that.

The number "6" is twice as big as "3", so to make the bottom numbers the same we can multiply the top and bottom of the first fraction by 2 , like this:

Important: you multiply both top and bottom by the same amount, to keep the value of the fraction the same

Now the fractions have the same bottom number ("6"), and our question looks like this:

The bottom numbers are now the same, so we can go to step 2.

Step 2 : Add the top numbers and put them over the same denominator:

2 6 + 1 6 = 2 + 1 6 = 3 6

Step 3 : Simplify the fraction:

In picture form the whole answer looks like this:

With Pen and Paper

And here is how to do it with a pen and paper (press the play button):

A Rhyme To Help You Remember

♫ "If adding or subtracting is your aim, The bottom numbers must be the same! ♫ "Change the bottom using multiply or divide, But the same to the top must be applied, ♫ "And don't forget to simplify, Before its time to say good bye"

Again, the bottom numbers are different (the slices are different sizes)!

But let us try dividing them into smaller sizes that will each be the same :

The first fraction: by multiplying the top and bottom by 5 we ended up with 5 15 :

The second fraction: by multiplying the top and bottom by 3 we ended up with 3 15 :

The bottom numbers are now the same, so we can go ahead and add the top numbers:

The result is already as simple as it can be, so that is the answer: 

1 3 + 1 5 = 8 15

Making the Denominators the Same

In the previous example how did we know to cut them into 1 / 15 ths to make the denominators the same? We simply multiplied the two denominators together (3 × 5 = 15).

Read about the two main ways to make the denominators the same here:

• Common Denominator Method , or the
• Least Common Denominator Method

They both work, use which one you prefer!

Example: Cupcakes

You want to make and sell cupcakes:

• A friend can supply the ingredients, if you give them 1 / 3 of sales
• And a market stall costs 1 / 4 of sales

How much is that altogether?

We need to add 1 / 3 and 1 / 4

First make the bottom numbers (the denominators) the same.

Multiply top and bottom of 1 / 3 by 4 :

And multiply top and bottom of 1 / 4 by 3 :

Now do the calculations:

Answer: 7 12 of sales go in ingredients and market costs.

Adding Mixed Fractions

We have a special (more advanced) page on Adding Mixed Fractions .

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Course: 4th grade   >   Unit 8

• Fraction word problem: piano
• Fraction word problem: pizza
• Fraction word problem: spider eyes

Add and subtract fractions word problems (same denominator)

• Your answer should be
• a proper fraction, like 1 / 2 ‍   or 6 / 10 ‍  
• an improper fraction, like 10 / 7 ‍   or 14 / 8 ‍  

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Fractions  - Adding and Subtracting Fractions

Fractions  -, adding and subtracting fractions, fractions adding and subtracting fractions.

Fractions: Adding and Subtracting Fractions

Lesson 3: adding and subtracting fractions.

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Adding and subtracting fractions

In the previous lessons, you learned that a fraction is part of a whole. Fractions show how much you have of something, like 1/2 of a tank of gas or 1/3 of a cup of water.

In real life, you might need to add or subtract fractions. For example, have you ever walked 1/2 of a mile to work and then walked another 1/2 mile back? Or drained 1/4 of a quart of gas from a gas tank that had 3/4 of a quart in it? You probably didn't think about it at the time, but these are examples of adding and subtracting fractions.

Click through the slideshow to learn how to set up addition and subtraction problems with fractions.

Let's imagine that a cake recipe tells you to add 3/5 of a cup of oil to the batter.

You also need 1/5 of a cup of oil to grease the pan. To see how much oil you'll need total, you can add these fractions together.

When you add fractions, you just add the top numbers, or numerators .

That's because the bottom numbers, or denominators , show how many parts would make a whole.

We don't want to change how many parts make a whole cup ( 5 ). We just want to find out how many parts we need total.

So we only need to add the numerators of our fractions.

We can stack the fractions so the numerators are lined up. This will make it easier to add them.

And that's all we have to do to set up an addition example with fractions. Our fractions are now ready to be added.

We'll do the same thing to set up a subtraction example. Let's say you had 3/4 of a tank of gas when you got to work.

If you use 1/4 of a tank to drive home, how much will you have left? We can subtract these fractions to find out.

Just like when we added, we'll stack our fractions to keep the numerators lined up.

This is because we want to subtract 1 part from 3 parts.

Now that our example is set up, we're ready to subtract!

Try setting up these addition and subtraction problems with fractions. Don't try solving them yet!

You run 4/10 of a mile in the morning. Later, you run for 3/10 of a mile.

You had 7/8 of a stick of butter and used 2/8 of the stick while cooking dinner.

Your gas tank is 2/5 full, and you put in another 2/5 of a tank.

Solving addition problems with fractions

Now that we know how to write addition problems with fractions, let's practice solving a few. If you can add whole numbers , you're ready to add fractions.

Click through the slideshow to learn how to add fractions.

Let's continue with our previous example and add these fractions: 3/5 of cup of oil and 1/5 of a cup of oil.

Remember, when we add fractions, we don't add the denominators.

This is because we're finding how many parts we need total. The numerators show the parts we need, so we'll add 3 and 1 .

3 plus 1 equals 4 . Make sure to line up the 4 with the numbers you just added.

The denominators will stay the same, so we'll write 5 on the bottom of our new fraction.

3/5 plus 1/5 equals 4/5 . So you'll need 4/5 of a cup of oil total to make your cake.

Let's try another example: 7/10 plus 2/10 .

Just like before, we're only going to add the numerators. In this example, the numerators are 7 and 2 .

7 plus 2 equals 9 , so we'll write that to the right of the numerators.

Just like in our earlier example, the denominator stays the same.

So 7/10 plus 2/10 equals 9/10 .

Try solving some of the addition problems below.

Solving subtraction problems with fractions

Subtracting fractions is a lot like regular subtraction. If you can subtract whole numbers , you can subtract fractions too!

Click through the slideshow to learn how to subtract fractions.

Let's use our earlier example and subtract 1/4 of a tank of gas from 3/4 of a tank.

Just like in addition, we're not going to change the denominators.

We don't want to change how many parts make a whole tank of gas. We just want to know how many parts we'll have left.

We'll start by subtracting the numerators. 3 minus 1 equals 2 , so we'll write 2 to the right of the numerators.

Just like when we added, the denominator of our answer will be the same as the other denominators.

So 3/4 minus 1/4 equals 2/4 . You'll have 2/4 of a tank of gas left when you get home.

Let's try solving another problem: 5/6 minus 3/6 .

We'll start by subtracting the numerators.

5 minus 3 equals 2 . So we'll put a 2 to the right of the numerators.

As usual, the denominator stays the same.

So 5/6 minus 3/6 equals 2/6 .

Try solving some of the subtraction problems below.

After you add or subtract fractions, you may sometimes have a fraction that can be reduced to a simpler fraction. As you learned in Comparing and Reducing Fractions , it's always best to reduce a fraction to its simplest form when you can. For example, 1/4 plus 1/4 equals 2/4 . Because 2 and 4 can both be divided 2 , we can reduce 2/4 to 1/2 .

Adding fractions with different denominators

On the last page, we learned how to add fractions that have the same denominator, like 1/4 and 3/4 . But what if you needed to add fractions with different denominators? For example, our cake recipe might say to blend 1/4 cup of milk in slowly and then dump in another 1/3 of a cup.

In Comparing and Reducing Fractions , we compared fractions with a different bottom number, or denominator. We had to change the fractions so their denominators were the same. To do that, we found the lowest common denominator , or LCD .

We can only add or subtract fractions if they have the same denominators. So we'll need to find the lowest common denominator before we add or subtract these fractions. Once the fractions have the same denominator, we can add or subtract as usual.

Click through the slideshow to learn how to add fractions with different denominators.

Let's add 1/4 and 1/3 .

Before we can add these fractions, we'll need to change them so they have the same denominator .

To do that, we'll have to find the LCD , or lowest common denominator, of 4 and 3 .

It looks like 12 is the smallest number that can be divided by both 3 and 4, so 12 is our LCD .

Since 12 is the LCD, it will be the new denominator for our fractions.

Now we'll change the numerators of the fractions, just like we changed the denominators.

First, let's look at the fraction on the left: 1/4 .

To change 4 into 12 , we multiplied it by 3 .

Since the denominator was multiplied by 3 , we'll also multiply the numerator by 3 .

1 times 3 equals 3 .

1/4 is equal to 3/12 .

Now let's look at the fraction on the right: 1/3 . We changed its denominator to 12 as well.

Our old denominator was 3 . We multiplied it by 4 to get 12.

We'll also multiply the numerator by 4 . 1 times 4 equals 4 .

So 1/3 is equal to 4/12 .

Now that our fractions have the same denominator, we can add them like we normally do.

3 plus 4 equals 7 . As usual, the denominator stays the same. So 3/12 plus 4/12 equals 7/12 .

Try solving the addition problems below.

Subtracting fractions with different denominators

We just saw that fractions can only be added when they have the same denominator. The same thing is true when we're subtracting fractions. Before we can subtract, we'll have to change our fractions so they have the same denominator.

Click through the slideshow to learn how to subtract fractions with different denominators.

Let's try subtracting 1/3 from 3/5 .

First, we'll change the denominators of both fractions to be the same by finding the lowest common denominator .

It looks like 15 is the smallest number that can be divided evenly by 3 and 5 , so 15 is our LCD.

Now we'll change our first fraction. To change the denominator to 15 , we'll multiply the denominator and the numerator by 3 .

5 times 3 equals 15 . So our fraction is now 9/15 .

Now let's change the second fraction. To change the denominator to 15 , we'll multiply both numbers by 5 to get 5/15 .

Now that our fractions have the same denominator, we can subtract like we normally do.

9 minus 5 equals 4 . As always, the denominator stays the same. So 9/15 minus 5/15 equals 4/15 .

Try solving the subtraction problems below.

Adding and subtracting mixed numbers

Over the last few pages, you've practiced adding and subtracting different kinds of fractions. But some problems will need one extra step. For example, can you add the fractions below?

In Introduction to Fractions , you learned about mixed numbers . A mixed number has both a fraction and a whole number . An example is 2 1/2 , or two-and-a-half . Another way to write this would be 5/2 , or five-halves . These two numbers look different, but they're actually the same.

5/2 is an improper fraction . This just means the top number is larger than the bottom number. Even though improper fractions look strange, you can add and subtract them just like normal fractions. Mixed numbers aren't easy to add, so you'll have to convert them into improper fractions first.

Let's add these two mixed numbers: 2 3/5 and 1 3/5 .

We'll need to convert these mixed numbers to improper fractions. Let's start with 2 3/5 .

As you learned in Lesson 2 , we'll multiply the whole number, 2 , by the bottom number, 5 .

2 times 5 equals 10 .

Now, let's add 10 to the numerator, 3 .

10 + 3 equals 13 .

Just like when you add fractions, the denominator stays the same. Our improper fraction is 13/5 .

Now we'll need to convert our second mixed number: 1 3/5 .

First, we'll multiply the whole number by the denominator. 1 x 5 = 5 .

Next, we'll add 5 to the numerators. 5 + 3 = 8 .

Just like last time, the denominator remains the same. So we've changed 1 3/5 to 8/5 .

Now that we've changed our mixed numbers to improper fractions, we can add like we normally do.

13 plus 8 equals 21 . As usual, the denominator will stay the same. So 13/5 + 8/5 = 21/5 .

Because we started with a mixed number, let's convert this improper fraction back into a mixed number.

As you learned in the previous lesson , divide the top number by the bottom number. 21 divided by 5 equals 4, with a remainder of 1 .

The answer, 4, will become our whole number.

And the remainder , 1, will become the numerator of the fraction.

So 2 3/5 + 1 3/5 = 4 1/5 .

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Fraction Addition Word Problems Worksheets

Let children work on our printable adding fractions word problems worksheets hammer and tongs! Whether it's sharing a meal with your friends or measuring the ingredients for a recipe, adding fractions is at the heart of it all, hence our worksheets. A wealth of real-life scenarios that involve addition of fractions with whole numbers and addition of two like fractions, two unlike fractions, and two mixed numbers, our pdf worksheets are indispensable for grade 3, grade 4, grade 5, and grade 6 students. The free fraction addition word problems worksheet is worth a try!

Adding Fractions with Whole Numbers

Dazzle 3rd grade kids with a gift of lifelike story problems! If you're a novice up against fraction addition, don't miss our pdf adding fractions word problems worksheets using whole numbers and fractions!

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Adding Like Fractions Word Problems

Gerald ate 5/9 of an apple, and Garry ate 4/9 of it. How many apples did they eat in all? Good going! They both ate one whole apple. Simply combine the numerators and solve the like fraction word problems here!

Adding Unlike Fractions

A potpourri of word problems that involve adding unlike fractions, these pdfs mean that 4th grade and 5th grade students will breeze through addition of fractions with different denominators in their day-to-day lives.

Adding Mixed Numbers | Same Denominators

See in your mind's eye adding mixed numbers with same denominators riding on the several real-life scenarios in our printable worksheets! Convert the mixed numbers to fractions, and add them as usual.

Adding Mixed Numbers | Different Denominators

Steal a march on your 5th grade and 6th grade peers by getting your act together to power through the real-world situations featured in this set of printable adding mixed numbers word problems worksheets.

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Adding & subtracting fractions word problems

Word problem worksheets: addition & subtraction of fractions.

Below are three versions of our grade 4 math worksheet on adding and subtracting fractions and mixed numbers.  All fractions have like denominators.  Some problems will include irrelevant data so that students have to read and understand the questions, rather than simply recognizing a pattern to the solutions.  These worksheets are pdf files .

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Addition and Subtraction of Fraction: Methods, Examples, Facts, FAQs

What is addition and subtraction of fractions, methods of addition and subtraction of fractions, addition and subtraction of mixed numbers, solved examples on addition and subtraction of fractions, practice problems on addition and subtraction of fractions, frequently asked questions on addition and subtraction of fractions.

Addition and subtraction of fractions are the fundamental operations on fractions that can be studied easily using two cases:

• Addition and subtraction of like fractions (fractions with same denominators)
• Addition and subtraction of unlike fractions (fractions with different denominators)

A fraction represents parts of a whole. For example, the fraction 37 represents 3 parts out of 7 equal parts of a whole. Here, 3 is the numerator and it represents the number of parts taken. 7 is the denominator and it represents the total number of parts of the whole.

Adding and subtracting fractions is simple and straightforward when it comes to like fractions. In the case of unlike fractions, we first need to make the denominators the same. Let’s take a closer look at both these cases.

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Before adding and subtracting fractions, we first need to make sure that the fractions have the same denominators. 

When the denominators are the same, we simply add the numerators and keep the denominator as it is. To add or subtract unlike fractions, we first need to learn how to make the denominators alike. Let’s learn how to add fractions and how to subtract fractions in both cases.

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Addition and Subtraction of Like Fractions

The rules for adding fractions with the same denominator are really simple and straightforward. 

Let’s learn with the help of examples and visual bar models.

Addition of Like Fractions

Here are the steps to add fractions with the same denominator:

Step 1: Add the numerators of the given fractions. 

Step 2: Keep the denominator the same. 

Step 3: Simplify.          

$\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}$  …$c \neq 0$

Example 1: Find $\frac{1}{4} + \frac{2}{4}$ .

$\frac{1}{4} + \frac{2}{4} = \frac{1 + 2}{4} = \frac{3}{4}$

We can visualize this addition using a bar model:

Example 2: $\frac{1}{8} + \frac{3}{8} = \frac{1 + 3}{8} = \frac{4}{8} = \frac{1}{2}$

Subtraction of Like Fractions

Here are the steps to subtract fractions with the same denominator:

Step 1: Subtract the numerators of the given fractions. 

Step 3: Simplify. 

$\frac{a}{c}\;-\;\frac{b}{c} = \frac{a \;-\; b}{c}$ …$c \neq 0$

Example 1: Find $\frac{4}{6} \;-\; \frac{1}{6}$.

$\frac{4}{6}\;-\;\frac{1}{6} = \frac{4-1}{6} = \frac{3}{6} = \frac{1}{2}$

Addition and Subtraction of Unlike Fractions

Addition and subtraction of fractions with unlike denominators can be a little bit tricky since the denominators are not the same. So, we need to first convert the unlike fractions into like fractions. Let’s look at a few ways to do this!

Addition of Unlike Fractions

We can make the denominators the same by finding the LCM of the two denominators. Once we calculate the LCM, we multiply both the numerator and the denominator with an appropriate number so that we get the LCM value in the denominator. 

Example: $\frac{3}{5} + \frac{3}{2}$

Step 1: Find the LCM (Least Common Multiple) of the two denominators.

The LCM of 5 and 2 is 10.

Step 2: Convert both the fractions into like fractions by making the denominators same.  

$\frac{3 \times 2}{5 \times 2} = \frac{6}{10}$  

$\frac{3 \times 5}{2 \times 5} = \frac{15}{10}$

Step 3: Add the numerators. The denominator stays the same.

$\frac{6}{10} + \frac{15}{10} = \frac{21}{10}$

Step 4: Convert the resultant fraction to its simplest form if the GCF of the numerator and denominator is not 1. 

In this case, GCF (21,10) $= 1$

The fraction $\frac{21}{10}$ is already in its simplest form. 

Thus, $\frac{3}{5} + \frac{3}{2} = \frac{21}{10}$

Subtraction of Unlike Fractions

Let’s learn how to subtract fractions when denominators are not the same. To subtract unlike fractions, we use the LCM method. The process is similar to what we discussed in the previous example.

Example: $\frac{5}{6} \;-\; \frac{2}{9}$

Step 1: Find the LCM of the two denominators.

LCM of 6 and $9 = 18$

Step 2: Convert both the fractions into like fractions by making the denominators same.

$\frac{5 \times 3}{6 \times 3} = \frac{15}{18}$   

$\frac{2 \times 2}{9 \times 2} = \frac{4}{18}$

Step 3: Subtract the numerators. The denominator stays the same.

$\frac{15}{18} \;-\; \frac{4}{18} = \frac{11}{18}$

In this case, the GCF (11,18) $= 1$

So, it is already in its simplest form. 

Thus, $\frac{5}{6}\;-\; 29 = \frac{11}{18}$

A mixed number is a type of fraction that has two parts: a whole number and a proper fraction. It is also known as a mixed fraction. Any mixed number can be written in the form of an improper fraction and vice-versa. 

Adding and subtracting mixed fractions is done by converting mixed numbers into improper fractions .

Addition and Subtraction of Mixed Fractions with Same Denominators

The steps of adding and subtracting mixed numbers with the same denominators are the same. The only difference is the operation.

Step 1: Convert the given mixed fractions to improper fractions.

Step 2: Add/Subtract the like fractions obtained in step 1.

Step 3: Reduce the fraction to its simplest form.

Step 4: Convert the resulting fraction into a mixed number.

Example 1: $2\frac{1}{5} + 1\frac{3}{5}$

$2\frac{1}{5} = \frac{(5 \times 2) + 1}{5} = \frac{11}{5}$

$1\frac{3}{5} = \frac{(5 \times 1) + 3}{5} = \frac{8}{5}$

Thus, $2\frac{1}{5} + 1\frac{3}{5} = \frac{11}{5} + \frac{8}{5} = \frac{19}{5}$

Converting $\frac{19}{5}$ into a mixed number, we get

$\frac{19}{5} = 3\frac{4}{5}$

Example 2: $2\frac{1}{5} + 1\frac{3}{5} = \frac{11}{5} \;-\; \frac{8}{5} = \frac{3}{5}$

Addition and Subtraction of Mixed Fractions with Unlike Denominators

Step 2: Convert both the fractions into like fractions by finding the least common denominator.

Step 3: Add the fractions. (or subtract the fractions.)

Step 4: Reduce the fraction if possible or convert back to a mixed number 

Let us understand the addition of mixed numbers with unlike denominators with the help of an example.

Example 1: Find the value of $1\frac{3}{5} + 2\frac{1}{2}$.

Convert the given mixed fractions to improper fractions.

$1\frac{3}{5} = \frac{8}{5}$ and $2\frac{1}{2} = \frac{5}{2}$

Step 2: Convert both the fractions into like fractions by making the denominators the same.

Here, LCM of 5 and 2 is 10.

Thus, $\frac{8 \times 2}{5 \times 2} = \frac{16}{10}$ and $\frac{5\times 5}{2 \times 5} = \frac{25}{10}$

Step 3: Add the fractions by adding the numerators.

$\frac{16}{10} + \frac{25}{10} = \frac{41}{10}$

Step 4: Convert back into a mixed number. 

Thus, $\frac{41}{10}$ will become  $4\frac{1}{10}$

Therefore, $1\frac{3}{5} + 2\frac{1}{2} =  4\frac{1}{10}$

Here’s an example for subtraction. It follows the same steps.

Example 2 : $6\frac{1}{2} \;-\; 1\frac{3}{4}$

Step 1: Convert the mixed numbers into improper fractions.

     $6\frac{1}{2} \;-\; 1\frac{3}{4} = \frac{13}{2} \;-\; \frac{7}{4}$

Step 2: Make the denominators equal.

LCM of 2 and 4 is 4. 

   $\frac{13 \times 2}{2 \times 2} = \frac{26}{4}$ 

Step 3: Subtract the fractions.

        $\frac{26}{4} \;-\;  \frac{7}{4} = \frac{19}{4}$

Step 4: Convert the fraction as a mixed number.

            $\frac{19}{4}  = 4\frac{3}{4}$  

Thus, $6\frac{1}{2} \;-\; 1\frac{3}{4}  =   4\frac{3}{4}$  

Facts about Addition and Subtraction of Fractions

• We cannot add or subtract fractions without converting them into like fractions.
• Like fractions are fractions that have the same denominator, and unlike fractions are fractions that have different denominators.
• Equivalent fractions are two different fractions that represent the same value.
• The LCD (least common denominator) of two fractions is the LCM of the denominators.

In this article, we have learned about addition and subtraction of fractions (like fractions, unlike fractions, mixed fractions), methods of addition and subtraction of these fractions along with the steps. Let’s solve some examples on adding and subtracting fractions to understand the concept better.

• Solve: $\frac{2}{4} + \frac{1}{4}$ .

Solution: 

Here, the denominators are the same.

Thus, we add the numerators by keeping the denominators as it is.

$\frac{2}{4} + \frac{1}{4} = \frac{2 + 1}{4}$ 

$\frac{2}{4} + \frac{1}{4} = \frac{3}{4}$

2. Find the sum of the fractions $\frac{3}{5}$ and $\frac{5}{2}$ by using the LCM method.

$\frac{3}{5}$ and $\frac{5}{2}$ are unlike fractions.

The LCM of 2 and 5 is 10.

Thus, we can write

$\frac{3}{5} + \frac{5}{2} = \frac{3 \times 2}{5 \times 2} + \frac{5 \times 5}{2 \times 5}$

$= \frac{6}{10} + \frac{25}{10}$

            $= \frac{6}{10} + \frac{25}{10}$

            $= \frac{31}{10}$

Thus, $\frac{3}{5} + \frac{5}{2} =  \frac{31}{10}$

3. Find $\frac{4}{16} + \frac{5}{8}$.

Solution:  

To add two fractions with different denominators, we first need to find the LCM of the denominators.

The LCM of 16 and 8 is 16.

$\frac{4}{16} + \frac{5}{8} = \frac{4 \times 1}{16\times 1} + \frac{5 \times 2}{8 \times 2}$ 

            $= \frac{10}{16} + \frac{4}{16}$ 

            $= \frac{14}{16}$

$= \frac{7}{8}$

4. From a rope $12\frac{1}{2}$ ft. long, a $7 \frac{6}{8}\;-$ ft-long piece is cut off. Find the length of the remaining rope.

Total length of the rope $= 12\frac{1}{2}$ ft.

Length of the rope that was cut off $= 7 \frac{6}{8}$ ft. 

The length of the remaining rope $= 12\frac{1}{2} \;-\; 7 \frac{6}{8}$

$12\frac{1}{2} \;-\; 7 \frac{6}{8} = \frac{25}{2} \;-\; \frac{62}{8}$

         $= \frac{25 \times 4}{2 \times 4} \;-\; \frac{62 \times 1}{8\times 1}$

         $= \frac{100}{8} \;-\; \frac{62}{8}$

         $= \frac{38}{8}$

         $= \frac{19}{4}$

Converting it into a mixed fraction, $\frac{19}{4}$ becomes $4 \frac{3}{4}$.

Thus, the length of the remaining rope is $4\frac{3}{4}$ ft.

Attend this quiz & Test your knowledge.

Find $\frac{2}{4} + \frac{2}{4}$.

$\frac{7}{24} + \frac{5}{16} =$, what is the least common denominator of $\frac{1}{2}$ and $\frac{1}{3}$, $\frac{3}{6} \;-\; \frac{1}{6} =$, what equation does the following figure represent.

How do we add and subtract negative fractions?

Negative fractions are simply fractions with a negative sign. The steps to add and subtract the negative fractions remain the same. We need to follow the rules for addition/subtraction with negative signs.

How can we convert an improper fraction into a mixed number?

To convert an improper fraction into a mixed number, we divide the numerator by the denominator. The denominator stays the same. The quotient represents the whole number part. The remainder represents the numerator of the mixed number.

Example: $\frac{14}{3} = 4\; \text{R}\; 2$

Quotient $= 4$

Remainder $= 2$

$\frac{14}{3} = 4\frac{2}{3}$

How do we divide two fractions?

To divide one fraction by another, we multiply the first fraction by the reciprocal of the second fraction.

$\frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \times \frac{D}{C}$

For example, $\frac{1}{2} \div \frac{3}{5} = \frac{1}{2} \times \frac{5}{3} = \frac{5}{6}$

What are the rules of adding and subtracting fractions?

• Before adding or subtracting, we check if the fractions have the same denominator.
• If the denominators are equal, then we add/subtract the numerators keeping the common denominator.
• If the denominators are different, then we make the denominators equal by using the LCM method. Once the fractions have the same denominator, we can add/subtract the numerators keeping the common denominator as it is.

How do we add and subtract fractions with whole numbers?

• Convert the whole number to a fraction. To do this, give the whole number a denominator of 1.
• Convert to fractions of like denominators. 
• Add/subtract the numerators. Now that the fractions have the same denominators, you can treat the numerators as a normal addition/subtraction problem.

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Fraction Addition Word Problems Worksheets

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Look to your laurels by answering our pdf fraction addition word problems worksheets, a collage of well-researched real-world word problems. Adding fractions or mixed numbers is not an alien concept. Whether it's cooking recipes; measuring lengths, weights, etc.; or sharing something among many, fraction addition and mixed-number addition are never too far away. Witness adding fractions and mixed numbers with like and unlike denominators leap into life as you solve this collection of word problems! Begin your learning journey with some of our free worksheets!

Adding Like Fractions

Pump up your practice with a pleasant potpourri of everyday situations in these adding fractions word problems worksheets for 3rd grade, 4th grade, and 5th grade. Keep at it, and summing up two like fractions will soon be a cakewalk!

Adding Unlike Fractions

An eclectic collection of word problems centering around fractions with unlike denominators, this pdf resource proves an imperative addition to your repertoire! Find equivalent like fractions and whizz through the problems!

Adding Fractions with Whole Numbers

Are you a novice wondering how to add fractions to whole numbers? Take a look at the real-life scenarios in these pdf worksheets and say goodbye to all your doubts! Put the numbers together as mixed numbers, and that's your sum.

Adding Mixed Numbers | Same Denominators

Natasha brewed a 1 1/2-ounce shot of espresso for latte and another 1 1/2-ounce shot for Americano. How much coffee did Natasha make in all? 3 shots! Keen to be explored in our printable set are a wealth of such situations!

Adding Mixed Numbers | Different Denominators

Evaluate 5th grade and 6th grade students' skills in adding mixed numbers with different denominators in this part of the fraction addition word problems worksheets. Convert to mixed numbers with the same denominators, and press on!

Related Printable Worksheets

▶ Adding Like Fractions

▶ Adding Unlike Fractions

▶ Fraction Word Problems

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Fraction Word Problems: Addition, Subtraction, and Mixed Numbers

In today’s post, we’re going to see how to solve some of the problems that we’ve introduced in Smartick: fraction word problems. They appear during the word problems section at the end of the daily session.

We’re going to look at how to solve problems involving addition and subtraction of fractions, including mixed fractions (the ones that are made up of a whole number and a fraction).

Try and solve the fraction word problems by yourself first, before you look for the solutions and their respective explanations below.

Fraction Word Problems

Problem nº 1.

Problem nº 2

Problem nº 3

Solution to Problem nº 1

This is an example of a problem involving the addition of a whole number and a fraction.

The simplest way to show the number of cookies I ate is to write it as a mixed number. And the data given in the word problem gives us the result: 9 biscuits and 5 / 6 of a biscuit = 9 5 / 6 biscuits.

Solution to Problem nº 2

In this example, we have to subtract two fractions with the same denominator.

To calculate how full the gas tank is, we have to subtract both fractions. Since we are given fractions, the best way to present the solution is in the form of a fraction. Additionally, we’re dealing with two fractions with the same denominator, so we just have to subtract the numerators of both fractions to get the result. 8 / 10 – 4 / 10 = 4 / 10

Solution to Problem nº 3

This problem requires us to subtract a mixed number and a fraction.

To solve this problem, we need to subtract the number of episodes that were downloaded this morning from the total number of episodes that are now downloaded.

To do this, we need to change the mixed number into a fraction: the 5 becomes 60 / 12 (5 x 12 = 60) and we add it to the fraction 60 / 12 + 8 / 12 = 68 / 12 .

We’ve converted the mixed number 5 8 / 12 to 68 / 12 . Now we just have to subtract the number of episodes that were downloaded yesterday ( 7 / 12 ),   68 / 12 – 7 / 12 = 61 / 12 .

Hopefully, you didn’t need the explanations and were able to solve them yourself without any help!

Fraction Video Tutorials

In the following video tutorials, you can learn a bit more about fractions. And if you would like to learn more math concepts, check out Smartick’s Youtube channel !

Simplifying Fractions

Simplification Using the GCD

Equivalent Fractions

If you would like to practice more fraction word problems like these and others, log in to Smartick and enjoy learning math.

Learn More:

• Word Problems with Fractions
• What Is a Fraction? Learn Everything There Is to Know!
• Using Mixed Numbers to Represent Improper Fractions
• Learning How to Subtract Fractions
• Learn How to Subtract Fractions
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Adding and Subtracting Fraction Word Problems

Subject: Mathematics

Age range: 7-11

Resource type: Worksheet/Activity

Last updated

16 June 2015

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Helpful sheet for practicing wordy fraction questions. Answers can sometimes be simplified further.

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Addition Word Problems 3rd Grade

Up to 4 digits.

Welcome to our Addition Word Problems 3rd Grade page. Here is our a range of addition word problem worksheets for third graders, which will help your child practice solving a range of addition problems using 3- and 4-digit numbers and up to 3 addends.

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• 3 Digit Addition Word Problems
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• Addition Word Problems up to 1000 Online Quiz

Addition Problems with 3- and 4-digit numbers

Each sheet consists of adding two or three 3- or 4-digit numbers.

There is a space on each sheet for working out, so that your child can write out the problem and solve it.

We have split the worksheets up into 3-digit word problems and 4-digit word problems.

Using these sheets will help your child to:

• add up two or three numbers with 3 or 4 digits;
• solve addition word problems.
• extract the relevant information to solve an addition problem

The questions on the 3-digit sheets and 4-digit sheets are very similar to each other, so 4-digit addition problems sheets are very similar to the 3-digit addition problems sheets but with larger numbers.

The sheets get gradually harder within each section:

• Sheet 1 and 2 consist of solving basic problems with 2 addends.
• Sheet 3 consists of solving basic problems with 3 addends.
• Sheet 4 consists of selecting the correct information from a table of information to solve the addition problem (with 2 or 3 addends)

3-Digit Addition Word Problems

• 3-Digit Addition Word Problems Sheet 1
• PDF version
• 3-Digit Addition Word Problems Sheet 2
• 3-Digit Addition Word Problems Sheet 3
• 3-Digit Addition Word Problems Sheet 4

4-Digit Addition Word Problems

• 4-Digit Addition Word Problems Sheet 1
• 4-Digit Addition Word Problems Sheet 2
• 4-Digit Addition Word Problems Sheet 3
• 4-Digit Addition Word Problems Sheet 4

Looking for some easier worksheets?

Take a look at our Addition word problems for first graders.

On this page, your child will learn to work out basic addition word problems with sums up to 20.

These sheets involve solving a range of addition word problems within 100.

• Addition Word Problems 2nd grade

More Recommended Math Worksheets

Take a look at some more of our worksheets similar to these.

More 3rd Grade Addition Worksheets

Here you will find some more of our 3rd Grade Addition Worksheets.

• Number Bonds to 50 and 100
• Third Grade Addition Facts Worksheets to 100+100
• 4-Digit Addition Worksheets

More 3rd Grade Math Word Problems

Here are a range of problems solving sheets for 3rd graders. Most of the sheets contain 'real-life' problems related to animal facts.

Using the sheets will help your child to:

• apply their addition, subtraction, and multiplication skills;
• apply their knowledge of rounding and place value;
• solve a range of 'real life' problems.

These sheets involve solving one or two more challenging longer problems.

• 3rd Grade Math Problems

These sheets involve solving many 'real-life' problems involving data.

• Math Word Problems for kids 3rd Grade
• 3rd Grade Addition and Subtraction Word Problems (3- and 4-digits)

These sheets involve solving 3-digit and 4-digit subtraction problems.

• Subtraction Word Problems 3rd Grade

These sheets involve solving a range of multiplciation problems.

• Multiplication Word Problem Worksheets 3rd Grade

These sheets involve solving a range of division problems.

• Division Worksheets Grade 3 Word Problems

Addition Word Problems to 1000 Online Quiz

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This will take you to a new webpage where your results will be shown. You can print a copy of your results from this page, either as a pdf or as a paper copy.

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This quick quiz tests your knowledge and skill at solving addition word problems within 1000.

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Problem Solving on Addition

Problem solving on addition will help us to get the idea on how to solve the basic addition statement problems.

1.  Three boys were playing cricket. Two more boys came along and played cricket with them. How many boys altogether were playing cricket?

Number of boys were playing cricket = 3

Number of boys came along and played cricket with them = 2

Therefore, total number of boys were playing cricket = 3 + 2 = 5

2.  Harry had 5 stamps. Mother gave him 4 more. How many stamps in all did Harry have?

Number of stamps Harry had = 5

Number of stamps his mother gave him = 4

Therefore, total number of stamps he have = 5 + 4 = 9

3. Rachael had 6 stamps. Father gave her 2 stamps. How many stamps does Rachael have now?

Number of stamps Rachael had = 6

Number of stamps her father gave her = 4

Therefore, total number of stamps she have now = 6 + 4 = 10

4. In a classroom, there were 2 brown tables, 1 red table and 3 green tables. How many tables were there altogether?

Number of brown tables = 2

Number of red table = 1

Number of green tables = 3

Therefore, total number of tables in a classroom = 2 + 1 + 3 = 6

5. Jack spent 5 dollars for a pen, 3 dollars for a color box, 2 dollars for a pencil box. How much did he spend altogether?

Amount of money Jack spent for a pen = $5

Amount of money he spent for a color box = $3

Amount of money he spent for a pencil box = $2

Therefore, total amount of money he spend altogether = $5 + $3 + $2 = 10

6. There were 6 yellow hats, 2 red hats and 6 blue hats. How many hats were there?

Number of yellow hats = 6

Number of red hats = 2

Number of blue hats = 6

Therefore, total number of hats were there = 6 + 2 + 6 = 14

7. Alex had 4 books on cars, 5 on airplanes and 7 on boats. How many books did he have in all?

Number of books on cars Alex had = 4

Number of books on airplanes = 5

Number of books on boats = 7

Therefore, total number of books he have in all = 4 + 5 + 7 = 16

More examples on statement problem solving on addition:

8. In a game Mary had the best score. She made 8 in her first turn, 6 in her second and 5 in her third. What was her total score?

Number of score Mary had in her first turn = 8

Number of score in her second turn = 5

Number of score in her third turn = 7

Therefore, total number of score Mary had in a game = 8 + 5 + 7 = 20

9. Sarah had 25 cards. She got 7 cards more. How many cards did she have then?

Number of cards Sarah had = 25

Number of cards she got more = 7

Therefore, total number of cards she have then = 25 + 7 = 32

10. A farmer had 45 sheep before he bought 25 more. How many sheep did he have then?

Number of sheep a farmer had = 45

Number of sheep he bought more = 25

Therefore, total number of sheep he have then = 45 + 25 = 70

Word Problems on Addition:

11. A basket had 28 mangoes in it. 14 more mangoes are kept in the basket. How many mangoes are there in the basket now?

The basket had                 2 8    mangoes

Mangoes kept in       (+)   1 4                      

Now, there are                 4 2 mangoes in all

Thus, there are 42 mangoes in the basket now.

12. Aaron scored 33 runs in the first cricket match and 29 runs in the second. How many runs did he score in all?

Aaron scored                  3 3  runs in the first match

He scored                (+)  2 9  runs in the second match

He scored                       6 2  runs in all                      

Thus, he scored 62 runs in all.

2nd Grade Math Practice

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Addition Worksheets

Welcome to the addition worksheets page at Math-Drills.com where we will add to your learning experience in many positive ways! On this page, you will find Addition worksheets from addition facts and two-digit addition to column addition and addition with games. In the first section, we've included a few addition printables that should help out the beginning student. Teaching addition facts is best done with some interesting teaching strategies.

Some teachers and parents use addition manipulatives to help students understand the basic addition facts. For example, adding groups of "Apple Jacks" (a breakfast cereal) by counting will quickly lead students to understand the concepts of addition. The sooner you can introduce base ten blocks to your students, the better. If you haven't already used them for counting, use them for basic addition and show students how regrouping works.

Most Popular Addition Worksheets this Week

Addition Facts Tables

Disputably not a great way to learn addition facts, but undeniably a great way to summarize, addition facts tables are an invaluable resource in any home or school classroom. Addition works very well as a table since the addends can be sequential. Encourage students to look for patterns and teach them a variety of strategies to learn the addition facts. For students who have not yet memorized their addition facts but need to know them for a more advanced math lesson such as adding two-digit numbers, provide an addition facts table to them, so they can quickly look up addition facts. After a while, they will most likely learn the facts through the use of the table and become less reliant on it. To make the tables more durable, print them on card stock and laminate them. They can be displayed on a screen or enlarged and printed on poster paper for whole class use.

• Addition Facts in One Square Table Addition Facts Table 1 to 10 (Filled In) Addition Facts Table 1 to 10 (Blank) Addition Facts Table 0 to 9 (Filled In) Addition Facts Table 0 to 9 (Blank) Left-Handed Addition Facts Table Left-Handed Blank Addition Facts Table All Addition Facts Tables Addition Facts Tables With One Fact at a time highlighted
• Addition Facts in Separate Tables Single Addition Facts Tables in Gray 1 to 12 Single Addition Facts Tables in Color 1 to 12 Single Addition Facts Tables in Montessori Colors 1 to 12

Five Minute Addition Frenzies

Five minute frenzy charts are 10 by 10 grids for addition fact practice. In each square, students write the sum of the column number and the row number.

Called mad minutes or timed drills by some, five minute frenzies are meant to be timed to add a little more excitement to practicing addition facts. They are ideally used to increase a student's ability to recall addition facts quickly which has all sorts of benefits later in their school life including preventing high school teachers from complaining about "how their students can't even add single-digit numbers without using a calculator."

A general goal to achieve would be to complete one chart in less than five minutes and score 98 percent or better, however, we recommend setting personal goals for students based on an initial test. If they are banging their head against the wall after a couple of minutes with only a few questions done, they really shouldn't be completing a timed addition facts drill at the moment. They still have some learning to do. We would recommend breaking out the manipulatives at this point. If they blast through the questions in 1.5 minutes and get almost all of them correct, they are probably ready for something a little more challenging.

One-per-page addition frenzies are not the most efficient use of paper resources, but they are a good starting point especially for younger students who have not quite mastered their penmanship enough to fit their numbers into a smaller chart. They are also great for displaying on screens or monitors for group activities. For example, you might use an interactive white board to fill out the chart.

• Five Minute Addition Frenzies Addition Frenzy ( 1 to 10 ) Addition Frenzy ( 11 to 20 ) Addition Frenzy ( 21 to 50 ) Addition Frenzy ( 51 to 100 )
• Left-handed Five Minute Addition Frenzies Left-handed Addition Frenzy ( 1 to 10 ) Left-handed Addition Frenzy ( 11 to 20 ) Left-handed Addition Frenzy ( 21 to 50 ) Left-handed Addition Frenzy ( 51 to 100 )

A wiser use of paper and photo-copy limits, having four charts on a page allows for multi-day practice, collaborative work or through the use of a paper-cutter, a quick stack of practice pages for students who finish early.

• Five Minute Addition Frenzies (Four Per Page) Four Addition Frenzies ( 1 to 10 ) Four Addition Frenzies ( 11 to 20 ) Four Addition Frenzies ( 21 to 50 ) Four Addition Frenzies ( 51 to 100 )
• Left-handed Five Minute Addition Frenzies (Four Per Page) Left-handed Four Addition Charts Per Page ( 1 to 10 ) Left-handed Four Addition Charts Per Page ( 11 to 20 ) Left-handed Four Addition Charts Per Page ( 21 to 50 ) Left-handed Four Addition Charts Per Page ( 51 to 100 )

Single-Digit Addition

Most people would agree that being able to add single-digit numbers quickly and in your head is an essential skill for success in math. The various addition worksheets in this section focus on skills that students will use their entire life. These worksheets will not magically make a student learn addition, but they are valuable for reinforcement and practice and can also be used as assessment tools.

• Single-Digit Addition with Some Regrouping 100 Single-Digit Addition Questions with Some Regrouping ✎ 81 Single-Digit Addition Questions with Some Regrouping ✎ 64 Single-Digit Addition Questions with Some Regrouping ✎ 50 Single-Digit Addition Questions with Some Regrouping ✎ 25 Single-Digit Addition Questions with Some Regrouping ✎ 12 Single-Digit Addition Questions with Some Regrouping ✎
• Single-Digit Addition with No Regrouping 100 Single-Digit Addition Questions with No Regrouping 64 Single-Digit Addition Questions with No Regrouping 25 Single-Digit Addition Questions with No Regrouping 12 Single-Digit Addition Questions with No Regrouping
• Single-Digit Addition with All Regrouping 100 Single-Digit Addition Questions with All Regrouping 64 Single-Digit Addition Questions with All Regrouping 25 Single-Digit Addition Questions with All Regrouping 12 Single-Digit Addition Questions with All Regrouping
• Horizontally Arranged Single-Digit Addition Horizontally Arranged Single-Digit Addition Facts (100 Questions) ✎ Horizontally Arranged Single-Digit Addition Facts (50 Questions) ✎ Horizontal Numbers that Add to 10 Horizontally Arranged Single-Digit Addition Facts up to 5 + 5 (100 Questions) ✎ Horizontally Arranged Single-Digit Addition Facts up to 6 + 6 (100 Questions) ✎ Horizontally Arranged Single-Digit Addition Facts up to 7 + 7 (100 Questions) ✎ Horizontally Arranged Single-Digit Addition Facts up to 8 + 8 (100 Questions) ✎
• Horizontally Arranged Single-Digit Addition of More than Two Addends Adding 3 Single-Digit Numbers Horizontally Adding 4 Single-Digit Numbers Horizontally Adding 5 Single-Digit Numbers Horizontally Adding 10 Single-Digit Numbers Horizontally
• Horizontally Arranged Single-Digit Addition with No Regrouping Horizontal Addition Facts with No Regrouping 100 per page Horizontal Addition Facts with No Regrouping and No Zeros 100 per page Horizontal Addition Facts with No Regrouping 50 per page
• Horizontally Arranged Single-Digit Addition with All Regrouping Horizontal Addition Facts with All Regrouping 100 per page Horizontal Addition Facts with All Regrouping 50 per page

The make ten addition strategy involves "spliting" the second addend into two parts. The first part combines with the first addend to make ten and the second part is the leftover amount. The strategy helps students quickly add amounts over ten in their head. For example, adding 8 + 7, students first recognize that they need to add 2 to 8 to get 10, so they split the 7 into 2 + 5. The 8 + 2 makes 10 and 5 more makes 15. The skill can be extended to many situations, for example adding 24 + 9, students recognize that they need 6 more to get to 30 and 9 can be split into 6 + 3, so 24 + 6 = 30 and 3 more makes 33. Continuing on, students can work on recognizing "complements" of other important numbers (see section further down) to develop this strategy further.

• Make 10 Addition Strategy Make 10 Addition Strategy Make 10 Addition Strategy Blanks Make 20 Addition Strategy Make 30 Addition Strategy Make 40 Addition Strategy Make 50 Addition Strategy Make 60 Addition Strategy Make 70 Addition Strategy Make 80 Addition Strategy Make 90 Addition Strategy Make Multiples of 10 Addition Strategy

Focusing on one number at a time is necessary for some students. Maybe they get overwhelmed with too much information and need to experience success in small steps.

• Adding Focus or Target Facts (50 Questions) Adding 0 to Single-Digit Numbers (50 Questions) ✎ Adding 1 to Single-Digit Numbers (50 Questions) ✎ Adding 2 to Single-Digit Numbers (50 Questions) ✎ Adding 1 or 2 to Single-Digit Numbers (50 Questions) ✎ Adding 3 to Single-Digit Numbers (50 Questions) ✎ Adding 4 to Single-Digit Numbers (50 Questions) ✎ Adding 5 to Single-Digit Numbers (50 Questions) ✎ Adding 6 to Single-Digit Numbers (50 Questions) ✎ Adding 7 to Single-Digit Numbers (50 Questions) ✎ Adding 8 to Single-Digit Numbers (50 Questions) ✎ Adding 9 to Single-Digit Numbers (50 Questions) ✎
• Adding Focus or Target Facts (25 Large Print Questions) Adding 0 to Single-Digit Numbers (25 Large Print Questions) ✎ Adding 1 to Single-Digit Numbers (25 Large Print Questions) ✎ Adding 2 to Single-Digit Numbers (25 Large Print Questions) ✎ Adding 3 to Single-Digit Numbers (25 Large Print Questions) ✎ Adding 4 to Single-Digit Numbers (25 Large Print Questions) ✎ Adding 5 to Single-Digit Numbers (25 Large Print Questions) ✎ Adding 6 to Single-Digit Numbers (25 Large Print Questions) ✎ Adding 7 to Single-Digit Numbers (25 Large Print Questions) ✎ Adding 8 to Single-Digit Numbers (25 Large Print Questions) ✎ Adding 9 to Single-Digit Numbers (25 Large Print Questions) ✎
• Adding Focus or Target Facts (25 Questions) with Sums Limited to 12 Adding 1 to Single-Digit Numbers With Sums Limited to 12 (25 Large Print Questions) ✎ Adding 2 to Single-Digit Numbers With Sums Limited to 12 (25 Large Print Questions) ✎ Adding 3 to Single-Digit Numbers With Sums Limited to 12 (25 Large Print Questions) ✎ Adding 4 to Single-Digit Numbers With Sums Limited to 12 (25 Large Print Questions) ✎ Adding 5 to Single-Digit Numbers With Sums Limited to 12 (25 Large Print Questions) ✎ Adding 6 to Single-Digit Numbers With Sums Limited to 12 (25 Large Print Questions) ✎ Adding 7 to Single-Digit Numbers With Sums Limited to 12 (25 Large Print Questions) ✎ Adding 8 to Single-Digit Numbers With Sums Limited to 12 (25 Large Print Questions) ✎ Adding 9 to Single-Digit Numbers With Sums Limited to 12 (25 Large Print Questions) ✎
• Horizontally-Arranged Adding Focus or Target Facts 100 Horizontal Adding 1s to Single-Digit Numbers Questions 100 Horizontal Adding 2s to Single-Digit Numbers Questions 50 Adding 1s and 2s to Single-Digit Numbers Questions 100 Horizontal Adding 3s to Single-Digit Numbers Questions 100 Horizontal Adding 4s to Single-Digit Numbers Questions 100 Horizontal Adding 5s to Single-Digit Numbers Questions 100 Horizontal Adding 6s to Single-Digit Numbers Questions 100 Horizontal Adding 7s to Single-Digit Numbers Questions 100 Horizontal Adding 8s to Single-Digit Numbers Questions 100 Horizontal Adding 9s to Single-Digit Numbers Questions

Multi-Digit Addition

A variety of strategies can be used to learn multi-digit addition; it isn't necessary to rely only on paper and pencil methods. Base ten blocks can help students conceptualize addition. Teaching students a mental left-to-right addition skill will help them in future math studies and life in general. E.g. 34 + 78 would be 30 + 70 = 100, 100 + 4 = 104, 104 + 8 = 112. Don't forget about using estimation with these worksheets.

• Multi-Digit Addition with Some Regrouping 2-Digit plus 1-Digit Addition (25 Questions) ✎ 2-Digit Plus 1-Digit Addition (36 Questions) ✎ 2-Digit plus 1-Digit Addition (64 Questions) ✎ 2-Digit plus 1-Digit Addition (100 Questions) ✎ 2-Digit plus 1-Digit Addition ( Sums Less Than 100 ) (25 Questions) 2-Digit Addition (25 Questions) ✎ 2-Digit Addition (36 Questions) ✎ 2-Digit Addition (64 Questions) ✎ 2-Digit Addition (100 Questions) ✎ 3-Digit Plus 1-Digit Addition (25 Questions) ✎ 3-Digit Plus 1-Digit Addition (36 Questions) ✎ 3-Digit Plus 2-Digit Addition (25 Questions) ✎ 3-Digit Plus 2-Digit Addition (36 Questions) ✎ 3-Digit Plus 2-Digit Addition (49 Questions) ✎ 3-Digit Plus 2-Digit Addition (100 Questions) ✎ 3-Digit Addition (25 Questions) ✎ 3-Digit Addition (36 Questions) ✎ 3-Digit Addition (49 Questions) ✎ 3-Digit Addition (100 Questions) ✎ 4-Digit Plus 1-Digit Addition (25 Questions) ✎ 4-Digit Plus 1-Digit Addition (36 Questions) ✎ 4-Digit Plus 2-Digit Addition (25 Questions) ✎ 4-Digit Plus 2-Digit Addition (36 Questions) ✎ 4-Digit Plus 3-Digit Addition (25 Questions) ✎ 4-Digit Plus 3-Digit Addition (36 Questions) ✎ 4-Digit Plus 3-Digit Addition (49 Questions) ✎ 4-Digit Plus 3-Digit Addition (100 Questions) ✎ 4-Digit Addition (25 Questions) ✎ 4-Digit Addition (36 Questions) ✎ 4-Digit Addition (49 Questions) ✎ 4-Digit Addition (100 Questions) ✎ 5-Digit Addition (25 Questions) ✎ Various 2-digit to 4-digit Addition (25 Questions) ✎ Various 2-Digit to 4-Digit Addition (36 Questions) ✎ Various 2-digit to 5-digit Addition (20 Questions) ✎ Various 2-Digit to 5-Digit Addition (36 Questions) ✎ Various 3-digit to 5-digit Addition (20 Questions) ✎ Various 3-Digit to 5-Digit Addition (36 Questions) ✎ 6-Digit Addition (20 Questions) ✎ 7-Digit Addition (15 Questions) ✎ 8-Digit Addition (15 Questions) ✎ 9-Digit Addition (15 Questions) ✎ 3-Digit Expanded Form Addition

Regrouping is what long addition is all about; these worksheets give students a lot of practice since every step requires regrouping.

• Multi-Digit Addition with All Regrouping 2-Digit Plus 1-Digit Addition with ALL Regrouping in the Ones Place (25 Questions) ✎ 2-Digit Addition with ALL Regrouping (25 Questions) ✎ 2-Digit Addition with ALL Regrouping (36 Questions) ✎ 3-Digit Addition with ALL Regrouping (25 Questions) ✎ 4-Digit Addition with ALL Regrouping (25 Questions) ✎ 5-Digit Addition with ALL Regrouping (20 Questions) ✎ 6-Digit Addition with ALL Regrouping (20 Questions) ✎ 7-Digit Addition with ALL Regrouping (15 Questions) ✎ 8-Digit Addition with ALL Regrouping (15 Questions) ✎ 9-Digit Addition with ALL Regrouping (15 Questions) ✎

If you haven't quite mastered all the addition facts or the long addition algorithm, these might be the worksheets for you. These worksheets don't require any regrouping, so they provide an extra in-between skill for students who require a little more guidance.

• Multi-Digit Addition with No Regrouping 2-Digit Plus 1-Digit Addition with NO Regrouping (25 Questions) ✎ 2-Digit Addition with NO Regrouping (25 Questions) ✎ 2-Digit Addition with NO Regrouping (36 Questions) ✎ 2-Digit Addition with NO Regrouping (64 Questions) ✎ 2-Digit Addition with NO Regrouping (100 Questions) ✎ 3-Digit Plus 1-Digit Addition with NO Regrouping (25 Questions) ✎ 3-Digit Plus 2-Digit Addition with NO Regrouping (25 Questions) ✎ 3-Digit Addition with NO Regrouping (25 Questions) ✎ 4-Digit Plus 1-Digit Addition with NO Regrouping (25 Questions) ✎ 4-Digit Plus 2-Digit Addition with NO Regrouping (25 Questions) ✎ 4-Digit Plus 3-Digit Addition with NO Regrouping (25 Questions) ✎ 4-Digit Addition with NO Regrouping (25 Questions) ✎ 5-Digit Addition with NO Regrouping (20 Questions) ✎ 6-Digit Addition with NO Regrouping (20 Questions) ✎ 7-Digit Addition with NO Regrouping (20 Questions) ✎ 8-Digit Addition with NO Regrouping (15 Questions) ✎ 9-Digit Addition with NO Regrouping (15 Questions) ✎

Horizontal addition can encourage students to use mental math or other strategies to add numbers. One of the most common mental math strategies for addition is a left-to-right (also called front end) addition strategy. This involves adding the greater place values first. Other strategies for adding multi-digit numbers include using base ten blocks or other manipulatives, number lines, decomposing numbers and adding the parts, and using a calculator.

• Horizontally Arranged Multi-Digit Addition Adding to 20 with the Second Addend Greater 2-Digit Plus 2-Digit Horizontal Addition with no Regrouping Horizontally Arranged 2-Digit Plus 2-Digit Addition ✎ Horizontally Arranged 3-Digit Plus 2-Digit Addition ✎ Horizontally Arranged 3-Digit Plus 3-Digit Addition ✎ Horizontally Arranged Various 2- and 3-Digit Addition ✎ Horizontally Arranged 4-Digit Plus 3-Digit Addition ✎ Horizontally Arranged 4-Digit Plus 4-Digit Addition ✎ Horizontally Arranged Various 2- to 4-Digit Addition ✎
• Horizontally Arranged Multi-Digit Addition of More Than Two Addends Adding 3 Two-Digit Numbers Horizontally Adding 4 Two-Digit Numbers Horizontally Adding 5 Two-Digit Numbers Horizontally Adding 10 Two-Digit Numbers Horizontally
• Adding Focus or Target Facts Greater Than 9 25 Adding 10s Questions ✎ 50 Adding 10s Questions ✎ 50 Adding 11s Questions ✎ 50 Adding 12s Questions ✎ 50 Adding 13s Questions ✎ 50 Adding 14s Questions ✎ 50 Adding 15s Questions ✎ 50 Adding 16s Questions ✎ 50 Adding 17s Questions ✎ 50 Adding 18s Questions ✎ 50 Adding 19s Questions ✎ 50 Adding 20s Questions ✎

Using a comma to separate thousands is the most common way to format large numbers in the English world.

• Multi-Digit Addition with Some Regrouping (Comma-Separated Thousands) Adding 4-Digit Numbers (Comma Separated) (25 Questions) ✎ Adding 5-Digit Numbers (Comma Separated) (20 Questions) ✎ Adding 6-Digit Numbers (Comma Separated) (20 Questions) ✎ Adding 7-Digit Numbers (Comma Separated) (15 Questions) ✎ Adding 8-Digit Numbers (Comma Separated) (15 Questions) ✎ Adding 9-Digit Numbers (Comma Separated) (15 Questions) ✎
• Multi-Digit Addition with All Regrouping (Comma-Separated Thousands) Adding 4-Digit Numbers with ALL Regrouping (Comma Separated) (25 Questions) ✎ Adding 5-Digit Numbers with ALL Regrouping (Comma Separated) (20 Questions) ✎ Adding 6-Digit Numbers with ALL Regrouping (Comma Separated) (20 Questions) ✎ Adding 7-Digit Numbers with ALL Regrouping (Comma Separated) (15 Questions) ✎ Adding 8-Digit Numbers with ALL Regrouping (Comma Separated) (15 Questions) ✎ Adding 9-Digit Numbers with ALL Regrouping (Comma Separated) (15 Questions) ✎
• Multi-Digit Addition with No Regrouping (Comma-Separated Thousands) Adding 4-Digit Numbers with NO Regrouping (Comma Separated) (25 Questions) ✎ Adding 5-Digit Numbers with NO Regrouping (Comma Separated) (20 Questions) ✎ Adding 6-Digit Numbers with NO Regrouping (Comma Separated) (20 Questions) ✎ Adding 7-Digit Numbers with NO Regrouping (Comma Separated) (15 Questions) ✎ Adding 8-Digit Numbers with NO Regrouping (Comma Separated) (15 Questions) ✎ Adding 9-Digit Numbers with NO Regrouping (Comma Separated) (15 Questions) ✎

Using a space to separate thousands in large numbers is common in some languages. In the English world, you will most likely find Canadians formatting their numbers in this way.

• Multi-Digit Addition with Some Regrouping (Space-Separated Thousands) Adding 4-Digit Numbers (Space Separated) (25 Questions) ✎ Adding 5-Digit Numbers (Space Separated) (20 Questions) ✎ Adding 6-Digit Numbers (Space Separated) (20 Questions) ✎ Adding 7-Digit Numbers (Space Separated) (15 Questions) ✎ Adding 8-Digit Numbers (Space Separated) (15 Questions) ✎ Adding 9-Digit Numbers (Space Separated) (15 Questions) ✎
• Multi-Digit Addition with All Regrouping (Space-Separated Thousands) Adding 4-Digit Numbers with ALL Regrouping (Space Separated) (25 Questions) ✎ Adding 5-Digit Numbers with ALL Regrouping (Space Separated) (20 Questions) ✎ Adding 6-Digit Numbers with ALL Regrouping (Space Separated) (20 Questions) ✎ Adding 7-Digit Numbers with ALL Regrouping (Space Separated) (15 Questions) ✎ Adding 8-Digit Numbers with ALL Regrouping (Space Separated) (15 Questions) ✎ Adding 9-Digit Numbers with ALL Regrouping (Space Separated) (15 Questions) ✎
• Multi-Digit Addition with No Regrouping (Space-Separated Thousands) Adding 4-Digit Numbers with NO Regrouping (Space Separated) (25 Questions) ✎ Adding 5-Digit Numbers with NO Regrouping (Space Separated) (20 Questions) ✎ Adding 6-Digit Numbers with NO Regrouping (Space Separated) (20 Questions) ✎ Adding 7-Digit Numbers with NO Regrouping (Space Separated) (15 Questions) ✎ Adding 8-Digit Numbers with NO Regrouping (Space Separated) (15 Questions) ✎ Adding 9-Digit Numbers with NO Regrouping (Space Separated) (15 Questions) ✎

Using a period as a thousands separator is not generally seen in English-speaking countries, but since there are people from around the world who use these addition worksheets, they are included.

• Multi-Digit Addition with Some Regrouping (Period-Separated Thousands) Adding 4-Digit Numbers (Period Separated) (25 Questions) ✎ Adding 5-Digit Numbers (Period Separated) (20 Questions) ✎ Adding 6-Digit Numbers (Period Separated) (20 Questions) ✎ Adding 7-Digit Numbers (Period Separated) (15 Questions) ✎ Adding 8-Digit Numbers (Period Separated) (15 Questions) ✎ Adding 9-Digit Numbers (Period Separated) (15 Questions) ✎
• Multi-Digit Addition with All Regrouping (Period-Separated Thousands) Adding 4-Digit Numbers with ALL Regrouping (Period Separated) (25 Questions) ✎ Adding 5-Digit Numbers with ALL Regrouping (Period Separated) (20 Questions) ✎ Adding 6-Digit Numbers with ALL Regrouping (Period Separated) (20 Questions) ✎ Adding 7-Digit Numbers with ALL Regrouping (Period Separated) (15 Questions) ✎ Adding 8-Digit Numbers with ALL Regrouping (Period Separated) (15 Questions) ✎ Adding 9-Digit Numbers with ALL Regrouping (Period Separated) (15 Questions) ✎
• Multi-Digit Addition with No Regrouping (Period-Separated Thousands) Adding 4-Digit Numbers with NO Regrouping (Period Separated) (25 Questions) ✎ Adding 5-Digit Numbers with NO Regrouping (Period Separated) (20 Questions) ✎ Adding 6-Digit Numbers with NO Regrouping (Period Separated) (20 Questions) ✎ Adding 7-Digit Numbers with NO Regrouping (Period Separated) (15 Questions) ✎ Adding 8-Digit Numbers with NO Regrouping (Period Separated) (15 Questions) ✎ Adding 9-Digit Numbers with NO Regrouping (Period Separated) (15 Questions) ✎

For various reasons, sometimes you need addition questions in a larger font. These should fit the bill.

• Large Print Multi-Digit Addition with Some Regrouping 2-digit Plus 1-digit Addition with SOME Regrouping (Large Print) (16 Questions) ✎ 3-digit Plus 1-digit Addition with SOME Regrouping (Large Print) (16 Questions) ✎ 4-digit Plus 1-digit Addition with SOME Regrouping (Large Print) (16 Questions) ✎ Various Plus 1-digit Addition with SOME Regrouping (Large Print) (16 Questions) ✎ 2-digit Plus 2-digit Addition with SOME Regrouping (Large Print) (16 Questions) ✎ 3-digit Plus 2-digit Addition with SOME Regrouping (Large Print) (16 Questions) ✎ 4-digit Plus 2-digit Addition with SOME Regrouping (Large Print) (16 Questions) ✎ Various Plus 2-digit Addition with SOME Regrouping (Large Print) (16 Questions) ✎ 3-digit Plus 3-digit Addition with SOME Regrouping (Large Print) (16 Questions) ✎ 4-digit Plus 3-digit Addition with SOME Regrouping (Large Print) (16 Questions) ✎ 4-digit Plus 4-digit Addition with SOME Regrouping (Large Print) (16 Questions) ✎ 5-digit Plus 5-digit Addition with SOME Regrouping (Large Print) (12 Questions) ✎ 6-digit Plus 6-digit Addition with SOME Regrouping (Large Print) (12 Questions) ✎
• Very Large Print Multi-Digit Addition with Some Regrouping 2-Digit Plus 1-Digit Addition with SOME Regrouping (Very Large Print) (9 Questions) ✎ 2-Digit Addition with SOME Regrouping (Very Large Print) (9 Questions) ✎ 3-Digit Plus 1-Digit Addition with SOME Regrouping (Very Large Print) (9 Questions) ✎ 3-Digit Plus 2-Digit Addition with SOME Regrouping (Very Large Print) (9 Questions) ✎ 3-Digit Addition with SOME Regrouping (Very Large Print) (9 Questions) ✎ 4-Digit Plus 1-Digit Addition with SOME Regrouping (Very Large Print) (9 Questions) ✎ 4-Digit Plus 2-Digit Addition with SOME Regrouping (Very Large Print) (9 Questions) ✎ 4-Digit Plus 3-Digit Addition with SOME Regrouping (Very Large Print) (9 Questions) ✎ 4-Digit Addition with SOME Regrouping (Very Large Print) (9 Questions) ✎ Various 2- to 4-Digit Addition with SOME Regrouping (Very Large Print) (9 Questions) ✎
• Large Print Multi-Digit Addition with All Regrouping 2-Digit Addition with ALL Regrouping (Large Print) (16 Questions) ✎ 3-Digit Addition with ALL Regrouping (Large Print) (16 Questions) ✎ 4-Digit Addition with ALL Regrouping (Large Print) (16 Questions) ✎ 5-Digit Addition with ALL Regrouping (Large Print) (12 Questions) ✎ 6-Digit Addition with ALL Regrouping (Large Print) (12 Questions) ✎
• Large Print Multi-Digit Addition with No Regrouping 2-Digit Addition with NO Regrouping (Large Print) (16 Questions) ✎ 3-Digit Plus 2-Digit Addition with NO Regrouping (Large Print) (16 Questions) ✎ 3-Digit Addition with NO Regrouping (Large Print) (16 Questions) ✎ 4-Digit Plus 3-Digit Addition with NO Regrouping (Large Print) (16 Questions) ✎ 4-Digit Addition with NO Regrouping (Large Print) (16 Questions) ✎ 5-Digit Addition with NO Regrouping (Large Print) (12 Questions) ✎ 6-Digit Addition with NO Regrouping (Large Print) (12 Questions) ✎ LP 2-Digit Addition with Sums up to 99 ( 25 Questions ) LP 2-Digit Addition with Sums up to 99 ( 12 Questions )

Multi-Digit Addition with Grid Support

Adding with grid support helps students who have trouble lining up place values themselves. Perhaps with a little practice, they might get a better understanding of not only lining up the place values, but why this is done. Pointing out that the 5 in 659 means 50, for example, is useful in helping students understand place value as it relates to addition.

• Adding 2 Addends With Grid Support Adding 2-Digit + 2-Digit Numbers on a Grid (2 Addends) Adding 3-Digit + 3-Digit Numbers on a Grid (2 Addends) Adding 3-Digit + 2-Digit Numbers on a Grid (2 Addends) Adding 4-Digit + 4-Digit Numbers on a Grid (2 Addends) Adding 4-Digit + 3-Digit Numbers on a Grid (2 Addends) Adding 4-Digit + 2-Digit Numbers on a Grid (2 Addends) Adding 5-Digit + 5-Digit Numbers on a Grid (2 Addends) Adding 5-Digit + 4-Digit Numbers on a Grid (2 Addends) Adding 5-Digit + 3-Digit Numbers on a Grid (2 Addends) Adding 5-Digit + 2-Digit Numbers on a Grid (2 Addends) Adding Various Digit Numbers on a Grid (2 Addends)
• Adding 3 Addends With Grid Support Adding 2-Digit Numbers on a Grid (3 Addends) Adding 3-Digit Numbers on a Grid (3 Addends) Adding 4-Digit Numbers on a Grid (3 Addends) Adding 5-Digit Numbers on a Grid (3 Addends) Adding Various-Digit Numbers on a Grid (3 Addends)
• Adding 4 Addends With Grid Support Adding 2-Digit Numbers on a Grid (4 Addends) Adding 3-Digit Numbers on a Grid (4 Addends) Adding 4-Digit Numbers on a Grid (4 Addends) Adding 5-Digit Numbers on a Grid (4 Addends) Adding Various-Digit Numbers on a Grid (4 Addends)
• Adding 5 Addends With Grid Support Adding 2-Digit Numbers on a Grid (5 Addends) Adding 3-Digit Numbers on a Grid (5 Addends) Adding 4-Digit Numbers on a Grid (5 Addends) Adding 5-Digit Numbers on a Grid (5 Addends) Adding Various-Digit Numbers on a Grid (5 Addends)

Various Other Addition Worksheets

Column addition is not just an exercise in accounting, it also develops mental addition skills that are useful in everyday life. Various strategies are available for adding columns of numbers. The traditional method is to use a pencil and paper approach, also known as right-to-left addition, where students add and regroup starting with the smallest place (ones in this case) and proceed up to the greatest place. A mental approach might involve students going from left-to-right where the greater place is added first. This is easier to keep track of in your head, but does require the occasional adjustment in previous answers. An example is to add 345 + 678 + 901. First add the 300, 600 and 900 to get 1800, then add 40, 70 and 0 in turn to get 1910, then deal with the 5, 8 and 1 to get 1924. Along the way you had to adjust your total, but keeping a running total in your head is a lot easier than transfering a pencil and paper method into your head.

• Column Addition with Single-Digit Numbers Adding Three Single-Digit Numbers Adding Four Single-Digit Numbers Adding Five Single-Digit Numbers Adding Six Single-Digit Numbers
• Column Addition with Two-Digit Numbers Adding Three Two-Digit Numbers Adding Four Two-Digit Numbers Adding Five Two-Digit Numbers Adding Six Two-Digit Numbers
• Column Addition with Three-Digit Numbers Adding Three Three-Digit Numbers Adding Four Three-Digit Numbers Adding Five Three-Digit Numbers Adding Six Three-Digit Numbers
• Column Addition with Four-Digit Numbers Adding Three Four-Digit Numbers Adding Four Four-Digit Numbers Adding Five Four-Digit Numbers Adding Six Four-Digit Numbers
• Column Addition with Various-Digit Numbers Adding Three Various-Digit Numbers Adding Four Various-Digit Numbers Adding Five Various-Digit Numbers Adding Six Various-Digit Numbers

Games help students develop mental addition skills but in a fun context. For the adding with playing cards worksheets, a Jack is counted as 11, a Queen as 12, a King as 13 and an Ace as 1. Playing math games while enjoying some social time with their friends is a great way to develop strategic thinking and math fluency in children.

• Adding With Games Adding 2 Playing Cards Adding 3 Playing Cards Adding 4 Playing Cards Adding 5 Playing Cards Adding 6 Playing Cards Adding 7 Playing Cards Adding 8 Playing Cards Counting Cribbage Hands Identify and Count Yahtzee! Combinations

Finding complements of numbers can help students a great deal in developing mental arithmetic skills and to further their understanding of number.

• Adding Complements of 9, 99 and 999 Adding Complements of 9 (Blanks in First or Second Position Mixed) Adding Complements of 9 (Blanks in First then Second Position) Adding Complements of 9 (Blanks in First Position Only) Adding Complements of 9 (Blanks in Second Position Only) Adding Complements of 9 (Blanks in Any Position, Including Sums) Adding Complements of 99 Adding Complements of 999
• Adding Complements of 10, 100 and 1000 Adding Complements of 10 Adding Complements of 100 Adding Complements of 1000
• Adding Complements of 11 Adding Complements of 11 (Blanks in First or Second Position Mixed) Adding Complements of 11 (Blanks in First then Second Position) Adding Complements of 11 (Blanks in First Position Only) Adding Complements of 11 (Blanks in Second Position Only) Adding Complements of 11 (Blanks in Any Position, Including Sums)

Using an adding doubles strategy can help students to process addition questions more quickly using mental math. To use this strategy, students must recognize that the two numbers are close to the same value (usually by one or two). They also must recognize by how much and whether it is greater or less than the first addend. A typical dialogue with the question, 15 + 16, might be, "I see that the second number is greater than the first number by 1. If I double the first number and add 1, I will get my answer. 15 doubled is 30 plus one is 31. 15 + 16, therefore, is 31."

• Adding Doubles Up to 9 Adding Doubles (Up to 9) Adding Doubles Plus One (Up to 9) Adding Doubles Plus Two (Up to 9) Adding Doubles Minus One (Up to 9) Adding Doubles Minus Two (Up to 9) Adding Doubles Mixed Variations (Up to 9)
• Adding Doubles Up to 15 Adding Doubles (Up to 15) Adding Doubles Plus One (Up to 15) Adding Doubles Plus Two (Up to 15) Adding Doubles Minus One (Up to 15) Adding Doubles Minus Two (Up to 15) Adding Doubles Mixed Variations (Up to 15)
• Adding Doubles Up to 30 Adding Doubles (Up to 30) Adding Doubles Plus One (Up to 30) Adding Doubles Plus Two (Up to 30) Adding Doubles Minus One (Up to 30) Adding Doubles Minus Two (Up to 30) Adding Doubles Mixed Variations (Up to 30)

Not commonly taught in modern schools, adding in other base number systems can stretch students' minds and have quite a few important applications, especially in technology. For example, you will find binary, octal and hexadecimal systems are quite often used in computer technology. Quaternary numbers can be used in genetics to store DNA sequences. The duodecimal system is sometimes suggested as a superior system to the decimal system

• Adding in Other Base Number Systems Adding Binary Numbers (Base 2) Adding Ternary Numbers (Base 3) Adding Quaternary Numbers (Base 4) Adding Quinary Numbers (Base 5) Adding Senary Numbers (Base 6) Adding Octal Numbers (Base 8) Adding Duodecimal Numbers (Base 12) Adding Hexadecimal Numbers (Base 16) Adding Vigesimal Numbers (Base 20) Adding Hexatrigesimal Numbers (Base 36) Adding Various Numbers (Various Bases)

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A.I. Can Write Poetry, but It Struggles With Math

A.I.’s math problem reflects how much the new technology is a break with computing’s past.

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By Steve Lohr

In the school year that ended recently, one class of learners stood out as a seeming puzzle. They are hard-working, improving and remarkably articulate. But curiously, these learners — artificially intelligent chatbots — often struggle with math.

Chatbots like Open AI’s ChatGPT can write poetry, summarize books and answer questions, often with human-level fluency. These systems can do math, based on what they have learned, but the results can vary and be wrong. They are fine-tuned for determining probabilities, not doing rules-based calculations. Likelihood is not accuracy , and language is more flexible, and forgiving, than math.

“The A.I. chatbots have difficulty with math because they were never designed to do it,” said Kristian Hammond, a computer science professor and artificial intelligence researcher at Northwestern University.

The world’s smartest computer scientists, it seems, have created artificial intelligence that is more liberal arts major than numbers whiz.

That, on the face of it, is a sharp break with computing’s past. Since the early computers appeared in the 1940s, a good summary definition of computing has been “math on steroids.” Computers have been tireless, fast, accurate calculating machines. Crunching numbers has long been what computers are really good at, far exceeding human performance.

Traditionally, computers have been programmed to follow step-by-step rules and retrieve information in structured databases. They were powerful but brittle. So past efforts at A.I. hit a wall.

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Is population decline a problem to solve or just one to rethink? | The Excerpt

On a special episode (first released on August 1, 2024) of The Excerpt podcast: Across the globe, populations are in precipitous decline. People are simply not having children at the fertility rate needed to sustain our global demography. Here in the U.S., the fertility rate in 2023 hit a historic low, according to the CDC. That means not enough new workers, on whose taxes the rest of us depend as we age. What else do we need to know about the issue, and importantly, what can be done about it? Jennifer Sciubba, President and CEO of the non-profit Population Reference Bureau, joins The Excerpt to offer a new perspective on the issue.

Hit play on the player below to hear the podcast and follow along with the transcript beneath it.  This transcript was automatically generated, and then edited for clarity in its current form. There may be some differences between the audio and the text.

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Taylor Wilson:

Hello, and welcome to The Excerpt. I'm Taylor Wilson, and today is Thursday, August 1st, 2024. This is a special episode of The Excerpt. Across the globe, populations are in precipitous decline. People are simply not having children at the fertility rate needed to sustain our global demography. Here in the U.S., the fertility rate in 2023 hit a historic low, according to the CDC. That means not enough new workers on whose taxes the rest of us depend as we age. So what else do we need to know about the issue? And importantly, what can be done about it? My guest today is Jennifer Sciubba, president and CEO of the nonprofit Population Reference Bureau. Jennifer, thanks for joining me.

Jennifer Sciubba:

Thank you so much, Taylor. I'm glad to be here.

So let's just get right to it, starting here. The generally recognized total fertility rate at which a population is stabilized is 2.1, meaning people should be having that many babies to maintain our population. But, Jennifer, that's not happening. What is our fertility rate now? And describe for us just how bad it is.

I wouldn't use the phrase bad or the word bad there because when we say 2.1, this is a math thing. Of course, because we're just talking about replacing two parents with a little margin of error to account for those who do not make it to reproductive age, unfortunately. But it's a theoretical number when we're thinking about actual population. So to try to aim for that as a number, I think, is where we make this mistake. There's no reason why we would ultimately end up at exactly that number as an individual country. So in the United States, we are below this replacement level rate. We're somewhere in the 1.6 range, between 1.6 and 1.7. Globally, we are actually at about 2.25. So globally, we actually see those numbers trending downwards towards replacement level as well.

So what are the biggest factors causing this drop?

So in the United States, we have achieved a big goal that we set out to do, which is to bring down teen pregnancy rates. So that's responsible for a good chunk of our decline from higher to lower fertility rates. Also, the U.S. really mirrors what we see as a global trend, which is having gone over a long period of time from higher numbers of children per woman on average to lower numbers, as we see education levels rise across the population, as we see women getting work opportunities outside the home to bring in incomes to their families. So we kind of follow that general trend. We also, though, follow the trend among our peers, other high-income countries that have seen a continued decline past that replacement level rate to even lower levels.

You touched on the global trends a bit here, Jennifer. Globally, where is the falling birth rate impacting people the most, and how does that compare to us here in the U.S.?

The United States actually fits pretty well with other peers in terms of our income level. We're all below replacement level, and it's just how far below that's interesting. If we really look at this, maybe this group of countries that we would consider ourselves in a club with, we perhaps at the higher end of that, maybe more like a France kind of situation than a Korea situation because Korea is at the lower end of that. We would call these ultra-low fertility rates, or that below 1.4 children per woman. There are actually quite a few of those as well. Many Eastern European countries fall in line there. We see Japan and South Korea. So if we're comparing ourselves to those group, we're not quite at that ultra-low fertility level yet.

Are there places where birth rates remain high?

There are a few places and a decreasing number of places where they remain high, but they're still important for us to keep in mind. So when we look at the countries in the world that have an average of five or more children per woman, there are only 8 out of around 200. But those eight will see, of course, a doubling of their populations over the next 30 years. So they're rapidly growing and kind of bring us back to these math principles here. We see that this becomes an exponential growth. It's really down everywhere. It's just how far and how fast it's down that differs around the world.

So, Jennifer, what kinds of negative impacts do we see from declining populations? I mentioned tax collection in the intro. Are there other big ones that would change the way we live?

I think they'll all change the way we live. So when we say, "Is this negative or positive?" The negative comes from if we don't try to change anything. If we just keep going about business as usual, thinking about every structure in our society that was built on the idea of infinite population growth, then we're really going to feel those crunches. But those countries that adapt to this first and best will come out on top. So we might think about the workforce, for example. We know that we have hit peak workforce in many economies. So what does that mean? How do you adjust to that? States have different options for this. Countries around the world can choose to increase immigration, but what kind of immigration? They can choose certain skill sets, for example.

Countries can also work to help keep people in the workforce longer by providing opportunities for them to upskill or reskill and by taking laws off the books that actually discriminate against having older workers. In some contexts, they can really work to bring more women into the workforce who have been excluded as well. So it's the idea that if we leave everything the same and we have, for example, just a pay-as-you-go social security system where today's workers pay into a pot for their current retirees as those proportions of the population shift, that becomes really unsustainable.

I'm curious, are there any upsides to some of these population shifts, these birth rate shifts that we've been talking about?

By itself and in and of itself, this is a good news story. If we're talking about shifting from high to low, I'm a student of population history and the foreign policy, and I know how much money, time, and effort was poured into helping bring education levels around the world, helping to bring those up, getting more women educated, having access to family planning, for example. So that actually worked, and now we see these lower fertility rates there. So we also see population aging because we're living longer, which is great. Thank goodness we're living longer. We really have worked hard for this.

Many experts have weighed in on how many people our planet can reasonably support with food and other resources. You've touched on this a bit. Some have put that number at 10 billion. Is an overall declining population across the globe part of this narrative about maximum lives the planet is able to sustain, and if so, how?

Absolutely. We think about what is the carrying capacity of the earth, and this has been a question for a really long time. If you go back to when Earth first had its 1 billion people, it hit that 1 billion mark. This is the same time that Malthus, famous for thinking about Malthusianism, said that there are too many people and we're not going to be able to feed everyone, and we see this echoed as we hit every other billion milestone. We will reach peak population this century, probably in the 2080s, and probably at that 10 billion mark. So it's just adding about two more billion people. If we're at the global level, it's not that big of a deal. In my lifetime, we've added four, and I'm not that old. So in that sense, it's not that big of a deal, but it's where they will be, and that's where this has more credence because there really is a relationship, of course, between population, resources, and the immediate environment.

So we know this population growth will be concentrated in some of the world's lowest-income countries, and their ability to handle that growth in a way that becomes an opportunity for them. This is the workforce of the future, for example, rather than spend their time really struggling to have stability and prosperity. That's the real challenge that we have today. Environmentally, eventually, it could be a good thing, but I worry that we will rest on our laurels and we will hope that deep population equals a cleaner environment. Forget how much consumption really matters to drive that environmental impact.

It's an interesting point. So various countries around the world have been trying to tackle this issue for a while. You mentioned South Korea earlier. The president of South Korea has called the rapidly falling birth rate in his country a national emergency. Very strong words recently creating a new ministry with the express mission of tackling this crisis. Are there any policies that have been particularly successful around the world?

We've often thought of the bar for success as a radical shift in those fertility rate trends. So success would be getting to this theoretical 2.1, for example. If that's the case, then no, not really. But if we change our definition of success and we think about what kind of policies make people's lives better, then yes, anytime we can alleviate some of the financial burden of having children, that can really help at the household level. Whenever we make maternity and paternity policy, that parental leave can really improve the experiences of individual families, and, of course, the same for childcare and childcare expenses, housing, jobs, and all of those things. They all are societal goods that they may not move the needle in terms of raising births, but they can really make having children a much smoother path for people.

Turning to the future, what will the U.S. population look like in 60, 80, 100 years from now?

Size-wise, the U.S. population depends on whether or not we continue immigration at these levels. So most of the U.S. population growth is these days from immigration. It's not from births, and that's not going to change. So I actually would expect those births to keep trending downwards a bit. So what do we end up doing on immigration? Now, I'm a political scientist, and I'll tell you, we're not going to do anything about it. I don't think we'll have any reform. We'll have a lot of talk about it, a lot of stress and bluster, but we won't really see any reform. So I think if U.S. immigration stays around the levels where it is now, we would expect to see population growth in the U.S. We, of course, also would see some different kinds of composition shifts. U.S. population will grow older. We know that that is the case. We also know that it will grow more racially and ethnically diverse, especially at younger generations. So that's the America that awaits us in a few decades.

I'm happy you've mentioned older folks. I'm curious, both in the U.S. and around the world, what leaders should be considering to properly plan for, really, this radical shift in age, given that there will be many more older people who are living longer. What's that mean functionally for the world we live in?

The first step for them is acceptance. I'm often really disturbed when I speak to policymakers that, rather than accepting that this is for sure the case, they're still trying to reverse it. What worries me about that is that we're wasting precious time to put in place the types of systems that can help us adjust to that. In the United States, health is a big key. So we can't work longer if we aren't healthier. Our peers in terms of something we call health spans, think about lifespan and health span, those years, we work healthy, our peers are Mexico, Russia, and China. We would like our peers to be Southern Europe, for example, or maybe Japan, because that allows us to work longer, healthier. Now, of course, not everyone wants to work longer, but we need to at least have this baseline system where those who want to can do so or those who have to can do so.

Finally, Jennifer, what is the biggest thing you want more people to know about population decline?

I think I want people to recognize that a lot of folks in the world, a lot of those in power, really worked for decades to help us live longer, healthier lives, which led to population aging, and to have greater opportunities for education and income-earning activities, which, in part, led to lower birth rates. So when we look at it as nothing but doom and gloom, I think we forget that intentionality behind it. But I think we also need to recognize that the world of the future is going to look different, but if we prepare for this and we make really tough changes, we can actually thrive in this world. Then also, that population aging is not necessarily the same as individual aging, where there'll be no innovation anymore, or we'll just be societies that are dusty and gray. There will always be young people, middle-aged people, and older people, just in different proportions. So we need to work towards thriving multi-generational communities.

Jennifer Sciubba, fascinating insight and perspective for us here. Thanks for joining us on The Excerpt.

Thanks to our senior producer, Shannon Rae Green, for her production assistance. Our executive producer is Laura Beatty. Let us know what you think of this episode by sending a note to [email protected]. Thanks for listening. I'm Taylor Wilson, and I'll be back tomorrow morning with another episode of The Excerpt.

Addition of Fractions Questions

The concept of fractional addition is one of the most fundamental topics taught in elementary and secondary schools. Students can easily understand a variety of addition of fractions questions with detailed explanations by using the resources below. These questions can be used by students to acquire a quick overview of the topics. Examine the detailed explanations for each question to double-check your answers. Click here for more information about the addition of fractions .

Addition of Fractions Questions with Solutions

1. Add the fraction: 5/7 + 2/7.

Given: 5/7 + 2/7

Here, the denominators of both fractions are the same, i.e. 7.

Hence, we can add the numerators and divide them by the common denominator.

5/7 + 2/7 = (5+2)/7

5/7 + 2/7 = 7/7

Further, the fractions can be simplified as follows:

5/7 + 2/7 = 1/1.

2. Find the sum: 10/2 + 5/2.

Given: 10/2 + 5/2.

Since the denominators of the given fractions are equal, we can add the numerators.

Hence, we get

10/2 + 5/2 = (10 + 5)/2

10/2 + 5/2 = 15/2

Now, the fraction 15/2 cannot be simplified further.

Therefore, 10/2 + 5/2 is 15/2.

3. Add the fractions: 71/9 + 19/9.

Given: 71/9 + 19/9.

Here, the denominators of the given two fractions are the same.

71/9 + 19/9 = (71+19)/9

71/9 + 19/9 = 90/9

Thus, it can be further simplified as:

71/9 + 19/9 = 10/1

4. Find the sum of 12/3 and 3/3.

Given two fractions are 12/3 and 3/3.

Here, both fractions have the same denominator, i.e. 3.

Hence, we can directly add the numerator and keep the common denominator as 3.

12/3 + 3/3 = (12+3)/3

12/3 + 3/3 = 15/3

Now, the simplified form is:

12/3 + 3/3 = 5/1.

Therefore, the sum of 12/3 and 3/3 is 5/1.

5. Add the fraction: 2/3 + ½.

Given fractions are 2/3 and 1/2.

Here, the denominators of the fractions are different. So, take the LCM of 2 and 3 to make denominators the same.

Thus, LCM of 2 and 3 is 6.

2/3 + 1/2 = 4/6 + 3/6

Now, the denominator of both fractions are the same, i.e. 6

2/3 + 1/2 = (4 + 3)/6

2/3 + 1/2 = 7/6, which cannot be simplified further.

Hence, 2/3 + 1/2 is 7/6.

Also, read : Like and Unlike Fractions .

6. Find the sum of the fractions 4/3 and 3/5.

Given fractions: 4/3 and 3/5.

Since the denominators are different, take the LCM of 3 and 5.

Thus, the LCM of 3 and 5 is 15.

Hence, 4/3 + 3/5 = 20/15 + 9/15

4/3 + 3/5 = (20 + 9)/15

4/3 + 3/5 = 29/15.

Now, the fraction 29/15 cannot be simplified further.

Hence, the sum of 4/3 and 3/5 is 29/15.

7. 2/7 + 5/4 = _____.

Given: 2/7 + 5/4

Take the LCM of 7 and 4 since the denominators are different.

As we know, the LCM of 4 and 7 is 28.

Therefore, 2/7 + 5/4 = 8/28 + 35/28

Now, add the numerators, we get

2/7 + 5/4 = (8 + 35) / 28

2/7 + 5/4 = 43/28

Hence, 2/7 + 5/4 = 43/28.

8. The sum of 8/3 and 4/9 is ____.

Here, the two fractions are 8/3 and 4/9.

Now, the LCM of 3 and 9 is 9.

Hence, 8/3 + 4/9 = 24/9 + 4/9

8/3 + 4/9 = (24 + 4) / 9

8/3 + 4/9 = 28/9

Therefore, the sum of 8/3 and 4/9 is 28/9.

9. Add: 7 + 2/3.

Given: 7 + 2/3.

Now, write 7 in the form of a fraction.

i.e ., 7 = 7/1

Hence, 7 + 2/3 = 7/1 + 2/3

Take, LCM of 1 and 3, which is equal to 3.

Hence, 7 + 2/3 = 21/3 + 2/3

7 + 2/3 = (21 + 2) / 3

7 + 2/3 = 23/3.

Therefore, the sum of 7 + 2/3 is 23/3.

10. The sum of 12 and 4/5 is ____.

Given: 12 + 4/5.

Thus, 12 + 4/5 = 12/1 + 4/5

Now, the LCM of 1 and 5 is 5.

Hence, 12 + 4/5 = 60/5 + 4/5

12 + 4/5 = (60 + 4)/5

12 + 4/5 = 64/5.

Hence, the sum of 12 and 4/5 is 64/5.

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Practice Questions

Answer the following questions:

• Find the sum of 26/5 and 12/5.
• The sum of 17/9 and 14/7 is ____.
• 12 + 16/9 = ____.

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MIT Technology Review

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How machines that can solve complex math problems might usher in more powerful AI

Google DeepMind’s AlphaProof and AlphaGeometry 2 are milestones for AI reasoning.

• Melissa Heikkilä archive page

This story originally appeared in The Algorithm, our weekly newsletter on AI. To get stories like this in your inbox first, sign up here .

It’s been another big week in AI. Meta updated its powerful new Llama model, which it’s handing out for free, and OpenAI said it is going to trial an AI-powered online search tool that you can chat with, called SearchGPT. 

But the news item that really stood out to me was one that didn’t get as much attention as it should have. It has the potential to usher in more powerful AI and scientific discovery than previously possible. 

Last Thursday, Google DeepMind announced it had built AI systems that can solve complex math problems. The systems—called AlphaProof and AlphaGeometry 2—worked together to successfully solve four out of six problems from this year’s International Mathematical Olympiad, a prestigious competition for high school students. Their performance was the equivalent of winning a silver medal. It’s the first time any AI system has ever achieved such a high success rate on these kinds of problems.  My colleague Rhiannon Williams has the news here . 

Math! I can already imagine your eyes glazing over. But bear with me. This announcement is not just about math.  In fact, it signals an exciting new development in the kind of AI we can now build.  AI search engines that you can chat with may add to the illusion of intelligence, but systems like Google DeepMind’s could improve the actual intelligence of AI. For that reason, building systems that are better at math has been a goal for many AI labs, such as  OpenAI .  

That’s because math is a benchmark for reasoning . To complete these exercises aimed at high school students, the AI system needed to do very complex things like planning to understand and solve abstract problems. The systems were also able to generalize, allowing them to solve a whole range of different problems in various  branches of mathematics. 

“What we’ve seen here is that you can combine [reinforcement learning] that was so successful in things like AlphaGo with large language models and produce something which is extremely capable in the space of text,” David Silver, principal research scientist at Google DeepMind and indisputably a pioneer of deep reinforcement learning, said in a press briefing. In this case, that capability was used to construct programs in the computer language Lean that represent mathematical proofs. He says the International Mathematical Olympiad represents a test for what’s possible and paves the way for further breakthroughs. 

This same recipe could be applied in any situation with really clear, verified reward signals for reinforcement-learning algorithms and an unambiguous way to measure correctness as you can in mathematics, said Silver. One potential application would be coding, for example. 

Now for a compulsory reality check:  AlphaProof and AlphaGeometry 2 can still only solve hard high-school-level problems. That’s a long way away from the extremely hard problems top human mathematicians can solve. Google DeepMind stressed that its tool did not, at this point, add anything to the body of mathematical knowledge humans have created. But that wasn’t the point. 

“We are aiming to provide a system that can prove anything,” Silver said. Think of an AI system as reliable as a calculator, for example, that can provide proofs for many challenging problems, or verify tests for computer software or scientific experiments. Or perhaps build better AI tutors that can give feedback on exam results, or fact-check news articles. 

But the thing that excites me most is what Katie Collins, a researcher at the University of Cambridge who specializes in math and AI (and was not involved in the project), told Rhiannon.  She says these tools create and evaluate new problems, motivate new people to enter the field, and spark more wonder. That’s something we definitely need more of in this world.

Now read the rest of The Algorithm

Deeper learning.

A new tool for copyright holders can show if their work is in AI training data

Since the beginning of the generative AI boom, content creators have argued that their work has been scraped into AI models without their consent. But until now, it has been difficult to know whether specific text has actually been used in a training data set. Now they have a new way to prove it: “copyright traps.” These are pieces of hidden text that let you mark written content in order to later detect whether it has been used in AI models or not. 

Why this matters:  Copyright traps tap into one of the biggest fights in AI. A number of publishers and writers are in the middle of litigation against tech companies, claiming their intellectual property has been scraped into AI training data sets without their permission. The idea is that these traps could help to nudge the balance a little more in the content creators’ favor.  Read more from me here . 

Bits and Bytes

AI trained on AI garbage spits out AI garbage New research published in Nature shows that the quality of AI models’ output gradually degrades when it’s trained on AI-generated data. As subsequent models produce output that is then used as training data for future models, the effect gets worse. ( MIT Technology Review ) 

OpenAI unveils SearchGPT  The company says it is testing new AI search features that give you fast and timely answers with clear and relevant sources cited. The idea is for the technology to eventually be incorporated into ChatGPT, and CEO  Sam Altman says  it’ll be possible to do voice searches. However, like many other AI-powered search services, including  Google’s , it’s already making errors, as  the Atlantic  reports.  ( OpenAI ) 

AI video generator Runway trained on thousands of YouTube videos without permission Leaked documents show that the company was secretly training its generative AI models by scraping thousands of videos from popular YouTube creators and brands, as well as pirated films. ( 404 media ) 

Artificial intelligence

How to opt out of meta’s ai training.

Your posts are a gold mine, especially as companies start to run out of AI training data.

Why does AI hallucinate?

The tendency to make things up is holding chatbots back. But that’s just what they do.

• Will Douglas Heaven archive page

How generative AI could reinvent what it means to play

AI-powered NPCs that don’t need a script could make games—and other worlds—deeply immersive.

• Niall Firth archive page

Synthesia’s hyperrealistic deepfakes will soon have full bodies

With bodies that move and hands that wave, deepfakes just got a whole lot more realistic.

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