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How to Solve Fraction Questions in Math

Last Updated: February 24, 2023 References Approved

This article was co-authored by Mario Banuelos, PhD and by wikiHow staff writer, Sophia Latorre . Mario Banuelos is an Assistant Professor of Mathematics at California State University, Fresno. With over eight years of teaching experience, Mario specializes in mathematical biology, optimization, statistical models for genome evolution, and data science. Mario holds a BA in Mathematics from California State University, Fresno, and a Ph.D. in Applied Mathematics from the University of California, Merced. Mario has taught at both the high school and collegiate levels. There are 7 references cited in this article, which can be found at the bottom of the page. wikiHow marks an article as reader-approved once it receives enough positive feedback. This article has 17 testimonials from our readers, earning it our reader-approved status. This article has been viewed 1,152,113 times.

Fraction questions can look tricky at first, but they become easier with practice and know-how. Start by learning the terminology and fundamentals, then pratice adding, subtracting, multiplying, and dividing fractions. [1] X Research source Once you understand what fractions are and how to manipulate them, you'll be breezing through fraction problems in no time.

Doing Calculations with Fractions

• For instance, to solve 5/9 + 1/9, just add 5 + 1, which equals 6. The answer, then, is 6/9 which can be reduced to 2/3.

• For instance, to solve 6/8 - 2/8, all you do is take away 2 from 6. The answer is 4/8, which can be reduced to 1/2.

• For example, if you need to add 1/2 and 2/3, start by determining a common multiple. In this case, the common multiple is 6 since both 2 and 3 can be converted to 6. To turn 1/2 into a fraction with a denominator of 6, multiply both the numerator and denominator by 3: 1 x 3 = 3 and 2 x 3 = 6, so the new fraction is 3/6. To turn 2/3 into a fraction with a denominator of 6, multiply both the numerator and denominator by 2: 2 x 2 = 4 and 3 x 2 = 6, so the new fraction is 4/6. Now, you can add the numerators: 3/6 + 4/6 = 7/6. Since this is an improper fraction, you can convert it to the mixed number 1 1/6.
• On the other hand, say you're working on the problem 7/10 - 1/5. The common multiple in this case is 10, since 1/5 can be converted into a fraction with a denominator of 10 by multiplying it by 2: 1 x 2 = 2 and 5 x 2 = 10, so the new fraction is 2/10. You don't need to convert the other fraction at all. Just subtract 2 from 7, which is 5. The answer is 5/10, which can also be reduced to 1/2.

• For instance, to multiply 2/3 and 7/8, find the new numerator by multiplying 2 by 7, which is 14. Then, multiply 3 by 8, which is 24. Therefore, the answer is 14/24, which can be reduced to 7/12 by dividing both the numerator and denominator by 2.

• For example, to solve 1/2 ÷ 1/6, flip 1/6 upside down so it becomes 6/1. Then just multiply 1 x 6 to find the numerator (which is 6) and 2 x 1 to find the denominator (which is 2). So, the answer is 6/2 which is equal to 3.

Practicing the Basics

• For instance, in 3/5, 3 is the numerator so there are 3 parts and 5 is the denominator so there are 5 total parts. In 7/8, 7 is the numerator and 8 is the denominator.

• If you need to turn 7 into a fraction, for instance, write it as 7/1.

• For example, if you have the fraction 15/45, the greatest common factor is 15, since both 15 and 45 can be divided by 15. Divide 15 by 15, which is 1, so that's your new numerator. Divide 45 by 15, which is 3, so that's your new denominator. This means that 15/45 can be reduced to 1/3.

• Say you have the mixed number 1 2/3. Stary by multiplying 3 by 1, which is 3. Add 3 to 2, the existing numerator. The new numerator is 5, so the mixed fraction is 5/3.

Tip: Typically, you'll need to convert mixed numbers to improper fractions if you're multiplying or dividing them.

• Say that you have the improper fraction 17/4. Set up the problem as 17 ÷ 4. The number 4 goes into 17 a total of 4 times, so the whole number is 4. Then, multiply 4 by 4, which is equal to 16. Subtract 16 from 17, which is equal to 1, so that's the remainder. This means that 17/4 is the same as 4 1/4.

Fraction Calculator, Practice Problems, and Answers

Community Q&A

• Take the time to carefully read through the problem at least twice so you can be sure you know what it's asking you to do. Thanks Helpful 2 Not Helpful 0
• Check with your teacher to find out if you need to convert improper fractions into mixed numbers and/or reduce fractions to their lowest terms to get full marks. Thanks Helpful 2 Not Helpful 0
• To take the reciprocal of a whole number, just put a 1 over it. For example, 5 becomes 1/5. Thanks Helpful 1 Not Helpful 0

You Might Also Like

• ↑ https://www.sparknotes.com/math/prealgebra/fractions/terms/
• ↑ https://www.bbc.co.uk/bitesize/articles/z9n4k7h
• ↑ https://www.mathsisfun.com/fractions_multiplication.html
• ↑ https://www.mathsisfun.com/fractions_division.html
• ↑ https://medium.com/i-math/the-no-nonsense-straightforward-da76a4849ec
• ↑ https://www.youtube.com/watch?v=PcEwj5_v75g
• ↑ https://sciencing.com/solve-math-problems-fractions-7964895.html

About This Article

To solve a fraction multiplication question in math, line up the 2 fractions next to each other. Multiply the top of the left fraction by the top of the right fraction and write that answer on top, then multiply the bottom of each fraction and write that answer on the bottom. Simplify the new fraction as much as possible. To divide fractions, flip one of the fractions upside-down and multiply them the same way. If you need to add or subtract fractions, keep reading! Did this summary help you? Yes No

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Home / United States / Math Classes / 5th Grade Math / Problem Solving using Fractions

Problem Solving using Fractions

Fractions are numbers that exist between whole numbers. We get fractions when we divide whole numbers into equal parts. Here we will learn to solve some real-life problems using fractions. ...Read More Read Less

Table of Contents

What are Fractions?

Types of fractions.

• Fractions with like and unlike denominators
• Operations on fractions
• Fractions can be multiplied by using
• Let’s take a look at a few examples

Solved Examples

• Frequently Asked Questions

Equal parts of a whole or a collection of things are represented by fractions . In other words a fraction is a part or a portion of the whole. When we divide something into equal pieces, each part becomes a fraction of the whole.

For example in the given figure, one pizza represents a whole. When cut into 2 equal parts, each part is half of the whole, that can be represented by the fraction  \(\frac{1}{2}\) . 

Similarly, if it is divided into 4 equal parts, then each part is one fourth of the whole, that can be represented by the fraction \(\frac{1}{4}\) .

Proper fractions

A fraction in which the numerator is less than the denominator value is called a  proper fraction.

For example ,  \(\frac{3}{4}\) ,  \(\frac{5}{7}\) ,  \(\frac{3}{8}\)   are proper fractions.

Improper fractions 

A fraction with the numerator higher than or equal to the denominator is called an improper fraction .

Eg \(\frac{9}{4}\) ,  \(\frac{8}{8}\) ,  \(\frac{9}{4}\)   are examples of improper fractions.

Mixed fractions

A mixed number or a mixed fraction is a type of fraction which is a combination of both a whole number and a proper fraction.

We express improper fractions as mixed numbers.

For example ,  5\(\frac{1}{3}\) ,  1\(\frac{4}{9}\) ,  13\(\frac{7}{8}\)   are mixed fractions.

Unit fraction

A unit fraction is a fraction with a numerator equal to one. If a whole or a collection is divided into equal parts, then exactly 1 part of the total parts represents a unit fraction .

Fractions with Like and Unlike Denominators

Like fractions are those in which two or more fractions have the same denominator, whereas unlike fractions are those in which the denominators of two or more fractions are different.

For example,  

\(\frac{1}{4}\)  and  \(\frac{3}{4}\)  are like fractions as they both have the same denominator, that is, 4.

\(\frac{1}{3}\)  and  \(\frac{1}{4}\)   are unlike fractions as they both have a different denominator.

Operations on Fractions

We can perform addition, subtraction, multiplication and division operations on fractions.

Fractions with unlike denominators can be added or subtracted using equivalent fractions. Equivalent fractions can be obtained by finding a common denominator. And a common denominator is obtained either by determining a common multiple of the denominators or by calculating the product of the denominators.

There is another method to add or subtract mixed numbers, that is, solve the fractional and whole number parts separately, and then, find their sum to get the final answer.

Fractions can be Multiplied by Using:

Division operations on fractions can be performed using a tape diagram and area model. Also, when a fraction is divided by another fraction then we can solve it by multiplying the dividend with the reciprocal of the divisor. 

Let’s Take a Look at a Few Examples

Addition and subtraction using common denominator

( \(\frac{1}{6} ~+ ~\frac{2}{5}\) )

We apply the method of equivalent fractions. For this we need a common denominator, or a common multiple of the two denominators 6 and 5, that is, 30.

\(\frac{1}{6} ~+ ~\frac{2}{5}\)

= \(\frac{5~+~12}{30}\)  

=  \(\frac{17}{30}\) 

( \(\frac{5}{2}~-~\frac{1}{6}\) )

= \(\frac{12~-~5}{30}\)

= \(\frac{7}{30}\)

Examples of Multiplication and Division

Multiplication:

(\(\frac{1}{6}~\times~\frac{2}{5}\))

= (\(\frac{1~\times~2}{6~\times~5}\))                                       [Multiplying numerator of fractions and multiplying denominator of fractions]

=  \(\frac{2}{30}\)

(\(\frac{2}{5}~÷~\frac{1}{6}\))

= (\(\frac{2 ~\times~ 5}{6~\times~ 1}\))                                     [Multiplying dividend with the reciprocal of divisor]

= (\(\frac{2 ~\times~ 6}{5 ~\times~ 1}\))

= \(\frac{12}{5}\)

Example 1: Solve \(\frac{7}{8}\) + \(\frac{2}{3}\)

Let’s add \(\frac{7}{8}\)  and  \(\frac{2}{3}\)   using equivalent fractions. For this we need to find a common denominator or a common multiple of the two denominators 8 and 3, which is, 24.

\(\frac{7}{8}\) + \(\frac{2}{3}\)

= \(\frac{21~+~16}{24}\)    

= \(\frac{37}{24}\)

Example 2: Solve \(\frac{11}{13}\) – \(\frac{12}{17}\)

Solution:  

Let’s subtract  \(\frac{12}{17}\) from \(\frac{11}{13}\)   using equivalent fractions. For this we need a common denominator or a common multiple of the two denominators 13 and 17, that is, 221.

\(\frac{11}{13}\) – \(\frac{12}{17}\)

= \(\frac{187~-~156}{221}\)

= \(\frac{31}{221}\)

Example 3: Solve \(\frac{15}{13} ~\times~\frac{18}{17}\)

Multiply the numerators and multiply the denominators of the 2 fractions.

\(\frac{15}{13}~\times~\frac{18}{17}\)

= \(\frac{15~~\times~18}{13~~\times~~17}\)

= \(\frac{270}{221}\)

Example 4: Solve \(\frac{25}{33}~\div~\frac{41}{45}\)

Divide by multiplying the dividend with the reciprocal of the divisor.

\(\frac{25}{33}~\div~\frac{41}{45}\)

= \(\frac{25}{33}~\times~\frac{41}{45}\)                            [Multiply with reciprocal of the divisor \(\frac{41}{45}\) , that is, \(\frac{45}{41}\)  ]

= \(\frac{25~\times~45}{33~\times~41}\)

= \(\frac{1125}{1353}\)

Example 5: 

Sam was left with   \(\frac{7}{8}\)  slices of chocolate cake and    \(\frac{3}{7}\)  slices of vanilla cake after he shared the rest with his friends. Find out the total number of slices of cake he had with him. Sam shared   \(\frac{10}{11}\)  slices from the total number he had with his parents. What is the number of slices he has remaining?

To find the total number of slices of cake he had after sharing we need to add the slices of each cake he had,

=   \(\frac{7}{8}\) +   \(\frac{3}{7}\)   

=   \(\frac{49~+~24}{56}\)

=   \(\frac{73}{56}\)

To find out the remaining number of slices Sam has   \(\frac{10}{11}\)  slices need to be deducted from the total number,

= \(\frac{73}{56}~-~\frac{10}{11}\)

=   \(\frac{803~-~560}{616}\)

=   \(\frac{243}{616}\)

Hence, after sharing the cake with his friends, Sam has  \(\frac{73}{56}\) slices of cake, and after sharing with his parents he had  \(\frac{243}{616}\)  slices of cake left with him.

Example 6: Tiffany squeezed oranges to make orange juice for her juice stand. She was able to get 25 ml from one orange. How many oranges does she need to squeeze to fill a jar of   \(\frac{15}{8}\) liters? Each cup that she sells carries 200 ml and she sells each cup for 64 cents. How much money does she make at her juice stand?

First  \(\frac{15}{8}\) l needs to be converted to milliliters.

\(\frac{15}{8}\)l into milliliters =  \(\frac{15}{8}\) x 1000 = 1875 ml

To find the number of oranges, divide the total required quantity by the quantity of juice that one orange can give.

The number of oranges required for 1875 m l of juice =  \(\frac{1875}{25}\) ml = 75 oranges

To find the number of cups she sells, the total quantity of juice is to be divided by the quantity of juice that 1 cup has

=  \(\frac{1875}{200}~=~9\frac{3}{8}\) cups

We know that, the number of cups cannot be a fraction, it has to be a whole number. Also each cup must have 200ml. Hence with the quantity of juice she has she can sell 9 cups,   \(\frac{3}{8}\) th  of a cup cannot be sold alone.

Money made on selling 9 cups = 9 x 64 = 576 cents

Hence she makes 576 cents from her juice stand.

What is a mixed fraction?

A mixed fraction is a number that has a whole number and a fractional part. It is used to represent values between whole numbers.

How will you add fractions with unlike denominators?

When adding fractions with unlike denominators, take the common multiple of the denominators of both the fractions and then convert them into equivalent fractions. 

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Algebra: Fraction Problems

Related Topics: More Algebra Word Problems

In these lessons, we will learn how to solve fraction word problems that deal with fractions and algebra. Remember to read the question carefully to determine the numerator and denominator of the fraction.

We will also learn how to solve word problems that involve comparing fractions, adding mixed numbers, subtracting mixed numbers, multiplying fractions and dividing fractions.

Fraction Word Problems using Algebra

Example: 2/3 of a number is 14. What is the number?

Answer: The number is 21.

Example: The numerator of a fraction is 3 less than the denominator. When both the numerator and denominator are increased by 4, the fraction is increased by fraction.

Solution: Let the numerator be x, then the denominator is x + 3, and the fraction is \(\frac{x}{{x + 3}}\) When the numerator and denominator are increased by 4, the fraction is \(\frac{{x + 4}}{{x + 7}}\) \(\frac{{x + 4}}{{x + 7}} - \frac{x}{{x + 3}} = \frac{{12}}{{77}}\) 77(x + 4)(x + 3) – 77x(x+7) = 12(x + 7)(x + 3) 77x 2 + 539x + 924 – 77x 2 – 539x = 12x 2 + 120x + 252 12x 2 + 120x – 672 = 0 x 2 + 10x – 56 = 0 (x – 4)(x + 14) = 0 x = 4 (negative answer not applicable in this case)

How to solve Fraction Word Problems using Algebra? Examples: (1) The denominator of a fraction is 5 more than the numerator. If 1 is subtracted from the numerator, the resulting fraction is 1/3. Find the original fraction. (2) If 3 is subtracted from the numerator of a fraction, the value of the resulting fraction is 1/2. If 13 is added to the denominator of the original fraction, the value of the new fraction is 1/3. Find the original fraction. (3) A fraction has a value of 3/4. When 14 is added to the numerator, the resulting fraction has a value equal to the reciprocal of the original fraction, Find the original fraction.

Algebra Word Problems with Fractional Equations Solving a fraction equation that appears in a word problem Example: One third of a number is 6 more than one fourth of the number. Find the number.

Fraction and Decimal Word Problems How to solve algebra word problems with fractions and decimals? Examples: (1) If 1/2 of the cards had been sold and there were 172 cards left, how many cards were printed? (2) Only 1/3 of the university students wanted to become teachers. If 3,360 did not wan to become teachers, how many university were there? (3) Rodney guessed the total was 34.71, but this was 8.9 times the total. What was the total?

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Represent and Simplify the Fractions: Type 1

Presented here are the fraction pdf worksheets based on real-life scenarios. Read the basic fraction word problems, write the correct fraction and reduce your answer to the simplest form.

Represent and Simplify the Fractions: Type 2

Before representing in fraction, children should perform addition or subtraction to solve these fraction word problems. Write your answer in the simplest form.

Adding Fractions Word Problems Worksheets

Conjure up a picture of how adding fractions plays a significant role in our day-to-day lives with the help of the real-life scenarios and circumstances presented as word problems here.

(15 Worksheets)

Subtracting Fractions Word Problems Worksheets

Crank up your skills with this set of printable worksheets on subtracting fractions word problems presenting real-world situations that involve fraction subtraction!

Multiplying Fractions Word Problems Worksheets

This set of printables is for the ardently active children! Explore the application of fraction multiplication and mixed-number multiplication in the real world with this exhilarating practice set.

Fraction Division Word Problems Worksheets

Gift children a broad view of the real-life application of dividing fractions! Let them divide fractions by whole numbers, divide 2 fractions, divide mixed numbers, and solve the word problems here.

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Step by Step Tutorial on How to Solve Fractions with Examples

Are you looking for the best method to lose weight? If yes!! Let’s know how to solve fractions. Sounds weird? Yes, you read it right! You need to know Body Mass Index (BMI) using fractions to lose weight effectively. 

Ladies, does the jeweler give you 18 or 24-karat jewelry? 24 karats are considered pure gold, whereas 18 karats mean 18/24, equal to 75% gold. This is how you can use fractions to know jewelry purity.

This blog will help you to understand how to solve fractions using various methods. Moreover, I have listed some useful tips to use fractions. So, without creating more confusion, let’s understand the concept of fractions in detail. 

Necessary things to know about Fraction!!

Table of Contents

First, understand what fraction is.  

Fraction is basically a numerical quantity or value that is not a whole number, for example, 4/5, 6/7, etc. 

The next thing to learn is the terms used for infractions. So each and every fraction has two parts. 

For example, In the 3/5 fraction, 3 is the numerator, and 5 is the denominator. 

Here 3 states that it is the 3rd part of the whole number, and 5 is signifying that the whole number has 5 parts. 

The second thing to learn about types of fractions. 

There are three types of fractions that we deal with. Let’s check each of them one by one.

• Proper fraction: In this fraction, the numerator has less value as compared to the denominator. 
• Improper fraction: In this fraction, the numerator is always greater than the denominator.
• Mixed fraction: In this fraction, the number is represented as the whole number, which is followed by the fractional numbers.

How to represent fractions in different forms?

Representing the mixed fraction to the whole number..

Let’s understand the steps to solve mixed fractions to the whole number with an example.

Change 7⅘ into the whole number.

Representing the fraction to the decimal number.

The easiest method to change the fraction into decimal is just by dividing the number. Here, you require to divide the numerator with the denominator.

Change the fraction 7/10 into decimal.

Representing the fraction to the percentage.

Three methods can be used for converting the fraction to a percentage. Below, I have given three different methods taking the example 7/20. 

Tutorial step-wise – How to solve Fractions 

Let’s first learn how to add two or more fractions with the help of an example. .

Suppose you have to add 3/4 with 1/4.

Here you can see the denominators are the same, so it is the simplest addition in fractions. 

So the first step in how to solve fractions by addition is to find the common denominators of the numbers. In this problem, both have the same denominator, so the common between them is 4 only. 

So you can write the equation as follows – 

3/4 + 1/4 

If you have different denominators, then also you easily solve the fractions. Let’s learn through different example – 

Suppose you have to add 3/4 with 2/5, then you will have the following equation – 

3/4 + 2/5 

Then the next step to solving this fraction in how to solve fractions is to find out any common denominator. 

Since it has no common denominator, we will multiply both the denominators and add the above numbers. Look at the below to understand better.

= (3+2) / 4×5 

= 5/20 

Now let’s take one more example where we can find the common denominators. 

Suppose the fractions we need to add are ¾ and ⅝

= 3/4 + 5/8 

Now we will find the least common factor that is LCM between the two denominators. 

Then we will get 8 as LCM, and thus the equation will be as follows –

= (3×2 + 5)/ 8 

Since the denominator is 8 and the denominator of the first fraction is 4, so we will multiply the first fraction with 2 to make the denominator of the fraction is 8; thus, the equation will be 

= 11/8 

So the answer is 11/8.

Now let’s learn the subtraction of fractions in how to solve fractions.

Suppose the equation is 3/2 – 1/2 

So we will follow the same process, and we will first bring out the common factor since the denominator is the same in this equation, so there is no such issue. 

Now you can directly put the 2 as a denominator and then subtract the 1 from 3. Follow the steps – 

We have got the answer that is 1. 

Another example of how to solve fractions

5/7 – 2/4 

We can see no common factor between the denominators, so we make the denominators the same by multiplying the first fraction with 4 and the second fraction with 7; then, we will get the following equation. 

= 5×4/7×4 – 2×7/4×7 

We have to make common denominators in order to solve the equation, as then only we can perform the Luther operations in the equation. Thus we will get the following answer – 

=(20-14)/7×4

Now we can see 2 is common to both numerator and denominator, so we can divide the whole fraction with 2 in the following way – 

= 3/14 

Thus the answer is 3/14.

Now let’s learn how to multiply two fractions. This is also very important to learn how to solve fractions. 

Let’s take the following example – 

¾ x ⅕ 

Multiplication is very easy in fractions as you just have to multiply numerators with each other and likewise denominators with each other. 

Then you will get the following result – 

= 3×1 / 4×5 

=3/20 

Thus the answer is 3/20.

Now let’s learn the method of division in fractions in how to solve fractions.

You can take the reciprocal of the fraction to divide the fraction. To reciprocate, you need to switch the denominator to the numerator and the numerator to the denominator.

Let’s take an example of it:

Solve the 1/2 ÷ 1/5.

First, take the reciprocal of 1/5 as 5/1.

Take the reciprocal fraction and multiply it with another number (s).

To solve it, multiply the denominators and numerators:

2 * 1 = 2 (denominator)

1 * 5 = 5 (numerator) 

That is : 1/2 * 5/1 = 5/2 = 2.5

Things to understand to avoid common mistakes in fractions!

It might not be easy to add and subtract the fractions with different denominators. That is why various students fail to solve the fractions with different denominators and make some mistakes. 

First, let’s an example of solving fractions with different denominators .

Now, understand what kind of mistakes students make.

• Misunderstand the requirements of the questions, such as dividing instead of multiplying and so on.
• When students need to add or subtract a fraction, they forget to change the fraction’s denominator. [Like in the above examples 4 and 6 change to 24].
• Moreover, it is noticeable that the numerator also needs to change as that of the denominator. [Like 3 * 6 =18 and 1 * 4 = 4].
• Finally, some students are unable to simplify the equation. [Like 22/24 also written as 11/12 after dividing the number by 2].

Many students are struggling with fractions, and fractions look tough at first instance but are easy when you practice them on a daily basis. 

If you are also searching for how to solve fractions , then we hope that this article would have helped you in understanding the process of solving fractions. If still, you are facing difficulty in solving fractions, then you can contact us anytime. We are always here to help you. Our professionals will give you 24/7 guidance. Get the best math homework help from the experts.

Frequently Asked Questions

Q1. what is the formula for fractions.

Fraction = selected number of parts / total number of parts Each fraction has a numerator, which is equal to the selected number of parts, and a denominator is equal to the total number of parts as a whole.

Q2. What is A and B in fraction?

In the fraction, A and B are considered as A/B. Where the number A is the numerator, whereas the B is the denominator.

Q3. What is 1/3 equivalent to as a fraction?

The 1/3 fraction is equivalent to: 2/6, 3/9, 4/12, 5/15, 6/18, 7/21, 8/24, 9/27, 10/30, and so on.

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5th Grade Math Word Problems Worksheets

Math word problem worksheets for grade 5.

These worksheets present students with real world word problems that students can solve with grade 5 math concepts. 

We encourage students to think about the problems carefully by:

• providing a number of mixed word problem worksheets
• including irrelevant data  so students need to understand the context before applying a solution

The four operations

Mixed 4 operations

Estimating and rounding word problems

Grade 5 fractions and decimal word problems

Addition and subtraction of fractions

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Explore all of our math word problem worksheets , from kindergarten through grade 5.

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The math problem: Kids are still behind. How can schools catch them up?

Aggie Gambino, left, helps one of her twin ten-year-old daughters, Giuliana, right, work on math worksheets as they go through homework from school at the dining room table in their home Wednesday, Aug. 23, 2023, in Spring, Texas. Aggie has often found herself searching YouTube for math videos to help her children with math. (AP Photo/Michael Wyke)

U.S. schools are scrambling to catch up students in math as post-pandemic test scores reveal the depth of missing skills. (Camilla Forte/The Hechinger Report via AP)

The Gambino family works on math worksheets as they go through homework from school at the dining room table in their home Wednesday, Aug. 23, 2023, in Spring, Texas. Across the country, schools are scrambling to catch up students in math as post-pandemic test scores reveal the depth of missing skills. (AP Photo/Michael Wyke)

Ayub Mohamed, left, 7, going into 2nd grade, gets help from Esmeralda Jimenez, 13, a volunteer tutor in a summer tutoring program with School Connect WA at Dearborn Park International Elementary School in Seattle on Friday, July 28, 2023. (Karen Ducey/The Seattle Times via AP)

Ayub Mohamed, left, 7 years, gets help from Esmeralda Jimenez, 13, left back, volunteer tutor; while Olivia Elaydo, 7, right back, and Eden Pollard, 6, right, work on math problems during a summer tutoring program with School Connect WA at Dearborn Park International Elementary School in Seattle on Friday, July 28, 2023. (Karen Ducey/The Seattle Times via AP)

Emeli Ousitu, 11, left and Diana Chen, right, 11, work on division and multiplication of fractions with Roy Chang, executive director for School Connect WA at Dearborn Park International Elementary School in Seattle on Friday, July 28, 2023. (Karen Ducey/The Seattle Times via AP)

Aggie Gambino, right, helps her ten-year-old daughter, Giada, left, work on math worksheets as twin sister Giuliana, center back, gets a snack in the kitchen as they do homework from school at the dining room table in their home Wednesday, Aug. 23, 2023, in Spring, Texas. “The more parents understand how they’re being taught,” she says, “the better participant they can be in their child’s learning.” (AP Photo/Michael Wyke)

Aggie Gambino, center, helps her twin ten-year-old daughters, Giada, left, and Giuliana, right, work on math worksheets as they go through homework from school at the dining room table in their home Wednesday, Aug. 23, 2023, in Spring, Texas. (AP Photo/Michael Wyke)

Giada Gambino, 10, left, becomes frustrated with a problem on a math worksheet from school as her mother helps her work through it at the dining room table in their home Wednesday, Aug. 23, 2023, in Spring, Texas. (AP Photo/Michael Wyke)

Giada Gambino, 10, left, uses her smartphone to consult a voice assistant about a math problem as her mother, Aggie, center, and twin sister Giuliana, right, look on while they work on math worksheets from school at the dining room table in their home Wednesday, Aug. 23, 2023, in Spring, Texas. (AP Photo/Michael Wyke)

Aggie Gambino, center, sits with her twin ten-year-old daughters, Giada, left, and Giuliana, right, at their home Wednesday, Aug. 23, 2023, in Spring, Texas. (AP Photo/Michael Wyke)

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On a breezy July morning in South Seattle, a dozen elementary-aged students ran math relays behind an elementary school.

One by one, they raced to a table, where they scribbled answers to multiplication questions before sprinting back to high-five their teammate. These students are part of a summer program run by the nonprofit School Connect WA, designed to help them catch up on math and literacy skills lost during the pandemic . There are 25 students in the program, and all of them are one to three grades behind.

One 11-year-old boy couldn’t do two-digit subtraction. Thanks to the program and his mother, who has helped him each night, he’s caught up. Now, he says math is challenging, but he likes it.

Other kids haven’t fared so well.

Across the country, schools are scrambling to catch up students in math as post-pandemic test scores reveal the depth of missing skills. On average, students’ math knowledge is about half a school year behind where it should be, according to education analysts.

Children lost ground on reading tests , too, but the math declines were particularly striking. Experts say virtual learning complicated math instruction, making it tricky for teachers to guide students over a screen or spot weaknesses in problem-solving skills. Plus, parents were more likely to read with their children at home than practice math.

The result: Students’ math skills plummeted across the board, exacerbating racial and socioeconomic inequities in math performance. And students aren’t bouncing back as quickly as educators hoped, supercharging worries about how they will fare in high school and whether science, tech and medical fields will be available to them.

The Education Reporting Collaborative, a coalition of eight newsrooms, is documenting the math crisis facing schools and highlighting progress. Members of the Collaborative are AL.com, The Associated Press, The Christian Science Monitor, The Dallas Morning News, The Hechinger Report, Idaho Education News, The Post and Courier in South Carolina, and The Seattle Times.

Students had been making incremental progress on national math tests since 1990. But over the past year, fourth and eighth grade math scores slipped to the lowest levels in about 20 years , according to data from the National Assessment of Educational Progress, known as the “Nation’s Report Card.”

“It’s a generation’s worth of progress lost,” said Andrew Ho, a professor at Harvard University’s Graduate School of Education.

At Moultrie Middle School in Mount Pleasant, South Carolina, Jennifer Matthews has seen the pandemic fallout in her eighth grade classes. Her students have shown indifference to understanding her pre-algebra and Algebra I lessons.

“They don’t allow themselves to process the material. They don’t allow themselves to think, ‘This might take a day to understand or learn,’” she said.

And recently students have been coming to her classes with gaps in their understanding of math concepts. Basic fractions, for instance, continue to stump many of them, she said.

Using federal pandemic relief money, some schools have added tutors or piloted new curriculum approaches in the name of academic recovery. But that money has a looming expiration date: The September 2024 deadline for allocating funds will arrive before many children have caught up.

Like other districts across the country, Jefferson County Schools in Birmingham, Alabama, saw students’ math skills take a nosedive from 2019 to 2021. Leveraging pandemic aid, the district placed math coaches in all of their middle schools.

The coaches help teachers learn new and better ways to teach students. About 1 in 5 public schools in the United States have a math coach, according to federal data . The efforts appear to be paying off: State testing shows math scores have started to inch back up for most of the Jefferson County middle schools.

In Pittsburgh’s school system, which serves a student population that is 53% African American, special education teacher Ebonie Lamb said it’s “emotionally exhausting” to see the inequities between student groups. But she believes those academic gaps can be closed through culturally relevant lessons, and targeting teaching to each student’s skill level.

Lamb said she typically asks students to do a “walk a mile in my shoes” project in which they design shoes and describe their lives. It’s a way she can learn more about them as individuals. Ultimately, those connections help on the academic front. Last year, she and a co-teacher taught math in a small group format that allowed students to master skills at their own pace.

“All students in the class cannot follow the same, scripted curriculum and be on the same problem all the time,” she said.

Adding to the challenge of catching kids up is debate over how math should be taught. Over the years, experts say, the pendulum has swung between procedural learning, such as teaching kids to memorize how to solve problems step-by-step, and conceptual understanding, in which students grasp underlying math relationships.

“Stereotypically, math is that class that people don’t like. ... For so many adults, math was taught just as memorization,” said Kevin Dykema, president of the National Council of Teachers of Mathematics. “When people start to understand what’s going on, in whatever you’re learning but especially in math, you develop a new appreciation for it.”

Teaching math should not be an either-or situation, said Sarah Powell, a professor at the University of Texas at Austin who researches math instruction. A shift too far in the conceptual direction, she said, risks alienating students who haven’t mastered the foundational skills.

“We actually do have to teach, and it is less sexy and it’s not as interesting,” she said.

In Spring, Texas, parent Aggie Gambino has often found herself searching YouTube for math videos. Giada, one of her twin 10-year-old daughters, has dyslexia and also struggles with math, especially word problems. Gambino says helping her daughter has proved challenging, given instructional approaches that differ from the way she was taught.

She wishes her daughter’s school would send home information on how students are being taught.

“The more parents understand how they’re being taught,” she said, “the better participant they can be in their child’s learning.”

Even at a nationally recognized magnet school, the lingering impact of the pandemic on students’ math skills is apparent. At the Townview School of Science and Engineering in Dallas, the incoming ninth graders in Lance Barasch’s summer camp course needed to relearn the meaning of words like “term” and “coefficient.”

“Then you can go back to what you’re really trying to teach,” he said.

Barasch wasn’t surprised that the teens were missing some skills after their chaotic middle school years.

The hope is that by taking a step back, students can begin to move forward.

Claire Bryan of The Seattle Times, Trisha Powell Crain of AL.com, Maura Turcotte of The Post and Courier, and Talia Richman of The Dallas Morning News contributed to this report.

The Associated Press education team receives support from the Carnegie Corporation of New York. The AP is solely responsible for all content.