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How to Solve Fraction Questions in Math

Last Updated: February 24, 2023 Fact Checked

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. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 1,187,165 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

Step 1 Add fractions with the same denominator by combining the numerators.

  • 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.

Step 2 Subtract fractions with the same denominator by subtracting the numerators.

  • 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.

Step 3 Find a common...

  • 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.

Step 4 Multiply fractions straight across.

  • 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.

Step 5 Divide fractions by flipping the second fraction upside down and multiplying straight across.

  • 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

Step 1 Note that the numerator is on the top and the denominator is on the bottom.

  • 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.

Step 2 Turn a whole number into a fraction by putting it over 1.

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

Step 3 Reduce fractions if you need to simplify them.

  • 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.

Step 4 Learn to turn...

  • 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.

Step 5 Figure out how...

  • 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

how to solve problem with fraction

Community Q&A

Community Answer

  • 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 2
  • 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 1
  • 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 1

how to solve problem with fraction

You Might Also Like

Solve Systems of Algebraic Equations Containing Two Variables

  • ↑ 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

Mario Banuelos, PhD

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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how to solve problem with fraction

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

how to solve problem with fraction

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 \(\frac{1}{4}\) .

new1

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 .

new2

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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Inverse operations

Solve equations with fractions

Here you will learn about how to solve equations with fractions, including solving equations with one or more operations. You will also learn about solving equations with fractions where the unknown is the denominator of a fraction.

Students will first learn how to solve equations with fractions in 7th grade as part of their work with expressions and equations and expand that knowledge in 8th grade.

What are equations with fractions?

Equations with fractions involve solving equations where the unknown variable is part of the numerator and/or denominator of a fraction.

The numerator (top number) in a fraction is divided by the denominator (bottom number).

To solve equations with fractions, you will use the “balancing method” to apply the inverse operation to both sides of the equation in order to work out the value of the unknown variable.

The inverse operation of addition is subtraction.

The inverse operation of subtraction is addition.

The inverse operation of multiplication is division.

The inverse operation of division is multiplication.

For example,

\begin{aligned} \cfrac{2x+3}{5} \, &= 7\\ \colorbox{#cec8ef}{$\times \, 5$} \; & \;\; \colorbox{#cec8ef}{$\times \, 5$} \\\\ 2x+3&=35 \\ \colorbox{#cec8ef}{$-\,3$} \; & \;\; \colorbox{#cec8ef}{$- \, 3$} \\\\ 2x & = 32 \\ \colorbox{#cec8ef}{$\div \, 2$} & \; \; \; \colorbox{#cec8ef}{$\div \, 2$}\\\\ x & = 16 \end{aligned}

What are equations with fractions?

Common Core State Standards

How does this relate to 7th grade and 8th grade math?

  • Grade 7: Expressions and Equations (7.EE.A.1) Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
  • Grade 8: Expressions and Equations (8.EE.C.7) Solve linear equations in one variable.
  • Grade 8: Expressions and Equations (8.EE.C.7b) Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

How to solve equations with fractions

In order to solve equations with fractions:

Identify the operations that are being applied to the unknown variable.

Apply the inverse operations, one at a time, to both sides of the equation.

Write the final answer, checking that it is correct.

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Solve equations with fractions examples

Example 1: equations with one operation.

Solve for x \text{: } \cfrac{x}{5}=4 .

The unknown is x.

Looking at the left hand side of the equation, the x is divided by 5.

\cfrac{x}{5}

2 Apply the inverse operations, one at a time, to both sides of the equation.

The inverse of “dividing by 5 ” is “multiplying by 5 ”.

You will multiply both sides of the equation by 5.

Solve equations with fractions example 1

3 Write the final answer, checking that it is correct.

The final answer is x=20.

You can check the answer by substituting the answer back into the original equation.

\cfrac{20}{5}=20\div5=4

Example 2: equations with one operation

Solve for x \text{: } \cfrac{x}{3}=8 .

Looking at the left hand side of the equation, the x is divided by 3.

\cfrac{x}{3}

The inverse of “dividing by 3 ” is “multiplying by 3 ”.

You will multiply both sides of the equation by 3.

Solve equations with fractions example 2

The final answer is x=24.

\cfrac{24}{3}=24\div3=8

Example 3: equations with two operations

Solve for x \text{: } \cfrac{x \, + \, 1}{2}=7 .

Looking at the left hand side of the equation, 1 is added to x and then divided by 2 (the denominator of the fraction).

\cfrac{x \, + \, 1}{2}

First, clear the fraction by multiplying both sides of the equation by 2.

Then, subtract 1 from both sides.

Solve equations with fractions example 3

The final answer is x=13.

\cfrac{13 \, +1 \, }{2}=\cfrac{14}{2}=14\div2=7

Example 4: equations with two operations

Solve for x \text{: } \cfrac{x}{4}-2=3 .

Looking at the left hand side of the equation, x is divided by 4 and then 2 is subtracted.

\cfrac{x}{4}-2

First, add 2 to both sides of the equation.

Then, multiply both sides of the equation by 4.

Solve equations with fractions example 4

\cfrac{20}{4}-2=20\div4-2=5-2=3

Example 5: equations with three operations

Solve for x \text{: } \cfrac{3x}{5}+1=7 .

Looking at the left hand side of the equation, x is multiplied by 3, then divided by 5 , and then 1 is added.

\cfrac{3x}{5}+1

First, subtract 1 from both sides of the equation.

Then, multiply both sides of the equation by 5.

Finally, divide both sides by 3.

Solve equations with fractions example 5

The final answer is x=10.

\cfrac{3 \, \times \, 10}{5}+1=\cfrac{30}{5}+1=6+1=7

Example 6: equations with three operations

Solve for x \text{: } \cfrac{2x-1}{7}=3 .

Looking at the left hand side of the equation, x is multiplied by 2, then 1 is subtracted, and the last operation is divided by 7 (the denominator).

\cfrac{2x-1}{7}

First, multiply both sides of the equation by 7.

Next, add 1 to both sides.

Solve equations with fractions example 6

The final answer is x=11.

\cfrac{2 \, \times \, 11-1}{7}=\cfrac{22-1}{7}=\cfrac{21}{7}=3

Example 7: equations with the unknown as the denominator

Solve for x \text{: } \cfrac{24}{x}=6 .

Looking at the left hand side of the equation, x is the denominator. 24 is divided by x.

\cfrac{24}{x}

You need to multiply both sides of the equation by x.

Then, you can divide both sides by 6.

Solve equations with fractions example 7

The final answer is x=4.

\cfrac{24}{4}=24\div4=6

Example 8: equations with the unknown as the denominator

Solve for x \text{: } \cfrac{18}{x}-6=3 .

Looking at the left hand side of the equation, x is the denominator. 18 is divided by x , and then 6 is subtracted.

\cfrac{18}{x}-6

First, add 6 to both sides of the equation.

Then, multiply both sides of the equation by x.

Finally, divide both sides by 9.

Solve equations with fractions example 8

The final answer is x=2.

\cfrac{18}{2}-6=9-6=3

Teaching tips for solving equations with fractions

  • When students first start working through practice problems and word problems, provide step-by-step instructions to assist them with solving linear equations.
  • Introduce solving equations with fractions with one-step problems, then two-step problems, before introducing multi-step problems.
  • Students will need lots of practice with solving linear equations. These standards provide the foundation for work with future linear equations in Algebra I and II.
  • Provide opportunities for students to explain their thinking through writing. Ensure that they are using key vocabulary, such as, absolute value, coefficient, equation, common factors, inequalities, simplify, etc.

Easy mistakes to make

  • The solution to an equation can be any type of number The unknowns do not have to be integers (whole numbers and their negative opposites). The solutions can be fractions or decimals. They can also be positive or negative numbers.
  • The unknown of an equation can be on either side of the equation The unknown, represented by a letter, is often on the left hand side of the equations; however, it doesn’t have to be. It could also be on the right hand side of an equation.

Solve equations with fractions image 2

  • Lowest common denominator (LCD) It is common to get confused between solving equations involving fractions and adding and subtracting fractions. When adding and subtracting, you need to work out the lowest/least common denominator (sometimes called the least common multiple or LCM). When you solve equations involving fractions, multiply both sides of the equation by the denominator of the fraction.

Related math equations lessons

  • Math equations
  • Rearranging equations
  • How to find the equation of a line
  • Substitution
  • Linear equations
  • Writing linear equations
  • Solving equations
  • Identity math
  • One step equations

Practice solve equations with fractions questions

1. Solve: \cfrac{x}{6}=3

GCSE Quiz False

You will multiply both sides of the equation by 6, because the inverse of “dividing by 6 ” is “multiplying by 6 ”.

Solve equations with fractions practice question 1

The final answer is x = 18.

\cfrac{18}{6}=18 \div 6=3

2. Solve: \cfrac{x \, + \, 4}{2}=7

Then subtract 4 from both sides.

Solve equations with fractions practice question 2

The final answer is x = 10.

\cfrac{10 \, + \, 4}{2}=\cfrac{14}{2}=14 \div 2=7

3. Solve: \cfrac{x}{8}-5=1

First, add 5 to both sides of the equation.

Then multiply both sides of the equation by 8.

Solve equations with fractions practice question 3

The final answer is x = 48.

\cfrac{48}{8}-5=48 \div 8-5=1

4. Solve: \cfrac{3x \, + \, 2}{4}=2

First, multiply both sides of the equation by 4.

Next, subtract 2 from both sides.

Solve equations with fractions practice question 4

The final answer is x = 2.

\cfrac{3 \, \times \, 2+2}{4}=\cfrac{6 \, + \, 2}{4}=\cfrac{8}{4}=8 \div 4=2

5. Solve: \cfrac{4x}{7}-2=6

Then multiply both sides of the equation by 7.

Finally, divide both sides by 4.

Solve equations with fractions practice question 5

The final answer is x = 14.

\cfrac{4 \, \times \, 14}{7}-2=\cfrac{56}{7}-2=56 \div 7-2=6

6. Solve: \cfrac{42}{x}=7

Then you divide both sides by 7.

Solve equations with fractions practice question 6

The final answer is x = 6.

\cfrac{42}{6}=42 \div 6=7

Solve equations with fractions FAQs

Yes, you still follow the order of operations when solving equations with fractions. You will start with any operations in the numerator and follow PEMDAS (parenthesis, exponents, multiply/divide, add/subtract), followed by any operations in the denominator. Then you will solve the rest of the equation as usual.

The next lessons are

  • Inequalities
  • Types of graphs
  • Coordinate plane
  • Number patterns
  • Algebraic expressions
  • Fractions operations

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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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4.12: Solve Equations with Fractions (Part 1)

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  • Page ID 6067

Learning Objectives

  • Determine whether a fraction is a solution of an equation
  • Solve equations with fractions using the Addition, Subtraction, and Division Properties of Equality
  • Solve equations using the Multiplication Property of Equality
  • Translate sentences to equations and solve

be prepared!

Before you get started, take this readiness quiz. If you miss a problem, go back to the section listed and review the material.

  • Evaluate \(x + 4\) when \(x = −3\) If you missed this problem, review Example 3.2.10 .
  • Solve: \(2y − 3 = 9\). If you missed this problem, review Example 3.5.2 .
  • Solve: \(y − 3 = −9\) If you missed this problem, review Example 4.2.10 .

Determine Whether a Fraction is a Solution of an Equation

As we saw in Solve Equations with the Subtraction and Addition Properties of Equality and Solve Equations Using Integers; The Division Property of Equality , a solution of an equation is a value that makes a true statement when substituted for the variable in the equation. In those sections, we found whole number and integer solutions to equations. Now that we have worked with fractions, we are ready to find fraction solutions to equations.

The steps we take to determine whether a number is a solution to an equation are the same whether the solution is a whole number, an integer, or a fraction.

HOW TO: DETERMINE WHETHER A NUMBER IS A Solution TO AN EQUATION

Step 1. Substitute the number for the variable in the equation.

Step 2. Simplify the expressions on both sides of the equation.

Step 3. Determine whether the resulting equation is true. If it is true, the number is a solution. If it is not true, the number is not a solution.

Example \(\PageIndex{1}\): find solution

Determine whether each of the following is a solution of \(x − \dfrac{3}{10} = \dfrac{1}{2}\).

  • \(x = \dfrac{4}{5}\)
  • \(x = − \dfrac{4}{5}\)

Since \(x = 1\) does not result in a true equation, \(1\) is not a solution to the equation.

Since \(x = \dfrac{4}{5}\) results in a true equation, \(\dfrac{4}{5}\) is a solution to the equation \(x − \dfrac{3}{10} = \dfrac{1}{2}\).

Since \(x = − \dfrac{4}{5}\) does not result in a true equation, \(− \dfrac{4}{5}\) is not a solution to the equation.

Exercise \(\PageIndex{1}\)

Determine whether each number is a solution of the given equation. \(x − \dfrac{2}{3} = \dfrac{1}{6}\)

  • \(x = \dfrac{5}{6}\)
  • \(x = − \dfrac{5}{6}\)

Exercise \(\PageIndex{2}\)

Determine whether each number is a solution of the given equation. \(y − \dfrac{1}{4} = \dfrac{3}{8}\)

  • \(y = - \dfrac{5}{8}\)
  • \(y = \dfrac{5}{8}\)

Solve Equations with Fractions using the Addition, Subtraction, and Division Properties of Equality

In Solve Equations with the Subtraction and Addition Properties of Equality and Solve Equations Using Integers; The Division Property of Equality , we solved equations using the Addition, Subtraction, and Division Properties of Equality. We will use these same properties to solve equations with fractions.

Addition, Subtraction, and Division Properties of Equality

For any numbers \(a\), \(b\), and \(c\),

In other words, when you add or subtract the same quantity from both sides of an equation, or divide both sides by the same quantity, you still have equality.

Example \(\PageIndex{2}\): solve

Solve: \(y + \dfrac{9}{16} = \dfrac{5}{16}\).

Since \(y = − \dfrac{1}{4}\) makes \(y + \dfrac{9}{16} = \dfrac{5}{16}\) a true statement, we know we have found the solution to this equation.

Exercise \(\PageIndex{3}\)

Solve: \(y + \dfrac{11}{12} = \dfrac{5}{12}\).

\(-\dfrac{1}{2}\)

Exercise \(\PageIndex{4}\)

Solve: \(y + \dfrac{8}{15} = \dfrac{4}{15}\).

\(-\dfrac{4}{15}\)

We used the Subtraction Property of Equality in Example \(\PageIndex{2}\). Now we’ll use the Addition Property of Equality.

Example \(\PageIndex{3}\): solve

Solve: a − \(\dfrac{5}{9}\) = \(− \dfrac{8}{9}\).

Since \(a = − \dfrac{1}{3}\) makes the equation true, we know that \(a = − \dfrac{1}{3}\) is the solution to the equation.

Exercise \(\PageIndex{5}\)

Solve: \(a − \dfrac{3}{5} = − \dfrac{8}{5}\).

Exercise \(\PageIndex{6}\)

Solve: \(n − \dfrac{3}{7} = − \dfrac{9}{7}\).

\(-\dfrac{6}{7}\)

The next example may not seem to have a fraction, but let’s see what happens when we solve it.

Example \(\PageIndex{4}\): solve

Solve: \(10q = 44\).

The solution to the equation was the fraction \(\dfrac{22}{5}\). We leave it as an improper fraction.

Exercise \(\PageIndex{7}\)

Solve: \(12u = −76\).

\(-\dfrac{19}{3}\)

Exercise \(\PageIndex{8}\)

Solve: \(8m = 92\).

\(\dfrac{23}{2}\)

Solve Equations with Fractions Using the Multiplication Property of Equality

Consider the equation \(\dfrac{x}{4} = 3\). We want to know what number divided by \(4\) gives \(3\). So to “undo” the division, we will need to multiply by \(4\). The Multiplication Property of Equality will allow us to do this. This property says that if we start with two equal quantities and multiply both by the same number, the results are equal.

Definition: The Multiplication Property of Equality

For any numbers \(a\), \(b\), and \(c\), if \(a = b\), then \(ac = bc\). If you multiply both sides of an equation by the same quantity, you still have equality.

Let’s use the Multiplication Property of Equality to solve the equation \(\dfrac{x}{7} = −9\).

Example \(\PageIndex{5}\): solve

Solve: \(\dfrac{x}{7} = −9\).

Exercise \(\PageIndex{9}\)

Solve: \(\dfrac{f}{5} = −25\).

Exercise \(\PageIndex{10}\)

Solve: \(\dfrac{h}{9} = −27\).

Example \(\PageIndex{6}\):solve

Solve: \(\dfrac{p}{−8} = −40\).

Here, \(p\) is divided by \(−8\). We must multiply by \(−8\) to isolate \(p\).

Exercise \(\PageIndex{11}\)

Solve: \(\dfrac{c}{−7} = −35\).

Exercise \(\PageIndex{12}\)

Solve: \(\dfrac{x}{−11} = −12\).

Solve Equations with a Coefficient of \(−1\)

Look at the equation \(−y = 15\). Does it look as if y is already isolated? But there is a negative sign in front of \(y\), so it is not isolated.

There are three different ways to isolate the variable in this type of equation. We will show all three ways in Example \(\PageIndex{7}\).

Example \(\PageIndex{7}\): solve

Solve: \(−y = 15\).

One way to solve the equation is to rewrite \(−y\) as \(−1y\), and then use the Division Property of Equality to isolate \(y\).

Another way to solve this equation is to multiply both sides of the equation by −1.

The third way to solve the equation is to read \(−y\) as “the opposite of \(y\).” What number has \(15\) as its opposite? The opposite of \(15\) is \(−15\). So \(y = −15\).

For all three methods, we isolated \(y\) is isolated and solved the equation.

Exercise \(\PageIndex{13}\)

Solve: \(−y = 48\).

Exercise \(\PageIndex{14}\)

Solve: \(−c = −23\).

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Parents are stumped over first grader’s math question - can you solve it?

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Math curriculums have been changing in the school system over the last few years, with many parents claiming that “new” math is too difficult.

A mother named Tiesha Sanders recently took to Facebook to share her first-grader’s homework problem that she had no idea how to solve. “The new Math is NOT IT!” her post’s caption began, next to a photo of the problem and a note that she wrote to the teacher .

“Disclaimer: I am not upset with the teacher, she’s just teaching what she’s supposed to. And #2, don’t come here like we’re the dumb ones, I taught elementary for the last six years, this question ain’t it! Also, this is 1st grade math,” the caption read.

The problem required the student to split the number 27 into tens and ones and then just into ones. Her daughter, Summer, wrote that there were two tens and seven ones, so she had assumed when she was asked for the ones again that it was still seven. After getting the question marked wrong, Sanders left a note for the teacher asking for the right answer to help her child next time.

“Hello!” the note began. “I just wanted to ask how Summer got #3 wrong? Her father and I were going over her mistakes and wanted to be sure we were on the right track.”

Her daughter’s teacher responded to the note, writing: “Hello! This is the new math they have us teaching.”

“It is 27 ones. It wants her to know that having two tens and seven ones is the same as 27 ones. If you have any other questions you can call or text me.”

After posting, many people took to the comments where they agreed with Sanders that the question was confusing.

“If they wanted her to decompose the number and show the ways in which the number can be logically made... why NOT = signs? The arrows make sense to who?” one comment read.

Another commenter agreed, writing: “I wonder why they wouldn’t word it differently? Like maybe what does the tens place + the ones place = that threw me for a loop because it’s really unclear.”

Others called out the flawed structure of the question, saying: “But if they have the box that labels ‘tens’ and ‘ones’ then only ask for the ‘ones’, how in the entire world is this math, mathing?”

Someone agreed, adding: “The question sets them up to fail.”

This isn’t the first time a parent has publicly voiced their confusion over an elementary school homework question. Back in December, one mother in Buckinghamshire, England, had become so confused helping her six-year-old with a worksheet that she posted the question in a private Facebook group . 

“At first I thought I was losing my mind. I was like, ‘What am I missing here?’” So I posted in a group with loads of moms hoping they would have the answer,” Laura Rathbone said in an interview with Today .

Rathbone’s daughter, Lilly-Mo, was asked on the worksheet to pick the odd item out based on the five items she was given. The items listed were: friend, toothbrush, desk, silver, and egg.

“So… my six-year-old daughter who’s in year one got this homework question,” her Facebook post read. “It’s confusing in my opinion, to say the least, especially considering the age it’s aimed at… but I’d love to hear your answers!”

She added: “I think it’s something you’d find in a Puzzler magazine personally but let me know your thoughts.”

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Title: geoeval: benchmark for evaluating llms and multi-modal models on geometry problem-solving.

Abstract: Recent advancements in Large Language Models (LLMs) and Multi-Modal Models (MMs) have demonstrated their remarkable capabilities in problem-solving. Yet, their proficiency in tackling geometry math problems, which necessitates an integrated understanding of both textual and visual information, has not been thoroughly evaluated. To address this gap, we introduce the GeoEval benchmark, a comprehensive collection that includes a main subset of 2000 problems, a 750 problem subset focusing on backward reasoning, an augmented subset of 2000 problems, and a hard subset of 300 problems. This benchmark facilitates a deeper investigation into the performance of LLMs and MMs on solving geometry math problems. Our evaluation of ten LLMs and MMs across these varied subsets reveals that the WizardMath model excels, achieving a 55.67\% accuracy rate on the main subset but only a 6.00\% accuracy on the challenging subset. This highlights the critical need for testing models against datasets on which they have not been pre-trained. Additionally, our findings indicate that GPT-series models perform more effectively on problems they have rephrased, suggesting a promising method for enhancing model capabilities.

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IMAGES

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  2. 4 Ways to Solve Fraction Questions in Math

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  3. 4 Ways to Solve Fraction Questions in Math

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COMMENTS

  1. 3 Ways to Solve Fraction Questions in Math

    1 Add fractions with the same denominator by combining the numerators. To add fractions, they must have the same denominator. If they do, simply add the numerators together. [2] 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. 2

  2. Fractions

    Fractions | Microsoft Math Solver Type a math problem Solve Examples 124 − 79 124 × 89 124 ÷ 89 124 + 89 124 + 89 × 315 − 1026 81 + 2(79) ÷ 315 Quiz 124 − 79 124 ÷ 89 124 + 89 × 315 − 1026 Learn about fractions using our free math solver with step-by-step solutions.

  3. Step-by-Step Math Problem Solver

    Simplify Factor Expand Graph GCF LCM New Example Help Tutorial Solve an equation, inequality or a system. Example: 2x-1=y,2y+3=x What can QuickMath do? QuickMath will automatically answer the most common problems in algebra, equations and calculus faced by high-school and college students.

  4. Problem Solving using Fractions (Definition, Types and Examples

    Problem Solving using Fractions Fractions are numbers that exist between whole numbers. We get fractions when we divide whole numbers into equal parts. ... Read More Select your child's grade in school: Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Grade 6 Grade 7 Grade 8 Schedule a free class Table of Contents What are Fractions? Types of Fractions

  5. 1.26: Solving Fractional Equations

    II. Multiple Fractions on Either Side of the Equation. Equations d) and e) in Example 24.1 fall into this category. We solve these equations here. We use the technique for combining rational expressions we learned in Chapter 23 to reduce our problem to a problem with a single fraction on each side of the equation. d) Solve \(\frac{3}{4}-\frac{1 ...

  6. Equation with variables on both sides: fractions

    To solve the equation (3/4)x + 2 = (3/8)x - 4, we first eliminate fractions by multiplying both sides by the least common multiple of the denominators. Then, we add or subtract terms from both sides of the equation to group the x-terms on one side and the constants on the other. Finally, we solve and check as normal.

  7. Understand fractions

    Arithmetic 19 units · 203 skills. Unit 1 Intro to multiplication. Unit 2 1-digit multiplication. Unit 3 Intro to division. Unit 4 Understand fractions. Unit 5 Place value through 1,000,000. Unit 6 Add and subtract through 1,000,000. Unit 7 Multiply 1- and 2-digit numbers. Unit 8 Divide with remainders.

  8. Solve Equations with Fractions

    Example 1: equations with one operation. Solve for x \text {: } \cfrac {x} {5}=4 x: 5x = 4. Identify the operations that are being applied to the unknown variable. The unknown is x. x. Looking at the left hand side of the equation, the x x is divided by 5. 5. \cfrac {x} {5} 5x. 2 Apply the inverse operations, one at a time, to both sides of the ...

  9. Evaluate Fractions

    Quiz. x−24 − x+15. x−67x ÷ 3(x−6)2x. Learn about evaluate fractions using our free math solver with step-by-step solutions.

  10. 4.9: Solving Equations with Fractions

    Solution. Multiply both sides of the equation by the least common denominator for the fractions that appear in the equation. − 8 9x = 5 18 Original equation. 18( − 8 9x) = 18( 5 18) Multiply both sides by 18. − 16x = 5 On each side, cancel and multiply. 18( − 8 9) = − 16 and 18( 5 18) = 5.

  11. Microsoft Math Solver

    Get math help in your language. Works in Spanish, Hindi, German, and more. Online math solver with free step by step solutions to algebra, calculus, and other math problems. Get help on the web or with our math app.

  12. Fractions Basic Introduction

    This math video tutorial provides a basic introduction into fractions. It explains how to add, subtract, multiply and divide fractions. It contains plenty ...

  13. Mathway

    Free math problem solver answers your algebra homework questions with step-by-step explanations.

  14. Learn How to Solve Fraction Word Problems with Examples and Interactive

    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?

  15. Exploring Fractions

    Exploring Fractions. Introduction. At NRICH, our aim is to offer rich tasks which develop deep understanding of mathematical concepts. Of course, by their very nature, rich tasks will also provide opportunities for children to work like a mathematician and so help them develop their problem-solving skills alongside this conceptual understanding.

  16. Algebra: Fraction Problems (solutions, examples, videos)

    Solution: Step 1: Assign variables : Let x = number Step 2: Solve the equation Isolate variable x 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,

  17. Solving for the missing fraction (video)

    Solving for the missing fraction (video) | Khan Academy 5th grade Course: 5th grade > Unit 4 Lesson 3: Adding and subtracting fractions with unlike denominators Adding fractions with unlike denominators Add fractions with unlike denominators Subtracting fractions with unlike denominators introduction Subtracting fractions with unlike denominators

  18. 4.12: Solve Equations with Fractions (Part 1)

    Solve: \(y − 3 = −9\) If you missed this problem, review Example 4.2.10. Determine Whether a Fraction is a Solution of an Equation As we saw in Solve Equations with the Subtraction and Addition Properties of Equality and Solve Equations Using Integers; The Division Property of Equality , a solution of an equation is a value that makes a ...

  19. Word Problems with Fractions

    Answer: Word problems with fractions: involving a fraction and a whole number Finally, we are going to look at an example of a word problem with a fraction and a whole number. Now we will have to convert all the information into a fraction with the same denominator (as we did in the example above) in order to calculate

  20. Step-by-Step Calculator

    Symbolab is the best step by step calculator for a wide range of physics problems, including mechanics, electricity and magnetism, and thermodynamics. It shows you the steps and explanations for each problem, so you can learn as you go. How to solve math problems step-by-step?

  21. CRASH MATH on Steam

    CRASH MATH You must solve math problems while trying to drive a car and avoid traps. This driving game has a lot of possibilities. The physics engine simulates every component of a vehicle in real-time, resulting in a funny behavior. While creating the excitement of solving math problems.

  22. Parents are stumped over first grader's math question

    The problem required the student to split the number 27 into tens and ones and then just into ones. Her daughter, Summer, wrote that there were two tens and seven ones, so she had assumed when she ...

  23. Understanding division of fractions (video)

    You have a very good question. Think about it this way: a fraction itself is a division problem, the numerator divided by the denominator.When you multiply by a value greater than 1, the original amount becomes greater; when you multiply by a value less than 1, the original value becomes smaller. You want to find a way to "move the dividend into the denominator of the divisor".

  24. A Nice Exponential Problem Math Olympiad

    A Nice Exponential Problem Math Olympiad | How to solve?If you like my video then please subscribe my channel 🥰🤗& Please Support Me 🥰🥰My social media li...

  25. Two-step equations with decimals and fractions

    In this equation we can see there are two fractions at the left and one at the right, as both fractions are divided by 2, you should convert the number of the right (in this case 3) into a fraction divided by 2, in this case, we can use 6/2 (that is equal to 3) and the equation would be: k/2 + 1/2 = 6/2. now, we can remove the /2 of all numbers ...

  26. [2402.10104] GeoEval: Benchmark for Evaluating LLMs and Multi-Modal

    Recent advancements in Large Language Models (LLMs) and Multi-Modal Models (MMs) have demonstrated their remarkable capabilities in problem-solving. Yet, their proficiency in tackling geometry math problems, which necessitates an integrated understanding of both textual and visual information, has not been thoroughly evaluated. To address this gap, we introduce the GeoEval benchmark, a ...