maths.free › Arithmetic › 4. Fractions › Solve Equations with Fractions

Solve Equations with Fractions

Determine whether a fraction is a solution of an equation

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.

Example

Try it.

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

ⓐ \(x=1\)
ⓑ \(x=\frac{4}{5}\)
ⓒ \(x=-\frac{4}{5}\)

Solution

ⓐ


Change to fractions with a LCD of 10.
Subtract.

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

ⓑ



Subtract.

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

ⓒ



Subtract.

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

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.

Example

Try it.

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

Solution


Subtract \(\frac{9}{16}\) from each side to undo the addition.
Simplify on each side of the equation.
Simplify the fraction.
Check:
Substitute \(y=-\frac{1}{4}\).
Rewrite as fractions with the LCD.
Add.

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

We used the Subtraction Property of Equality in . Now we’ll use the Addition Property of Equality.

Example

Try it.

Solve: \(a-\frac{5}{9}=-\frac{8}{9}.\)

Solution


Add \(\frac{5}{9}\) from each side to undo the subtraction.
Simplify on each side of the equation.
Simplify the fraction.
Check:
Substitute \(a=-\frac{1}{3}\).
Change to common denominator.
Subtract.

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

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

Example

Try it.

Solve: \(10q=44.\)

Solution

	\(10q=44\)
Divide both sides by 10 to undo the multiplication.	\(\frac{10q}{10}=\frac{44}{10}\)
Simplify.	\(q=\frac{22}{5}\)
Check:
Substitute \(q=\frac{22}{5}\) into the original equation.	\(10(\frac{22}{5})\overset{?}{=}44\)
Simplify.	\(\overset{2}{10}(\frac{22}{5})\overset{?}{=}44\)
Multiply.	\(44=44\ ✓\)

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

Solve Equations with Fractions Using the Multiplication Property of Equality

Consider the equation \(\frac{x}{4}=3.\) 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.

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

Example

Try it.

Solve: \(\frac{x}{7}=-9.\)

Solution


Use the Multiplication Property of Equality to multiply both sides by \(7\). This will isolate the variable.
Multiply.
Simplify.

The equation is true.

Example

Try it.

Solve: \(\frac{p}{-8}=-40.\)

Solution

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


Multiply both sides by \(-8\)
Multiply.
Simplify.
Check:
Substitute \(p=320\).
The equation is true.

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

Try it.

Solve: \(-y=15.\)

Solution

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


Rewrite \(-y\) as \(-1y\).
Divide both sides by −1.
Simplify each side.

Another way to solve this equation is to multiply both sides of the equation by \(-1.\)


Multiply both sides by −1.
Simplify each side.

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.

Check:


Substitute \(y=-15\).
Simplify. The equation is true.

Condensed — the full section is in OpenStax Prealgebra 2e.

Translate Sentences to Equations and Solve

Now we have covered all four properties of equality—subtraction, addition, division, and multiplication. We’ll list them all together here for easy reference.

Subtraction Property of Equality: For any real numbers \(\text{a, b,}\) and \(\text{c,}\) if \(a=b,\) then \(a-c=b-c.\)	Addition Property of Equality: For any real numbers \(\text{a, b,}\) and \(\text{c,}\) if \(a=b,\) then \(a+c=b+c.\)
Division Property of Equality: For any numbers \(\text{a, b,}\) and \(\text{c,}\) where \(\text{c}\ne 0\) if \(a=b,\) then \(\frac{a}{c}=\frac{b}{c}\)	Multiplication Property of Equality: For any real numbers \(\text{a, b,}\) and \(\text{c}\) if \(a=b,\) then \(ac=bc\)

When you add, subtract, multiply or divide the same quantity from both sides of an equation, you still have equality.

In the next few examples, we’ll translate sentences into equations and then solve the equations. It might be helpful to review the translation table in Evaluate, Simplify, and Translate Expressions.

Example

Try it.

Translate and solve: \(n\) divided by \(6\) is \(-24.\)

Solution

Translate.
Multiply both sides by \(6\).
Simplify.
Check:	Is \(-144\) divided by \(6\) equal to \(-24\)?
Translate.
Simplify. It checks.

Example

Try it.

Translate and solve: The quotient of \(q\) and \(-5\) is \(70.\)

Solution

Translate.
Multiply both sides by \(-5\).
Simplify.
Check:	Is the quotient of \(-350\) and \(-5\) equal to \(70\)?
Translate.
Simplify. It checks.

Example

Try it.

Translate and solve: Two-thirds of \(f\) is \(18.\)

Solution

Translate.
Multiply both sides by \(\frac{3}{2}\).
Simplify.
Check:	Is two-thirds of \(27\) equal to \(18\)?
Translate.
Simplify. It checks.

Condensed — the full section is in OpenStax Prealgebra 2e.

Key Concepts

Determine whether a number is a solution to an equation.
1. Substitute the number for the variable in the equation.
2. Simplify the expressions on both sides of the equation.
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.
Addition, Subtraction, and Division Properties of Equality
- For any numbers a, b, and c,
  if \(a=b\), then \(a+c=b+c\). Addition Property of Equality
- if \(a=b\), then \(a-c=b-c\). Subtraction Property of Equality
- if \(a=b\), then \(\frac{a}{c}=\frac{b}{c}\), \(c\ne 0\). Division Property of Equality
The Multiplication Property of Equality
- For any numbers \(ab\) and \(c,a=b\), then \(ac=bc\).
- If you multiply both sides of an equation by the same quantity, you still have equality.

Chapter Practice Test

Convert the improper fraction to a mixed number.

Convert the mixed number to an improper fraction.

Locate the numbers on a number line.

In the following exercises, simplify.

Evaluate.

In the following exercises, solve the equation.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Evaluate \(x+4\) when \(x=-3\)
If you missed this problem, review .

Fi àwọn àgbèwọlé hàn

\(1\)
Solve: \(2y-3=9.\)
If you missed this problem, review .

Fi àwọn àgbèwọlé hàn

\(y=6\)
Solve: \(y-3=-9\)
If you missed this problem, review .

Fi àwọn àgbèwọlé hàn

\(-6\)
Determine whether each of the following is a solution of \(x-\frac{3}{10}=\frac{1}{2}.\)
1. ⓐ \(x=1\)
2. ⓑ \(x=\frac{4}{5}\)
3. ⓒ \(x=-\frac{4}{5}\)
Fi àwọn àgbèwọlé hàn

ⓐ
Change to fractions with a LCD of 10.
Subtract.
Since \(x=1\) does not result in a true equation, \(1\) is not a solution to the equation.
ⓑ
Subtract.
Since \(x=\frac{4}{5}\) results in a true equation, \(\frac{4}{5}\) is a solution to the equation \(x-\frac{3}{10}=\frac{1}{2}.\)
ⓒ
Subtract.
Since \(x=-\frac{4}{5}\) does not result in a true equation, \(-\frac{4}{5}\) is not a solution to the equation.
Determine whether each number is a solution of the given equation.
\(x-\frac{2}{3}=\frac{1}{6}\):
1. ⓐ \(x=1\)
2. ⓑ \(x=\frac{5}{6}\)
3. ⓒ \(x=-\frac{5}{6}\)
Fi àwọn àgbèwọlé hàn
1. ⓐ no
2. ⓑ yes
3. ⓒ no
Determine whether each number is a solution of the given equation.
\(y-\frac{1}{4}=\frac{3}{8}\):
1. ⓐ \(y=1\)
2. ⓑ \(y=-\frac{5}{8}\)
3. ⓒ \(y=\frac{5}{8}\)
Fi àwọn àgbèwọlé hàn
1. ⓐ no
2. ⓑ no
3. ⓒ yes
Solve: \(y+\frac{9}{16}=\frac{5}{16}.\)

Fi àwọn àgbèwọlé hàn

Subtract \(\frac{9}{16}\) from each side to undo the addition.
Simplify on each side of the equation.
Simplify the fraction.
Check:
Substitute \(y=-\frac{1}{4}\).
Rewrite as fractions with the LCD.
Add.
Since \(y=-\frac{1}{4}\) makes \(y+\frac{9}{16}=\frac{5}{16}\) a true statement, we know we have found the solution to this equation.
Solve: \(y+\frac{11}{12}=\frac{5}{12}.\)

Fi àwọn àgbèwọlé hàn

\(-\frac{1}{2}\)
Solve: \(y+\frac{8}{15}=\frac{4}{15}.\)

Fi àwọn àgbèwọlé hàn

\(-\frac{4}{15}\)
Solve: \(a-\frac{5}{9}=-\frac{8}{9}.\)

Fi àwọn àgbèwọlé hàn

Add \(\frac{5}{9}\) from each side to undo the subtraction.
Simplify on each side of the equation.
Simplify the fraction.
Check:
Substitute \(a=-\frac{1}{3}\).
Change to common denominator.
Subtract.
Since \(a=-\frac{1}{3}\) makes the equation true, we know that \(a=-\frac{1}{3}\) is the solution to the equation.
Solve: \(a-\frac{3}{5}=-\frac{8}{5}.\)

Fi àwọn àgbèwọlé hàn

−1
Solve: \(n-\frac{3}{7}=-\frac{9}{7}.\)

Fi àwọn àgbèwọlé hàn

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

Solve: \(10q=44.\)

Fi àwọn àgbèwọlé hàn

	\(10q=44\)
Divide both sides by 10 to undo the multiplication.	\(\frac{10q}{10}=\frac{44}{10}\)
Simplify.	\(q=\frac{22}{5}\)
Check:
Substitute \(q=\frac{22}{5}\) into the original equation.	\(10(\frac{22}{5})\overset{?}{=}44\)
Simplify.	\(\overset{2}{10}(\frac{22}{5})\overset{?}{=}44\)
Multiply.	\(44=44\ ✓\)

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

Solve: \(12u=-76.\)

Fi àwọn àgbèwọlé hàn

\(-\frac{19}{3}\)
Solve: \(8m=92.\)

Fi àwọn àgbèwọlé hàn

\(\frac{23}{2}\)
Solve: \(\frac{x}{7}=-9.\)

Fi àwọn àgbèwọlé hàn

Use the Multiplication Property of Equality to multiply both sides by \(7\). This will isolate the variable.
Multiply.
Simplify.
The equation is true.
Solve: \(\frac{f}{5}=-25.\)

Fi àwọn àgbèwọlé hàn

−125
Solve: \(\frac{h}{9}=-27.\)

Fi àwọn àgbèwọlé hàn

−243
Solve: \(\frac{p}{-8}=-40.\)

Fi àwọn àgbèwọlé hàn

Here, \(p\) is divided by \(-8.\) We must multiply by \(-8\) to isolate \(p.\)
Multiply both sides by \(-8\)
Multiply.
Simplify.
Check:
Substitute \(p=320\).
The equation is true.
Solve: \(\frac{c}{-7}=-35.\)

Fi àwọn àgbèwọlé hàn

245
Solve: \(\frac{x}{-11}=-12.\)

Fi àwọn àgbèwọlé hàn

132
Solve: \(-y=15.\)

Fi àwọn àgbèwọlé hàn

One way to solve the equation is to rewrite \(-y\) as \(-1y,\) and then use the Division Property of Equality to isolate \(y.\)
Rewrite \(-y\) as \(-1y\).
Divide both sides by −1.
Simplify each side.
Another way to solve this equation is to multiply both sides of the equation by \(-1.\)
Multiply both sides by −1.
Simplify each side.
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.
Check:
Substitute \(y=-15\).
Simplify. The equation is true.
Solve: \(-y=48.\)

Fi àwọn àgbèwọlé hàn

−48
Solve: \(-c=-23.\)

Fi àwọn àgbèwọlé hàn

23
Solve: \(\frac{3}{4}x=24.\)

Fi àwọn àgbèwọlé hàn

Multiply both sides by the reciprocal of the coefficient.
Simplify.
Multiply.
Check:
Substitute \(x=32\).
Rewrite \(32\) as a fraction.
Multiply. The equation is true.
Notice that in the equation \(\frac{3}{4}x=24,\) we could have divided both sides by \(\frac{3}{4}\) to get \(x\) by itself. Dividing is the same as multiplying by the reciprocal, so we would get the same result. But most people agree that multiplying by the reciprocal is easier.
Solve: \(\frac{2}{5}n=14.\)

Fi àwọn àgbèwọlé hàn

35
Solve: \(\frac{5}{6}y=15.\)

Fi àwọn àgbèwọlé hàn

18
Solve: \(-\frac{3}{8}w=72.\)

Fi àwọn àgbèwọlé hàn

The coefficient is a negative fraction. Remember that a number and its reciprocal have the same sign, so the reciprocal of the coefficient must also be negative.
Multiply both sides by the reciprocal of \(-\frac{3}{8}\).
Simplify; reciprocals multiply to one.
Multiply.
Check:
Let \(w=-192\).
Multiply. It checks.
Solve: \(-\frac{4}{7}a=52.\)

Fi àwọn àgbèwọlé hàn

−91
Solve: \(-\frac{7}{9}w=84.\)

Fi àwọn àgbèwọlé hàn

−108
Translate and solve: \(n\) divided by \(6\) is \(-24.\)

Fi àwọn àgbèwọlé hàn

Translate.
Multiply both sides by \(6\).
Simplify.
Check: Is \(-144\) divided by \(6\) equal to \(-24\)?
Translate.
Simplify. It checks.
Translate and solve: \(n\) divided by \(7\) is equal to \(-21.\)

Fi àwọn àgbèwọlé hàn

\(\frac{n}{7}=-21;n=-147\)
Translate and solve: \(n\) divided by \(8\) is equal to \(-56.\)

Fi àwọn àgbèwọlé hàn

\(\frac{n}{8}=-56;n=-448\)
Translate and solve: The quotient of \(q\) and \(-5\) is \(70.\)

Fi àwọn àgbèwọlé hàn

Translate.
Multiply both sides by \(-5\).
Simplify.
Check: Is the quotient of \(-350\) and \(-5\) equal to \(70\)?
Translate.
Simplify. It checks.
Translate and solve: The quotient of \(q\) and \(-8\) is \(72.\)

Fi àwọn àgbèwọlé hàn

\(\frac{q}{-8}=72;q=-576\)
Translate and solve: The quotient of \(p\) and \(-9\) is \(81.\)

Fi àwọn àgbèwọlé hàn

\(\frac{p}{-9}=81;p=-729\)
Translate and solve: Two-thirds of \(f\) is \(18.\)

Fi àwọn àgbèwọlé hàn

Translate.
Multiply both sides by \(\frac{3}{2}\).
Simplify.
Check: Is two-thirds of \(27\) equal to \(18\)?
Translate.
Simplify. It checks.
Translate and solve: Two-fifths of \(f\) is \(16.\)

Fi àwọn àgbèwọlé hàn

\(\frac{2}{5}f=16;f=40\)
Translate and solve: Three-fourths of \(f\) is \(21.\)

Fi àwọn àgbèwọlé hàn

\(\frac{3}{4}f=21;f=28\)

Translate and solve: The quotient of \(m\) and \(\frac{5}{6}\) is \(\frac{3}{4}.\)

Fi àwọn àgbèwọlé hàn

	The quotient of \(m\) and \(\frac{5}{6}\) is \(\frac{3}{4}\).
Translate.	\(\frac{\ m\ }{\frac{5}{6}}=\frac{3}{4}\)
Multiply both sides by \(\frac{5}{6}\) to isolate \(m\).	\(\frac{5}{6}(\frac{\ m\ }{\frac{5}{6}})=\frac{5}{6}(\frac{3}{4})\)
Simplify.	\(m=\frac{5\cdot 3}{6\cdot 4}\)
Remove common factors and multiply.	\(m=\frac{5}{8}\)
Check:
Is the quotient of \(\frac{5}{8}\) and \(\frac{5}{6}\) equal to \(\frac{3}{4}\)?	\(\frac{\ \frac{5}{8}\ }{\ \frac{5}{6}\ }\overset{?}{=}\frac{3}{4}\)
Rewrite as division.	\(\frac{5}{8}\div \frac{5}{6}\overset{?}{=}\frac{3}{4}\)
Multiply the first fraction by the reciprocal of the second.	\(\frac{5}{8}\cdot \frac{6}{5}\overset{?}{=}\frac{3}{4}\)
Simplify.	\(\frac{3}{4}=\frac{3}{4}✓\)

Our solution checks.

Symbols used here

\neq

not equal
The two sides are different.

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\approx

approximately equal
Equal to the precision shown, not exactly.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

a \bmod n

remainder
What is left after dividing a by n.

per cent
Per hundred: 15% = 15/100.

a : b,\ \frac{a}{b}

ratio, fraction
a for every b; a divided by b.

How to: Solve Equations with Fractions

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
Substitute the number for the variable in the equation.
Simplify the expressions on both sides of the equation.
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.

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

Wárá

Parts of this page are adapted from OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

Diẹ̀ nínú Arithmetic

Order of operations Fractions Percentages Powers and roots

if \(a=b,\) then \(a+c=b+c.\)	Addition Property of Equality
\(\) if \(a=b,\) then \(a-c=b-c.\)	Subtraction Property of Equality
if \(a=b,\) then \(\frac{a}{c}=\frac{b}{c},c\ne 0.\)	Division Property of Equality


Multiply both sides by the reciprocal of the coefficient.
Simplify.
Multiply.
Check:
Substitute \(x=32\).
Rewrite \(32\) as a fraction.
Multiply. The equation is true.


Multiply both sides by the reciprocal of \(-\frac{3}{8}\).
Simplify; reciprocals multiply to one.
Multiply.
Check:
Let \(w=-192\).
Multiply. It checks.