Solve Equations with Fractions

Determine whether a fraction is a solution of an equation

A solution of an equation is a value that makes a true statement when substituted for the variable. The steps for checking whether a number is a solution are the same whether the solution is a whole number, an integer, or a fraction.

Example. Determine whether each of the following is a solution of $x - \tfrac{3}{10} = \tfrac{1}{2}$: (a) $x=1$; (b) $x=\tfrac{4}{5}$; (c) $x=-\tfrac{4}{5}$.

(a) Substitute $1$ for $x$: $1 - \tfrac{3}{10} \stackrel{?}{=} \tfrac{1}{2}$. Rewriting with LCD $10$: $\tfrac{10}{10} - \tfrac{3}{10} = \tfrac{7}{10}$, which is not equal to $\tfrac{5}{10}$. So $x=1$ is not a solution.

(b) Substitute $\tfrac{4}{5}$ for $x$: $\tfrac{4}{5} - \tfrac{3}{10} \stackrel{?}{=} \tfrac{1}{2}$. Rewriting: $\tfrac{8}{10} - \tfrac{3}{10} = \tfrac{5}{10} = \tfrac{5}{10}$. True, so $x=\tfrac{4}{5}$ is a solution.

(c) Substitute $-\tfrac{4}{5}$ for $x$: $-\tfrac{8}{10} - \tfrac{3}{10} = -\tfrac{11}{10}$, which is not equal to $\tfrac{5}{10}$. So $x=-\tfrac{4}{5}$ is not a solution.

Solve equations with fractions using the Addition, Subtraction, and Division Properties of Equality

We use the same properties of equality with fractions as we did with whole numbers and integers.

Example. Solve: $y + \tfrac{9}{16} = \tfrac{5}{16}$.

Subtract $\tfrac{9}{16}$ from each side to undo the addition: $y = \tfrac{5}{16} - \tfrac{9}{16} = -\tfrac{4}{16}$. Simplify:

Checking by substituting $-\tfrac{1}{4}$ back in confirms the solution.

Example. Solve: $a - \tfrac{5}{9} = -\tfrac{8}{9}$.

Add $\tfrac{5}{9}$ to each side to undo the subtraction: $a = -\tfrac{8}{9} + \tfrac{5}{9} = -\tfrac{3}{9}$. Simplify:

The next example may not look like it has a fraction, but see what happens when we solve it.

Example. Solve: $10q = 44$.

Divide both sides by $10$ to undo the multiplication: $q = \tfrac{44}{10}$. Simplify:

The solution is a fraction — we leave it in that form rather than converting to a mixed number, since it came from solving an equation.

Solve equations using the Multiplication Property of Equality

Consider the equation $\tfrac{x}{4} = 3$. To “undo” the division, we multiply both sides by $4$ — the Multiplication Property of Equality.

Example. Solve: $\tfrac{x}{7} = -9$.

Multiply both sides by $7$ to isolate $x$: $7 \cdot \tfrac{x}{7} = 7(-9)$. Simplify:

Example. Solve: $\tfrac{p}{-8} = -40$.

Here, $p$ is divided by $-8$; multiply both sides by $-8$ to isolate $p$: $-8 \cdot \tfrac{p}{-8} = -8(-40)$. Simplify:

Solve equations with a coefficient of $-1$

Look at the equation $-y = 15$. It might look like $y$ is already isolated, but there’s a negative sign in front, so it’s not. There are three ways to isolate the variable here: rewrite $-y$ as $-1y$ and divide both sides by $-1$; multiply both sides by $-1$; or read $-y$ as “the opposite of $y$” and ask what number has $15$ as its opposite.

Example. Solve: $-y = 15$.

Rewriting $-y$ as $-1y$ and dividing both sides by $-1$: $\tfrac{-1y}{-1} = \tfrac{15}{-1}$, so $y = -15$. (Multiplying both sides by $-1$, or reasoning that the opposite of $-15$ is $15$, gives the same answer.)

Solve equations with a fraction coefficient

When an equation has a fraction coefficient, we use the Multiplication Property of Equality to make the coefficient equal to $1$ — by multiplying both sides by the reciprocal of the coefficient.

Example. Solve: $\tfrac{3}{4}x = 24$.

The coefficient of $x$ is $\tfrac{3}{4}$. Multiply both sides by its reciprocal, $\tfrac{4}{3}$: $\tfrac{4}{3} \cdot \tfrac{3}{4}x = \tfrac{4}{3} \cdot 24$. Simplify:

Notice we could instead have divided both sides by $\tfrac{3}{4}$ — the same result, since dividing by a fraction is the same as multiplying by its reciprocal — but multiplying by the reciprocal is usually easier.

Example. Solve: $-\tfrac{3}{8}w = 72$.

The coefficient is a negative fraction, so its reciprocal is also negative. Multiply both sides by $-\tfrac{8}{3}$: $-\tfrac{8}{3}\left(-\tfrac{3}{8}w\right) = -\tfrac{8}{3}(72)$. Simplify:

Translate sentences to equations and solve

Now we’ve covered all four properties of equality — addition, subtraction, multiplication, and division. When you add, subtract, multiply, or divide the same quantity on both sides of an equation, you still have equality.

Example. Translate and solve: “$n$ divided by $6$ is $-24$.”

Translate: $\tfrac{n}{6} = -24$. Multiply both sides by $6$: $n = 6(-24)$. Simplify:

Checking: is $-144$ divided by $6$ equal to $-24$? Yes.

Example. Translate and solve: “The quotient of $q$ and $-5$ is $70$.”

Translate: $\tfrac{q}{-5} = 70$. Multiply both sides by $-5$: $q = -5(70)$. Simplify:

Example. Translate and solve: “Two-thirds of $f$ is $18$.”

Translate: $\tfrac{2}{3}f = 18$. Multiply both sides by the reciprocal, $\tfrac{3}{2}$: $f = \tfrac{3}{2} \cdot 18$. Simplify:

Example. Translate and solve: “The quotient of $m$ and $\tfrac{5}{6}$ is $\tfrac{3}{4}$.”

Translate: $\tfrac{m}{\tfrac{5}{6}} = \tfrac{3}{4}$. Multiply both sides by $\tfrac{5}{6}$ to isolate $m$: $m = \tfrac{5}{6} \cdot \tfrac{3}{4}$. Removing common factors:

Example. Translate and solve: “The sum of three-eighths and $x$ is three and one-half.”

Translate: $\tfrac{3}{8} + x = 3\tfrac{1}{2}$. Subtract $\tfrac{3}{8}$ from both sides: $x = 3\tfrac{1}{2} - \tfrac{3}{8}$. Convert $3\tfrac{1}{2}$ to the improper fraction $\tfrac{7}{2}$, then to LCD $8$: $\tfrac{28}{8} - \tfrac{3}{8} = \tfrac{25}{8}$. Write as a mixed number:

We write the answer as a mixed number since the original problem used a mixed number.

Key terms

solution of an equation — a value that, substituted for the variable, makes the equation a true statement. Multiplication Property of Equality — for any numbers $a$, $b$, $c$: if $a=b$, then $ac=bc$.

This section is adapted from Prealgebra 2e, Section 4.7: Solve Equations with Fractions by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: presented the two-column worked examples as prose walkthroughs with typeset math; omitted the Be Prepared quiz, media links, and end-of-section exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.