Lesson 8.2 · 20 min

The Unknown in the Denominator

Solving equations such as $\frac{20}{x} = 4$ and $3 = \frac{15}{x + 2}$ by multiplying both sides by the denominator.

In this lesson you will learn to…

Solve an equation with the unknown in the denominator
Use brackets when the denominator is an expression
Solve equations with a fraction on both sides
Construct and solve equations like this from a problem

Why this matters

Sharing problems — a bill split between a group, a journey at an unknown speed — lead to equations where the unknown ends up on the bottom of a fraction. One extra step turns them back into the equations you already know how to solve.

Enri’s mistake

Enri tried to solve $2 = \frac{22}{x + 4}$ like this:

$2 \times x + 4 = 22 \;\Rightarrow\; 2x = 18 \;\Rightarrow\; x = 9$

He left out the brackets. The whole of $x + 4$ must be multiplied by 2:

$2(x + 4) = 22 \;\Rightarrow\; x + 4 = 11 \;\Rightarrow\; x = 7$

Check: $\frac{22}{7 + 4} = \frac{22}{11} = 2$ ✓

A harder one

$\frac{3x - 2}{x - 2} = 2 \;\Rightarrow\; 3x - 2 = 2(x - 2) = 2x - 4 \;\Rightarrow\; x = -2$

Check it. Always substitute back in. As well as catching slips, it guards against an answer that would make the denominator zero — which is never allowed.

Key rules and formulas

Multiply by the denominator first
$\frac{20}{x} = 4 \Rightarrow 20 = 4 x \Rightarrow x = 5$
Multiplying both sides by the denominator lifts the unknown out of the bottom of the fraction. What is left is an ordinary linear equation.
Keep the brackets
$3 = \frac{15}{x + 2} \Rightarrow 3 (x + 2) = 15$
When the denominator is an expression like $x + 2$ , it must stay in brackets. Writing $3 \times x + 2$ is the most common mistake here.
Fraction on both sides
Multiply by both denominators, one at a time: $\frac{18}{x - 3} = \frac{2}{5}$ becomes $18 \times 5 = 2 (x - 3)$ .

Worked examples

Worked example 1

An expression in the denominator

Solve

3 = \frac{15}{x + 2}

Try it yourself first, then reveal the solution one step at a time.

Worked example 2

A problem

A class has

x

students. One day 3 are absent, and the others share 120 pencils equally, getting 4 each. How many students are in the class?

Try it yourself first, then reveal the solution one step at a time.

Worked example 3

Fractions on both sides

Solve

\frac{18}{x - 3} = \frac{2}{5}

Try it yourself first, then reveal the solution one step at a time.

Practice: the unknown in the denominator

9 questions

Nine questions. Multiply by the denominator first — and keep the brackets.

Question 1 of 9Complete the tableFoundation
Solve each equation.
Equation x =
12/x = 6 Row 1, x =
40/x = 4 Row 2, x =
4 = 2/x Row 3, x =
4 = 10/x Row 4, x =
Question 2 of 9Multiple choice
Enri solved $2 = \frac{22}{x + 4}$ by writing $2 \times x + 4 = 22$ and got $x = 9$ . What is the correct answer?
Question 3 of 9Type the answer
Solve $25 = \frac{50}{x - 5}$ .
Your answer
Question 4 of 9Type the answer
Solve $\frac{2}{x + 3} = 0.25$ .
Your answer
Question 5 of 9Step by step
Solve step by step.
Solve $\frac{36}{x + 2} - 1 = 5$ .
1. Add 1 to both sides: $\frac{36}{x + 2} = □$
  Step 1 answer
Question 6 of 9Type the answer
A group of $x$ people is joined by 7 more. They share 570 dollars and each person receives 15 dollars. How many people were in the group to start with?
Your answer
Question 7 of 9Multiple choice
A rectangle has an area of $145 cm^{2}$ . One side is $(x - 8)$ cm and the other is 5 cm. Which equation is correct?
Question 8 of 9Type the answer
Solve $\frac{3}{10} = \frac{6}{x - 5}$ .
Your answer
Question 9 of 9Type the answerChallenge
Solve $\frac{4 y - 5}{2 y - 1} = 3$ .
Your answer

Equation	x =
12/x = 6	Row 1, x =
40/x = 4	Row 2, x =
4 = 2/x	Row 3, x =
4 = 10/x	Row 4, x =

In this lesson you will learn to…

Why this matters

Enri’s mistake

A harder one

Key rules and formulas

Multiply by the denominator first

Keep the brackets

Fraction on both sides

Worked examples

Practice: the unknown in the denominator