# How to Cross Multiply to Solve Equations with Fractions?

How to cross multiply and solve algebraic equations including fractions? Examples and questions with their detailed solutions are presented.

## What is cross multiplication in maths and where is it used?

Example: Let us consider the equation: $\dfrac{a}{b} = \dfrac{c}{d}$
Equations with fractions such as the above are challenging to solve because of the denominators. Let us find an equivalent equation to the above but without denominators.
1 - Multiply the two sides of the equation by the product of the denominators $b \times d$ so that the given equation becomes:
$b \times d \times \dfrac{a}{b} = b \times d \times \dfrac{c}{d}$
2 - Simplify the two sides in the above equation.
$\cancel{b} \times d \times \dfrac{a}{\cancel{b}} = b \times \cancel{d} \times \dfrac{c}{\cancel{d}}$
3 - Rewrite without denominator
$a \times d = b \times c$
The above method of transforming an equation with fractions into an equation without fractions is called "cross multiplication".

## How is cross multiplication used?

It can be used to solve equations.
Example 2: Solve the equation $\dfrac{x}{3} = \dfrac{10}{6}$.
1 - Cross multiply the denominator of one with the numerator of the other obtain the equation.
$6 \times x = 3 \times 10$
2 - Divide both sides by the coefficient of $x$ which is 6
$\dfrac{6 \times x}{6} = \dfrac{3 \times 10}{6}$
3 - Simplify to find x.
$x = 5$

It can be used to verify if two fractions are equivalent.
Example 3: Are the fractions $\dfrac{4}{3}$ and $\dfrac{12}{9}$ equivalent?
1 - Cross multiply the denominator of one with numerator of the other to obtain two quantities.
$4 \times 9 = 36$ and $3 \times 12 = 36$
2 - Compare the two quantities. If they are equal, then the fractions are equivalent which is the case in the above example and we can write.
$\dfrac{4}{3} = \dfrac{12}{9}$

The exercises below with solutions and explanations are all about using cross multiplication.

1. Solve the equations
a) $\dfrac{x}{6} = \dfrac{3}{2}$
b) $\dfrac{1}{3x} = \dfrac{2}{24}$
c) $\dfrac{3}{2} = \dfrac{12}{4x}$
d) $\dfrac{4}{6} = \dfrac{x}{9}$
e) $2 = \dfrac{x}{14}$
f) $\dfrac{2}{x+2} = \dfrac{1}{7}$

2. Which of the following pairs of fractions are equivalent (equal)?
a) $\dfrac{5}{6}$ and $\dfrac{15}{18}$
b) $\dfrac{5}{3}$ and $\dfrac{20}{13}$
c) $\dfrac{25}{35}$ and $\dfrac{5}{7}$
d) $\dfrac{23}{7}$ and $\dfrac{46}{17}$

3. solutions and explanations