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.
\( \dfrac{4}{3} = \dfrac{12}{9} \)

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) \( \dfrac{2}{x+2} = \dfrac{1}{7} \)

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 \( \dfrac{46}{17} \)