Solving proportions of complex fractions with cross multiplication

Using cross multiplication on a proportion

When you have complex fractions as a proportion you can solve for the variable or rewrite them by using cross multiplication.

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What is cross multiplication?

Remember that

$\frac{a}{b}=\frac{c}{d}$

can be rewritten as $ad=bc$.

Also remember these rules from the complex fraction section:

A reciprocal is a number “flipped upside down.”

The reciprocal of $\frac{a}{b}$ is $\frac{b}{a}$

The reciprocal of $\frac{x}{1}$ is $\frac{1}{x}$

A fraction bar can be thought of like a division sign.

$\frac{x}{y}=x\div y$

To divide by a fraction you can multiply by its reciprocal.

$\frac{x}{\frac{a}{b}}=x \div \frac{a}{b}=x \cdot \frac{b}{a}$

Any number or variable can be divided by $1$ and remain the same number.

$x = \frac{x}{1}$

How to use cross multiplication to solve a proportion of complex fractions

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Cross multiplying to solve for x

Example

Solve for the variable.

$\frac{ \frac{1}{3} }{x} = \frac{ \frac{1}{6} }{ \frac{1}{7} }$

We’ll cross multiply.

$\frac{1}{3} \cdot \frac{1}{7}=x \cdot \frac{1}{6}$

Now we can simplify by multiplying the fractions.

$\frac{1 \cdot 1}{3 \cdot 7}=\frac{x}{6}$

$\frac{1}{21} = \frac{x}{6}$

Multiply both sides of this equation by $6$ to solve for $x$.

$\frac{6}{21}=x$

$x=\frac{2}{7}$

When you have complex fractions as a proportion you can solve for the variable or rewrite them by using cross multiplication.

Example

Solve for the variable.

$\frac{\frac{x}{4}}{\frac{8}{3}}=\frac{\frac{4}{3}}{\frac{5}{4}}$

Instead of dividing by the fractions in the denominators, we can multiply by their reciprocals.

$\frac{x}{4}\cdot\frac{3}{8}=\frac{4}{3}\cdot \frac{4}{5}$

After multiplying you get

$\frac{3x}{32}=\frac{16}{15}$

Multiply both sides by $32$.

$3x=32\cdot \frac{16}{15}$

Divide by $3$ to solve for $x$. Then multiply fractions to simplify.

$x=\frac{32}{3} \cdot \frac{16}{15}$

$x=\frac{512}{45}$

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