Solving proportions of complex fractions with cross multiplication

 
 
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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}???

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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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