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.
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}???
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}???