Proportion Solving

Proportion Solving

A proportion says two ratios are equal: $\dfrac{a}{b} = \dfrac{c}{d}$. This pattern gives you three of the four values and asks you to find the missing one. Cross-multiply and solve.

Worked Examples

Example 1. A recipe uses $3$ cups of oil for every $5$ cups of vinegar. How many cups of oil are needed for $20$ cups of vinegar?

Set up: $\dfrac{3}{5} = \dfrac{x}{20}$
Cross-multiply: $3 \times 20 = 5x$, so $60 = 5x$, $x = 12$.
Gotcha: Make sure the same quantities are in the same positions. Oil goes with oil, vinegar with vinegar. If you flip one ratio, you'll get the wrong answer.

Example 2. A map scale is $1$ inch $= 25$ miles. Two cities are $3.5$ inches apart on the map. What is the actual distance?

$\dfrac{1}{25} = \dfrac{3.5}{x}$ → $x = 25 \times 3.5 = 87.5$ miles.

Example 3. If $7$ out of every $20$ students prefer math, how many students prefer math in a school of $500$?

$\dfrac{7}{20} = \dfrac{x}{500}$ → $x = \dfrac{7 \times 500}{20} = \dfrac{3500}{20} = 175$.

Example 4. In a survey, $3$ out of $8$ people chose Brand A. If $120$ people were surveyed, how many chose Brand A?

$\dfrac{3}{8} = \dfrac{x}{120}$ → $x = \dfrac{3 \times 120}{8} = 45$.

What to Do on Test Day

• Cross-multiply: $\dfrac{a}{b} = \dfrac{c}{d}$ becomes $ad = bc$. Solve for the unknown.
• Keep ratios consistent. If the left fraction is oil/vinegar, the right fraction must also be oil/vinegar — not vinegar/oil.
• Scale factor shortcut: If the denominator multiplied by some factor gives the other denominator, multiply the numerator by that same factor. $\dfrac{3}{5} = \dfrac{?}{20}$: since $5 \times 4 = 20$, the answer is $3 \times 4 = 12$.
• Check reasonableness. If $3$ out of $8$ chose Brand A from $120$ people, the answer should be less than $120$ and roughly $\dfrac{3}{8}$ of it.