Pre-Algebra
Ratios and rates
Solving proportions
20 practice questions
0 video lessons
Theory + worked examples
Solving Proportions
California Pre-Algebra • Standard 7.RP.2 • Ratios & Rates
Solving Proportions is a topic in Ratios & Rates in the California Common Core State Standards. It is aligned to Standard 7.RP.2, which requires students to set up and solve proportions using cross-multiplication.
A proportion sets two ratios equal and is solved by cross-multiplying.
Theory
A proportion states that two ratios are equal, \(\dfrac{a}{b}=\dfrac{c}{d}\).
Cross-multiply: \(ad=bc\), then solve for the unknown.
Solving a proportion.
A worked example.
Cross products:
\[\dfrac{a}{b}=\dfrac{c}{d}\iff ad=bc\]
Keep quantities in the same order on both sides.
How to solve a proportion
- Write the two equal ratios.
- Cross-multiply to clear fractions.
- Solve the resulting equation.
- Check the units and answer.
Example 1 — Cross-multiply
Solve \(\dfrac{3}{4}=\dfrac{x}{20}\).
Solution
Cross-multiply.
| \(4x\) | \(=\) | \(60\) |
| \(x\) | \(=\) | \(15\) |
Example 2 — Unknown on top
Solve \(\dfrac{x}{6}=\dfrac{10}{15}\).
Solution
Cross-multiply.
| \(15x\) | \(=\) | \(60\) |
| \(x\) | \(=\) | \(4\) |
Example 3 — Word problem
If \(2\) pens cost \(\$3\), what do \(8\) pens cost?
Solution
Set up a proportion.
| \(\dfrac{2}{3}=\dfrac{8}{x}\) | \(\Rightarrow\) | \(x=\$12\) |
Example 4 — Recipe
A recipe uses \(3\) cups for \(4\) people. How many for \(10\)?
Solution
Proportion.
| \(\dfrac{3}{4}=\dfrac{x}{10}\) | \(\Rightarrow\) | \(x=7.5\) |
Common pitfalls
Line up like quantities in numerator and denominator.
Cross-multiply, don't add across.
Check the answer is reasonable.
Frequently asked questions
What is a proportion?
An equation setting two ratios equal.
How do I solve one?
Cross-multiply and solve.
Solve \(\dfrac{3}{4}=\dfrac{x}{20}\).
\(x=15\).
What does cross-multiply mean?
Set the cross products equal: \(ad=bc\).
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