Key takeaway: A proportion is an equation stating that two ratios are equal, such as 7/21 = 15/45, and it is used to find a missing value when two quantities change together at a steady rate.
Grade 77.RP.A.2Ratios, Rates and Percents
Definition of Proportion
A proportion says two ratios name the same relationship. If a runner completes 7 laps in 21 minutes, and keeps that pace, then 15 laps take 45 minutes. Written as fractions, \(\frac{7}{21} = \frac{15}{45}\) is a true proportion.
Proportions solve problems with one unknown. Set up two ratios in the same order, with the same units in matching positions, put an equal sign between them, and let a letter stand for the missing amount. Then solve with scaling or by multiplying across. The setup is the hardest part: compare laps with laps, and minutes with minutes.
Formula and notation
A proportion is written as two equal fractions, or with colons:
\[\frac{a}{b} = \frac{c}{d} \qquad\text{or}\qquad a : b = c : d\]
The four numbers are called the terms. In a true proportion, the products \(a \times d\) and \(b \times c\) are equal.
Key facts
- Each ratio must compare the same two quantities in the same order.
- Both fractions in a proportion simplify to the same lowest terms.
- The cross products of a true proportion are equal.
- Units in the numerators must match each other, and so must those in the denominators.
Proportion example
A runner completes 7 laps in 21 minutes. At the same pace, how many laps does she complete in 45 minutes?
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Put laps over minutes in both ratios: \(\frac{7}{21} = \frac{x}{45}\).
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Multiply both sides by 45: \(x = \frac{7 \times 45}{21}\).
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Compute: \(\frac{315}{21} = 15\).
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Check: \(\frac{7}{21} = \frac{1}{3}\) and \(\frac{15}{45} = \frac{1}{3}\).
She completes 15 laps in 45 minutes.
Common mistake
Watch out
Beto writes \(\frac{7}{21} = \frac{45}{x}\) for the same problem and gets \(x = 135\).
Why it is wrong
The second ratio has minutes on top and laps on the bottom, but the first has laps on top. The setup must keep the same order: \(\frac{7}{21} = \frac{x}{45}\). A check helps: 135 laps in 45 minutes is clearly faster than 7 laps in 21 minutes.
Related terms and lessons
Browse the glossary: A–Z list · Grade 7 vocabulary · Ratios, Rates and Percents