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Math foundations

SAT ratios and rates: set up proportions and convert units without errors

Which jar of peanut butter is the better deal? How many feet per second is 60 miles per hour? Learn to set up ratios, unit rates, and conversions so the units guide you to the answer.

SPSAT Practice Editorial7 min read

The short version

What you'll take away

  • Compare options by computing a unit rate: the amount per one unit.
  • A part-to-part ratio of 3:5 means the parts are 3/8 and 5/8 of the whole.
  • Convert units by multiplying by factors equal to 1 and canceling units like fractions.
In this article

Unit rates make comparisons fair

A rate compares two different quantities, such as dollars and ounces. A unit rate scales it to one unit of the second quantity, so different package sizes can be compared directly. Divide the cost by the number of ounces to find the cost per ounce.

See the skill

Cost per ounce for three jar sizes (illustrative prices)

The biggest jar is not the best deal here. The 20-ounce jar costs the least per ounce: $4.40 ÷ 20 = $0.22.

Part-to-part versus part-to-whole

If the ratio of juniors to seniors in a club is 3:5, there are 3 juniors for every 5 seniors. Together that is 8 parts, so juniors make up \(\displaystyle \frac{3}{8}\) of the club and seniors make up \(\displaystyle \frac{5}{8}\). If the club has 40 members, there are \(\displaystyle 40 \times \frac{3}{8} = 15\) juniors.

Proportions: keep the units aligned

A recipe uses 3 cups of flour for 4 servings. For 10 servings, write cups over servings on both sides: \(\displaystyle \frac{3}{4} = \frac{c}{10}\). Cross-multiply to get \(\displaystyle 4c = 30\), so \(\displaystyle c = 7.5\) cups.
If the units on top match each other and the units on the bottom match each other, the proportion is set up correctly.

Your turn

Quick practice

A car travels 150 miles on 6 gallons of gas. At the same rate, how many gallons does it need to travel 400 miles?

Hint: Find miles per gallon first.

Choose one answer

Convert units by canceling

Unit conversions are chains of multiplication by fractions that equal 1, such as \(\displaystyle \frac{5280 \text{ ft}}{1 \text{ mi}}\). Arrange each fraction so the unwanted unit cancels.

See the skill

Convert 60 miles per hour to feet per second

  1. Write what you have, with units

    60 miles / 1 hour

  2. Cancel miles

    Multiply by 5,280 feet / 1 mile. Miles cancel, leaving feet per hour: 316,800 ft/h.

  3. Cancel hours

    Multiply by 1 hour / 3,600 seconds. Hours cancel: 316,800 ÷ 3,600 = 88 ft/s.

  4. Sanity check

    A second is much shorter than an hour, and a foot is much shorter than a mile. 88 is a reasonable size.

If a unit does not cancel, the fraction is upside down. Flip it.

Practice: try these before checking the answers

Write units on every number.
  • 1. The ratio of red to blue marbles is 2:7, and there are 36 marbles. How many are red?
  • 2. A train goes 45 km in 30 minutes. What is its speed in km per hour?
  • 3. How many square feet are in 3 square yards?
  • 4. Five pencils cost $1.80. How much do 12 pencils cost at the same price?

Sources & further reading

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