LEMMA

Foundations

Ratios and proportions

ALEKS · foundation repairTEAS math

A proportion is two equal ratios, and cross-multiplying turns it into a linear equation.

What this covers

ba​=dc​⇒ad=bc
  • Write both ratios with the same units in the same position.
  • Cross-multiply to clear both denominators.
  • Solve the linear equation that results.
  • Mental route and check: ask what multiplies the known pair across. 8 boys became 40, that is times 5, so 7 girls become 35.
  • For a three-part ratio, add a total column: 3 to 4 to 7 totals 14, so 112 coins means every part is times 8.
  • Check that the answer's units match what was asked for.

Worked example

Worked example

Solve 3/8 = x/20.

  1. 3⋅20=8xCross-multiply.
  2. 60=8xSimplify the left side.
  3. x=215​Divide both sides by 8 and reduce.
Another worked example

A recipe uses 2 cups for 3 servings. Find the cups needed for 7 servings.

  1. 3 servings2 cups​=32​ cup per servingKeep the units in the same order; the unit rate need not be a whole number.
  2. 7⋅32​=314​=432​ cupsMultiply the rate by the requested servings.

A common mistake

A mistake Lemma catches

From 3 servings2 cups​=32​ cup per serving

7⋅23​7⋅32​

“The rate was flipped. The recipe gives cups per serving: 2 cups ÷ 3 servings = 2/3 cup per serving. 3 ÷ 2 = 3/2 is servings per cup, and servings times servings-per-cup is not a number of cups. Keep cups on top: 7 · 2/3 = 14/3 cups.”

Try one

Sample problem

Solve the proportion for x.

42​=16x​

Practice proportions free

No account, no email. Every line you type is checked by the same computer algebra system as the worked example above.

Where this fits

Before this

After this