Ratios and Proportions • Topic 3 of 7

Proportions

A proportion is a statement that two ratios are equal, a/b = c/d. The most reliable way to solve one is cross-multiplication: a·d = b·c, then solve for the unknown. Proportions model scaling problems — if 3 pens cost $6, then 7 pens cost x, set up 3/6 = 7/x and solve. Keep the same quantities in the same positions (pens with pens, dollars with dollars). Proportions are everywhere on the SAT: similar triangles, map scales, currency conversion and "best value" comparisons all reduce to setting two ratios equal and cross-multiplying.

✅ Solved examples

1. Solve 3/4 = x/20.
Cross-multiply: 4x = 60, so x = 15.
2. If 4 notebooks cost $10, how much do 10 notebooks cost?
Set 4/10 = 10/x; cross-multiply: 4x = 100, x = $25.
3. Solve 5/x = 15/9.
Cross-multiply: 15x = 45, so x = 3.
4. A 6 m pole casts a 4 m shadow. How long is the shadow of a 9 m pole at the same time?
6/4 = 9/x; cross-multiply: 6x = 36, x = 6 m.

✏️ Practice — try these, take hints as needed

1. Solve 2/5 = x/30.
Cross-multiply: 5x = 2·30.
5x = 60.
Divide by 5.
12.
2. If 5 apples cost $2, how much do 20 apples cost?
Set 5/2 = 20/x.
Cross-multiply: 5x = 40.
Divide.
$8.
3. Solve 7/x = 21/12.
Cross-multiply: 21x = 84.
Divide both sides by 21.
x = 4.
4.
4. A map uses 1 cm to 5 km. How many km is 8 cm?
Set 1/5 = 8/x.
Cross-multiply: x = 40.
Units are km.
40 km.
5. Solve 9/12 = 6/x.
Cross-multiply: 9x = 72.
Divide by 9.
x = 8.
8.

📝 Topic test — 8 questions

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