Apply it

How to set up proportion word problems

Translate units into matching positions before you calculate.

The hardest part of a proportion word problem is often choosing the equation. Before multiplying anything, write labels for the quantities and decide whether their relationship is actually proportional.

Example: a constant unit price

If 4 notebooks cost $10, how much do 7 notebooks cost at the same unit price? Put notebooks on top and dollars below on both sides: 4/10 = 7/x. Cross multiplication gives 4x = 70, so x = 17.5. The cost is $17.50.

The unit-rate method gives the same answer: $10 ÷ 4 = $2.50 per notebook, and 7 × $2.50 = $17.50. Discounts, fixed delivery fees, or different notebook prices would invalidate the constant-price assumption.

Example: a map scale

A map uses 2 centimeters for 5 kilometers. A route measures 7 centimeters. Set up 2/5 = 7/x, where each ratio is map centimeters per real kilometer. Then 2x = 35 and x = 17.5 kilometers. The units within each ratio differ, but the same type of quantity occupies each matching position.

Example: scaling ingredients

A recipe uses 3 cups of flour for 8 servings. For 12 servings, write 3/8 = x/12, which gives x = 4.5 cups. This assumes the ingredient amount scales directly with servings. Baking time and pan depth need separate judgment; see recipe scaling.

Know when a proportion does not apply

A taxi fare with a starting charge is not directly proportional to distance. If the fare is $4 plus $2 per mile, a 2-mile trip costs $8 but a 4-mile trip costs $12, not $16. The fixed charge prevents a constant fare-per-mile ratio.

Inverse relationships also need a different setup. At a fixed distance, doubling speed halves travel time; it does not double it. Do not use a direct proportion simply because a question contains four quantities.

A short setup checklist

1. Name both quantities and their units.
2. Check the constant-rate assumption.
3. Keep matching units in matching positions.
4. Solve, then include the unit in the answer.
5. Ask whether the magnitude makes sense.

Return to the solving method if the algebra needs a refresher.