Proportional Reasoning Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Proportional Reasoning.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The ability to recognize and work with multiplicative relationships between quantities. If one quantity doubles, a proportional quantity also doubles — the ratio stays constant.

If 3 pizzas feed 12 people, how many feed 20? Think multiplication, not addition.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Proportional quantities grow by the same factor, so you stretch one to match the other instead of adding to it.

Common stuck point: The procedure for proportional reasoning is the easy part; the trap is adding the difference instead of multiplying by the factor. Asking "When one quantity multiplies by a factor, does the other multiply by the same factor?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: When one quantity multiplies by a factor, does the other multiply by the same factor?

Worked Examples

Example 1

easy

If 3 notebooks cost \$7.50, how much do 7 notebooks cost?

Answer

\$17.50

First step

Because the relationship is proportional, first find the cost of 1 notebook.

Full solution

2
Find the unit price: $\frac{7.50}{3} = \$2.50$ per notebook.
3
Multiply by 7: $2.50 \times 7 = \$17.50$ .

Proportional reasoning often starts by finding the unit rate, then scaling to the desired quantity.

Example 2

medium

A recipe calls for 2 cups of flour for every 3 cups of sugar. If you use 9 cups of sugar, how many cups of flour do you need?

Example 3

medium

A 5 ft tall person casts a 7 ft shadow. A nearby tree casts a 28 ft shadow. How tall is the tree?

Example 4

hard

If 9 workers can finish a job in 12 days, how many days for 4 workers (inverse proportion)?

Example 5

medium

Tip is proportional to bill at a flat 18% rate. If the bill is \$45, find the tip.

Example 6

hard

a : b : c = 2 : 3 : 5

and the total

a + b + c = 80

, find each value.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium

A map uses a scale of 1 cm : 25 km. Two cities are 7.5 cm apart on the map. What is the actual distance?

Example 2

easy

A recipe that serves 4 people uses 10 cups of flour. How many cups are needed to serve 10 people?

Example 3

easy

If 2 pencils cost

\$3

, how much do 4 pencils cost?

Example 4

easy

A map scale is 1 cm : 5 km. How many km is 4 cm?

Example 5

easy

If 3 pizzas feed 12 people, how many people do 6 pizzas feed?

Example 6

easy

Solve the proportion

\frac{2}{5} = \frac{x}{20}

Example 7

easy

If 5 notebooks cost

\$10

, how much does 1 notebook cost?

Example 8

easy

A recipe ratio of sugar to flour is 1 : 3. If you use 2 cups of sugar, how much flour?

Example 9

easy

If a photo is enlarged so width goes from 4 in to 12 in, by what factor did it scale?

Example 10

easy

If 4 liters of paint cover 60 square meters, how many square meters do 8 liters cover?

Example 11

medium

Solve

\frac{3}{4} = \frac{x}{12}

by cross-multiplication.

Example 12

medium

A 6-foot person casts a 9-foot shadow. A nearby tree casts a 30-foot shadow. How tall is the tree?

Example 13

medium

If 7 machines produce 350 widgets per day, how many widgets do 10 machines produce per day?

Example 14

medium

A recipe for 8 servings uses 200 g of pasta. How much pasta for 5 servings?

Example 15

medium

Two quantities are in proportion: when

x = 4

y = 10

. Find

y

when

x = 14

Example 16

medium

A car uses 8 liters of fuel per 100 km. How much fuel for a 250 km trip?

Example 17

medium

In a class the ratio of boys to girls is 3 : 4. If there are 21 boys, how many girls are there?

Example 18

medium

On a blueprint, 2 inches represents 8 feet. What length does 5 inches represent?

Example 19

medium

If 9 identical books weigh 12 kg, how much do 15 such books weigh?

Example 20

challenge

A recipe serves 6 and uses 3 eggs and 2 cups of milk. To serve 15, how many eggs and cups of milk are needed?

Example 21

challenge

y

is directly proportional to

x

and

y = 18

when

x = 6

, write the equation and find

x

when

y = 30

Example 22

challenge

A 1:50 scale model of a building is 0.4 m tall. How tall is the real building, in meters?

Example 23

easy

Solve the proportion

\frac{3}{4} = \frac{x}{12}

Example 24

easy

If 4 pens cost \$6, how much do 10 pens cost?

Example 25

easy

If a recipe for 4 servings uses 2 cups of milk, how many cups for 6 servings?

Example 26

medium

A car uses 5 gallons to travel 150 miles. How many gallons to travel 240 miles?

Example 27

medium

A scale model: 2 in represents 50 ft. What does 7 in represent?

Example 28

medium

Solve:

\frac{x}{15} = \frac{8}{20}

Example 29

medium

Three workers paint 60 ft of fence in a day. How much can 5 workers paint in a day (same pace)?

Example 30

easy

\frac{6}{x} = \frac{2}{5}

, find

x

Example 31

hard

A blueprint scale is

1 \text{ in} : 8 \text{ ft}

. If a room is 24 ft long, how long is it on the blueprint?

Example 32

medium

A drink mix uses 3 scoops for 4 cups of water. How many scoops for 10 cups?

Example 33

medium

Solve:

\frac{x + 1}{6} = \frac{5}{3}

Example 34

hard

A car travels 180 miles using 12 gallons. At this rate, how far can it go on 20 gallons?

Example 35

medium

A rectangle is 4 in by 6 in. If you enlarge to width 10 in keeping the same proportions, what is the new length?

Example 36

hard

y

is proportional to

x

, and

y = 30

when

x = 10

, find

y

when

x = 25

Example 37

medium

A typist types 360 words in 6 minutes. How many words in 11 minutes at the same rate?

Example 38

medium

If 4 oranges cost \$2.40, how much do 7 oranges cost?

Example 39

challenge

On a map, 3 cm represents 45 km. Two cities are 105 km apart in reality. How far apart on the map?

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Related Concepts

Ratios Multiplication Proportions Similar Figures

Background Knowledge

These ideas may be useful before you work through the harder examples.

ratiosmultiplication