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

Thinking about multiplicative relationships between quantities that scale together.

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 thinking is multiplicative—'how many times' not 'how many more.'

Common stuck point: Using additive thinking when multiplicative is needed: doubling a recipe means multiplying, not adding 2 cups.

Sense of Study hint: Set up two equivalent fractions side by side and use cross-multiplication to find the missing value.

Worked Examples

Example 1

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

Solution

  1. 1
    Because the relationship is proportional, first find the cost of 1 notebook.
  2. 2
    Find the unit price: \frac{7.50}{3} = \2.50$ per notebook.
  3. 3
    Multiply by 7: 2.50 \times 7 = \17.50$.

Answer

\$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?

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?
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Background Knowledge

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

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