Grade 7 Ratio & Proportion: Scale Factors, Unit Rates & Relationships (US Common Core)

GRADE 7 MATHEMATICS • LESSON 4

Ratio & Proportion

النسبة والتناسب

Learn to compare quantities, simplify ratios, share amounts fairly, and solve proportions — then take 15 exams with full step-by-step solutions.

What is a ratio?

A ratio compares two amounts — it tells you how much of one thing there is compared to another. If a recipe uses 2 cups of flour and 1 cup of sugar, the ratio of flour to sugar is 2 : 1 (read “two to one”).

Order matters. 2 : 1 is not the same as 1 : 2. Always write the terms in the order the question gives them.

A ratio behaves just like a fraction, so you can simplify it the same way. Try the tool below.

🟪 Ratio Simplifier
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Simplest form
First part of whole
Second part of whole

Words you must know

Short, exact definitions to memorise.

Ratio
A comparison of two quantities of the same kind, written a : b.
Term
Each number in a ratio. In 3 : 5 the terms are 3 and 5.
Equivalent ratios
Ratios that have the same value, like equivalent fractions. 2 : 3 = 4 : 6 = 6 : 9.
Simplest form
A ratio whose terms share no common factor except 1.
Proportion
A statement that two ratios are equal: a : b = c : d.
Rate
A ratio comparing quantities of different units, like km per hour or price per item.
Unit rate
A rate written for exactly one unit (“per 1”), found by dividing.

Part-to-part vs part-to-whole

In a class with 3 boys and 2 girls, the part-to-part ratio of boys to girls is 3 : 2. The part-to-whole fraction of boys is 3/5 of the class, because there are 5 students in total.

Working with ratios

Your main study reference. Every method is broken into numbered steps, each one explained, with a worked example and a teacher hack.

1) Simplifying a ratio

  1. Find the GCD. Work out the largest number that divides both terms exactly — this is what makes the numbers as small as possible.
  2. Divide both terms by the GCD. Whatever you divide one term by, you must divide the other by the same number, or the ratio changes.
  3. Write the new ratio. The two smaller numbers are the simplest form.
Example: simplify 8 : 12
GCD of 8 and 12 = 4
8 ÷ 4 = 2   and   12 ÷ 4 = 3
= 2 : 3
Hack: a ratio is just a fraction lying on its side. Simplify a : b exactly like you simplify a/b.

2) Equivalent ratios

  1. Find the scale factor. Divide a known new term by the matching old term to see what the ratio was multiplied by.
  2. Apply it to the other term. Multiply (or divide) the other term by that same scale factor.
  3. Write the equivalent ratio. The value has not changed, only the size of the numbers.
Example: 2 : 3 = 8 : ?
scale factor = 8 ÷ 2 = 4
other term = 3 × 4 = 12
= 8 : 12
Hack — do the same to both: whatever you do to one term, do to the other. Multiply both or divide both by the same number.

3) Proportion & cross-multiplication

  1. Write it as two equal fractions. a : b = c : x becomes a/b = c/x.
  2. Cross-multiply. Multiply each top by the opposite bottom: a × x = b × c. This works because the two fractions are equal.
  3. Solve for the unknown. Divide both sides by the number in front of x.
Example: 3 : 4 = 9 : x
3 × x = 4 × 9
3x = 36
x = 36 ÷ 3 = 12
Hack: cross-multiply means “corner times corner”. The two products are always equal in a true proportion.

4) Sharing in a ratio

  1. Add the ratio numbers. This tells you the total number of equal parts the amount is split into.
  2. Find one part. Divide the total amount by the number of parts.
  3. Multiply for each share. Multiply one part by each ratio number to get every person's share.
  4. Check. The shares should add back up to the total amount.
Example: share 30 in the ratio 2 : 3
parts = 2 + 3 = 5
one part = 30 ÷ 5 = 6
shares = 2×6 and 3×6 = 12 and 18
check: 12 + 18 = 30 ✓
Hack: “add, divide, multiply.” Add the parts, divide the total, multiply for each share.

5) Unit rate

  1. Spot the two quantities. A rate compares two different things, like cost and number of items.
  2. Divide to reach 1. Divide the total by the quantity to find the value for exactly one (“per 1”).
Example: 5 pens cost 20. Cost of 1 pen?
20 ÷ 5 = 4
1 pen costs 4
Hack: “per” means divide. Per 1 item, per 1 hour, per 1 kg — always divide by that quantity.
🧮 Share-in-Ratio Calculator
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Test Yourself — 15 Exams

Each exam has 10 questions, and every answer comes with a step-by-step solution — even when you get it right.

Key takeaways

1. A ratio compares amounts and the order matters: 2 : 1 is not 1 : 2.

2. Simplify a ratio like a fraction: divide both terms by their GCD.

3. Equivalent ratios: do the same thing to both terms.

4. To solve a proportion, cross-multiply: a : b = c : x means a×x = b×c.

5. To share an amount: add the parts, divide the total, multiply for each share.