Prairie Minds
MATH · GRADE 7Number

Equivalent Ratios and Proportional Reasoning

A ratio is a fraction in disguise — and equivalents are the whole point.

Grade 7
3 : 5 = 6 : 10 = 9 : 15SAME RATIO, THREE SIZES3:56:109:15×2, ×3

What a ratio is

A ratio compares two quantities. 3 : 5 means "for every 3 of the first thing, there are 5 of the second." The first term and the second term can be any numbers — integers, decimals, or fractions — and either term can be larger.

Two ratios are equivalent when they describe the same comparison at a different scale. 3 : 5 and 6 : 10 and 9 : 15 are all the same proportion: 3 of one for every 5 of the other.

See it as scaled bars

Two stacked bars: a short orange bar for the first term, a longer prairie bar for the second term. Slide the factor and both bars scale together. The ratio at the bottom updates as you go.

Try it

Drag the slider. Both terms scale together.

FIRST TERM6SECOND TERM103 : 5 = 6 : 10
18

3:5=6:10(× 2)

Iteration multiplies both terms by the same factor. Partition divides both terms by a common factor. Either way, the ratio is equivalent.

Two ways to generate equivalent ratios. The curriculum names both:

  • Iteration — multiply both terms by the same factor. 3 : 5 becomes 12 : 20 by multiplying by 4.
  • Partition — divide both terms by a common factor. 12 : 20 becomes 3 : 5 by dividing by 4. Partition is only available when the two terms share a factor.

Iteration grows the ratio; partition reduces it. Either way the proportion stays the same.

The cross-product property

A ratio can be written as a fraction. 3 : 5 is 3/5. So two equivalent ratios are also equivalent fractions, and that gives a useful identity:

The diagonal arrows below show why. For 3/4 = 9/12, both diagonals (3 × 12 and 4 × 9) land on the same number, 36. That single matching product is the test.

Cross-multiplication: a × d = b × c

39412=3 × 12 = 364 × 9 = 36= 36

When two fractions are equivalent, both diagonals land on the same product.

This is also how to find an unknown term. If 3/5 = ?/20, the cross product 3 × 20 = 60 has to equal 5 × ?, so ? = 12. Type a guess into the widget below and watch both products update — when they match, the proportion is balanced.

Try it

Type the missing term. The proportion is balanced when the two cross products match.

35
=
?20

Cross products

3 × 20 = —vs5 × 0 = —

Cross-multiply: a × d should equal b × c. Adjust your guess until both products land on the same number. A proportion a over b equals c over d, with one of the four terms blank. The student types a value into the blank; the widget evaluates a times d and b times c live, and turns the proportion green when both products match.

Generating equivalents by adding terms

A subtle property worth knowing: if a/b = c/d, then adding the two first terms and the two second terms also gives an equivalent ratio.

ab=cd  ⟹  a+cb+d=ab\frac{a}{b} = \frac{c}{d} \implies \frac{a + c}{b + d} = \frac{a}{b}

Concretely: 3/5 = 6/10. Add the numerators (3 + 6 = 9) and the denominators (5 + 10 = 15). The result 9/15 is also 3/5. This is the same identity from the curriculum's "the first terms and the second terms of two equivalent ratios can be added or subtracted to generate another equivalent ratio."

It works because (a + c) / (b + d) = (a + ka) / (b + kb) = (1 + k) a / (1 + k) b = a/b. The shared factor (1 + k) cancels.

Where it shows up in real life

A class trip needs chaperones. Alberta school regulations recommend about one adult per twelve students. So a small group of 12 students needs 1 adult; a class of 24 needs 2 adults; a grade of 60 needs 5. The ratio 1 : 12 runs through every group size.

Class trip — chaperone ratio

Base ratio

2:12

Twice the trip

4:24(× 2)

Three times the trip

6:36(× 3)

Per Alberta school regulations: 1 chaperone per ~12 students.

This is iteration in plain clothes. The base ratio doesn't change; only the size of the trip does. Other Alberta examples: scaling up a bannock recipe for a community feast, mixing fertilizer at a quarter-section in southern Alberta (parts product : parts water), converting AB-grade beef hanging weights from kilograms to pounds. Same arithmetic, different units.

Worksheet

These aren't graded. Try one approach per question; the goal is fluency.

Question 1 of 4

Try it

Which ratio is equivalent to 3 : 5?

Multiple choice: which ratio is equivalent to 3 to 5? Four cards: 9 to 15, 5 to 7, 8 to 10, 6 to 5.

Equivalent ratios

Base 3 : 5

Scaled 12 : 20

Same proportion. × 4 on both terms.

Going further

The next lesson takes equivalent ratios to a specific application: percentages. A percentage is just a ratio with 100 as the second term — 35% is 35 : 100, which is also 7 : 20. Discounts, taxes, tips, and tipping points all reduce to this single idea.

In Grade 8, ratios extend to proportional functions — y = kx is the algebraic shape of a ratio. Plotting the equivalent pairs (3, 5), (6, 10), (9, 15) produces a straight line through the origin, and the slope 5/3 is the unchanging ratio.