Equivalent Ratios and Proportional Reasoning

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

The two bars represent the first and second terms of the ratio. Sliding the factor scales both bars by the same amount.

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 only works when the two terms share a factor.

Iteration scales the ratio up; partition scales it down. Either way the proportion is the same. Every move on the slider is a multiplication or a division acting on both terms.

The most common ratio mistake, and why it's wrong

Given 2 : 3 = 4 : ?, a common wrong answer is 5. Because 2 + 2 = 4, students add 2 to the second term too: 3 + 2 = 5. That's additive thinking, and it gives the wrong ratio. The correct answer is 6, because the move from 2 to 4 is multiplication by 2 (not addition of 2), and the same multiplication has to act on the second term: 3 × 2 = 6.

The bars make this concrete. The ratio 2 : 3 and the ratio 4 : 5 have different shapes. In 4 : 5, the second bar is barely taller than the first; in 2 : 3, the second bar is half again as tall. They are not the same proportion. Adding the same number to both terms changes the ratio; multiplying both terms by the same factor keeps it.

The cross-product property

The curriculum frames the whole strand this way: multiplicative relationships are foundational to proportional reasoning. Every equivalent ratio is a multiplication acting on both terms; every percent in the next lesson is a multiplier; every unit-rate conversion is an iteration of one fixed factor. The cross-product property below is the algebraic shape of that one idea.

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.

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.

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.

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.

Comparing ratios with common terms

Cross-multiplying always works for comparing two ratios, but when two ratios share a first or second term, there's a faster route: scale both ratios to a common term, then compare what's left.

Compare 3 : 5 and 4 : 7. Neither pair shares a term as written. Scale both to second term 35 (since 35 = 5 × 7):

Both have second term 35, so just compare first terms: 21 > 20, which means 3 : 5 > 4 : 7. No cross-multiplication needed once the denominators match.

The same trick uses a common first term when that's easier. Compare 6 : 7 and 6 : 11: first terms already match, second terms differ. The smaller second term ( 7) means a larger ratio (since the same 6 is divided across fewer parts), so 6 : 7 > 6 : 11.

Unit rates: a ratio with second term 1

A unit rate is an equivalent ratio whose second term has been scaled to 1. It answers "how much of the first thing per one of the second?"

A car driving 90 km in 1 h has a unit rate of 90 km/h. Scale the second term down by 2 and 90 : 1 becomes 45 : 0.5: 45 km in half an hour, same speed. Scale up by 4 and the same rate reads 360 : 4, or 360 km in 4 h. Same speed at three time scales.

To convert any ratio into a unit rate, partition until the second term is 1:

Once a ratio is in unit-rate form, scaling to any other amount is a single multiplication.

Converting between units

Unit conversions are equivalent ratios. 1 kilogram is about 2.2 pounds, so 1 : 2.2 is the conversion ratio between kg and lb. Iterating that ratio gives the equivalent at any other size:

5 kg ≈ 11 lb. 10 kg ≈ 22 lb. The conversion ratio doesn't change. Only how much you've scaled it.

The same idea works inside Alberta agriculture. A bushel of wheat weighs about 27.2 kg, so 1 : 27.2 is the bushel-to-kg ratio. A 100-bushel load comes in around 2,720 kg: multiply the ratio's second term by 100. Conversion is just iteration with a specific factor.

Worksheet

Try these to put the idea into practice. Work one approach per question; the goal is fluency.

Going further

The next lesson applies equivalent ratios to percentages. A percentage is a ratio with 100 as the second term: 35% is 35 : 100, which is also 7 : 20. Discounts, taxes, and tips are all built on that one 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.