Chapter 5.1 — Introduction to Ratios and Proportion

When Two Ratios Are in Proportion

If two ratios are equal, the four numbers involved are said to be in proportion. Writing a : b = c : d means a, b, c and d are in proportion, and this equality has a useful cross-multiplied form:

A special case shows up often enough to name separately: if a : b = b : c, then a, b and c are in continued proportion, and b² = a × c — here b is called the mean proportional between a and c.

Compound Ratio

Two simple ratios can be combined into a single ratio by multiplying their antecedents together and their consequents together. The compound ratio of a : b and c : d is (a × c) : (b × d).

The inverse ratio of a : b is simply b : a — swapping which term comes first. Compound ratios and inverse ratios both appear frequently in problems that give you a combined result and ask you to work backward to an unknown term.

Percentage — A Ratio Out of 100

A percentage is nothing more than a ratio with 100 fixed as the second term — "percent" literally means "per hundred." Any percentage can be treated as a fraction with denominator 100, which is what makes it combine so easily with ordinary arithmetic:

Turning that around, if a part x comes out of a whole y, the percentage it represents is (x/y) × 100%. This two-way relationship — percentage to fraction, and fraction to percentage — is used constantly and is worth being equally comfortable running in either direction.

Increasing and Decreasing by a Percentage

Two formulas cover every percentage-change situation this chapter deals with:

• Increase: a quantity x increased by y% becomes x × (100 + y)/100.
• Decrease: a quantity x decreased by y% becomes x × (100 − y)/100.

Both formulas do the same underlying thing: they express the new amount as a single percentage of the original — 100% plus (or minus) the change — then convert that percentage back into an actual value in one multiplication, rather than calculating the change and the new total as two separate steps.

A Ratio Says Nothing About Actual Size

It's worth being clear about what a ratio does and doesn't tell you. Knowing that two quantities are in the ratio 3:4 says nothing about how large either one actually is — the quantities could be 3 kg and 4 kg, or 300 tonnes and 400 tonnes, and the ratio 3:4 describes both equally well. A ratio captures only the relative relationship between two quantities, never their absolute size; recovering the actual size requires one more piece of information, such as the total, or one of the two actual values — the ratio alone can never supply that missing piece by itself.

Why the Same Idea Underlies This Whole Chapter

Ratio, proportion, and percentage might look like three separate topics, but they're really one idea viewed from different angles: comparing one quantity to another. A ratio compares two quantities directly; a percentage compares a quantity specifically to 100; and proportion is what happens when two such comparisons are declared equal to each other. Every exercise ahead — splitting a profit fairly, applying a discount, computing GST, or working out interest — is this same comparison idea applied to a specific real-world context.

Why Cross-Multiplication Works

The rule "product of extremes equals product of means" isn't an arbitrary trick — it follows directly from what it means for two fractions to be equal. Saying a : b = c : d is the same as saying a/b = c/d. Multiplying both sides of that equation by b and by d (to clear both denominators at once) gives a × d = b × c exactly. Seeing proportion this way — as two equal fractions, not a separate new kind of statement — is what makes the cross-multiplication shortcut something you can rebuild from scratch rather than a rule to memorise blindly.

Mean Proportional in Practice

The continued-proportion case, b² = ac, is worth a concrete example. If a : b = b : c with a = 4 and c = 9, then b² = 4 × 9 = 36, so b = 6 — and checking, 4 : 6 does indeed simplify to the same ratio as 6 : 9 (both equal 2:3). The number 6 here is called the mean proportional between 4 and 9, and finding it is really just solving b² = ac for the one positive value of b that keeps both ratios equal, discarding the negative square root since lengths and quantities in these problems are never negative.

Percentage Change Isn't Symmetric

A subtlety worth flagging early: increasing a quantity by 20% and then decreasing the result by 20% does not return the original value. Take 100: increased by 20% gives 120; decreasing 120 by 20% gives 120 × 0.8 = 96, not 100. The second percentage is always taken of a different (already-changed) amount, which is why equal-and-opposite percentage changes never fully cancel out — a detail that matters the moment a problem chains an increase and a decrease together, as several in this chapter do — always apply each percentage change to the most recent value in the chain, never back to the original starting value, no matter how tempting the shortcut of simply adding the two percentages together might look at first glance.

Common Misconceptions to Avoid

• Comparing quantities in different units directly. Before forming a ratio, convert both quantities to the same unit — 8 litres to 750 millilitres only compares sensibly once both are in millilitres (or both in litres).
• Assuming a ratio must reduce to whole numbers. A ratio like 32 : 3 is already in simplest form — there's no rule that the terms must both be small.
• Confusing a compound ratio with adding two ratios. Combining a : b and c : d means multiplying antecedents and consequents separately (ac : bd), not adding the two ratios term by term.

Put to Work Immediately

Ratio and proportion are put to direct use immediately in Exercise 5.1, and the percentage formulas here become the backbone of Exercise 5.2 (discounts, profit and loss, GST) and Exercise 5.3 (simple and compound interest), where the same increase/decrease formulas reappear applied repeatedly over time.