EQN-02-01

Ratios — The Basics

Read, simplify, and split quantities by a ratio. Ratios appear in arithmetic, geometry, and data questions alike.

~5 min read•Difficulty 2/5

What a ratio says

A ratio a : b compares two quantities by division, not by amount. "3 : 2" means for every 3 of the first there are 2 of the second. Ratios can be scaled up or down freely, exactly like fractions.

Simplifying

Divide both terms by their greatest common factor.

12 : 18 \to (\div 6) \to 2 : 3

The "parts" method (the key skill)

To split a total in a given ratio, add the ratio terms to get the number of equal parts, then find the value of one part.

Split 200 in the ratio 3 : 2

Total parts = 3 + 2 = 5

One part = 200 \div 5 = 40

Shares = 3 \times 40 : 2 \times 40 = 120 and 80

Three-term ratios

A : B : C works the same way. If $A : B = 2 : 3$ and $B : C = 3 : 4,$ the shared term B already matches, so $A : B : C = 2 : 3 : 4.$ If it doesn’t match, scale one ratio so the common term is equal.

Equivalent ratios solve "missing value" problems

3 : 5 = 12 : x \to cross-multiply \to 3 x = 60 \to x = 20

Worked examples

Example 1 (difficulty 2/5)

A sum of 450 is divided between two people in the ratio 4 : 5. What is the larger share?

◎ Predict the answer first:

Example 2 (difficulty 1/5)

2 : 7 = x : 28,

what is x ?

◎ Predict the answer first:

Tips

Add the ratio terms first — the "number of parts" unlocks almost every ratio question.
A ratio has no units of its own; 3 : 2 could be 3 cm : 2 cm or 30 : 20.
Keep the order: "the ratio of cats to dogs is 3 : 2" is not the same as 2 : 3.

Common mistakes

Treating a ratio term as the actual amount (reading 3 : 2 as 3 and 2 items)

Why: The numbers look like quantities.

Fix: A ratio gives proportions; multiply by the value of one part to get amounts.

Reversing the order of the ratio

Why: Under time pressure the wording is skimmed.

Fix: Match each term to the noun it describes, in the stated order.

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