Ratios & Proportions • Topic 1 of 4

Basic & Compound Ratios

A ratio compares quantities of the same kind by division: 12 : 18 means 12/18, which reduces to 2 : 3. Always simplify to lowest terms and keep the order the language gives you — "boys to girls is 3 : 2" is not the same as 2 : 3. Ratios have no units, so before comparing you must convert to a common unit (50 cm : 1 m becomes 50 : 100 = 1 : 2). To combine two ratios that share a middle term, scale them so that shared term matches: if a : b = 2 : 3 and b : c = 4 : 5, rewrite as 8 : 12 and 12 : 15 to read a : b : c = 8 : 12 : 15. A compound (combined) ratio multiplies term by term — the ratio of (2 : 3) and (4 : 5) is 8 : 15 — and it is the engine behind rate and work problems. On the GRE, converting a three-part ratio into a single "parts" count is usually the fastest route to the answer.

✅ Solved examples

1. Simplify the ratio 45 : 60 to lowest terms.
Divide both by their GCD, 15: 45/15 : 60/15 = 3 : 4.
2. If a : b = 2 : 3 and b : c = 6 : 7, find a : b : c.
Make b match. First ratio ×2 gives 4 : 6; second is already 6 : 7. So a : b : c = 4 : 6 : 7.
3. Quantitative Comparison. The ratio of two numbers is 5 : 7. Quantity A: the smaller number as a fraction of the larger. Quantity B: 0.7. Which is greater?
Smaller/larger = 5/7 ≈ 0.714 > 0.70, so Quantity A is greater. (No actual values needed — a ratio is all you need.)
4. The compound ratio of 3 : 4 and 8 : 9 is what, in lowest terms?
Multiply term by term: (3×8) : (4×9) = 24 : 36 = 2 : 3.

✏️ Practice — try these, take hints as needed

1. Reduce 84 : 108 to lowest terms.
Find the GCD of 84 and 108.
GCD = 12.
Divide both by 12.
7 : 9
2. Express 750 g to 2 kg as a ratio in lowest terms.
Use one common unit.
2 kg = 2000 g.
Reduce 750 : 2000.
3 : 8
3. If x : y = 3 : 4 and y : z = 8 : 9, find x : y : z.
Make y the same in both.
Scale 3 : 4 up to have y = 8.
3 : 4 → 6 : 8; keep 8 : 9.
6 : 8 : 9
4. Find the compound ratio of 2 : 5 and 15 : 8.
Multiply corresponding terms.
(2×15) : (5×8).
Then simplify 30 : 40.
3 : 4
5. Two numbers are in the ratio 4 : 7. If both are multiplied by 3, what is the new ratio?
Scaling both terms by the same factor does not change a ratio.
(4×3) : (7×3).
Reduce 12 : 21.
4 : 7 (unchanged)

📝 Topic test — 8 questions

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Dividing in a Ratio →