Compound ratios and changing ratios

Learning outcome

Form compound ratios, connect shared quantities, and solve changing-ratio problems by tracking changes precisely.

Understanding compound and changing ratios

The compound ratio of a:b and c:d is (a × c):(b × d). This follows from multiplying the fractions a/b and c/d. Multiply corresponding terms, not crosswise; cross-products serve comparison or proportion testing.

If A:B and B:C are known, multiplying A/B by B/C cancels the shared B and gives A/C. This gives the endpoint ratio, not the full three-term ratio. For A:B:C, scale until the B terms match.

Write initial amounts as au and bu, where u > 0 is their common scale and au means a × u. Apply additions or removals to the specified quantities before forming the new ratio. Match units first.

Equal positive scaling preserves a ratio. Equal additions generally change it because relative increases differ; equal initial amounts are an exception. A transfer subtracts from one quantity and adds to the other, preserving the total. Remaining quantities must stay positive.

Worked example 1

Positive lengths satisfy A:B = 2:3 and B:C = 4:5. Find A:C and A:B:C.

Compounding gives A:C = (2 × 4):(3 × 5) = 8:15. For A:B:C, multiply 2:3 by 4 to get 8:12, and 4:5 by 3 to get 12:15. Matching B now gives A:B:C = 8:12:15.

Worked example 2

Two containers hold water in ratio 3:5. Add 8 L to the first only; the second stays unchanged. The new ratio is 5:7. Find the initial amounts.

Let initial amounts be 3u L and 5u L. Then (3u + 8)/(5u) = 5/7. Cross-multiplication gives 21u + 56 = 25u, so 56 = 4u and u = 14. Initially there were 42 L and 70 L. After addition, 50:70 simplifies to 5:7.

Worked example 3

Two boxes have beads in ratio 7:5. Transfer 10 beads from the first to the second; nothing else changes. The new ratio is 4:5. Find the initial counts.

Initially, take 7u and 5u. Then (7u − 10)/(5u + 10) = 4/5. Thus 35u − 50 = 20u + 40, giving 15u = 90 and u = 6. Initially there were 42 and 30 beads. Afterwards there are 32 and 40, giving 4:5. Both totals are 72.

Common mistakes

Identify what a compound ratio represents. Equal additions are not equal scaling. Transfers change both quantities; check that remaining amounts are positive.

Practice questions

1. Find the compound ratio of 3:8 and 4:9.
2. For positive quantities, P:Q = 3:4 and Q:R = 6:7. Find P:Q:R and P:R.
3. Two containers hold water in ratio 2:3. Adding 6 L to each changes it to 3:4. Find the initial amounts.
4. Two containers hold water in ratio 5:8. Remove 9 L from only the second; the first stays unchanged. The new ratio is 5:6. Find initial amounts.

Worked answers

1. Multiply corresponding terms: (3 × 4):(8 × 9) = 12:72 = 1:6.
2. Scale the ratios to 9:12 and 12:14. Hence P:Q:R = 9:12:14 and P:R = 9:14.
3. Write (2u + 6)/(3u + 6) = 3/4. Then 8u + 24 = 9u + 18, so u = 6. Initially: 12 L and 18 L. Afterwards: 18 L and 24 L, giving 3:4.
4. Write (5u)/(8u − 9) = 5/6. Then 30u = 40u − 45, so u = 4.5. Initially: 22.5 L and 36 L. Afterwards: 22.5 L and 27 L, both positive, giving 5:6.