If \(A:B = \frac{1}{2}:\frac{3}{8}\), \(B:C = \frac{1}{3}:\frac{5}{9}\) and…
2023
If \(A:B = \frac{1}{2}:\frac{3}{8}\), \(B:C = \frac{1}{3}:\frac{5}{9}\) and \(C:D = \frac{5}{6}:\frac{3}{4}\), then the ratio A:B:C:D is
Answer: B. 8:6:10:9 — ConceptWhen several ratios share variables, first reduce each pair and then multiply whole ratios by scale factors so that every common variable has the same…
- A.
6:8:9:10
- B.
8:6:10:9
- C.
4:6:8:10
- D.
More than one of the above
- E.
None of the above
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Correct answer: B
Concept
When several ratios share variables, first reduce each pair and then multiply whole ratios by scale factors so that every common variable has the same value.
Multiplying every term of a ratio by the same non-zero factor preserves the ratio.
Application
Reduce A:B: (1/2):(3/8) gives (1/2) ÷ (3/8) = 4/3, so A:B = 4:3.
Reduce B:C: (1/3):(5/9) gives (1/3) ÷ (5/9) = 3/5, so B:C = 3:5. The common value B = 3 gives A:B:C = 4:3:5.
Reduce C:D: (5/6):(3/4) gives (5/6) ÷ (3/4) = 10/9, so C:D = 10:9.
Multiply 4:3:5 by 2 to make C = 10. Thus A:B:C = 8:6:10, and adjoining D = 9 gives A:B:C:D = 8:6:10:9.
Cross-check
In 8:6:10:9, the adjacent ratios reduce to 4:3, 3:5, and 10:9, exactly the three reduced pairwise ratios.
Therefore, A:B:C:D = 8:6:10:9.