Three positive numbers are in the ratio 2 : 3 : 4. If the sum of their squares…
2024
Three positive numbers are in the ratio 2 : 3 : 4. If the sum of their squares is 2349, then the numbers respectively are
Answer: B. 18, 27 and 36 — ConceptWhen positive numbers are in the ratio a : b : c, they can be represented as ak, bk and ck for one common positive scale factor k. Their squared sum is…
- A.
14, 21 and 28
- B.
18, 27 and 36
- C.
16, 24 and 32
- D.
12, 18 and 24
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Correct answer: B
Concept
When positive numbers are in the ratio a : b : c, they can be represented as ak, bk and ck for one common positive scale factor k.
Their squared sum is k2(a2 + b2 + c2); solving this equation determines the common scale factor.
Application
Let the numbers be 2k, 3k and 4k, where k > 0.
Using the squared-sum condition: (2k)2 + (3k)2 + (4k)2 = 2349.
Therefore, 4k2 + 9k2 + 16k2 = 29k2 = 2349.
Thus k2 = 2349 ÷ 29 = 81. Since the numbers are positive, k = 9.
Hence the numbers are 2 × 9 = 18, 3 × 9 = 27 and 4 × 9 = 36.
Cross-check
The ratio 18 : 27 : 36 reduces to 2 : 3 : 4.
Also, 182 + 272 + 362 = 324 + 729 + 1296 = 2349, so both conditions are satisfied. Therefore, the numbers are 18, 27 and 36.