Discussion
Three numbers are in the ratio 2 : 3 : 4 and the sum of their cubes is 33957. What is the sum of the three numbers ?
Answer: B
The sum of the three numbers is 63.
Let the numbers be 2k, 3k and 4k. Then:
\[ \begin{aligned} (2k)^{3} + (3k)^{3} + (4k)^{3} &= 99k^{3} = 33957 \\ k^{3} &= 343 \Rightarrow k = 7 \\ 2k + 3k + 4k &= 9k = 63 \end{aligned} \]
The numbers are 14, 21 and 28 (check: 2744 + 9261 + 21952 = 33957). 54, 72 and 81 are 9k for k = 6, 8 and 9, whose cubes do not give 33957.
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