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23 Sep 2021, 08:07
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The ratio of the number of marbles in boxes X and Y was 5 to 8. After some number of marbles were transferred from box Y to box X, the ratio of the number of marbles in boxes X and Y became 7 to 6. What is the total number of marbles in the two boxes?
(1) Initially the number of marbles in box Y was between 45 and 100.
(2) After the transfer, the number of marbles in box X became between 45 and 100.
(1) Initially the number of marbles in box Y was between 45 and 100.
(2) After the transfer, the number of marbles in box X became between 45 and 100.
23 Sep 2021, 08:07
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Official Solution:
The ratio of the number of marbles in boxes X and Y was 5 to 8. After some number of marbles were transferred from box Y to box X, the ratio of the number of marbles in boxes X and Y became 7 to 6. What is the total number of marbles in the two boxes?
Initial ratio: \(\frac{X}{Y}=\frac{5a}{8a}\), for some positive integer \(a\).
Final ratio: \(\frac{X'}{Y'}=\frac{5a+k}{8a-k}=\frac{7b}{6b}\), for some positive integers \(b\) and \(k\).
The total number of marbles \(= 5a+8a=7b+6b\), which gives \(a=b\).
Thus:
Initial ratio: \(\frac{X}{Y}=\frac{5a}{8a}\)
Final ratio: \(\frac{X'}{Y'}=\frac{7a}{6a}\) (the number of marbles transferred from Y to X is \(2a\))
The question asks to find the number of total marbles, which is \(5a+8a=13a\)
(1) Initially the number of marbles in box Y was between 45 and 100.
\(45 < 8a < 100\)
\(a\) could be 6, 7, 8, 9, 10, 11, or 12. Not sufficient.
(2) After the transfer, the number of marbles in box X became between 45 and 100.
\(45 < 7a < 100\)
\(a\) could be 7, 8, 9, 10, 11, 12, 13, or 14. Not sufficient.
(1)+(2) \(a\) still could take more than one value: 7, 8, 9, 10, 11, or 12. Not sufficient.
Answer: E
The ratio of the number of marbles in boxes X and Y was 5 to 8. After some number of marbles were transferred from box Y to box X, the ratio of the number of marbles in boxes X and Y became 7 to 6. What is the total number of marbles in the two boxes?
Initial ratio: \(\frac{X}{Y}=\frac{5a}{8a}\), for some positive integer \(a\).
Final ratio: \(\frac{X'}{Y'}=\frac{5a+k}{8a-k}=\frac{7b}{6b}\), for some positive integers \(b\) and \(k\).
The total number of marbles \(= 5a+8a=7b+6b\), which gives \(a=b\).
Thus:
Initial ratio: \(\frac{X}{Y}=\frac{5a}{8a}\)
Final ratio: \(\frac{X'}{Y'}=\frac{7a}{6a}\) (the number of marbles transferred from Y to X is \(2a\))
The question asks to find the number of total marbles, which is \(5a+8a=13a\)
(1) Initially the number of marbles in box Y was between 45 and 100.
\(45 < 8a < 100\)
\(a\) could be 6, 7, 8, 9, 10, 11, or 12. Not sufficient.
(2) After the transfer, the number of marbles in box X became between 45 and 100.
\(45 < 7a < 100\)
\(a\) could be 7, 8, 9, 10, 11, 12, 13, or 14. Not sufficient.
(1)+(2) \(a\) still could take more than one value: 7, 8, 9, 10, 11, or 12. Not sufficient.
Answer: E
22 Oct 2025, 21:58
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I like the solution - it’s helpful.
