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26 Aug 2021, 08:00
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E
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Difficulty:
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Question Stats:
63% (02:40) correct
37%
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wrong
based on 355
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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.
M36-12
(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.
M36-12
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23 Sep 2021, 08:09
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Bunuel
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
General Discussion
26 Aug 2021, 08:32
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My answer is E).
Assuming that X initially has 5k marbles and Y 8k. Notice that all it takes is for Y to transfer 2k marbles to X to make the ratio 7:6 (7k vs. 6k).
To know the total # of marbles is to know the value of k.
Statement 1: we know that 8k is in the range of [45,100], so k could be [6,12]. Insufficient.
Statement 2: we know that 7k is in the range of [45,100], so k could be [7,14]. Insufficient.
Statement 1+2: notice that the 2 ranges above for k overlaps at [7,12], so still exist multiple values of k to make both statements hold. Insufficient.
So E).
Assuming that X initially has 5k marbles and Y 8k. Notice that all it takes is for Y to transfer 2k marbles to X to make the ratio 7:6 (7k vs. 6k).
To know the total # of marbles is to know the value of k.
Statement 1: we know that 8k is in the range of [45,100], so k could be [6,12]. Insufficient.
Statement 2: we know that 7k is in the range of [45,100], so k could be [7,14]. Insufficient.
Statement 1+2: notice that the 2 ranges above for k overlaps at [7,12], so still exist multiple values of k to make both statements hold. Insufficient.
So E).
