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06 Aug 2026, 05:15
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Elena and Riley each bought notebooks, pens, and folders from the same supplier. For each item type, the unit price was the same for Elena and Riley. If Riley’s total spending was twice Elena’s total spending, what fraction of Elena’s total spending was on folders?
(1) Elena bought 6 notebooks, 9 pens, and 10 folders.
(2) Riley bought 18 notebooks, 27 pens, and 15 folders.
(1) Elena bought 6 notebooks, 9 pens, and 10 folders.
(2) Riley bought 18 notebooks, 27 pens, and 15 folders.
06 Aug 2026, 05:15
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Official Solution:
Elena and Riley each bought notebooks, pens, and folders from the same supplier. For each item type, the unit price was the same for Elena and Riley. If Riley’s total spending was twice Elena’s total spending, what fraction of Elena’s total spending was on folders?
(1) Elena bought 6 notebooks, 9 pens, and 10 folders.
This gives Elena’s quantities, but not the unit prices. So we cannot determine what fraction of Elena’s spending was on folders.
Not sufficient.
(2) Riley bought 18 notebooks, 27 pens, and 15 folders.
This gives Riley’s quantities, but not Elena’s quantities or the unit prices. So we cannot determine what fraction of Elena’s spending was on folders.
Not sufficient.
(1)+(2) From the two statements, Riley bought 3 times as many notebooks and 3 times as many pens as Elena. So Riley spent 3 times as much as Elena on notebooks and pens combined.
Riley bought 15 folders, while Elena bought 10 folders, so Riley spent \(1.5\) times as much as Elena on folders.
Let \(b\) be Elena’s spending on notebooks and pens combined, and let \(f\) be Elena’s spending on folders.
Then Riley’s total spending was:
\(3b + 1.5f\)
Since Riley’s total spending was twice Elena’s total spending:
\(3b + 1.5f = 2(b + f)\)
\(3b + 1.5f = 2b + 2f\)
\(b = 0.5f\)
\(f = 2b\)
Therefore, Elena’s total spending was:
\(b + f = b + 2b = 3b\)
So the fraction of Elena’s spending that was on folders was:
\(\frac{f}{b + f} = \frac{2b}{3b} = \frac{2}{3}\)
Sufficient.
Answer: C.
Alternatively, after combining the two statements, treat notebooks and pens as one category. Riley spent 3 times as much as Elena on notebooks and pens combined, since Riley bought 3 times as many of each. Riley spent \(1.5\) times as much as Elena on folders, since Riley bought 15 folders and Elena bought 10 folders. Overall, Riley spent 2 times as much as Elena.
So we have a weighted-average situation: one category is \(1.5\) times, the other is 3 times, and the overall value is 2 times.
Since 2 is between \(1.5\) and 3, compare the distances:
\(3 - 2 = 1\)
\(2 - 1.5 = 0.5\)
So Elena’s spending on notebooks and pens compared with her spending on folders is in the reverse ratio:
\(0.5:1 = 1:2\)
Therefore, folders accounted for:
\(\frac{2}{1 + 2} = \frac{2}{3}\)
of Elena’s total spending.
Answer: C
Elena and Riley each bought notebooks, pens, and folders from the same supplier. For each item type, the unit price was the same for Elena and Riley. If Riley’s total spending was twice Elena’s total spending, what fraction of Elena’s total spending was on folders?
(1) Elena bought 6 notebooks, 9 pens, and 10 folders.
This gives Elena’s quantities, but not the unit prices. So we cannot determine what fraction of Elena’s spending was on folders.
Not sufficient.
(2) Riley bought 18 notebooks, 27 pens, and 15 folders.
This gives Riley’s quantities, but not Elena’s quantities or the unit prices. So we cannot determine what fraction of Elena’s spending was on folders.
Not sufficient.
(1)+(2) From the two statements, Riley bought 3 times as many notebooks and 3 times as many pens as Elena. So Riley spent 3 times as much as Elena on notebooks and pens combined.
Riley bought 15 folders, while Elena bought 10 folders, so Riley spent \(1.5\) times as much as Elena on folders.
Let \(b\) be Elena’s spending on notebooks and pens combined, and let \(f\) be Elena’s spending on folders.
Then Riley’s total spending was:
\(3b + 1.5f\)
Since Riley’s total spending was twice Elena’s total spending:
\(3b + 1.5f = 2(b + f)\)
\(3b + 1.5f = 2b + 2f\)
\(b = 0.5f\)
\(f = 2b\)
Therefore, Elena’s total spending was:
\(b + f = b + 2b = 3b\)
So the fraction of Elena’s spending that was on folders was:
\(\frac{f}{b + f} = \frac{2b}{3b} = \frac{2}{3}\)
Sufficient.
Answer: C.
Alternatively, after combining the two statements, treat notebooks and pens as one category. Riley spent 3 times as much as Elena on notebooks and pens combined, since Riley bought 3 times as many of each. Riley spent \(1.5\) times as much as Elena on folders, since Riley bought 15 folders and Elena bought 10 folders. Overall, Riley spent 2 times as much as Elena.
So we have a weighted-average situation: one category is \(1.5\) times, the other is 3 times, and the overall value is 2 times.
Since 2 is between \(1.5\) and 3, compare the distances:
\(3 - 2 = 1\)
\(2 - 1.5 = 0.5\)
So Elena’s spending on notebooks and pens compared with her spending on folders is in the reverse ratio:
\(0.5:1 = 1:2\)
Therefore, folders accounted for:
\(\frac{2}{1 + 2} = \frac{2}{3}\)
of Elena’s total spending.
Answer: C
