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30 May 2026, 11:15
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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.
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M47-178
07 Jun 2026, 01:30
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GMAT Club 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:
f/(b + f) = 2b/(3b) = 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:
2/(1 + 2) = 2/3
of Elena’s total spending.
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:
f/(b + f) = 2b/(3b) = 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:
2/(1 + 2) = 2/3
of Elena’s total spending.
General Discussion
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30 May 2026, 22:41
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Let notebook, pen, and folder prices be N, P, and F.
We want:
Elena's folder spending ÷ Elena's total spending.
(1) Elena bought 6 notebooks, 9 pens, and 10 folders.
Total spending = 6N + 9P + 10F
Folder spending = 10F
Required fraction = 10F / (6N + 9P + 10F)
Since prices are unknown, we cannot find a unique value.
(1) Insufficient.
(2) Riley bought 18 notebooks, 27 pens, and 15 folders.
We know Riley spent twice as much as Elena, but we don't know Elena's purchases.
(2) Insufficient.
Together:
Elena's spending = 6N + 9P + 10F
Riley's spending = 18N + 27P + 15F
Given Riley spent twice as much as Elena:
18N + 27P + 15F = 2(6N + 9P + 10F)
18N + 27P + 15F = 12N + 18P + 20F
6N + 9P = 5F
Substitute into Elena's spending:
E = (6N + 9P) + 10F = 5F + 10F = 15F
So the fraction spent on folders is:
10F / 15F = 2/3
A unique value is obtained.
Answer: C — Both statements together are sufficient, neither alone is sufficient.
We want:
Elena's folder spending ÷ Elena's total spending.
(1) Elena bought 6 notebooks, 9 pens, and 10 folders.
Total spending = 6N + 9P + 10F
Folder spending = 10F
Required fraction = 10F / (6N + 9P + 10F)
Since prices are unknown, we cannot find a unique value.
(1) Insufficient.
(2) Riley bought 18 notebooks, 27 pens, and 15 folders.
We know Riley spent twice as much as Elena, but we don't know Elena's purchases.
(2) Insufficient.
Together:
Elena's spending = 6N + 9P + 10F
Riley's spending = 18N + 27P + 15F
Given Riley spent twice as much as Elena:
18N + 27P + 15F = 2(6N + 9P + 10F)
18N + 27P + 15F = 12N + 18P + 20F
6N + 9P = 5F
Substitute into Elena's spending:
E = (6N + 9P) + 10F = 5F + 10F = 15F
So the fraction spent on folders is:
10F / 15F = 2/3
A unique value is obtained.
Answer: C — Both statements together are sufficient, neither alone is sufficient.
