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12 Days of Christmas 2024 - 2025 Competition with $40,000 of Prizes
A business exports 2 products, A and B. If 300 total units of these 2 products were exported yesterday, what fraction of the product B units were exported to Country X?
(1) Half of the units exported yesterday were product A, and one-third of the units exported yesterday were exported to countries other than Country X.
(2) Yesterday, there were twice as many units of product A exported to Country X as there were units of B exported to other countries.
A business exports 2 products, A and B. If 300 total units of these 2 products were exported yesterday, what fraction of the product B units were exported to Country X?
(1) Half of the units exported yesterday were product A, and one-third of the units exported yesterday were exported to countries other than Country X.
(2) Yesterday, there were twice as many units of product A exported to Country X as there were units of B exported to other countries.
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23 Dec 2024, 10:00
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Bunuel
Manhattan Prep Official Explanation:
This Data Sufficiency problem is testing Overlapping Sets: each of the 300 units is either Type A or Type B, and each is either exported to Country X or exported elsewhere. Set up a Double-Set Matrix to track what you know:
| A | B | Total | |
| X | |||
| Other | |||
| Total | 300 |
The question stem asks for the ratio of the number of units of product B exported to Country X to the total number of units of product B.
Statement (1): INSUFFICIENT. This allows you to fill in all of the Totals as follows:
| A | B | Total | |
| X | 200 | ||
| Other | 100 | ||
| Total | 150 | 150 | 300 |
In particular, there were 150 total units of product B. However, the four cells in the upper left remain unknown, so you cannot solve for the number of units of B exported to Country X.
Statement (2): INSUFFICIENCT. This establishes a 2:1 ratio between two of the cells in the matrix, which can be represented using the Unknown Multiplier:
| A | B | Total | |
| X | 2x | ||
| Other | x | ||
| Total | 300 |
However, without more information, the ratio asked for in the question stem cannot be determined.
Statements (1) and (2) together: SUFFICIENT. Combining the information from the two tables above yields:
| A | B | Total | |
| X | 2x | 200 | |
| Other | x | 100 | |
| Total | 150 | 150 | 300 |
There were 150 – x units of product B exported to Country X and the top row creates an equation that can be solved for x:
2x + (150 – x) = 200
x + 150 = 200
x = 50
Therefore 150 – 50 = 100 units of the 150 total units of product B were exported to Country X and the desired ratio is 2⁄3.
The correct answer is (C): Both statements together are sufficient, but neither statement alone is sufficient.
General Discussion
23 Dec 2024, 09:34
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A business exports 2 products, A and B. If 300 total units of these 2 products were exported yesterday, what fraction of the product B units were exported to Country X?
(1) Half of the units exported yesterday were product A, and one-third of the units exported yesterday were exported to countries other than Country X.
The fraction of the product B units were exported to Country X = x/150
Since x is unknown
NOT SUFFICIENT
(2) Yesterday, there were twice as many units of product A exported to Country X as there were units of B exported to other countries.
With the information provided, the fraction of the product B units were exported to Country X can not be derived.
NOT SUFFICIENT
(1) + (2)
(1) Half of the units exported yesterday were product A, and one-third of the units exported yesterday were exported to countries other than Country X.
(2) Yesterday, there were twice as many units of product A exported to Country X as there were units of B exported to other countries.
300 - 2x + x = 200
x = 100
The fraction of the product B units were exported to Country X = 100/150 = 2/3
SUFFICIENT
IMO C
(1) Half of the units exported yesterday were product A, and one-third of the units exported yesterday were exported to countries other than Country X.
| Product A | Product B | Total | |
| Country X | x | 200 | |
| Other than Country X | 150-x | 100 | |
| Total | 150 | 150 | 300 |
The fraction of the product B units were exported to Country X = x/150
Since x is unknown
NOT SUFFICIENT
(2) Yesterday, there were twice as many units of product A exported to Country X as there were units of B exported to other countries.
| Product A | Product B | Total | |
| Country X | 2x | ||
| Other than Country X | x | ||
| Total | 300 |
With the information provided, the fraction of the product B units were exported to Country X can not be derived.
NOT SUFFICIENT
(1) + (2)
(1) Half of the units exported yesterday were product A, and one-third of the units exported yesterday were exported to countries other than Country X.
(2) Yesterday, there were twice as many units of product A exported to Country X as there were units of B exported to other countries.
| Product A | Product B | Total | |
| Country X | 2(150-x) = 300-2x | x | 200 |
| Other than Country X | x -50 | 150-x | 100 |
| Total | 150 | 150 | 300 |
300 - 2x + x = 200
x = 100
| Product A | Product B | Total | |
| Country X | 2(150-x) = 300-2x = 100 | x = 100 | 200 |
| Other than Country X | x -50 = 50 | 150-x = 50 | 100 |
| Total | 150 | 150 | 300 |
The fraction of the product B units were exported to Country X = 100/150 = 2/3
SUFFICIENT
IMO C
