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27 Dec 2024, 09:00
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B
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12 Days of Christmas 2024 - 2025 Competition with $40,000 of Prizes
A certain electronics company produces and sells two products: a fitness tracker and a smart ring. Each fitness tracker requires 4 hours of assembly and 6 hours of software calibration, while each smart ring requires 4 hours of assembly and 2 hours of software calibration. The company’s profit on each fitness tracker is $40 and its profit on each smart ring is $30. If the facility the company uses to produce these products has limited weekly capacities for assembly and software calibration, how many fitness trackers should the company produce each week to maximize profit for these two products?
(1) The cost of the components for each fitness tracker is twice that for each smart ring.
(2) The facility has the capacity for 200 hours of assembly time each week and 200 hours of software calibration each week.
A certain electronics company produces and sells two products: a fitness tracker and a smart ring. Each fitness tracker requires 4 hours of assembly and 6 hours of software calibration, while each smart ring requires 4 hours of assembly and 2 hours of software calibration. The company’s profit on each fitness tracker is $40 and its profit on each smart ring is $30. If the facility the company uses to produce these products has limited weekly capacities for assembly and software calibration, how many fitness trackers should the company produce each week to maximize profit for these two products?
(1) The cost of the components for each fitness tracker is twice that for each smart ring.
(2) The facility has the capacity for 200 hours of assembly time each week and 200 hours of software calibration each week.
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27 Dec 2024, 10:00
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Bunuel
Manhattan Prep Official Explanation:
This Data Sufficiency problem is testing Translations. Let f be the number of fitness trackers produced in a week and r be the number of smart rings produced in a week. This is a Value question asking for the value of f that will maximize profit, subject to the given constraints. Jot down the givens before heading to the statements.
each f: 4 hrs assembly + 6 hrs calibration
each r: 4 hrs assembly + 2 hrs calibration
profit = 40f + 30r
max profit → f = ?
Statement (1): INSUFFICIENT. The cost of the components is irrelevant because those costs are already accounted for by the profit margins of $40 and $30 per fitness tracker and per smart ring, respectively, given in the question stem.
Statement (2): SUFFICIENT. Translate these time constraints into inequalities.
200 hours assembly: 4f + 4r ≤ 200
200 hours calibration: 6f + 2r ≤ 200
Suppose the company utilized all available time at the facility. Then these inequalities would become a system of equations:
4f + 4r = 200
6f + 2r = 200
The first equation can be divided by 4 to become f + r = 50, or r = 50 – f. The second equation can be divided by 2 to become 3f + r = 100. You can then substitute the first equation into the second to yield:
3f + r = 100
3f + (50 – f) = 100
2f + 50 = 100
2f = 50
f = 25
So when f = r = 25, all available assembly and calibration time will be used. The profit in this scenario will be
profit = 40f + 30r = 40(25) + 30(25) = 1,000 + 750 = $1,750
But, is this the maximum value of profit?
Suppose the company tries to produce more than 25 smart rings. Because each fitness tracker requires the same number of assembly hours to produce as each smart ring, if the company produces 25 + n smart rings (for some positive integer n) it will only have the assembly capacity to produce 25 – n fitness trackers. This would reduce the total profit, as each fitness tracker brings in $10 more profit than each smart ring.
On the other hand, suppose the company tries to produce more than 25 fitness trackers. Because each smart ring requires only 2 hours of software calibration and each fitness tracker requires 6 hours (3 times as many hours), if the company produces 25 + n fitness trackers it will only have the calibration capacity to produce 25 – 3n smart rings. One fitness tracker brings in $40 of profit but the company would have to forego the 3($30) = $90 of profit from three smart rings, so this would also reduce profit.
The profit will be maximized when the company produces 25 of each product per week.
The correct answer is (B): Statement (2) alone is sufficient, but statement (1) is not sufficient.
General Discussion
27 Dec 2024, 09:18
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x = number of fitness trackers
y = number of smart rings
Statement 1: The cost of the components for each fitness tracker is twice that for each smart ring.
This is irrelevant as component costs are already factored into profit margins. Even if not so, this provides information about costs, but it is irrelevant to the calculation of profit (already given as $40 and $30 per product).
Statement 2: The facility has the capacity for 200 hours of assembly time each week and 200 hours of software calibration each week.
Assembly: 4x + 4y ≤ 200
Calibration: 6x + 2y ≤ 200
Profit function: P = 40x + 30y
P = 40x + 30(100 - 3x)
P = 40x + 3000 - 90x
P = -50x + 3000
Since the coefficient of x is negative, profit is maximized at x = 0.
Thus, Statement 2 ALONE is sufficient.
Answer: B
y = number of smart rings
Statement 1: The cost of the components for each fitness tracker is twice that for each smart ring.
This is irrelevant as component costs are already factored into profit margins. Even if not so, this provides information about costs, but it is irrelevant to the calculation of profit (already given as $40 and $30 per product).
Statement 2: The facility has the capacity for 200 hours of assembly time each week and 200 hours of software calibration each week.
Assembly: 4x + 4y ≤ 200
Calibration: 6x + 2y ≤ 200
Profit function: P = 40x + 30y
P = 40x + 30(100 - 3x)
P = 40x + 3000 - 90x
P = -50x + 3000
Since the coefficient of x is negative, profit is maximized at x = 0.
Thus, Statement 2 ALONE is sufficient.
Answer: B
27 Dec 2024, 09:30
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Bunuel
Let:
x = number of fitness trackers
y = number of smart rings
Profit Function: P = 40x + 30y
Constraints (from the problem statement):
Assembly: 4x + 4y ≤ Assembly Capacity
Calibration: 6x + 2y ≤ Calibration Capacity
Analyzing the Statements:
(1) The cost of the components for each fitness tracker is twice that for each smart ring.
- This statement gives us information about cost, which is irrelevant to maximizing profit given the time constraints already provided. We already know the profit per unit. This statement is insufficient.
- This statement gives us the missing constraints:
- 4x + 4y ≤ 200 (Assembly)
- 6x + 2y ≤ 200 (Calibration)
IMO B
