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First Batch: 20%
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
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100% (03:32) correct
0%
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Two-Part Analysis - Math/Medium
A metallurgist melts together three metal batches to make one copper mixture: 240 kilograms from the first batch, 150 kilograms from the second batch, and 120 kilograms from the third batch. To meet the required standard, the final mixture must contain at least 93 kilograms of copper.
The exact copper concentration of each batch is not known, but testing shows that it can be any value from 0% up to its stated maximum, inclusive. The maximum copper concentrations are 25% for the first batch, 18% for the second batch, and 15% for the third batch.
Select for First Batch the lowest possible copper concentration that the first batch can have while the final mixture still meets the 93-kilogram requirement, and select for Third Batch the lowest possible copper concentration that the third batch can have while the final mixture still meets the 93-kilogram requirement. Make only two selections, one in each column.
| First Batch | |
| 3% | |
| 5% | |
| 10% | |
| 15% | |
| 20% | |
| 25% |
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Official Answer
First Batch: 20%
20 Apr 2026, 22:09
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Bunuel
Let F, S, and T be the copper concentrations of the first, second, and third batches, respectively, expressed as decimals.
Then the total copper content must satisfy
240 * F + 150 * S + 120 * T >= 93
General rule for such problems:
To maximize one quantity, minimize the others.
To minimize one quantity, maximize the others.
Minimizing F:
Use the maximum possible values for the other two concentrations:
S = 0.18 and T = 0.15
240 * F + 150 * 0.18 + 120 * 0.15 >= 93
240 * F + 27 + 18 >= 93
240 * F >= 48
F >= 48/240
F >= 0.2
So the lowest possible copper concentration for the first batch is 20%.
Minimizing T:
Use the maximum possible values for the other two concentrations:
F = 0.25 and S = 0.18
240 * 0.25 + 150 * 0.18 + 120 * T >= 93
60 + 27 + 120 * T >= 93
120 * T >= 6
T >= 6/120
T >= 0.05
So the lowest possible copper concentration for the third batch is 5%.
Correct selections:
First Batch: 20%
Third Batch: 5%
Takeaway
In a two-part analysis question, remember that you are answering two separate questions, one in each column, and both selections must be correct for the overall answer to be correct.
In optimization questions like this, where the overall total is fixed or lower or upper bounds are given, use the standard optimization rule: to maximize one quantity, minimize the others; to minimize one quantity, maximize the others.
What This Question Tests
This question tests mixture-based two-part analysis reasoning with concentration constraints, weighted totals, and optimization under upper bounds. It also tests whether you can translate the setup into an inequality and determine the minimum required value of one variable by maximizing the others.
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