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08 Jan 2025, 05:48
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A factory produced 12 shipments of bottles, with each shipment containing 15 bottles. If the average number of defective bottles per shipment was 1.5, did any shipment contain at least 4 defective bottles?
(1) One-third of the shipments had no defective items, and one-third of the shipments had exactly 1 defective item each.
(2) Two-thirds of the shipments had at most 1 defective item each.
(1) One-third of the shipments had no defective items, and one-third of the shipments had exactly 1 defective item each.
(2) Two-thirds of the shipments had at most 1 defective item each.
08 Jan 2025, 05:48
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Official Solution:
A factory produced 12 shipments of bottles, with each shipment containing 15 bottles. If the average number of defective bottles per shipment was 1.5, did any shipment contain at least 4 defective bottles?
An average of 1.5 defective bottles per shipment for 12 shipments implies a total of 1.5 * 12 = 18 defective bottles.
(1) One-third of the shipments had no defective items, and one-third of the shipments had exactly 1 defective item each.
This means that 4 shipments had 0 defective items, and another 4 shipments had exactly 1 defective item each. Together, these 8 shipments accounted for 0 * 4 + 1 * 4 = 4 defective items, leaving 18 - 4 = 14 defective items to be distributed among the remaining 4 shipments.
Can each of these 4 shipments have fewer than 4 defective items? No, because if each shipment had fewer than 4 defective items, the maximum total would be 3 * 4 = 12, which is less than 14. Therefore, at least one of these 4 shipments must have at least 4 defective items. Sufficient.
(2) Two-thirds of the shipments had at most 1 defective item each.
This implies that 8 shipments had 1 or fewer defective items each. If those 8 shipments had exactly 1 defective item each, the remaining 4 shipments must account for 10 defective bottles.
We can distribute the 10 defective bottles across the 4 shipments in such a way that none of them has 4 or more defective bottles: for example, 2, 2, 3, 3. However, we can also distribute the 10 defective bottles such that one shipment does have at least 4 defective bottles: for example, 2, 2, 2, 4. Since we have two different outcomes, the information provided is not sufficient.
Answer: A
A factory produced 12 shipments of bottles, with each shipment containing 15 bottles. If the average number of defective bottles per shipment was 1.5, did any shipment contain at least 4 defective bottles?
An average of 1.5 defective bottles per shipment for 12 shipments implies a total of 1.5 * 12 = 18 defective bottles.
(1) One-third of the shipments had no defective items, and one-third of the shipments had exactly 1 defective item each.
This means that 4 shipments had 0 defective items, and another 4 shipments had exactly 1 defective item each. Together, these 8 shipments accounted for 0 * 4 + 1 * 4 = 4 defective items, leaving 18 - 4 = 14 defective items to be distributed among the remaining 4 shipments.
Can each of these 4 shipments have fewer than 4 defective items? No, because if each shipment had fewer than 4 defective items, the maximum total would be 3 * 4 = 12, which is less than 14. Therefore, at least one of these 4 shipments must have at least 4 defective items. Sufficient.
(2) Two-thirds of the shipments had at most 1 defective item each.
This implies that 8 shipments had 1 or fewer defective items each. If those 8 shipments had exactly 1 defective item each, the remaining 4 shipments must account for 10 defective bottles.
We can distribute the 10 defective bottles across the 4 shipments in such a way that none of them has 4 or more defective bottles: for example, 2, 2, 3, 3. However, we can also distribute the 10 defective bottles such that one shipment does have at least 4 defective bottles: for example, 2, 2, 2, 4. Since we have two different outcomes, the information provided is not sufficient.
Answer: A
