Bunuel wrote:
A carpenter has to build 71 wooden boxes in one week (7 days). He can build as many per day as he wants but he has decided that the number of boxes he builds on any one day should be within 4 off the number he builds on any other day. What is the least number of boxes that he could have build on Saturday?
A. 4
B. 5
C. 6
D. 7
E. 8
Solution:We can check the answer choices starting from choice A since we want the least number of boxes for Saturday.
A. If the number of boxes built on Saturday is 4, then he will have 71 - 4 = 67 boxes to be built for the other 6 days. The average for those 6 days would be 67/6 = 11 ⅙. We see that 11 (or 12) boxes exceeds the daily limit of 4 + 4 = 8 boxes. Eliminate Choice A.
B. If the number of boxes built on Saturday is 5, then he will have 71 - 5 = 66 boxes to be built for the other 6 days. The average for those 6 days would be 66/6 = 11. We see that 11 boxes exceeds the daily limit of 5 + 4 = 9 boxes. Eliminate Choice B.
C. If the number of boxes built on Saturday is 6, then he will have 71 - 6 = 65 boxes to be built for the other 6 days. The average for those 6 days would be 65/6 = 10 5/6. We see that on more than one of the days he would have to build 11 boxes, and this would exceed the daily limit of 4 + 6 = 10 boxes. Eliminate Choice C.
D. If the number of boxes built on Saturday is 7, then he will have 71 - 7 = 64 boxes to be built for the other 6 days. The average for those 6 days would be 64/6 = 10 2/3. We see that on more than one of the days he would have to build 11 boxes, but this would be acceptable because 11 boxes does not exceed the daily limit of 4 + 7 = 11 boxes. Choice D is correct.
Alternate Solution:Let x be the number of boxes built on Saturday. To minimize x, we should maximize the number of boxes built on every other day. According to the question, the number of boxes built in one day cannot exceed the number of boxes built in any other day by more than 4; therefore, let’s assume that x + 4 boxes were built on every day besides Saturday. Then, the number of boxes built in one week is:
(x + 4) + (x + 4) + (x + 4) + (x + 4) + (x + 4) + x + (x + 4)
7x + 24
We want this number to equal 71, so we get the following equation:
7x + 24 = 71
7x = 47
x ≈ 6.7
We see that if x is 6 or less, then the carpenter won’t be able to build 71 boxes in one week. Therefore, the minimum value of x is 7.
Answer: D _________________
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