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The below question appeared on one of my Cat exams:
In a certain game, a large bag is filled with blue, green, purple and red chips worth 1, 5, x and 11 points each, respectively. The purple chips are worth more than the green chips, but less than the red chips. A certain number of chips are then selected from the bag. If the product of the point values of the selected chips is 88,000, how many purple chips were selected?
1
2
3
4
5
I believe there can be multiple answers to this question as x can equal 8 and there can be 1 purple chip and 16 green. or even 2 purple, 1 blue, 1 red and 12 green...in addition to others. How do you know how to solve this problem?
The answer presented is below, but it still doesn't answer my question. Thanks.
ANSWER:
88,000 is the product of an unknown number of 1's, 5's, 11's and x's. To figure out how many x’s are multiplied to achieve this product, we have to figure out what the value of x is. Remember that a number's prime box shows all of the prime factors that when multiplied together produce that number: 88,000's prime box contains one 11, three 5’s, and six 2’s, since 88,000 = 11 × 53 × 26.
The 11 in the prime box must come from a red chip, since we are told that 5 < x < 11 and therefore x could not have 11 as a factor. In other words, the factor of 11 definitely did not come from the selection of a purple chip, so we can ignore that factor for the rest of our solution.
So, turning to the remaining prime factors of 88,000: the three 5’s and six 2’s. The 2’s must come from the purple chips, since the other colored chips have odd values and thus no factor of two. Thus, we now know something new about x: it must be even. We already knew that 5 < x < 11, so now we know that x is 6, 8, or 10.
However, x cannot be 6: 6 = 2 × 3, and our prime box has no 3’s.
x seemingly might be 10, because 10 = 2 × 5, and our prime box does have 2’s and 5’s. However, our prime box for 88,000 only has three 5’s, so a maximum of three chips worth 10 points are possible. But that leaves three of the six factors of 2 unaccounted for, and we know those factors of two must have come from the purple chips.
So x must be 8, because 8 = 23 and we have six 2’s, or two full sets of three 2’s, in the prime box. Since x is 8, the chips selected must have been 1 red (one factor of 11), 3 green (three factors of 5), 2 purple (two factors of 8, equivalent to six factors of 2), and an indeterminate number of blue chips.
The correct answer is B.
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