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In a certain game, a large bag is filled with blue, green,

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In a certain game, a large bag is filled with blue, green,

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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?

A. 1
B. 2
C. 3
D. 4
E. 5

OA B

Source: Manhattan Prep

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BTGmoderatorDC wrote:
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?

A. 1
B. 2
C. 3
D. 4
E. 5

OA B

Source: Manhattan Prep
This question begs for some prime factorization.
88,000 = (2)(2)(2)(2)(2)(2)(5)(5)(5)(11)

First, we can see that there must be one (11-point) red chip.
Now, what role do these 2's play? Since there are no 2's hiding among the 5-point chips or the 11-point chips, the 2's must be associated with the x-point chips.
Since we know that each purple chip is worth 6, 7, 8, 9 or 10 points, we know that x must equal 6, 8 or 10.

x cannot equal 6, because we don't have any 3's in the prime factorization.
If x were to equal 10, we'd need six 5's to go with our six 2's. Since we don't have six 5's in the prime factorization of 88,000, we can rule out the possibility that x equals 10.

By the process of elimination, x MUST equal 8.
Since 8 = (2)(2)(2), we can see that the six 2's can be used to create two products of 8.

Answer: B

Cheers,
Brent

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BTGmoderatorDC wrote:
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?

A. 1
B. 2
C. 3
D. 4
E. 5
Let’s break 88,000 into its prime factors:

88,000 = 88 x 1000 = 11 x 8 x 10 x 100 = 11 x 2^3 x 5 x 2 x 5^2 x 2^2 = 2^6 x 5^3 x 11^1

We see that there could be any number of blue chips since they are worth 1 point each. The prime factor 5^3 tells us that the number of green chips must be 3 since they are worth 5 points each. The prime factor 11^1 indicates that the number of red chips must be 1 since each red chip is worth 11 points. Thus, the product of the point values of purple chips must be 2^6. Since each purple chip is worth between 5 and 11 points, and the value of a purple chip must be a power of 2, each purple chip must be worth 2^3 = 8 points. Since 2^6 = 8^2, there must be 2 purple chips.

Answer: B

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Scott Woodbury-Stewart Founder and CEO

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