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chips

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j_shreyans Legendary Member Default Avatar
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chips

Post Wed Oct 22, 2014 11:09 am
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

OAB

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Post Mon Nov 13, 2017 10:18 am
j_shreyans 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 8 is the only power of 2 between 5 and 11. Since 2^6 = 8^2, there must be 2 purple chips.

Answer: B

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Matt@VeritasPrep GMAT Instructor
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Post Mon Oct 27, 2014 10:56 am
Brent's solution is great. Let me add an equation-based approach, as some students like those.

Let's suppose we have g green chips, p purple chips, and r red chips. The product of all the chip values is thus 5ᵍ * x- * 11ʳ. Since we know the product is also 88,000, we have the equation

5ᵍ * x- * 11ʳ = 88,000 = 11 * 5³ * 2⁶

Since each side is an integer, we know that each side must have the SAME number of each prime factor. Hence r = 1, for instance, and we have one 11 on both sides.

Now let's deal with the 5s. g doesn't necessarily = 3, but let's suppose that it does. That leaves us with x- = 2⁶. This gives us THREE possibilities:

x = 2, p = 6
x = 4, p = 3
x = 8, p = 2

The prompt told us that x must be between 5 and 11, so we know x = 8, p = 2, and the answer is B.

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Post Thu Oct 23, 2014 7:20 am
Quote:
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
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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Post Wed Oct 22, 2014 9:50 pm
Hi j_shreyans,

This question involves a bit of logical thinking and factoring skills. You have to take notes and stay organized though, if you want to answer this question correctly.

We're told:
Blue chips = 1 point each
Green chips = 5 points each
Purple chips = X points each (more than Green, less than Red, so X = 6, 7, 8, 9 or 10)
Red chips = 11 points each

We're told that taking an unknown number of chips gives us a product equal to 88,000; we need to factor 88,000 and we should look specifically for 5s, 11s and some mystery number between 6 and 10, inclusive….

88,000 =
(11)(8,000) =
(11)(5)(1600) =
(11)(5)(5)(320) =
(11)(5)(5)(5)(64)

Now, we KNOW that there's a mystery number that is between 6 and 10 (inclusive) and MUST account for that 64….

64 = (8)(8)

This gives us 2 purple chips.

Final Answer: B

GMAT assassins aren't born, they're made,
Rich

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