At an amusement park, tom bought a number of red tokens

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Mo2men: Legendary Member; Posts: 712; Joined: Fri Sep 25, 2015 4:39 amFollowed by: 5 members; Thanked: 14 times

by Mo2men » Fri Mar 24, 2017 5:08 pm

At an amusement park, tom bought a number of red tokens and green tokens. Each red token costs $0.09, and each green token costs $0.14. If Tom spent a total of exactly $2.06, how many token in total did Tom buy?

A. 16
B. 17
C. 18
D. 19
E. 20

OA:D

Source: Manhattan

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Fri Mar 24, 2017 6:10 pm

Mo2men wrote:At an amusement park, tom bought a number of red tokens and green tokens. Each red token costs $0.09, and each green token costs $0.14. If Tom spent a total of exactly $2.06, how many token in total did Tom buy?

A. 16
B. 17
C. 18
D. 19
E. 20

One approach:

9r + 14g = 206.

Since the two values in blue are even, the term in red must also be even, implying that the term in red is a MULTIPLE OF 18.
Thus, the equation can be rephrased as follows:

9(2x) + 14g = 206
9x + 7g = 103
7g = 103 - 9x
g = (103 - 9x)/7.

Since g is a INTEGER, 103-9x must be a MULTIPLE OF 7.
Keep subtracting 9 from 103 until a multiple of 7 is yielded:
103-9 = 94
94-9 = 85
85-9 = 76
76-9 = 67
67-9 = 58
58-9 = 49.

The value in green is a multiple of 7.
Implication:
103-9x = 49, with the result that g = 49/7 = 7.

Substituting g=7 into 9r-14g = 206, we get:
9r - 14(7) = 206
9r - 98 = 206
9r = 108
r = 12.

Thus, r+g = 12+7 = 19.

The correct answer is D.

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Mo2men: Legendary Member; Posts: 712; Joined: Fri Sep 25, 2015 4:39 amFollowed by: 5 members; Thanked: 14 times

by Mo2men » Sat Mar 25, 2017 5:50 am

GMATGuruNY wrote:
Mo2men wrote:At an amusement park, tom bought a number of red tokens and green tokens. Each red token costs $0.09, and each green token costs $0.14. If Tom spent a total of exactly $2.06, how many token in total did Tom buy?

A. 16
B. 17
C. 18
D. 19
E. 20
One approach:

9r + 14g = 206.

Since the two values in blue are even, the term in red must also be even, implying that the term in red is a MULTIPLE OF 18.
Thus, the equation can be rephrased as follows:

9(2x) + 14g = 206
9x + 7g = 103
7g = 103 - 9x
g = (103 - 9x)/7.

Since g is a INTEGER, 103-9x must be a MULTIPLE OF 7.
Keep subtracting 9 from 103 until a multiple of 7 is yielded:
103-9 = 94
94-9 = 85
85-9 = 76
76-9 = 67
67-9 = 58
58-9 = 49.

The value in green is a multiple of 7.
Implication:
103-9x = 49, with the result that g = 49/7 = 7.

Substituting g=7 into 9r-14g = 206, we get:
9r - 14(7) = 206
9r - 98 = 206
9r = 108
r = 12.

Thus, r+g = 12+7 = 19.

The correct answer is D.

Dear Mitch,

Thanks for you approach above.

Could the above be treated as weighted average and solved through alligation method? if so, can you kindly present this approach in the question above.

Thanks in advance

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by [email protected] » Sat Mar 25, 2017 10:52 am

Hi Mo2men,

If you don't immediately see an 'elegant' approach to solving this problem, then you can still solve it relatively quickly with some 'brute force' and a bit of arithmetic. From the answer choices, you can see that the total number of coins is no more than 20, so there aren't that many potential calculations that you would have to do to find the exact number of each type of coins that 'fit' this situation.

In basic terms, we're told that a certain number of .09s + a certain number of .14s total 2.06.... There are some Number Property rules that we can use to save some time:

.14 multiplied by an integer will end in an EVEN digit
2.06 ends in an even digit
Since (even) + (even) = (even), .09 multiplied by an integer MUST end in an EVEN digit for the sum to equal 2.06

The number of red tokens MUST be EVEN, so that significantly cuts down the number of options to consider....

IF... we have....
2 red tokens, then the remaining value is $1.88. Can that be evenly divided by .14? Try it... (the answer is NO).
4 red tokens, then the remaining value is $1.70. Can that be evenly divided by .14? Try it... (the answer is NO).
6 red tokens, then the remaining value is $1.52. Can that be evenly divided by .14? Try it... (the answer is NO).
8 red tokens, then the remaining value is $1.34. Can that be evenly divided by .14? Try it... (the answer is NO).
10 red tokens, then the remaining value is $1.16. Can that be evenly divided by .14? Try it... (the answer is NO).
12 red tokens, then the remaining value is $0.98. Can that be evenly divided by .14? Try it... (the answer is YES and the remaining 7 tokens are green).

Thus, the total number of tokens is 12 red + 7 green = 19 total tokens

Final Answer: D

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

Contact Rich at [email protected]

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Mo2men: Legendary Member; Posts: 712; Joined: Fri Sep 25, 2015 4:39 amFollowed by: 5 members; Thanked: 14 times

by Mo2men » Thu Jul 11, 2019 7:58 pm

Mo2men wrote:At an amusement park, tom bought a number of red tokens and green tokens. Each red token costs $0.09, and each green token costs $0.14. If Tom spent a total of exactly $2.06, how many token in total did Tom buy?

A. 16
B. 17
C. 18
D. 19
E. 20

OA:D

Source: Manhattan

0.09R + 0.14G = 2.06
9R + 14G = 206 ---- 1

Let R + G = X
R = X - G ---- 2

Substitute (2) in (1)
9 (X - G) + 14G = 206
9X + 5G = 206
5G = 206 - 9X.....then last digit of (206 - 9X) should be 5 or 0
9*19 (ANSWER D) = 171
So, 5G = 206 - 191 = 35 i.e. G = 7 and R = 12.
Thus it satisfies the equation = (9*12) + (14*7) = 206

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