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Arithmetic, Word Problems

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by swerve » Thu Feb 13, 2020 8:01 am
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 tokens in total did Tom buy?

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

The OA is D

Source: Economist GMAT
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Re: Arithmetic, Word Problems

by Jay@ManhattanReview » Thu Feb 13, 2020 9:36 pm
swerve wrote: ↑
Thu Feb 13, 2020 8:01 am
 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 tokens in total did Tom buy?

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

The OA is D

Source: Economist GMAT
Say Tom bought x red and y green tokens, so he spent (9x + 14y) cents.

=> 9x + 14y = 206

Though this is a linear equation with two variables, it seems that we cannot have a unique solution; however, it is not necessary. We must keep in mind that there's one more condition that can work for us in getting the unique values of x and y and that is: both x and y are positive integers.

Again, we have 9x + 14y = 206.

We can rewrite 9x + 14y = 206 as 9x + 9y + 5y = 206 => 9(x + y) + 5y = 206.

Say x + y = z. Thus, we have 9z + 5y = 206; where y and z are positive integers.

We have to get the value of z.

Let's get the value of z from 9z + 5y = 206

=> 9z = 206 – 5y
z = (206 – 5y)/9 = 22 + (8 – 5y)/9

Since y is a positive integer, the only qualified value for y is 7 such that z is also a positive integer.

Again, z = 22 + (8 – 5*7)/9 = 22 – 3 = 19.

The correct answer: D

Hope this helps!

-Jay
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