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

I solved it this way:

Starting with the equation 9X+14Y = 206

=> 9(X+Y) + 5Y = 206

5Y = 206 - 9(X+Y)

we need to find X+Y. The RHS has to be a multiple of 5

Substituting the answers for X+Y abive, only 19 gives a multiple of 5.
You don't need to actually multiply all the answers with 9, just look for the units digit of the difference. (it has to be either 5 or 0)
When 19 is substituted, we get a units digit of 5 in the difference.
So D.

0.09x + 0.14y = 2.06
=> 9x + 14y = 206

To solve this remember that x must be even because 14y, when subtracted from 206, will yield an even number (even - even = even). The solution comes out to be x=12, y=7.

Therefore the total number of tokens bought = 12+7 = 19

Option (D).

Given:
0.09R + 0.14G = 2.06;
9R + 14G = 206.

Now, it's special type of equations as G and R must be a non-negative integers, so there might be only one solution to it. After some trial and error you'll get (actually there are several ways of doing it):
R = 12 and G = 7;
R + G = 19.

Answer: D.

For more on this type of questions check:
eunice-sold-several-cakes-if-each-cake-sold-for-either-109602.html
martha-bought-several-pencils-if-each-pencil-was-either-a-100204.html
a-rental-car-agency-purchases-fleet-vehicles-in-two-sizes-a-105682.html
joe-bought-only-twenty-cent-stamps-and-thirty-cent-stamps-106212.html
a-certain-fruit-stand-sold-apples-for-0-70-each-and-bananas-101966.html
joanna-bought-only-0-15-stamps-and-0-29-stamps-how-many-101743.html
at-an-amusement-park-tom-bought-a-number-of-red-tokens-and-126814.html
collections-confused-need-a-help-81062.html

Hope it helps.
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Is there any way to solve for it other than trial and error?

Trial and error along with some common sense is pretty much the only way you should approach such kind of problems on the GMAT. You won't get some very tough numbers to manipulate with or there will be some shortcut available, based on multiples concept or on the answer choices. So generally you would have to try just couple of values to get the answer.

Check the links in my previous post to practice similar problems and you'll see that getting the answer is not that hard for the realistic GMAt question.
_________________

Hi calreg11,

supporting the explanations of Bunuel and karishma above, I can show you shortest way of solving it by some amalgamation of trial and error with the algebraic approach, though, as mentioned by karishma, there is no pure algebraic solution of this problem. Let's start:

first, we have to formulate the premise in an algebraic way through expressing red and green tokens by any letters we think convenient to us--> assuming, e.g., red tokens as 'r' and green tokens as 'g';

secondly, for the sake of convenience we can take the prices of red and greem tokens and also the total cost in cents, i.e., $0,09 as 9 cents, $0.14 as 14 cents, and $2.06 as 206 cents;

then, thirdly, we do formulate it --> 9r + 14g = 206;

fourth, now we can refer to the point that red tokens and green tokens make up the total number of tokens which is unknown to us and this is why formula can be --> r + g = x

fifth, we have to apply trial and error approach through replacing x by each answer choice and we do it this way:

r + g = 16
r = 16 - g
replace 'r' in the original formula --> 9r + 14g = 206and we get 9(16-g) + 14g = 206 --> 5g=62 --> 62 is not divisible by 5, and hence, we cannot derive the number of g (green tokens), consequently, that of red tokens' also.

only 19 can satisfy the condition drawn from the formulae r = x - g and 9(x-g) + 14g = 206

Hope, it helps!

What I like to do in this questions is the following

We have 9x + 14y = 206

First always try to simplify, in this case we can't

Now look for a number that is the same for both and will be close to 206

In this case 9 is our best choice (You can quickly ballpark with 10 but you will realize it is >206)

So with 9 for both x and y we get 207 which is one more. Now the fun part starts
We need to play with this 9,9 combination to try to get one less, How so?

Well, let see we need to be one lower so if we get rid of one 14 and add one 9 we be further down.

If we subtract to 14's though we are down 28 and if we add 3 9's we are up 27

That perfect just to match our +1 difference!

So in total we have 12+7 = 19

Hence our correct answer is D

Hope it helps
Cheers!
J

PROBLEM:
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

SOLUTION:

0.09R + 0.14G = 2.06
i.e. 9R + 14G = 206 ---- (i)
R + G = ?

Plug in answer choices for solving. The challenge is how to narrow down without too much calculation. Here is what I did:

Let X be our answer choice:
R + G = X
i.e. R = X - G ---- (ii)

Substitute (ii) in (i)
9 (X - G) + 14G = 206
9X + 5G = 206
5G = 206 - 9X
i.e. 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

ANSWER: D

Hi All,

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:

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

I knew I had to do trial and error as x+y or in this case r+g is not given ….
I knew to check for the divisibility of 9, that the sum of digit should be a multiple of 9 (ie, sum of digit should be divisible by 9)….
So I started like this…
206 -14 = 192
-14 =178
-14 = 164
-14 = 150
-14 = 136
-14 = 122
-14 = 108
After that I counted no. of 14 = 7 => g
I did 108/9 = 12 => r
Which was the answer