"Greek Tokens" question (700-800)

[AndresR810]

Hello,

For the question below, the Manhattan Prep explanation tells you to find the lowest common denominator between the two ratios. I was wondering if there is a way to solve this algebraically.

Thank you.

"Narcisse and Aristide have numbers of arcade tokens in the ratio 7 : 3, respectively. Narcisse gives Aristide some of his tokens, and the new ratio is 6 : 5. What is the least number of tokens that Narcisse could have given to Aristide?"

A. 9
B. 17
C. 21
D. 27
E. 53

[RonPurewal]

the problem asks for the least number that will solve the problem—so, you know that you definitely WON'T be able to solve algebraically for a unique solution.

just think about it for a sec: if you could solve the problem algebraically, and get a numerical solution, then there would be no need to ask for "the least number".

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if you want to do algebra, the only way that's really going to work is if you do algebra for each individual answer choice. since you want the LEAST number, you'll want to start from choice A and move up.

the opening ratio is 7:3. so, for EVERY choice, you can represent these numbers as 7x and 3x.

here's how you'd test choice A:
after the trade, narcisse has 7x – 9 tokens, an aristide has 3x + 9 tokens.
these are supposed to be in the ratio 6:5 now, so... (7x – 9)/(3x + 9) = 6/5
cross-multiply... 35x – 45 = 18x + 54
17x = 99
x = 99/17... this isn't an integer, so choice A is wrong.

...then keep testing choices until you actually get an integer for x.