dflorez wrote:
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
Do you guys have a suggestion on how to approach the problem?
Good question to learn factors, multiples and the related concepts.
Let N and A stand for tokens with Narcisse and Aristide.
Thus per the original ratio:
N/A = 7/3 ---> N = (7/3)A ...(1)
Let x be the number of tokens given to Aristide by Narcisse.
Thus after redistribution of the tokens you get,
\(\frac{N-x}{A+x} = \frac{6}{5}\) ---> rearranging you get, 5N=6A+11x ... (2)
When you subsititue N in terms of A from (1) in (2), you get,
11x = (17/3)A ---> x = (17/33)A....(3)
Now as tokens can only take INTEGER values and so does 'x', A must be a multiple of 33 in order to get an integer value for 'x' in (3).
Once you realize this, you will see that x will then become a multiple of 17 and out of all the options provided, only option B is a multiple of 17 and is hence the correct answer.
x = (17/33)A, as A= 33p, where p is a positive integer, you get,
x = (17/33)*33p = 17p
Hope this helps.