Bunuel wrote:
Every member of a local chess club is assigned one of two skill levels: A or B. At the beginning of the month, there were 'a' level A members and 'b' level B members. The ratio of a to b was 5:4. At the end of the month, the ratio of level A members to level B members has changed to 7:8. If nobody left the club and no new members joined during the month, how many total members are in the club?
(1) 49 < a < 79
(2) 29 < b <59
Are You Up For the Challenge: 700 Level Questions Analyzing the question:Since the ratios are 5:4 and 7:8, we know that the total amount of club members must be able to be divided by 5 + 4 and 7 + 8. Hence the total amount must have a factor of 9 and 15. The LCM of 9 and 15 is 9 * 15 / 3 = 45. Therefore the total amount of members is a multiple of 45. 'a' is equal to 5/9 of this multiple of 45, and 'b' is equal to 4/9 of this multiple of 45. Hence 'a' must be a multiple of 25 and 'b' must be a multiple of 20, and they must have the same multiplier.
Statement 1:'a' must be a multiple of 25, here we can have either 50 or 75 so insufficient.
Statement 2:'b' must be a multiple of 20, we can only have b = 40 here so this is sufficient.
Ans: B
Note: If we change statement 2 to 19 < b < 59. Then we can select either b = 20 or b = 40 so it would be insufficient on its own. However combining with the original statement 1, we can only choose a = 50 paired with b = 40 since they must have the 5:4 ratio (or the same case of total of 90). In that case, we would choose C.