Problem Solving

A salesman sells two equally priced brands of icecream, for which he receives a sales commission of 5 percent on one brand and 7 percent on the other. If the salesman sold a total of $10,000 in icecream and collected $660 in commissions, what fractional part of the total icecream sold did he sell at the lower commission rate?

A) 7/8
B) 4/5
C) 1/2
D) 1/5
E) 1/8


Answer : D

please give a quick method to solve this one...thanks!

A,B -> price=p
x,y (number sold)
x+y = 10000
5% of px + 7% of py = 660
p(5x+7y) = 66000
p(5(x+y) + 2y) = 66000
50000p + 2py = 66000
p=2 is not possible.
if p=1, 2y=16000 or y=8000 then x=2000
thus 2000/10000 = 1/5
IMO D

Cheese12 wrote:A salesman sells two equally priced brands of icecream, for which he receives a sales commission of 5 percent on one brand and 7 percent on the other. If the salesman sold a total of $10,000 in icecream and collected $660 in commissions, what fractional part of the total icecream sold did he sell at the lower commission rate?

A) 7/8
B) 4/5
C) 1/2
D) 1/5
E) 1/8


Answer : D

please give a quick method to solve this one...thanks!

This is a weighted average question.
660/10,000 = 6.6%.

A 5% commission is being combined with a 7% commission to yield a mixture with a 6.6% commission.
We can use alligation:

The proportion of each element in the mixture is equal to the distance between the OTHER two percentages.

Proportion of 5% commission = distance between 7% and 6.6% = |7-6.6| = .4
Proportion of 7% commission = distance between 5% and 6.6% = |5-6.6| = 1.6.
Ratio of 5% to 7% in the mixture = (.4) : (1.6) = 4:16= 1:4.

The sum of the parts of the ratio = 1+4 = 5
Thus, for every $5 earned, $1 comes from the lower rate, $4 comes from the higher rate.
Fraction from the lower rate = 1/5.

The correct answer is D.