Problem Solving

An optometrist charges $150 per pair for soft contact lenses and $85 per pair for hard contact lenses. Last week sh sold 5 more pairs of soft lenses than hard lenses. If her total sales for pairs of contact lenses last week were $1,690, what was the total number of pairs of contact lenses that she sold?

A) 11

B) 13

C) 15

D) 17

E) 19

The solution below is taken from the GMATFix App.



-Patrick

BlueDragon2010 wrote:An optometrist charges $150 per pair for soft contact lenses and $85 per pair for hard contact lenses. Last week sh sold 5 more pairs of soft lenses than hard lenses. If her total sales for pairs of contact lenses last week were $1,690, what was the total number of pairs of contact lenses that she sold?

A) 11

B) 13

C) 15

D) 17

E) 19

The total amount of revenue -- $1690 -- is a multiple of 10.
Since the price of each pair of soft lenses is a multiple of 10, the total amount of revenue from the soft lenses must be a multiple of 10.
Implication:
The total amount of revenue from the hard lenses must also be a multiple of 10.
Since the price of each pair of hard lenses is $85, the number of pairs of hard lenses sold must be an EVEN integer.

Case 1: 2 pairs of hard lenses sold
Since 5 more pairs of soft lenses are sold, the number of pairs of soft lenses sold = 2+5 = 7.
Total pairs sold = 2+5 = 7.
Since 7 is not among the answer choices, Case 1 is not viable.

Case 2: 4 pairs of hard lenses sold
Since 5 more pairs of soft lenses are sold, the number of pairs of soft lenses sold = 4+5 = 9.
Total pairs sold = 4+9 = 13.
This is the value in answer choice B.
Check whether Case 2 yields $1690 in revenue:
(4*85) + (9*150) = 340 + 1350 = 1690.
Success!

The correct answer is B.