Mira is making telescopes, each consisting of 2 lenses, 1 tube, and 1 eyepiece. Lenses can be purchased only in packs of

[BTGmoderatorDC]

Mira is making telescopes, each consisting of 2 lenses, 1 tube, and 1 eyepiece. Lenses can be purchased only in packs of 50, tubes only in packs of 10, and eyepieces only in packs of 30. However, half of the lenses in each pack are not usable for telescopes. If all parts are used only for the telescopes, what is the minimum number of lenses Mira must purchase to make a set of telescopes with no leftover components other than the unusable lenses?

A. 75
B. 150
C. 300
D. 600
E. 7,500


OA D

Source: Manhattan Prep

[Scott@TargetTestPrep]

Mira is making telescopes, each consisting of 2 lenses, 1 tube, and 1 eyepiece. Lenses can be purchased only in packs of 50, tubes only in packs of 10, and eyepieces only in packs of 30. However, half of the lenses in each pack are not usable for telescopes. If all parts are used only for the telescopes, what is the minimum number of lenses Mira must purchase to make a set of telescopes with no leftover components other than the unusable lenses?

A. 75
B. 150
C. 300
D. 600
E. 7,500


OA D

Source: Manhattan Prep

Solution:

Let’s assume that Mira is making n telescopes, thus she need 2n lenses, n tubes and n eyepieces. Let’s also assume that she bought a packs of lenses, b packs of tubes and c packs of eyepieces, thus she bought 50a lenses, 10b tubes and 30c eyepieces.

Since she can only use half of the lenses, she used 25a lenses. So we have:

2n = 25a, n = 10b and n = 30c

Since n, a, b, and c have to be integers, we see that the smallest value of a is 2. If that is the case, then n will be 25, but then b will be 2.5, which is not an integer. So a can’t be 2.

The next smallest value of a is 4. If that is the case, then n will be 50 and b will be 5, but then c will be 5/3, which is not an integer. So a can’t be 4.

Notice that a has to be even in order for n to be an integer and we can keep trying values for a (and make sure that b and c will be integers also):

If a = 6, then n = 75 and b = 7.5.

If a = 8, then n = 100 and c = 10/3.

If a = 10, then n = 125 and b = 12.5.

If a = 12, then n = 150 and b = 15 and c = 5.

We’ve founded the value of a (a = 12) so that n, b and c are all integers. Thus the minimum number of lenses she’s used is 25 x 12 = 300. However, she needed to buy twice as many lenses, or 600 lenses, since half of them are unusable.

Alternate Solution:

Let’s test each answer choice, starting from the smallest:

Answer Choice A: 75 lenses cannot be purchased since lenses come in packs of 50

Answer Choice B: If 150 lenses are purchased, only 75 of them can be used for making a telescope and since each telescope uses two lenses, one usable lens will be leftover.

Answer Choice C: If 300 lenses are purchased, only 150 of them can be used for making a telescope and since each telescope uses two lenses, a total of 75 telescopes can be made. But to make 75 telescopes, Mira will have to buy at least 8 packs of tubes for a total of 8 x 10 = 80 tubes; 5 of which will be leftover.

Answer Choice D: If 600 lenses are purchased, only 300 of them can be used for making a telescope and since each telescope uses two lenses, a total of 150 telescopes can be made. To make 150 telescopes, Mira can purchase 15 packs of tubes and 5 packs of eyepieces with no leftover parts besides unusable eyepieces.

Answer: D