## Shoe size

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### Shoe size

by josh80 » Mon Dec 09, 2013 4:59 pm
A manufacturer produces a certain men's athletic shoe in integer sizes from 8 to 17. For this particular shoe, each unit increase in size corresponds to a 1/4-inch increase in the length of the shoe. If the largest size of this shoe is 20% longer than the smallest size, how long, in inches, is the shoe in size 15?

12

12.25

12.5

12.75

13

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by sahilchaudhary » Mon Dec 09, 2013 9:05 pm
Good question josh80.

Let the length of size 8 shoe = x.
So, the length of size 17 shoe = x + 20% of x = x + x/5. (1)

Since, it is given that each unit increase in size corresponds to a 1/4 inch increase in size.
So, the length of size 9 shoe = x + 1/4.
Similarly, the length of size 10 shoe = x + 2/4.
Similarly, the length of size 15 shoe = x + 7/4.
Similarly, the length of size 17 shoe = x + 9/4. (2)

Equating (1) and (2), since they both represent length of size 17 shoe.
x + x/5 = x + 9/4
x = 45/4.

So, the length of size 15 shoe = x + 7/4 = 45/4 + 7/4 = 52/4 = 13.

Sahil Chaudhary
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by GMATGuruNY » Tue Dec 10, 2013 3:53 am
josh80 wrote:A manufacturer produces a certain men's athletic shoe in integer sizes from 8 to 17. For this particular shoe, each unit increase in size corresponds to a 1/4-inch increase in the length of the shoe. If the largest size of this shoe is 20% longer than the smallest size, how long, in inches, is the shoe in size 15?

12

12.25

12.5

12.75

13
An alternate approach is to PLUG IN THE ANSWERS, which represent a size 15 shoe.

Answer choices A and E -- both integer values -- are the easiest to plug in.
Every shoe size is 1/4-inch greater than the next smallest shoe size.
The average test-taker will be more attracted to A than to E, since an increase of 1/4 seems more relevant to a length of 12.

Since a size 17 shoe is 20% greater than a size 8 shoe, the correct answer choice must yield the following ratio:
(size 17)/(size 8) = 120/100 = 6/5.

Answer choice E: size 15 = 13 inches = 52/4 inches
Size 17 shoe = 52/4 + (2)(1/4) = 54/4.
Size 8 shoe = 52/4 - (7)(1/4) = 45/4.
(size 17)/(size 8) = (54/4) / (45/4) = 54/45 = 6/5.
Success!

Algebraic approach:

Let x = a size 8 shoe.

Each increase in shoe size is equal to 1/4 inch.
Thus, an increase of 9 shoe sizes -- from size 8 to size 17 -- is equal to 9/4 inches.
Since these 9/4 inches represent a 20% increase from a size 8, we get:
9/4 = 0.2x
9/4 = (1/5)x
x = 45/4.

A size 15 shoe -- an increase of 7 sizes from a size 8 shoe -- must be 7/4 inches longer than a size 8 shoe:
45/4 + 7/4 = 52/4 = 13.