In a certain clothing store, the most expensive pair of socks sells fo

Question

In a certain clothing store, the most expensive pair of socks sells for one dollar less than twice the price of the cheapest pair of socks. A customer notices that for exactly $18, she can buy three fewer pairs of the most expensive socks than the cheapest socks. What could be the number of pairs of the cheapest socks she could have purchased?
 
(A) 3 
(B) 5 
(C) 6 
(D) 12 
(E) 36

u1983 · Accepted Answer

Let the Price of Cheapest pair is x . Hence price of expensive pair is 2x-1
For $18 , we get n cheapest pair & N expensive pair . Hence , n=18/x & N=18/(2x-1)
As per the Q steam :  n-N=3 
i.e., 18/x - 18/(2x-1) =3
or, 6/x - 6/(2x-1) =1
or ,  6(2x-1)-6x=x(2x-1) 
or , 2x^2 - 7x +6 =0 
or,  (2x-3)(x-2)=0  =====>  x=2,1.5

When x=2, 2x-1=3 , n=9 & N=6
When x=1.5, 2x-1=2 , n=12 & N=9....................... Hence the ans is 12 (D)

BrentGMATPrepNow · Answer

Let  C  = price of one pair of the CHEAPEST socks
So,  2C - 1  = price of one pair of the MOST EXPENSIVE socks

A customer notices that for exactly $18, she can buy three fewer pairs of the most expensive socks than the cheapest socks.   
As a word equation, we can write: ( # of pairs of EXPENSIVE socks purchased with $18 ) = ( # of pairs of CHEAP socks purchased with $18 ) - 3

With $18, the number of pairs of CHEAP socks we can purchase = 18/ C 
With $18, the number of pairs of EXPENSIVE socks we can purchase = 18/ (2C - 1)

So, we can write: ( 18/(2C - 1) ) = ( 18/C ) - 3
Multiply both sides by C to get: 18C/(2C - 1) = 18 - 3C
Multiply both sides by (2C - 1) to get: 18C = 18(2C - 1) - 3C(2C - 1)
Simplify: 18C = 36C - 18 - 6C² + 3C
Simplify: 18C = 39C - 18 - 6C² 
Rearrange to get: 6C² - 21C + 18 = 0
Divide both sides by 3 to get: 2C² - 7C + 6 = 0
Factor to get: (2C - 3)(C - 2) = 0
So, EITHER 2C - 3 = 0 OR C - 2 = 0
In other words, EITHER C = 1.5 OR C = 2

Let's check each answer choice.
If  C = $1.5 (the price of one pair of the CHEAPEST socks) , then  2C - 1 = $2 (the price of one pair of the MOST EXPENSIVE socks) 
So, the number of CHEAP socks purchased for $18 = 18/$1.5 = 12
And the number of EXPENSIVE socks purchased for $18 = 18/$2 = 9
Perfect, this meets the condition that the customer can buy three fewer pairs of the most expensive socks than the cheapest socks.

If  C = $2 (the price of one pair of the CHEAPEST socks) , then  2C - 1 = $3 (the price of one pair of the MOST EXPENSIVE socks) 
So, the number of CHEAP socks purchased for $18 = 18/$2 = 9
And the number of EXPENSIVE socks purchased for $18 = 18/$3 = 6
This ALSO meets the condition that the customer can buy three fewer pairs of the most expensive socks than the cheapest socks.

So, the customer bought EITHER 12 pairs of cheap socks or 9 pairs of cheap socks.
Check the answer choices, we have 12

Answer: D

Cheers, 
Brent

GMATGuruNY · Answer

We can PLUG IN THE ANSWERS, which represent the number of cheap pairs that can be purchased for $18.
When the correct answer is plugged in, the number of expensive pairs that can be purchased will be 3 less than the number of cheap pairs that can be purchased.
B will yield an ugly price for each cheap pair (18/5 = 3.60) and thus is unlikely to be correct.

D:  12  pairs, implying that the price for each cheap pair = 18/12 = $1.50
Since each expensive pair costs $1 less than twice each cheap pair, each expensive pair = 2(1.50) - 1 = 2.
Number of $2 expensive pairs that be purchased for $18 = 18/2 =  9 .
Success!
The value in blue is 3 less than the value in red.

D .

ScottTargetTestPrep · Answer

We can let c = the price of the cheapest socks and e = the price of the expensive socks, and thus:

e = 2c - 1

Letting n = the number of pairs of the cheapest socks that she could buy, we have:
 
nc = 18

and

(n - 3)e = 18

Equating the second and the third equations, we have:

nc = (n - 3)e

nc = (n - 3)(2c - 1)

nc = 2nc - n - 6c + 3

18 = 2(18) - n - 6(18/n) + 3

18 = 36 - n - 108/n + 3

n - 21 + 108/n = 0

n^2 - 21n + 108 = 0

(n - 9)(n - 12) = 0

n = 9 or n = 12

We see that n is either 12 or 9, but since only 12 is one of the answer choices. 12 is the correct answer.

Answer: D

akadiyan · Answer

We can solve this by plugging in numbers

First step in plugging numbers is using the smart numbers to try. In the question, i look for keywords that helps me in selecting the first number to try with. Important info given is "most expensive is 1 dollar less than twice the pair of cheapest", so atleast one of them is even number. So i will look or even numbers in answer choices to try first.

Next important information is total amount given is $18 , so its always safe to try numbers that gives results in numbers > 1. 36 pair for $18 is less than $1, so we can start with option D. $12.

12 pair of cheap socks for $18. so 1 pair is 1.5$ so 1 pair of expensive pair is $2 ($1 less than twice the cheap one) . For 18$ we could buy 9 pair of expensive socks (which is 3 less than 12 pair of cheap socks he could have bought )

Ans   D

Kinshook · Answer

dimmak 
Given: In a certain clothing store, the most expensive pair of socks sells for one dollar less than twice the price of the cheapest pair of socks. A customer notices that for exactly $18, she can buy three fewer pairs of the most expensive socks than the cheapest socks.

Asked: What could be the number of pairs of the cheapest socks she could have purchased?
Let the price of cheapest socks be x and number of pair of the cheapest socks purchased be y.
The price of the most expensive pair of socks = 2x - 1
The number of pair of the most expensive socks purchased = y-3

18/x = y; 
18/(2x-1) = y-3 = 18/x - 3
6x = 6(2x-1) - x(2x-1) = 12x - 6 - 2x^2 + x
2x^2 -  7x + 6 = 0
2x^2 - 3x - 4x + 6 = 0
(2x-3)(x-2) = 0

x = 1.5 or 2
y = 18/x = 12 or 9

IMO D
since only 12 is provided as an answer

Adarsh_24 · Answer

IMO setting the answer we need as x and keeping them as denominator similar to below equation makes it much easier.

X is the number of cheaper sock pairs.
We will get

18/(x-3) = 2(18/x) -1
Easier to plug and play the options and final answer won’t need any further calculation.
12 fits.

HarshavardhanR · Answer

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