Question: A store currently charges the same price for each towel that it sells. If the current price of each towel were to be increased by $1, 10 fewer of the towels could be bought for $120, excluding sales tax. What is the current price of each towel?
(A) $ 1
(B) $ 2
(C) $ 3
(D) $ 4
(E) $12
“A store currently charges the same price for each towel that it sells.”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “The Official Guide for GMAT Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.
Solution and Explanation:
Approach Solution 1:
It is asked What is the current price of each towel? If the current price of each towel were to be increased by $1, 10 fewer of the towels could be bought for $120, excluding sales tax.
Evaluating further -
The first step is to establish some variables.
Q is the number of towels sold.
P is the cost per sold towel.
The next step is to create some equations.
We are aware that given the present cost:
PQ = 120
It is given that if the current price of each towel were to be increased by $1, 10 fewer of the towels could be bought for $120
This means
(P + 1)(Q – 10) = 120
The second equation should only be expressed in terms of P because we need to know the value of P. The PQ = 120 equation can be changed to achieve this. Consequently, we can state:
Q = 120/P
The equation (P + 1)(Q - 10) = 120 can now be solved by substituting 120/P for Q. There are now:
\((P + 1)(\frac{120}{P} – 10) = 120\)
Substituting values, we get:
\(120 – 10P + \frac{120}{P} – 10 = 120\)
\(–10P + \frac{120}{P} – 10 = 0\)
The denominators can be removed from the equation by multiplying the entire thing by P. This results in:
\(–10P^2 + 120 – 10P = 0\)
\(10P^2 + 10P – 120 = 0\)
\(P^2 + P – 12 = 0\)
(P + 4)(P – 3) = 0
P = -4 or P = 3
Since P can’t be negative, P = 3.
The answer is 3
Correct Answer: C
Approach Solution 2:
There is another approach to answering this question: It is asked What is the current price of each towel? If the current price of each towel were to be increased by $1, 10 fewer of the towels could be bought for $120, excluding sales tax.
Let Us assume the original cost of towels = x
This means
\(\frac{120}{x}-\frac{120}{[x+1]}= 10\)
\(120 = 10[x^2+x]\)
\([x^2+x-12]=0\)
[x + 4 ] [x - 3 ]
x = -4, 3
Since x can’t be negative, x = 3.
The answer is 3
Correct Answer: C
Approach Solution 3:
⇒ Let number of towels bought for $120 be n.
⇒ So price of single towel =$n120
⇒ Now price of 1 towel increases by $1
⇒ So, new price of single towel =$n120+1
⇒ Number of towel that could be bought at this price = n−10
⇒ So, new price of single towel = (n−10)120
So, by equating both new price of single towel,
⇒ n120+1=(n−10)120
⇒ n(120+n)=(n−10)120
⇒ n2−10n−1200=0
⇒ (n−40)(n+30)=0
⇒ n=40 or n=−30
⇒ Number of towels is 40.
⇒ Current price per towel = 40120=$3
Correct Answer: C
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