OG2016- PS 208

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amina.shaikh309: Senior | Next Rank: 100 Posts; Posts: 31; Joined: Sat Apr 23, 2016 9:06 pm

OG2016- PS 208

by amina.shaikh309 » Sun May 22, 2016 6:21 am

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Source: Beat The GMAT — Problem Solving | Sun May 22, 2016 6:21 am

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pm; Location: New York, NY; Thanked: 13060 times; Followed by:1906 members; GMAT Score:790

by GMATGuruNY » Sun May 22, 2016 6:41 am

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

We can PLUG IN THE ANSWERS, which represent the current price.
When the correct answer choice is plugged in, increasing the price by $1 will allow for the purchase of 10 fewer towels.

Answer choice D: 4
At $4 per towel, the number of towels that can be purchased for $120 = 120/4 = 30.
If the price is increased by $1 to $5 per towel, the number of towels that can be purchased for $120 = 120/5 = 24.
Decrease in the number of towels that can be purchased = 30-24 = 6.
Here, the decrease is too small.
Eliminate D.

Answer choice B: $2
At $2 per towel, the number of towels that can be purchased for $120 = 120/2 = 60.
If the price is increased by $1 to $3 per towel, the number of towels that can be purchased for $120 = 120/3 = 40.
Decrease in the number of towels that can be purchased = 60-40 = 20.
Here, the decrease is too great.
Eliminate B.

Since D yields a decrease that is TOO SMALL, and B yields a decrease that is TOO GREAT, the correct answer must be BETWEEN B AND D.

The correct answer is C.

Answer choice C: $3
At $3 per towel, the number of towels that can be purchased for $120 = 120/3 = 40.
If the price is increased by $1 to $4 per towel, the number of towels that can be purchased for $120 = 120/4 = 30.
Decrease in the number of towels that can be purchased = 40-30 = 10.
Success!

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pm; Location: Vancouver, BC; Thanked: 5254 times; Followed by:1268 members; GMAT Score:770

by Brent@GMATPrepNow » Sun May 22, 2016 6:54 am

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

I prefer Mitch's PLUG-IN method, but for those interested, here's an algebraic approach:

Let P = CURRENT price (in dollars)
So, P + 1 = NEW price (in dollars)

At P dollars apiece, the # of towels we can buy = 120/P
At (P+1) dollars apiece, the # of towels we can buy = 120/(P+1)
We're told we can buy 10 fewer towels at the NEW price.
So, (# of towels we can buy at CURRENT price) - (# of towels we can buy at NEW price) = 10
So, we can write: 120/P - 120/(P+1) = 10
SOLVE FOR P
Divide both sides by 10 to get: 12/P - 12/(P + 1) = 1
Multiply both sides by (P)(P+1) to get: 12(P+1) - 12P = P(P + 1)
Expand: 12P + 12 - 12P = PÂ² + P
Rearrange: PÂ² + P - 12 = 0
Factor: (P + 4)(P - 3) = 0
So, P = -4 or P = 3
P cannot be negative (since we can't have a negative price)
So, P must equal 3
Answer: C

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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ceilidh.erickson: GMAT Instructor; Posts: 2095; Joined: Tue Dec 04, 2012 3:22 pm; Thanked: 1443 times; Followed by:247 members

by ceilidh.erickson » Mon Oct 09, 2017 8:16 am

One common source of confusion on this question: the question doesn't explicitly state that current total cost is $120. Many students wonder how we're allowed to set the equation equal to that amount. But we are allowed to infer based on inverse proportionality (and common sense!). The relationship between price and quantity is always:
pq = total cost

If we want to keep the total cost constant, then when the price goes up, the quantity will go down, and vice versa. Any time you have a product of two variables equaling a constant, they will be inversely proportional: (wage)(hours)=income, etc.

In this problem, we're told that if price goes up by a certain amount, then the quantity would have to go down by a specified amount to equal $120. By saying "10 fewer of the towels could be bought for $120," it's implying "10 fewer than before, when the total was also $120."

All the experts here are in agreement: we would Work Backwards from the answer choices. Whenever a PS question asks for the value of an unknown (variable), one option is to plug the answer choices back into the problem to see which one works.

A slightly different spin on working backwards (rather than Mitch's strategy of going 1 answer choice at a time) is to set up a table.
Here are the answer choice options for p, and the corresponding value of q that would equal $120:
p x q
1 x 120
2 x 60
3 x 40
4 x 30
12 x 10

Now add 1 to each price and subtract 10 from each quantity:
p x q
1 x 120 --> 2 x 110 --> 220. no
2 x 60 --> 3 x 50 --> 150. no.
3 x 40 --> 4 x 30 --> 120. yes!
4 x 30 --> 5 x 20 --> 100. no.
12 x 10 --> 13 x 0 --> 0. no.

For which of these new pairs does pq = 120? Only 4 x 30. The answer must be C.

Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education

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[email protected]: Elite Legendary Member; Posts: 10392; Joined: Sun Jun 23, 2013 6:38 pm; Location: Palo Alto, CA; Thanked: 2867 times; Followed by:511 members; GMAT Score:800

by [email protected] » Mon Oct 09, 2017 2:32 pm

Hi All,

Most GMAT questions can be solved in a variety of ways, so you should look for alternatives to "math" approaches (in many cases, the math approach takes the longest to set up and complete).

Here, we're essentially asked to spend $120 on towels. We're then asked to figure out the price point at which ADDING $1 to the price of a towel results in 10 FEWER towels purchased. Since the answers are NUMBERS (and almost all consecutive integers), we can TEST THE ANSWERS....

IF....
Towels are....
$1 each, then we can buy 120 towels
$2 each, then we can buy 60 towels
$3 each, then we can buy 40 towels
$4 each, then we can buy 30 towels
$5 each, then we can buy 24 towels

Now, stop and look at the progression. We're looking for a point at which the DIFFERENCE is 10 towels. That only happens in one "spot" - when the price is increased from $3 to $4. The question asks for the current (re: lower) price.

Final Answer: C

GMAT assassins aren't born, they're made,
Rich

Contact Rich at [email protected]

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Jeff@TargetTestPrep: GMAT Instructor; Posts: 1462; Joined: Thu Apr 09, 2015 9:34 am; Location: New York, NY; Thanked: 39 times; Followed by:22 members

by Jeff@TargetTestPrep » Sat Jul 14, 2018 6:41 pm

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
We can let Q = quantity of towels sold and P = price per towel sold and create the equations:

PQ = 120

Q = 120/P

And

(P + 1)(Q - 10) = 120

Now we can plug in 120/P for Q in the equation (P + 1)(Q - 10) = 120. We now have:

(P + 1)(120/P - 10) = 120

120 - 10P + 120/P - 10 = 120

-10P + 120/P - 10 = 0

-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.

Alternate Solution:

Let's test each answer choice.

A) If the price of a towel was $1, then 120/1 = 120 towels can be bought before the increase and 120/2 = 60 towels can be bought after the increase, which is 60 fewer. A is not correct.

B) If the price of a towel was $2, then 120/2 = 60 towels can be bought before the increase and 120/3 = 40 towels can be bought after the increase, which is 20 fewer. B is not correct.

A) If the price of a towel was $3, then 120/3 = 40 towels can be bought before the increase and 120/4 = 30 towels can be bought after the increase, which is 10 fewer. C is correct.

Answer: C