Impossible question? PS 203

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vlp700: Newbie | Next Rank: 10 Posts; Posts: 2; Joined: Tue Jan 27, 2015 4:53 pm

by vlp700 » Tue Jan 27, 2015 5:08 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

OA: C

Ok, two questions here:
1. The book's explanation says "let p be the current price per towel, and let n be the number of towels that can be bought for $120." What? We're told that if the price and quantity change it will be $120. But it doesn't say it's 120 currently. I thought we're not allowed to assume?
2. There's no way I could do all that math in 2 min. Am I supposed to?

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Source: Beat The GMAT — Problem Solving | Tue Jan 27, 2015 5:08 pm

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

by Brent@GMATPrepNow » Tue Jan 27, 2015 5:14 pm

vlp700 wrote: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

OA: C

The answer choices all divide nicely into $120, so it might be fastest to PLUG in the answer choices.

A) $1
If towels are $1 each, $120 will buy 120 towels
Increase by $1 so that towels are $2 each. At this price, we can buy 60 towels
Difference = 120 - 60 = 60 fewer towels
We want 10 fewer
ELIMINATE A

B) $2
If towels are $2 each, $120 will buy 60 towels
Increase by $1 so that towels are $3 each. At this price, we can buy 40 towels
Difference = 60 - 40 = 20 fewer towels
We want 10 fewer
ELIMINATE B

c) $3
If towels are $3 each, $120 will buy 40 towels
Increase by $1 so that towels are $4 each. At this price, we can buy 30 towels
Difference = 40 - 30 = 10 fewer towels
PERFECT!

Answer: C

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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

by GMATGuruNY » Tue Jan 27, 2015 5:24 pm

vlp700 wrote: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!

Last edited by GMATGuruNY on Tue Jan 27, 2015 5:25 pm, edited 1 time in total.

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

by Brent@GMATPrepNow » Tue Jan 27, 2015 5:24 pm

vlp700 wrote: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

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 pmFollowed by: 247 members; Thanked: 1443 times

by ceilidh.erickson » Tue Jan 27, 2015 5:24 pm

I understand your confusion - the question doesn't explicitly say that current total cost is $120. But we are allowed to infer based on inverse proportionality. 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 equally 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."

As to your second question - I agree, the long algebraic solution would likely take more than 2 minutes for even the savviest test takers. Here's an easier solution:

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. Here, it helps to set up a table:

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
2 x 60 --> 3 x 50
3 x 40 --> 4 x 30
4 x 30 --> 5 x 20
12 x 10 --> 13 x 0

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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vlp700: Newbie | Next Rank: 10 Posts; Posts: 2; Joined: Tue Jan 27, 2015 4:53 pm

by vlp700 » Tue Jan 27, 2015 5:31 pm

Thanks ceilidh.erickson for answering my first question. It's annoying but I get it. Will there be a lot of questions where the total isn't given, just implied?

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

by ceilidh.erickson » Tue Jan 27, 2015 5:39 pm

vlp700 wrote:Thanks ceilidh.erickson for answering my first question. It's annoying but I get it. Will there be a lot of questions where the total isn't given, just implied?

I wouldn't say that this is a common construction, but it's certainly something that you might see again. The GMAT will assume that you understand direct and inverse proportionality.

Consider another similar scenario with inverse proportionality:
"If Sally's wage were to increase by $2 per hour, she would be able to work 10 fewer hours and still make $600." --> It's implied that her current total is $600.

Here is a scenario with direct proportionality:
"If 30 new students are enrolled in the school, 2 new teachers will have to be hired to maintain a ratio of 1 teacher for every 15 students." --> It's implied that the current ratio is 1:15.

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

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8138; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

by Scott@TargetTestPrep » Tue Jul 28, 2015 1:56 pm

vlp700 wrote: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

OA: C

Ok, two questions here:
1. The book's explanation says "let p be the current price per towel, and let n be the number of towels that can be bought for $120." What? We're told that if the price and quantity change it will be $120. But it doesn't say it's 120 currently. I thought we're not allowed to assume?
2. There's no way I could do all that math in 2 min. Am I supposed to?

Solution:

We can start by defining some variables.

Q = quantity of towels sold

P = price per towel sold

Next we set up some equations.

We know that at the current price:

PQ = 120

We are next 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. From this we can say:

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

Since we need to determine the value of P, we can get the second equation in terms of P only. We can do this by manipulating the equation PQ = 120. So we can say:

Q = 120/P

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

FOILing this, we get:

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

-10P + 120/P - 10 = 0

We can multiply the entire equation by P to eliminate the denominators. Doing so gives us:

-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.

Answer is C

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nikhilgmat31: Legendary Member; Posts: 518; Joined: Tue May 12, 2015 8:25 pm; Thanked: 10 times

by nikhilgmat31 » Wed Jul 29, 2015 12:34 am

1 way to solve is

pn = 120 i.e. n = 120/p
(p+1)(n-10) = 120

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

now you can either solve for p or plugin values of p from 5 options.

Answer is 3.

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