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A store currently charges the same price for each towel that [#permalink]

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27 Dec 2012, 04:47

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C

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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?

Re: A store currently charges the same price for each towel that [#permalink]

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18 Oct 2013, 08:32

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I thought it would be easier to just use a smart number. I picked the middle number option (C) $3 per towel and proceeded to divide $120 with $3 to make 40 towels. Then I tried with (D) $4 which gave me 30 towels. Therefore the current price must be $3

Re: A store currently charges the same price for each towel that [#permalink]

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15 May 2014, 09:27

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Bunuel wrote:

Walkabout 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

Let the current price be \(p\) and the # of towels sold at this price be \(n\). Then we would have two equations:

\(pn=120\) amd \((p+1)(n-10)=120\) at this point you can solve the system of equations for \(p\) (you'll get quadratic equation to solve) or try to substitute answer choices.

When substituting answer choices it's good to start with the middle value, so in our case $3. So, if \(p=3\) then \(3n=120\) --> \(n=40\) --> \((3+1)(40-10)=4*30=120\), so this answer works.

Answer: C.

Hope it helps.

Bunuel

Instead of two variables p,n cant it be solved in 1 variable as below

Say p = original price of 1 towel

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

putting values from options gives the value of p=3

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

Let the current price be \(p\) and the # of towels sold at this price be \(n\). Then we would have two equations:

\(pn=120\) amd \((p+1)(n-10)=120\) at this point you can solve the system of equations for \(p\) (you'll get quadratic equation to solve) or try to substitute answer choices.

When substituting answer choices it's good to start with the middle value, so in our case $3. So, if \(p=3\) then \(3n=120\) --> \(n=40\) --> \((3+1)(40-10)=4*30=120\), so this answer works.

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

Solution:

We can start by creating some variables.

Q = quantity of towels sold

P = price per towel sold

Next we can 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 should 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 get rid of the denominators. This 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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GMAT Quant Self-Study Course 500+ lessons 3000+ practice problems 800+ HD solutions

Re: A store currently charges the same price for each towel that [#permalink]

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16 Oct 2016, 18:19

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Walkabout 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?

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.

I thought it would be easier to just use a smart number. I picked the middle number option (C) $3 per towel and proceeded to divide $120 with $3 to make 40 towels. Then I tried with (D) $4 which gave me 30 towels. Therefore the current price must be $3

Answer: C.

I agree. I think this problem is a textbook example for why reverse plugging in is a valuable strategy.
_________________

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

Let the current price be \(p\) and the # of towels sold at this price be \(n\). Then we would have two equations:

\(pn=120\) amd \((p+1)(n-10)=120\) at this point you can solve the system of equations for \(p\) (you'll get quadratic equation to solve) or try to substitute answer choices.

When substituting answer choices it's good to start with the middle value, so in our case $3. So, if \(p=3\) then \(3n=120\) --> \(n=40\) --> \((3+1)(40-10)=4*30=120\), so this answer works.

Answer: C.

Hope it helps.

Bunuel

Instead of two variables p,n cant it be solved in 1 variable as below

Say p = original price of 1 towel

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

putting values from options gives the value of p=3

Re: A store currently charges the same price for each towel that [#permalink]

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17 Sep 2014, 09:07

Bunuel wrote:

Walkabout 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

Let the current price be \(p\) and the # of towels sold at this price be \(n\). Then we would have two equations:

\(pn=120\) amd \((p+1)(n-10)=120\) at this point you can solve the system of equations for \(p\) (you'll get quadratic equation to solve) or try to substitute answer choices.

When substituting answer choices it's good to start with the middle value, so in our case $3. So, if \(p=3\) then \(3n=120\) --> \(n=40\) --> \((3+1)(40-10)=4*30=120\), so this answer works.

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

Let the current price be \(p\) and the # of towels sold at this price be \(n\). Then we would have two equations:

\(pn=120\) amd \((p+1)(n-10)=120\) at this point you can solve the system of equations for \(p\) (you'll get quadratic equation to solve) or try to substitute answer choices.

When substituting answer choices it's good to start with the middle value, so in our case $3. So, if \(p=3\) then \(3n=120\) --> \(n=40\) --> \((3+1)(40-10)=4*30=120\), so this answer works.

Answer: C.

Hope it helps.

Hi Bunnel

How is pn=120 first equation ??

If the current price of each towel were to be increased by $1 (the current price p, new price p+1), 10 fewer of the towels (n for the current number of towels, n-10 for new number of towels) could be bought for $120.

So, for $120 for the current price p, we can buy n towels: pn=120.
_________________

Re: A store currently charges the same price for each towel that [#permalink]

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30 Sep 2017, 23:59

I understand pn = 120 but doesn't (p+1)(n−10)=120 assume that the new price divides 120 evenly? What if that is not the case?

Obviously in this problem it does divide evenly but what about a similar problem where the new price leaves us with some some money left over < p? Shouldn't the second equation be an inequality or something to do with remainders?

I understand pn = 120 but doesn't (p+1)(n−10)=120 assume that the new price divides 120 evenly? What if that is not the case?

Obviously in this problem it does divide evenly but what about a similar problem where the new price leaves us with some some money left over < p? Shouldn't the second equation be an inequality or something to do with remainders?

We are told that "If the current price of each towel were to be increased by $1, 10 fewer of the towels could be bought for $120", so $120 is exactly how much you need to buy 10 fewer of the towels if the price of each towel were to be increased by $1.
_________________