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A store currently charges the same price for each towel that

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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?

(A) $ 1
(B) $ 2
(C) $ 3
(D) $ 4
(E) $12
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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?

(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.
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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

Answer: C.

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New post 19 Oct 2013, 08:23
psychedelictwirl wrote:
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.
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N = total no. of towel
P = Price of towel

NP=120…(1), ; N = 120/p
(N-10)(P+1) = 120…(2)

From 2,
NP+N-10P-10=120
120/P*P + 120/P -10P = 130 (Plugging from 1)
120/P -10P = 10
P^2+P-12 = 0
(P+4)(P-3)=0
Hence, P=3 (As price cannot be negative)

*This is indeed the long method, but I am comfortable with algebra, than plugging no.)

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New post 20 Nov 2013, 04:29
(P+1)(N-10)=NP
N-10P-10=0

NP=120 => N=P/120
N-10P-10=0

P^2+P-12=0

P=3
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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

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New post 16 May 2014, 00:26
himanshujovi wrote:
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


Yes, that's correct.
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New post 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.

Answer: C.

Hope it helps.


Hi Bunnel

How is pn=120 first equation ??

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New post 17 Sep 2014, 10:22
SunthoshiTejaswi wrote:
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.


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.
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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?

(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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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 original cost of towels = x

\(\frac{120}{x}- \frac{120}{(x+1)} = 10\)
\(120 = 10(x^2 + x)\)
(\(x^2 + x -12) = 0\)
\((x+4)(x-3)=0\)
x=3
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New post 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?

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New post 01 Oct 2017, 03:19
ThisandThat wrote:
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.
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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:
[Reveal] Spoiler:
C


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New post 06 Jan 2018, 03:52
Bunuel niks18

similar query as Walkabout

Quote:
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.


is this because of: A store currently charges the same price for each towel that it sells.
What am I missing?
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New post 06 Jan 2018, 09:18
adkikani wrote:
Bunuel niks18

similar query as Walkabout

Quote:
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.


is this because of: A store currently charges the same price for each towel that it sells.
What am I missing?


Hi adkikani

Here important point to note is that total expenses before and after changing the price remain same i.e $120.

so if earlier price was p and total quantities bought was n, then net cost will be pn=120

after change in price and quantity also the expense remain same, hence we get (p+1)*(n-10)=120

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New post 06 Jan 2018, 14:57
niks18


Quote:
Here important point to note is that total expenses before and after changing the price remain same i.e $120.

Is this distinctly mentioned anywhere or you inferred the same?

Let me dissect sentence wise to understand:

Quote:
A store currently charges the same price for each towel that it sells.


Let there be x towels in store, this sentence means that if p is price for each towel
then px is price for total no of towels.

Quote:
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

(p+1)*(x-11) = 120
$120 is the amount after the decrease in quantity and price increase.

From where did you infer px=120 from the two statements?

Should not correct way be:
A store currently charges total of $120and 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

I am assuming sales tax is simply given for confusion.
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adkikani wrote:
niks18


Quote:
Here important point to note is that total expenses before and after changing the price remain same i.e $120.

Is this distinctly mentioned anywhere or you inferred the same?

Let me dissect sentence wise to understand:

Quote:
A store currently charges the same price for each towel that it sells.


Let there be x towels in store, this sentence means that if p is price for each towel
then px is price for total no of towels.

Quote:
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

(p+1)*(x-11) = 120
$120 is the amount after the decrease in quantity and price increase.

From where did you infer px=120 from the two statements?

Should not correct way be:
A store currently charges total of $120and 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

I am assuming sales tax is simply given for confusion.


Hi adkikani

Here's the question stem

"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?"

now read this part carefully -

"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."

Why is 120 mentioned here? what is the constraint with 120? for example price is increased so could be the quantity resulting in a different total cost. But why 120?

For eg. if the price of 1 pen is 9 and you bought 10 pens, your total cost is 90. Now you increase the price of pen by 1 i.e new price of pen is 10 so with this 90 you will be able to buy 9 pens only. This is same as saying "If the current price of each pen were to be increased by $1, 1 fewer of the pens could be bought for $90, excluding sales tax."

Kindly read the sentence multiple times to understand the language. the current price is not directly mentioned but you can surely arrive at it from the second sentence.

Kudos [?]: 380 [1], given: 42

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A store currently charges the same price for each towel that

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