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Word Trans (Alg Trans): A store currently charges the same p

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II Master | Next Rank: 500 Posts
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Word Trans (Alg Trans): A store currently charges the same p

Post Mon May 12, 2008 4:41 am
A store currently charges the same price for each towel 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 used the standard approach by solving algebraically:
Let n = number of towels
Let p = price of each towel
So np = total price.

"current price of each towel were to be increased by $1" can be written as:
p + 1
"10 fewer of the towels could be bought" can be written as:
n - 10

So we have to equations:
(1) np = 120
(2) (n-10)(p+1) = 120

Simplify (2):
np + n - 10p - 10 = 120
np + n - 10p = 130

We can now substitute for "np" using equation (1: np=120), so
np + n - 10p = 130
120 + n - 10p = 130
n - 10p = 10
n = 10 + 10p

From equation 1: we know np=120 ... so n = 120/p. We can subsititute this for n. So ...
n = 10 + 10p
120/p = 10 + 10p
120 = p (10 + 10p)
120 = 10p + 10p^2
0 = 10p^2 + 10p - 120

We can divide everything by 10 to simplify:

0 = p^2 + p - 12

This quadratic equation can be further simplified:

0 = (p+4) (p-3), and has the solutions p=-4, p=3.

The answer to this must be a positive value, so p=3. And answer is (C).


Interested in learning about the VARIOUS OTHER approaches to solving this.

Thanks
II[/u]

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codesnooker Legendary Member
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Post Mon May 12, 2008 5:11 am
We can solve it with Plugging values. To reduce the numbers to be plugged we will use divide and conquer approach.

We have first four choices as 1, 2, 3, & 4 (each increasing by 1)

You can ignore 12 as it is too high.

So pick any middle number (Hope you have heard of Divide and conquer approach, if not then search it on Google).

Lets say we picked 2.

So for 2 how many towels we can purchase? Ans = 60
Increment it by one. Now for 3 how many towels we can purchase? Ans = 40

As difference between both of them is greater than 10, so we increase the price to decrease the difference otherwise vice-versa)

For 3 you have already calculated.

Now calculate it for 4? Ans = 30

So your correct answer is (3)

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AleksandrM Legendary Member
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Post Mon May 12, 2008 9:04 am
GMAC suggests that you use algebra - namely, quadratic equation - to solve this problem. There is nothing wrong with this approach, if you earned extra points for showing your work. However, when you are facing GMAT time constraints, you need to work through this problem as quickly as possible.

First of all, the sentence about tax is just there to confuse you, so ignore it... as I did. Then, notice that the question is asking you for the CURRENT price of a towel. Now, just move on to the answer choices.

You need a one-dollar difference and a ten-towel spread:

12 is out right away, 13 does not divide evenly into 120. A is easy to eliminate as well. Now, if you look at B: 120/2 = 60 and 120/3 = 40, which is a 20-towel spread. Next move on to C, and try the same: 120/3 = 40 and 120/4 = 30, which is a 10-towel spread...BINGO!!!

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II Master | Next Rank: 500 Posts
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Post Mon May 12, 2008 9:15 am
AleksandrM wrote:
GMAC suggests that you use algebra - namely, quadratic equation - to solve this problem. There is nothing wrong with this approach, if you earned extra points for showing your work. However, when you are facing GMAT time constraints, you need to work through this problem as quickly as possible.

First of all, the sentence about tax is just there to confuse you, so ignore it... as I did. Then, notice that the question is asking you for the CURRENT price of a towel. Now, just move on to the answer choices.

You need a one-dollar difference and a ten-towel spread:

12 is out right away, 13 does not divide evenly into 120. A is easy to eliminate as well. Now, if you look at B: 120/2 = 60 and 120/3 = 40, which is a 20-towel spread. Next move on to C, and try the same: 120/3 = 40 and 120/4 = 30, which is a 10-towel spread...BINGO!!!
Excellent alternative approach. Thanks guys ... I think its VERY important to have alternative approaches in your toolbox for beating the GMAT.

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pbanavara Master | Next Rank: 500 Posts Default Avatar
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Post Sat Jan 24, 2009 2:27 pm
I think the codesnooker approach is the fastest .. here's one more ..

Get the first equations algebraically

so you have

x * p = px
(x-10) * (p+1) = 120

Now plug the values for P. 12 is out right away and if you proceed backwards - you'll end up in C

Let's take D

(x -10)*4 =120
x-10 = 30
x=40

Now use this in equation 1 : 40*4 should be equal to 120 it's not.

3 satisfies the values.

- pradeep

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coffee5251 Senior | Next Rank: 100 Posts Default Avatar
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Post Wed Jan 28, 2009 5:55 pm
You could also test the answer choices. Take each answer choice (the current price of the towel) and divide it by 120. Then take each one again, add 1 for the price increase of $1 and then divide that number by 120. Then, see which two numbers have a difference of 10.

for example, a:
120/1 = 120. then, add 1+1=2
120/2 = 60
120-60=60, not 10

b:
120/2= 60. then add 2+1=3
120/3= 40
60-40=20, not 10

c:
120/3= 40. then add 3+1=4
120/4= 30
40-30= 10. There you go.

Since the computation is simple this method took me under a minute.

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