Problem Solving

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
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