Question

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. $\$ 4$

Answer 1

Hint: We will first let the number of towels bought from $\$ 120$ were $x$. Then, find the price of each towel originally. Then, find the new price of each towel. Then, use the given condition that the difference in the prices of each towel is $\$ 1$ to form an equation. Solve the formed quadratic equation to find the numbers of towels bought for $\$ 120$. Finally, divide the total cost by the number of towels to find the price of each towel.

Complete step-by-step answer:
We are given that the price of each towel is increased by $\$ 1$ and now the person can buy 10 towels less for $\$ 120$.
Let the number of towels bought initially for $\$ 120$ were $x$.
Then, the price of one towel can be calculated by dividing the cost of towels by number of towels.
That is, the price of one towel was $\dfrac{{120}}{x}$
Now, the person can buy 10 towels less, that the person can buy $x - 10$ towels for $\$ 120$.
Hence, the new price of single towel is $\dfrac{{120}}{{x - 10}}$
And the difference between the original price and the new price is $\dfrac{{120}}{{x - 10}} - \dfrac{{120}}{x}$
Also, we are given that the price is increased by $\$ 1$, which implies,
$\dfrac{{120}}{{x - 10}} - \dfrac{{120}}{x} = 1$
Take the LCM of the denominator and cross multiply to simplify the equation.
$
  \dfrac{{120x - 120\left( {x - 10} \right)}}{{\left( {x - 10} \right)x}} = 1 \\
   \Rightarrow 120x - 120x + 1200 = {x^2} - 10x \\
   \Rightarrow {x^2} - 10x - 1200 = 0 \\
$
Factorise the above equation
$
  {x^2} - 40x + 30x - 1200 = 0 \\
   \Rightarrow x\left( {x - 40} \right) + 30\left( {x - 30} \right) = 0 \\
   \Rightarrow \left( {x + 30} \right)\left( {x - 40} \right) = 0 \\
$
Equate each factor to 0 to find the value of $x$
$
  x + 30 = 0 \\
  x = - 30 \\
$
And
$
  x - 40 = 0 \\
  x = 40 \\
$
But, the number of towels cannot be negative, hence, the number of towels are 40.
We have to find the original price of the towel.
Therefore, we will substitute the value of $x$ in the original price of the towel, that is,
$\dfrac{{120}}{{40}} = \$ 3$
Hence, the original price of the towel is $\$ 3$.
Thus, option C is correct.

Note: The product of price of one article and the number of articles give the total cost of the product. When we divide the cost of all the items by the number of items to find the price of one item, it is known as the unitary method.