Question

Alma bought a laptop computer at a store that gave a 20% discount off its original price. The total amount she paid to the cashier was p dollars, including an 8 percent sales tax on the discounted price. Which of the following represents the original price of the computer in terms of p?
(a) 0.88p
(b) $\dfrac{p}{0.88}$
(c) (0.8)(1.08)p
 (d) $\dfrac{p}{\left( 0.8 \right)\left( 1.08 \right)}$

Answer 1

Hint: Firstly, we suppose the original value of the laptop will be x dollars. Then, we have to find the value of the laptop after giving a 20% discount over it. Then, we find the value after the sale tax of 8% is imposed on the discounted price. Then, we will equate the calculated to p and get the desired result in terms of P.

Complete step-by-step answer:
In this question, we are supposed to find the original price of the computer in terms of p dollars.
Firstly, we suppose the original value of the laptop will be x dollars.
Now, for getting the 20% discount on the laptop, we need to calculate the 20% of the value x.
So, the above expression is as 20% of x:
$\dfrac{20}{100}\times x=\dfrac{20x}{100}$
Now, for discount the above calculated value is subtracted from the original value which is x as:
$\begin{align}
  & x-\dfrac{20x}{100}=\dfrac{100x-20x}{100} \\
 & \Rightarrow \dfrac{80x}{100} \\
\end{align}$
Now, the sale tax of 8% is imposed on the discounted price.
So, 8% of $\dfrac{80x}{100}$ is given by:
$\dfrac{8}{100}\times \dfrac{80x}{100}=\dfrac{64x}{1000}$
So, by increasing the discounted price with the value $\dfrac{64x}{1000}$ and get the total price as p dollars.
Now, firstly the increased price is:
$\begin{align}
  & \dfrac{80x}{100}+\dfrac{64x}{1000}=\dfrac{800x+64x}{1000} \\
 & \Rightarrow \dfrac{864x}{1000} \\
 & \Rightarrow 0.864x \\
\end{align}$
Now, according to the question it is the total price paid which is p dollars.
Then, equating the above equation with p and getting the desired result as:
$0.864x=p$
So, 0.864 can be written as multiplication of the 0.8 and 1.08 as:
$x=\dfrac{p}{\left( 0.8 \right)\left( 1.08 \right)}$
So, the above expression gives the final result as the original cost of the laptop in terms of p as $\dfrac{p}{\left( 0.8 \right)\left( 1.08 \right)}$.
Hence, option (d) is correct.

Note: The most common mistake we make in these type of the questions is that we take the original price x to calculate the sale tax imposed over it and this will lead to the increased value of the amount to be paid for laptop as (100+8)% of x as:
$\dfrac{\left( 100+8 \right)}{100}\times x=\dfrac{108x}{100}$
So, the mistake we make here is that there is discount of 20% and increase due to sale tax of 8% and some students most often take it as 12% overall decrement in the original value which leads to the following answer as:
$\begin{align}
  & x-\dfrac{12x}{100}=\dfrac{100x-12x}{100} \\
 & \Rightarrow \dfrac{88x}{100} \\
\end{align}$
It will give x as 0.88p and we will mark option (a) as correct which is a wrong choice.