Chapter 7.1, Problem 4CYP

To determine

Find out the better price:taking the discount before the rebate or taking the discount after the rebate.

Answer to Problem 4CYP

The price of the digital audio player is less when the discount is applied before the rebate.

Explanation of Solution

Given:

Sounds-to-Go offers both an in-store $35 rebate and a 15% discount on a digital audio player that normally sells for $300.

Calculation:

Let the original price of the audio player = x=$300

Discount = 15%

So, the price after discount = original price − discount = $x-15\%x=x\cdot \left(1-\frac{15}{100}\right)=\frac{85}{100}x=0.85x$

So, the function for the price after discount :

  $d(x)=x\cdot \left(1-\frac{15}{100}\right)=0.85x$

Rebate = $35

So, the price after rebate = original price - rebate = $x-35$

So, the function for the price after rebate:

  $r(x)=x-35$

If the discount is applied before the rebate, then the final price of the digital audio player is represented by $\left[r\circ d\right](300)$ :

  $\begin{array}{l}\left[r\circ d\right](x)=r\left[d(x)\right]\\ \Rightarrow \left[r\circ d\right](300)\\ =r\left[d(300)\right]\\ =r\left(0.85\times 300\right)\\ =r\left(255\right)\\ =255-35\\ =220\end{array}$

So, the price of the digital audio player if the discount is applied before the rebate is $220.

If the discount is applied after the rebate, then the final price of the digital audio player is represented by $\left[d\circ r\right](300)$ :

  $\begin{array}{l}\left[d\circ r\right](x)=d\left[r(x)\right]\\ \Rightarrow \left[d\circ r\right](300)\\ =d\left[r(300)\right]\\ =d\left(300-35\right)\\ =d\left(265\right)\\ =265\times 0.85\\ =225.25\end{array}$

So, the price of the digital audio player if the discount is applied after the rebate is $225.25.

Hence, the price of the digital audio player is less when the discount is applied before the rebate.