Problem 72 - Introductory Algebra for College Students

Question

The price of a color printer is reduced by \(30 \%\) of its original price. When it still does not sell, its price is reduced by \(20 \%\) of the reduced price. The salesperson informs you that there has been a total reduction of \(50 \% .\) Is the salesperson using percentages properly? If not, what is the actual percent reduction from the original price?

Step-by-Step Solution

Answer

No, the salesperson is not using percentages properly. The actual percent reduction from the original price is \(44\%\), not \(50\%\).

Step 1: Calculate the price after the first discount

To find the price after the first \(30\%\) reduction, subtract \(30\%\) from \(100\%\) to get \(70\%.\) Denote the original price as \(p,\) the price after the first reduction will be \(0.7 \cdot p.\)

Step 2: Calculate the price after the second discount

The second discount of \(20\%\) applies to the reduced price, i.e., \(0.7 \cdot p.\) Subtracting \(20\%\) from \(100\%,\) we get \(80\%.\) So, the final price after both reductions is \(0.7 \cdot p \cdot 0.8 = 0.56 \cdot p.\)

Step 3: Calculate the actual percent reduction

To find the actual percentage reduction, compute the difference between the original price and the price after both reductions, in terms of the original price. This is given by \((1 - 0.56) \cdot 100\% = 44\%,\) which is actual total reduction from the original price.