AndrewN wrote:
John purchases a gaming system and no other items at a retailer during a 10 percent storewide sale. He hands the cashier D dollars, including a 5 percent sales tax on the purchase. Which of the following expressions gives the original price of the gaming system in terms of D?
(A) \(0.95D\)
(B) \((0.9)(1.05)D\)
(C) \(\frac{D}{0.95}\)
(D) \(\frac{D}{(0.9)(1.05)}\)
(E) \(\frac{D}{(0.95)(1.05)}\)
Source: Self-made question.
Hello, everyone. I had fun exploring the notion of a question that dealt with discounts in an unconventional manner. That is, the majority of the time you encounter questions on the concept, you are looking to find the
final price, not the initial one in terms of the discounted price. Anyway, the algebraic approach outlined above by
Fdambro294 is flawless. I will put it into my own step-by-step words below.
Let P = Price (the original price)
10 percent off means 90 percent was paid, and the 5 percent tax would be on top of 100 percent of the discounted price, D. Thus,
\(D=(0.9)(1.05)P\)
To get to P, we just need to divide the numbers out of the right-hand side:
\(\frac{D}{[(0.9)(1.05)]} = P\)
The answer must be (D), since that is exactly where the algebra has led us.
An alternative approach: You could set an original price and work backwards. I would recommend a simple number such as 100 to make the percents easier to work with (even though a $100 gaming system may not be the best representation of reality).
If P = 100, then
\(D=(0.9)(1.05)*100\)
\(D = 0.945*100\)
\(D = 94.5\)
Run through the answers to test whether you would derive 100. You might find it more efficient
not to figure out the exact numbers in each case.
(A) \(0.95(94.5)<100\)
Analysis: This answer, in which a number less than 100 is multiplied by a number less than 1, is obviously not going to increase the value of 94.5. Get rid of it.
(B) \((0.9)(1.05)(94.5)<100\)
Analysis: We already figured out that 0.9 * 1.05 is 0.945, so again, the product of the three numbers on the left-hand side cannot yield 100. Another pretty simple elimination.
(C) \(\frac{(94.5)}{0.95}≠100\)
Analysis: This one would be close, but we can see that the top and bottom do not match exactly, so we will not derive 100. (A little logic can go a long way here.)
(D) \(\frac{(94.5)}{(0.9)(1.05)}=\frac{94.5}{0.945}=100\)
Analysis: The top and bottom match perfectly, and the value in the numerator has to increase, since we are dividing by a positive number less than 1.
(E) \(\frac{(94.5)}{(0.95)(1.05)}≠100\)
Analysis: There would be no need to work through this one, but if you were working with the answers from the bottom up, as I sometimes do, you might start here.
\((0.95)(1.05)=0.9975\)
Therefore,
\(\frac{(94.5)}{0.9975}<100\)
The numerator will increase marginally, but the numbers do not match, so we know we will not derive 100 anyway.
In the end,
the answer must be (D), either way you look at it. I hope you had fun with this one. As always, good luck with your studies.
- Andrew