# 2012 AMC 8 Problems/Problem 8

## Problem

A shop advertises everything is "half price in today's sale." In addition, a coupon gives a 20% discount on sale prices. Using the coupon, the price today represents what percentage off the original price? $\textbf{(A)}\hspace{.05in}10\qquad\textbf{(B)}\hspace{.05in}33\qquad\textbf{(C)}\hspace{.05in}40\qquad\textbf{(D)}\hspace{.05in}60\qquad\textbf{(E)}\hspace{.05in}70$

## Solution 1: With Algebra

Let the original price of an item be $x$.

First, everything is half-off, so the price is now $\frac{x}{2} = 0.5x$.

Next, the extra coupon applies 20% off on the sale price, so the price after this discount will be $100\% - 20\% = 80\%$ of what it was before. (Notice how this is not applied to the original price; if it were, the solution would be applying 50% + 20 % = 70% off the original price.) $80\% \cdot 0.5 x = \frac{4}{5} \cdot 0.5 x = 0.4x$

The price of the item after all discounts have been applied is $0.4x = 40\% \cdot x$. However, we need to find the percentage off the original price, not the current percentage of the original price. We then subtract $40\% x$ from $100\% x$ (the original price of the item), to find the answer, $\boxed{\textbf{(D)}\ 60}$.

## Solution 2: Fakesolving

Since the problem implies that the percentage off the original price will be the same for every item in the store, fakesolving is applicable here. Say we are buying an item worth 10 dollars, a convenient number to work with. First, it is clear that we'll get 50% off, which makes the price then 5 dollars. Taking 20% off of 5 dollars gives us 4 dollars. Therefore, we have saved a total of $\frac{10-4}{10} = \frac{6}{10} = \frac{60}{100} = \boxed{\textbf{(D)}\ 60} \%$.

## Solution 3:

The percent off would be 50% of 20% because it has an additional 20% off of half the price. The answer to that would be 10%. You could do 10% + 50% to find the answer. This answer is (D)60

~ SmartGrowth

## Solution 4:

The price is $50\%$ off, which means the price would be $100\% - 50\% = 50\%$ of what it was before. Then, you get $20\%$ off. Sidenote: A common mistake is finding $20\%$ of the price, but forgetting to subtract it. $20\%$ off the price is $100\% - 20\% = 80\%.$ Now, we can multiply the two discounts as decimals. $0.5 \times 0.8 = 0.4 = 40\%.$ Our final answer is $100\% - 40\% = \boxed{\textbf{(D)}\ 60\%}$

~ ENDERGAMER25/Endy

## Video Solution

https://youtu.be/SYou4OZcq1E ~savannahsolver

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 