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Double Discounts
Task
Emily has a coupon for 20 percent off of her purchase at the store. She finds a backpack that she likes on the discount rack. Its original price is \$60 but everything on the rack comes with a 30 percent discount. Emily says
Thirty percent and twenty percent make fifty percent so it will cost $30.

Is Emily correct? Explain.

What price will Emily pay for the backpack?
IM Commentary
The goal of this problem is to calculate percent decreases in the context of several (sequential) discounts. The task addresses a common misconception about successive discounts: many people think that a twenty percent discount on top of a thirty percent discount will result in a net fifty percent discount. One way to help see that this is not correct is to change the numbers. A 50 percent discount cuts the price in half. Two 50 percent discounts do not end up in giving an item away for free but rather the price is cut in half a second time, resulting in a new price that is 25 percent of the original price. The teacher may wish to hold back on question (b) until students have adequately reasoned through and discussed part (a). If students read both parts of the question they may look right away for a flaw in the reasoning for (a) whereas the reasoning in (a) might appear solid on its own.
Sales tax should be ignored for this problem and the teacher may wish to address this if it comes up. An interesting followup question would be this: what percent discounts (for the rack price and the coupon) does Emily need in order to end up paying 50% of the original price? This could be done with the 30% general discount or with the 20% off of her purchase or with different numbers for either. This is also a multistep ratio problem and students may want to know the answer to this question after working through the problem.
Solution
 It is true that 20% and 30% make 50%. But in the context of sale prices it is essential to keep track of the wholes to which these percents apply. For the backpack, the 30% discount applies to the original \$60 price: 30% of \$60 is $0.3 \times 60 = 18$ making the discount on the backback \$18. So after using the coupon, the backpack price becomes \$42. Emily's additional 20% coupon applies not to the original backpack price but to the discounted price of \$42: 20% of \$42 is \$8.40. Emily would need to save an additional \$12 off the \$42 price in order to buy the backpack for \$30 so her calculations are not correct.
 As seen in part (a), Emily's coupon lowers the discount rack price by \$8.40 so she will pay $$42  8.40 = 33.60$$ or \$33.60.
Double Discounts
Emily has a coupon for 20 percent off of her purchase at the store. She finds a backpack that she likes on the discount rack. Its original price is \$60 but everything on the rack comes with a 30 percent discount. Emily says
Thirty percent and twenty percent make fifty percent so it will cost $30.

Is Emily correct? Explain.

What price will Emily pay for the backpack?
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