Chapter 7.6, Problem 23P

* Frank’s Jewelers runs the following advertisement: “Come to our 40%-off sale on Saturday. We’re not like the competition, who raise prices by 30% and then have a 70%-off sale!”

a. If two items start off with the same price, which gives you the lower price in the end: taking off 40% or raising the price by 30% and then taking off 70% (of the raised price)?

b. Consider the same problem more generally, with other numbers. For example, if you raise prices by 20% and then take off 50% (of the raised price), how does this compare with taking 30% off of the original price? If you raise a price by 30% and then lower the raised price by 30%, how does that compare with the original price? Try at least two other pairs of percentages by which to raise and then lower a price. Describe what you observe. Predict what happens in general: If you raise a price by A% and then take 8% off of the raised price, does that have the same result as if you had lowered the original price by (B — A) %? If not, which produces the lower final price?

c. Use the distributive property or FOIL to explain the pattern you discovered in part (b). Remember that to raise a price by 15%, for example, you multiply the price by $1+0.15,$ whereas to lower a price by 15%, you multiply the price by $1—\mathrm{0.15.}$