# Sale!

Alignments to Content Standards: 7.RP.A

Four different stores are having a sale. The signs below show the discounts available at each of the four stores.

 Two for the price of one Buy one and get 25% off the second Buy two and get 50% off the second one Three for the price of two

1. Which of these four different offers gives the biggest price reduction? Explain your reasoning clearly.
2. Which of these four different offers gives the smallest price reduction? Explain your reasoning clearly.

## IM Commentary

The Standards for Mathematical Practice focus on the nature of the learning experiences by attending to the thinking processes and habits of mind that students need to develop in order to attain a deep and flexible understanding of mathematics. Certain tasks lend themselves to the demonstration of specific practices by students. The practices that are observable during exploration of a task depend on how instruction unfolds in the classroom. While it is possible that tasks may be connected to several practices, the commentary will spotlight one practice connection in depth. Possible secondary practice connections may be discussed but not in the same degree of detail.

The purpose of this task is to engage students in Standard for Mathematical Practice 4, "Model with mathematics," and as such, the question as it is worded cannot be answered without making some assumptions. For example, if the items that are purchased do not have the same value, then the price reduction depends on the cost of the items. The answer also depends on how you interpret the meaning of “price reduction” which could be either the absolute reduction or the relative reduction. Consider the four scenarios for purchasing pairs of shoes below.

“Two for the price of one”

Pair 1 Pair 2 Money saved Fraction of purchase saved
\$36 \$12 \$12$\frac{1}{4}$\$36 \$36 \$36 $\frac{1}{2}$

“Three for the price of two”

Pair 1 Pair 2 Pair 2 Money saved Fraction of purchase saved
\$60 \$48 \$18 \$18 $\frac{1}{7}$
\$12 \$12 \$12 \$12 $\frac{1}{3}$

Which has the greatest price reduction? It depends, and a complete answer to this question requires a mathematical argument beyond the expectations of 7th grade. On the other hand, students need opportunities to evaluate the relative savings of advertised sales, so realizing that the best sale depends on what you are buying is a good insight to develop. The solutions below assume that you are comparing the sales for purchasing items of the same price.

It is also worth pointing out that there is a very important, although non-mathematical, issue related to whether a particular sale will save you money: you do not save money by buying things you do not need. So, for example, 3 for the price of 2 is not a better deal than buy one get the second at 25% off if you do not need three of the item.l.

The teacher might use this task after formally teaching 7.RP.1-3.  Students could be given the task and asked to collaborate in small groups to solve the questions posed using all the formal instruction on ratio and proportional reasoning.  The teacher might ask questions such as; “What if the price of each item is different?  Does that change which discount is biggest?”  “What if the price of each item is the same? Does that make a difference in which discount is biggest?”  Depending on the level of the students, the teacher could direct the students by giving them a specific value or use a more abstract approach by having them solve using a variable.  The students could share out their findings and compare/contrast the answers and discuss why the results vary.

## Solutions

Solution: Starting with a specific value

Assume that you are comparing the sales for purchasing items of the same price (it is a much harder question to answer if you don’t). Let’s first look at the answer for a specific value. Suppose the regular price for all items is $60. Then the following table shows how much you will pay per item. Cost for 3 items Cost for 2 items Cost per item Total savings 2 for 1 60 60 ÷ 2 = 30 \$60
25% off the 2nd 60 + 45 = 105 105 ÷ 2 = 52.50 \$15 50% off the 2nd 60 + 30 = 90 90 ÷ 2 = 45 \$30
3 for 2 60 + 60 = 120 120 ÷ 3 = 40 \$60 In general, suppose that a single item costs$x$dollars. Cost for 3 items Cost for 2 items Cost per item Total savings 2 for 1$xx \div2 = \frac12 xx$25% off the 2nd$x+ \frac34 x = \frac74 x\frac74 x \div 2 = \frac78 x\frac14 x$50% off the 2nd$ x + \frac12 x = \frac32 x  \frac32 x \div 2 = \frac34 x  \frac12 x $3 for 2$x + x2x \div 3 = \frac23 xx$So “Two for the price of one” gives the biggest price reduction per item but the total savings is the same as the “Three for the price of two” sale. Also, “Buy one and get 25% off the second” has both the highest price per item and the lowest total savings, which means it offers the smallest price reduction. Solution: A more abstract approach Assume we are comparing the sales for buying identically priced items and that we are comparing the reduction in price per item (as opposed to price for the entire purchase). The sale price (per item) is the total cost divided by the number of items. The reduction in price is equal to the original price minus the sale price. We need to calculate the price reduction for every case in order to answer the two questions. Since we don't know what the regular price per item is let's just call it p. Two for the price of one: The reduction is:$ p-\frac{1p}{2} = \frac{1}{2}p$Buy one and get 25% off the second: The reduction is:$p -\frac{(1+0.75)p}{2} = p-\frac{1.75p}{2} = p-\frac{7/4}{2}p = p -\frac{7}{4}\cdot \frac{1}{2}p = p-\frac{7}{8}p = \frac{1}{8}p$Buy two and get 50% off the second one: The reduction is:$ p-\frac{(1+0.50)p}{2} = p-\frac{1.5p}{2} = p -\frac{3/2}{2}p = p -\frac{3}{2}\cdot \frac{1}{2}p = p-\frac{3}{4}p =\frac{1}{4}p$Three for the price of two: The reduction is:$ p-\frac{2p}{3} = \frac{1}{3}p\$

Now we can answer the questions

Which of these four different offers gives the biggest price reduction?

Two for the price of one with a price reduction of one half.

Which of these four different offers gives the smallest price reduction?

Buy one and get 25% off the second with a price reduction of one eighth.