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# Sale!

## Task

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 |

- Which of these four different offers gives the biggest price reduction? Explain your reasoning clearly.
- 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 | $x$ | $x \div2 = \frac12 x$ | $x$ | |

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 | $x + x$ | $2x \div 3 = \frac23 x$ | $x$ |

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.

## Sale!

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 |

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

## Comments

Log in to comment## Peter Norman says:

over 5 yearsThe original question asks for the biggest and smallest percentage price reductions. The tables, however, only find total savings as a dollar figure. Wouldn't the student need to calculate the total price without the sale and compare it to the actual sale price to compute the actual percent decrease? For example, 2 for 1 will always result in paying 50% of normal full price while 2 for 3 is always 66.7% of full price. Likewise, 25% off of second results in paying 87.5% of original price for two pairs and 50% off of second results is paying 75% of original price for two pairs. Perhaps I misunderstood the question. Also, I think only taking into account sales of two pairs first would be better and introduce the 2 for 3 later and compare/ask questions surrounding that concept at that point. Also, I would add the assumption of same priced items in the original problem and then as a follow-up introduce the idea of lesser priced items.

## Kristin says:

about 5 yearsThanks for your comments. I agree that the solution shown doesn't make the connection to percentages explicit, as students would need to do. As for your suggestions for focusing on just two pairs first and adding the assumption of same-priced items--those are pedagogical choices that are worth exploring, but I can see an argument for either the one presented in the problem and the one you suggest, and the real test of which works best probably depends on the teacher and the student (in an instructional situation).

## Kristin says:

about 5 yearsI'm not sure why, but on further study the stem was changed. I reverted the stem and now the task matches the commentary and solution again.

## llessard says:

over 6 yearsThis problem takes students so much further than a comparison of two sale prices, which I have traditionally challenged students to make.

Since the problem includes percents as well as money, I would encourage students to "solve a simpler problem" by making all products cost $1 originally. Many 7th graders could possibly solve the problem then with mental math.

## Kristin says:

over 6 yearsThe strategy you suggest is an excellent one. It would be interesting and informative to hear how students do with this task--let us know if you do use it!