Cider versus Juice - Variation 1

Task

The price of a gallon of apple cider is \$7.00. The price of eight 4.23-ounce juice boxes is \$2.39.

1. Suppose the juice was instead packaged like the cider. Approximately what is the cost per gallon of the juice?
    
2. Suppose the cider was instead packaged like the juice. Approximately what is the cost per eight 4.23-ounce boxes of cider?
    
3. Peter wants to have at least a gallon of either only cider or only juice. Which product is the better deal?
    
4. State the unit rate(s) you used to compare the cost of cider versus juice in your answer to Question c.
    

5. List two or more additional unit rates that could be used to make this comparison. 

IM Commentary

This task asks students to compute multiple unit rates, aligning with standard 7.RP.A.1. The problem also has a real-world context, which requires students to compare two rates in different units in order to reach a conclusion on buying two different products.

There are a couple issues that may need to be addressed as students work through this problem. The first is that not all needed information is given for this task, namely, the conversion rate between gallons and ounces. Instructors can either let students look this up themselves or have it available for them, but ideally students should come to the conclusion themselves that they need to know this conversion rate in order to continue. The second issue is that by "ounces" we mean "fluid ounces". If students struggle with this distinction, it could be a good opportunity to discuss mass versus volume measurements, and how to tell by context which one is appropriate to use. There are 128 fluid ounces in 1 gallon.

Question e could be extended by asking students to compute some or all of the additional unit rates they listed, and to explain how these further support their answer to Question c. 

This task could be implemented on its own as a handout for students, or, ideally, via the blog post which inspired it, which can be found at http://christopherdanielson.wordpress.com/2011/11/29/cider-price-rules/. The post could be displayed/distributed first to the whole class in order to generate a conversation prior to administering the task. This allows for students to connect with the real-world scenario they’re being put in, as well as to formulate their own guesses as to which product would be a better monetary buy and why.

Solution

1. We wish to find the price per gallon of the juice. There are a total of $(8)(4.23)=33.84$ ounces in the juice boxes and the price for the boxes is \$2.39. We can set up the proportion 
   
   $$\frac{2.39\text{ dollars}}{33.84 \text{ ounces}} = \frac{x \text{ dollars}}{1 \text{ gallon}}$$
   
   and solve for $x$. We see that we will need to convert ounces to gallons, and as there are 128 ounces in a gallon, we have
   
   $$\frac{2.39\text{ dollars}}{33.84 \text{ ounces}} \cdot \frac{128\text{ ounces}}{1\text{ gallon}} \approx \frac{9.04 \text{ dollars}}{1\text{ gallon}},$$
   
   so $x\approx 9.04$, and the price of the juice is about \$9.04 per gallon. 
    
2. We wish to find the price per a package of boxes of the cider. The price for the gallon of cider is \$7.00, and there are 128 ounces in a gallon, giving the proportion
   
   $$\frac{7\text{ dollars}}{128\text{ ounces}}=\frac{x\text{ dollars}}{1 \text{package of boxes}}.$$
   
   To solve for $x$, we see that we need to convert ounces into "package of boxes", which refers to the eight 4.23-ounces boxes that the juice is sold in. So, 
   
   $$\frac{7\text{ dollars}}{128\text{ ounces}} \cdot \frac{33.84 \text{ ounces}}{1 \text{ package of boxes}} \approx \frac{1.85\text{ dollars}}{1 \text{ package of boxes}},$$
   
   and $x\approx 1.85$ and the price of the cider is about \$1.85 per eight 4.23-ounce boxes. 
    
3. Utilizing our previous answers to part a and b, we see that the cider is the more cost-effective choice.  (In fact, Peter would have to buy four 8-packs of juice to get at least a gallon, for a cost of \$9.56, much more than the \$7 it would take to get a gallon of cider.)
    
4. The unit rates we used in parts a and b are dollars per gallon and dollars per pack of eight 4.23-ounce boxes. 
    
5. Additional unit rates could be dollars per ounce, cents per ounce, ounces per dollar, dollars per 4.23-ounce box, etc.