Fizzy Juice

Task

When Sam and his friends get together, Sam makes orange soda by mixing orange juice with soda (sparkling water). On Friday, Sam makes 7 liters of orange soda by mixing 3 liters of orange juice with 4 liters of soda. On Saturday, Sam makes 9 liters of orange soda by mixing 4 liters of orange juice with 5 liters of soda.

Make a prediction about how the two orange sodas will compare in taste.  Explain your reasoning. 

IM Commentary

The goal of this task is to provide an engaging context for students to work with ratios. If orange juice and soda are available, it would be interesting to have students test the mixtures described in the problem (ideally in a ''blind'' taste test) to see if they can taste a difference: note that even if the volumes are poured and measured accurately, the ratios chosen are close enough that it might not be possible to taste the difference.

This task is designed to elicit a variety of student responses.  If given at the beginning of the unit, students will likely reason directly with the numbers, as in the fifth provided solution.  If given in the middle of the unit, students will have the strategies of creating a graph, using a double number line, equivalent ratios, ratio tables, or comparisons with fractions or percents.  If given at the end of the unit, student should be solving this task with the most efficient strategy.

Note that the final solution given is similar in spirit to an incorrect argument: 'I added equal amounts (one liter each) of orange juice and soda to the first mixture so the second mixture will taste the same as the first.' What is true is that if we add, to the first mixture, another amount of orange juice and soda mixed in the same ratio then the new mixture will taste the same. For example, if we added $\frac{3}{2}$ liters of orange juice and 2 liters of soda to get $\frac{9}{2}$ liters of orange juice and 6 liters of soda, this fizzy juice will taste the same as 3 liters of juice mixed with 4 liters of soda.

This task is based on this lesson from Mathematics Assessment Project:

http://map.mathshell.org/materials/download.php?fileid=1502

This task was written as part of a collaborative project between Illustrative Mathematics, the Smarter Balanced Digital Library, the Teaching Channel, and Desmos.

Solutions

Solution: Finding a Unit Rate

The two mixtures will taste the same if the ratios of juice to soda are equivalent. One way to check this is by finding unit rates. For example, we can determine how much juice is added per liter of soda. For the first mixture, there are 3 liters of juice and 4 liters of soda. This means that there are $\frac{3}{4}$ liters of juice for each 1 liter of soda. For the second mixture, there are 4 liters of juice and 5 liters of soda. This means that there are $\frac{4}{5}$ liters of juice for each liter of soda. Comparing the two, there is more juice in the second mixture, per liter of soda, so the second mixture will taste more orangey. 

Another unit rate that can be used to solve this problem would be the amount of soda per liter of juice. For the first mixture, there is $\frac{4}{3}$ liters of soda per liter of juice while the second mixture has $\frac{5}{4}$ liters of soda per liter of juice. Since $\frac{4}{3}$ is greater than $\frac{5}{4}$, this means that the first mixture has more soda per liter of juice. So it will be more diluted and it is the second mixture that will taste more orangey.

A third unit rate that can be use to solve the problem is the amount of orange juice per liter of the mixture. For the first mixture, 3 parts out of 7 of the mixture are orange juice so $\frac{3}{7}$ of the first orange soda is orange juice. For the second mixture, 4 parts out of 9 of the orange soda are orange juice so $\frac{4}{9}$ of the orange soda is orange juice. Since $\frac{4}{9} \gt \frac{3}{7}$ the second mixture tastes more orangey.

Solution: Using Ratio Tables/Scaling

We can make ratio tables for the two orange sodas. The goal is to get a common amount in one of the three columns. The tables below show a few ways to do this. Here is a table for mixture 1:

Orange Juice (liters)	Soda (liters)	Orange Soda (liters)
3	4	7
12	16	28
15	20	35
27	36	63

Here is a table for mixture 2:

Orange Juice (liters)	Soda (liters)	Orange Soda (liters)
4	5	9
12	15	27
16	20	36
28	35	63

Comparing row 2 of the tables, we see that the first mixture has an extra liter of soda and so the second mixture will taste more orangey. Comparing row 3 of the tables we see that the second mixture has an extra liter of orange juice and so it will taste more orangey. Finally, comparing the last rows of both tables we see that when we make 63 liters of each orange soda, the second mixture has 1 liter more juice than the first so it will taste more orangey.

This method is closely tied to the idea of scaling. For example, in the first table, the successive rows show what happens when we scale the recipe for the first orange soda by 4, 5, and 9. In other words, we can imagine making 4, 5, or 9 batches of this orange soda. Similarly the second ratio table shows what happens when we apply a scale factor of 3, 4, and 7 to the second orange soda recipe.

Solution: Using a Double Number Line

Double number lines can be used in parallel with  ratio tables. In fact, the two present similar information in slightly different ways. The ratio table is a better choice when the numbers are complex while the double number lines offer a geometric interpretation of the quantities being compared. Here is a double number line showing orange juice and soda for the first mixture:

       

Here is a double number line showing juice and soda for the second mixture:

    

Looking at the two double number lines, we see that with 12 liters of orange juice, the second mixture has less soda and so it is more orangey. With 20 liters of soda, the second mixture has one extra liter of juice so again it is more orangey.

We can also make double number lines with orange juice and orange soda or with soda and orange soda to make the same conclusions.

Solution: Comparing Fractions using Percents

This method of solution is similar to the first though now the fractions are converted to percents. For the first orange soda mixture, there are 3 liters of orange juice and 7 liters of orange soda. Since $3 \div 7$ is between 0.42 and 0.43, this means that the first orange soda mixture has between 42 percent and 43 percent orange juice. For the second orange soda, there are 4 liters of orange juice and 9 liters of orange soda. Since $4 \div 9$ is between 0.44 and 0.45 this mixture has between 44 percent and 45 percent orange juice. The second orange soda mixture is more orangey by about 1.5 percent. It will be a very interesting experiment to see if it is possible to taste such a small difference!

Solution: Direct Reasoning with Quantities

The mixture with 4 liters of orange juice and 5 liters of soda has one extra liter of juice and one extra liter of soda compared to the mixture with 3 liters of orange juice and 4 liters of soda. This mixture with 3 liters of orange juice and 4 liters of soda has more soda than orange juice and so will taste less orangey than 1 liter of orange juice and 1 liter of soda. Adding this more orangey mixture will make the orange soda more orangey. So the mixture with 4 liters of orange juice and 5 liters of soda will taste more orangey.