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Sore Throats, Variation 1
Alignments to Content Standards:
7.RP.A.2
Task
Nia and Trey both had a sore throat so their mom told them to gargle with warm salt water.
$\hskip30pt$ Nia mixed 1 teaspoon salt with 3 cups water.
$\hskip30pt$ Trey mixed $\frac12$ teaspoon salt with $1 \frac12$ cups of water.
Nia tasted Trey’s salt water. She said,
“I added more salt so I expected that mine would be more salty, but they taste the same.”

Explain why the salt water mixtures taste the same.

Which of the following equations relates $s$, the number of teaspoons of salt, with $w$, the number of cups of water, for both of these mixtures? Choose all that apply.
 $s = \frac13 w$
 $s = 3w$
 $s = 1\frac12 w$
 $w = 3s$
 $w = \frac13 s$
 $w = \frac12 s$
IM Commentary
There is a nonmathematical fact that students must know about mixtures in order to answer this question. When salt is dissolved in water, the salt disperses evenly through the mixture, so any sample from the mixture that has the same volume will have the same amount of salt. This is not something that kids could know a priori or by reasoning about it. For example, the same is not true when you mix sand and water. In general, it is important to know what facts about the world warrant applying a particular mathematical structure in a given context. In this particular case, teachers may need to provide some background knowledge or help students explain why a ratio is an appropriate mathematical tool in this context.
There is an eighth grade version of this task; see 8.EE Sore Throats, Variation 2.
Solutions
Solution:
Finding equivalent ratios and equations

The ratio of the number of teaspoons of salt to the number of cups of water is 1:3 in Nia's solution. If we divide the amount of salt and the amount of water by 3, the ratio will be the same.
$$1\div 3 = \frac13$$
$$3 \div 3 = 1$$
Thus 1:3 is equivalent to the ratio $\frac13 : 1$, which means that Nia's solution has $\frac13$ teaspoon of salt for every cup of water.
The ratio of the number of teaspoons of salt to the number of cups of water is $\frac12 : 1 \frac12$ in Trey’s solution. If we divide the amount of salt and the amount of water by $1\frac12$, the ratio will be the same.
$$\frac12 \div 1 \frac12 = \frac13$$
$$1\frac12 \div 1\frac12 = 1$$
So Trey’s ratio is also equivalent to the ratio $\frac13 : 1$.
Since each mixture has the same amount of salt for every $1$ cup of water, they are equally salty.

The two correct equations are (i) $s = \frac13 w$ and (iv) $w = 3 s$.
Solution:
Another way to find equivalent ratios
 Another, simpler way to solve the first part is to note that if you divide both quantities in Nia's ratio by 2, you get Trey's ratio.
 The two correct equations are (i) $s = \frac13 w$ and (iv) $w = 3 s$.
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