Let’s write equations describing proportional relationships.
Find each quotient mentally.
$645\div100$
$645\div50$
$48.6\div30$
$48.6\div x$
A recipe says that 2 cups of dry rice will serve 6 people. Complete the table as you answer the questions. Be prepared to explain your reasoning.
How many people will 1 cup of rice serve?
How many people will 3 cups of rice serve? 12 cups? 43 cups?
How many people will $x$ cups of rice serve?
| cups of dry rice | number of people | |
|---|---|---|
| row 1 | 1 | |
| row 2 | 2 | 6 |
| row 3 | 3 | |
| row 4 | 12 | |
| row 5 | 43 | |
| row 6 | $x$ |
A recipe says that 6 spring rolls will serve 3 people. Complete the table as you answer the questions. Be prepared to explain your reasoning.
| number of spring rolls | number of people | |
|---|---|---|
| row 1 | 1 | |
| row 2 | 6 | 3 |
| row 3 | 10 | |
| row 4 | 16 | |
| row 5 | 25 | |
| row 6 | $n$ |
A plane flew at a constant speed between Denver and Chicago. It took the plane 1.5 hours to fly 915 miles.
Map of the Midwest from Denver to Chicago
Copyright Owner:
American Fact Finder
License:
Public Domain
Via:
American Fact Finder
| row 1 | time (hours) | distance (miles) | speed (miles per hour) |
|---|---|---|---|
| row 2 | 1 | ||
| row 3 | 1.5 | 915 | |
| row 4 | 2 | ||
| row 5 | 2.5 | ||
| row 6 | $t$ |
A rocky planet orbits Proxima Centauri, a star that is about 1.3 parsecs from Earth. This planet is the closest planet outside of our solar system.
A bakery uses 8 tablespoons of honey for every 10 cups of flour to make bread dough. Some days they bake bigger batches and some days they bake smaller batches, but they always use the same ratio of honey to flour.
How much flour is needed for 15 tablespoons of honey? 17 tablespoons? Explain or show your reasoning.
| honey (tbsp) | flour (c) | |
|---|---|---|
| Row 1 | 1 | |
| Row 2 | 8 | 10 |
| Row 3 | 16 | |
| Row 4 | 30 | |
| Row 5 | $h$ |
The table shows the amount of red paint and blue paint needed to make a certain shade of purple paint, called Venusian Sunset.
Note that “parts” can be any unit for volume. If we mix 3 cups of red with 12 cups of blue, you will get the same shade as if we mix 3 teaspoons of red with 12 teaspoons of blue.
| red paint (parts) |
blue paint (parts) |
|
|---|---|---|
| row 1 | 3 | 12 |
| row 2 | 1 | 4 |
| row 3 | 7 | 28 |
| row 4 | $\frac14$ | 1 |
| row 5 | $r$ | $4 r$ |
The last row in the table says that if we know the amount of red paint needed, $r$, we can always multiply it by 4 to find the amount of blue paint needed, $b$, to mix with it to make Venusian Sunset. We can say this more succinctly with the equation $b=4 r$. So the amount of blue paint is proportional to the amount of red paint and the constant of proportionality is 4.
We can also look at this relationship the other way around.
If we know the amount of blue paint needed, $b$, we can always multiply it by $\frac14$ to find the amount of red paint needed, $r$, to mix with it to make Venusian Sunset. So $r=\frac14 b$. The amount of blue paint is proportional to the amount of red paint and the constant of proportionality $\frac14$.
| blue paint (parts) |
red paint (parts) |
|
|---|---|---|
| row 1 | 12 | 3 |
| row 2 | 4 | 1 |
| row 3 | 28 | 7 |
| row 4 | 1 | $\frac14$ |
| row 5 | $b$ | $\frac14 b$ |
In general, when $y$ is proportional to $x$, we can always multiply $x$ by the same number $k$—the constant of proportionality—to get $y$. We can write this much more succinctly with the equation $y=k x$.
