Exercise \(\PageIndex<1>\): NUmber Talk: Division Find each quotient mentally. \(645\div 100\) \(645\div 50\) \(48.6\div 30\) \(48.6\div x\)
| cups of dry rice | number of people |
|---|---|
| 1 | |
| 2 | 6 |
| 3 | |
| 12 | |
| 43 | |
| \(x\) |
2. 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 |
|---|---|
| 1 | |
| 6 | 3 |
| 10 | |
| 16 | |
| 25 | |
| \(n\) |
3. How was completing this table different from the previous table? How as it the same?
Exercise \(\PageIndex\): Denver to Chicago
A plane flew at a constant speed between Denver and Chicago. It took the plane 1.5 hours to fly 915 miles.
| time (hours) | distance (miles) | speed (miles per hour) |
|---|---|---|
| 1 | ||
| 1.5 | 915 | |
| 2 | ||
| 2.5 | ||
| \(t\) |
Are you ready for more?
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.
Exercise \(\PageIndex\): Revisiting a Bread Dough
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.
| honey (tbsp) | flour (c) |
|---|---|
| 1 | |
| 8 | 10 |
| 16 | |
| 20 | |
| \(h\) |
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) |
|---|---|
| \(3\) | \(12\) |
| \(1\) | \(4\) |
| \(7\) | \(28\) |
| \(\frac\) | \(1\) |
| \(r\) | \(4r\) |
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 \(b=4r\). 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 \(\frac\) to find the amount of red paint needed, \(r\), to mix with it to make Venusian Sunset. So \(r=\fracb\). The amount of blue paint is proportional to the amount of red paint and the constant of proportionality \(\frac\).
| blue paint (parts) | red paint (parts) |
|---|---|
| \(12\) | \(3\) |
| \(4\) | \(1\) |
| \(28\) | \(7\) |
| \(1\) | \(\frac\) |
| \(b\) | \(\fracb\) |
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 \(y=kx\).
A certain ceiling is made up of tiles. Every square meter of ceiling requires 10.75 tiles. Fill in the table with the missing values.
| square meters of ceiling | number of tiles |
|---|---|
| \(1\) | |
| \(10\) | |
| \(100\) | |
| \(a\) |
On a flight from New York to London, an airplane travels at a constant speed. An equation relating the distance traveled in miles, \(d\), to the number of hours flying, \(t\), is \(t=\fracd\). How long will it take the airplane to travel 800 miles?
Each table represents a proportional relationship. For each, find the constant of proportionality, and write an equation that represents the relationship.
| \(s\) | \(P\) |
|---|---|
| \(2\) | \(8\) |
| \(3\) | \(12\) |
| \(5\) | \(20\) |
| \(10\) | \(40\) |
Constant of proportionality:
| \(d\) | \(C\) |
|---|---|
| \(2\) | \(6.28\) |
| \(3\) | \(9.42\) |
| \(5\) | \(15.7\) |
| \(10\) | \(31.4\) |
Constant of proportionality:
A map of Colorado says that the scale is 1 inch to 20 miles or 1 to 1,267,200. Are these two ways of reporting the scale the same? Explain your reasoning.
Here is a polygon on a grid.
This page titled 2.2.1: Proportional Relationships and Equations is shared under a CC BY license and was authored, remixed, and/or curated by Illustrative Mathematics.
