5.1: Missing Figures
Here are the second and fourth figures in a pattern.
- What do you think the first and third figures in the pattern look like?
- Describe the 10th figure in the pattern.
Let’s investigate the equations that represent proportional relationships.
Here are the second and fourth figures in a pattern.
There are 100 centimeters (cm) in every meter (m).
row 1 | length (m) | length (cm) |
---|---|---|
row 2 | 1 | 100 |
row 3 | 0.94 | |
row 4 | 1.67 | |
row 5 | 57.24 | |
row 6 | $x$ |
row 1 | length (cm) | length (m) |
---|---|---|
row 2 | 100 | 1 |
row 3 | 250 | |
row 4 | 78.2 | |
row 5 | 123.9 | |
row 6 | $y$ |
It took Priya 5 minutes to fill a cooler with 8 gallons of water from a faucet that was flowing at a steady rate. Let $w$ be the number of gallons of water in the cooler after $t$ minutes.
Which of the following equations represent the relationship between $w$ and $t$? Select all that apply.
What does 1.6 tell you about the situation?
What does 0.625 tell you about the situation?
Priya changed the rate at which water flowed through the faucet. Write an equation that represents the relationship of $w$ and $t$ when it takes 3 minutes to fill the cooler with 1 gallon of water.
At an aquarium, a shrimp is fed $\frac{1}{5}$ gram of food each feeding and is fed 3 times each day.
How much food does a shrimp get fed in one day?
Complete the table to show how many grams of food the shrimp is fed over different numbers of days.
number of days | food in grams | |
---|---|---|
Row 1 | 1 | |
Row 2 | 7 | |
Row 3 | 30 |
Shrimp in aquarium Copyright Owner: uzilday License: Public Domain Via: Pixabay
If Kiran rode his bike at a constant 10 miles per hour, his distance in miles, $d$, is proportional to the number of hours, $t$, that he rode. We can write the equation $$d = 10 t$$ With this equation, it is easy to find the distance Kiran rode when we know how long it took because we can just multiply the time by 10.
We can rewrite the equation:
\(\begin{align} d &= 10 t\\ \left( \frac{1}{10} \right) d &= t \\ t &= \left( \frac{1}{10} \right) d \end{align}\)
This version of the equation tells us that the amount of time he rode is proportional to the distance he traveled, and the constant of proportionality is $\frac{1}{10}$. That form is easier to use when we know his distance and want to find how long it took because we can just multiply the distance by $\frac{1}{10}$.
When two quantities $x$ and $y$ are in a proportional relationship, we can write the equation $$y = k x$$ and say, “$y$ is proportional to $x$.” In this case, the number $k$ is the corresponding constant of proportionality. We can also write the equation $$x = \frac{1}{k} y$$ and say, “$x$ is proportional to $y$.” In this case, the number $\frac{1}{k}$ is the corresponding constant of proportionality. Each one can be useful depending on the information we have and the quantity we are trying to figure out.