In this resource, we will look more closely at one method for solving problems that involve proportional relationships—multiplication by a constant.
Read the following problem and answer the questions below. Record your answers on a separate piece of paper or using your notes.
A man is planning to grow chili peppers in his garden. He knows that each plant will produce eight chili peppers. He wants to know how many peppers he will grow based on the number of plants.
Copy the following table into your notes and fill in the missing information.
Plants

Process

Peppers

1

1(8)

8

2

2(8)

16

3



4



5



8



10



Click here to see the completed table.
1. What do all the expressions in the 'Process' column have in common?
2. Write a rule for determining the number of peppers, if you know the number of plants.
3. Now write your rule as an equation. Since both "peppers" and "plants" start with the letter p, you will have to use different letters for your variables. In your equation, use y for the number of peppers and x for the number of plants.
In this problem we solved a proportional relationship by multiplying by a constant:
Instead of writing out the word "constant," mathematicians use the letter k.
The constant that you use does not have to be a whole number. It can be any number including fractions, decimals, percents, or irrational numbers like √2.
On track and field day at an elementary school, the teacher wants to give her class a treat, so she buys popsicles. The popsicles cost $0.45 each. Fill in the following table to determine the equation for the cost c, of buying p, popsicles.
Popsicles

Process

Cost

1

1(0.45)

$0.45

2


3


5


10


p

Click here to see the completed table.
1. What is the rule for filling in the table?
2. What is the constant in this problem?
3. Write an equation that could be used to solve this problem.
4. The teacher has 22 students, how much will she pay for the popsicles?
Both of these problems could be modeled using proportional relationships because when one variable changed, the other changes by the same factor. We solved them by finding a constant of proportionality between the two variables and set up the equation y = constant • x. The equation may also be written in the form, y = kx where k represents the constant of proportionality.