# Lesson 2: Introducing Proportional Relationships with Tables

Let’s solve problems involving proportional relationships using tables.

## 2.1: Notice and Wonder: Paper Towels by the Case

Here is a table that shows how many rolls of paper towels a store receives when they order different numbers of cases.

What do you notice about the table? What do you wonder?

## 2.2: Feeding a Crowd

1. 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.

1. How many people will 10 cups of rice serve?

2. How many cups of rice are needed to serve 45 people?
row 1 cups of rice number of people
row 2 2 6
row 3 3 9
row 4 10
row 5   45
2. A recipe says that 6 spring rolls will serve 3 people. Complete the table.
row 1 number of spring rolls number of people
row 2 6 3
row 3 30
row 4 40
row 5   28

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. Complete the table as you answer the questions. Be prepared to explain your reasoning.

1. How many cups of flour do they use with 20 tablespoons of honey?

2. How many cups of flour do they use with 13 tablespoons of honey?

3. How many tablespoons of honey do they use with 20 cups of flour?

row 1 honey (tbsp) flour (c)
row 2 8 10
row 3 20
row 4 13
row 5   20
1. What is the proportional relationship represented by this table?

## 2.4: Quarters and Dimes

4 quarters are equal in value to 10 dimes.

1. How many dimes equal the value of 6 quarters?
2. How many dimes equal the value of 14 quarters?
3. What value belongs next to the 1 in the table? What does it mean in this context?
row 1 number of
quarters
number of
dimes
row 2 1
row 3 4 10
row 4 6
row 5 14

## Summary

If the ratios between two corresponding quantities are always equivalent, the relationship between the quantities is called a proportional relationship.

This table shows different amounts of milk and chocolate syrup. The ingredients in each row, when mixed together, would make a different total amount of chocolate milk, but these mixtures would all taste the same.

Notice that each row in the table shows a ratio of tablespoons of chocolate syrup to cups of milk that is equivalent to $4:1$.

About the relationship between these quantities, we could say:

row 1 tablespoons of
chocolate syrup
cups of
milk
row 2 4 1
row 3 6 $1\frac{1}{2}$
row 4 8 2
row 5 $\frac{1}{2}$ $\frac{1}{8}$
row 6 12 3
row 7 1 $\frac{1}{4}$
• The relationship between amount of chocolate syrup and amount of milk is proportional.
• The relationship between the amount of chocolate syrup and the amount of milk is a proportional relationship.
• The table represents a proportional relationship between the amount of chocolate syrup and amount of milk.
• The amount of milk is proportional to the amount of chocolate syrup.

We could multiply any value in the chocolate syrup column by $\frac14$ to get the value in the milk column. We might call $\frac14$ a unit rate, because $\frac14$ cups of milk are needed for 1 tablespoon of chocolate syrup. We also say that $\frac14$ is the constant of proportionality for this relationship. It tells us how many cups of milk we would need to mix with 1 tablespoon of chocolate syrup.

## Glossary

proportional relationship

#### proportional relationship

If there is a positive constant $k$ so that the quantities $x$ and $y$ are related by the equation $y = kx$, then we say that $y$ and $x$ are in a proportional relationship, and that $y$ is proportional to $x$. The constant $k$ is called the constant of proportionality.

constant of proportionality

#### constant of proportionality

See proportional relationship.