- Tape diagram is a method of solving problems related to ratios.
- In this visual model (tape diagram), we use rectangles to represent the part of a ratio.

**For example:**

Jason and David together have \(35\) marbles in the ratio of \(3:2\).

Now, we want to know how many marbles does each one have?

- To solve this problem, we will use tape diagram to represent the ratio, as shown in the figure.

- It is clear from the figure that Jason has \(3\) units and David has \(2\) units.
- Thus, the total number of units they both have is \(5\). Also, the total number of marbles is \(35\).
- Hence, \(1\) unit represents \(\dfrac{35}{5}=7\) marbles
- Jason has \(3\) units and \(1\) unit represents \(7\) marbles.

Thus, the total number of marbles Jason has \(=3\times7=21\) marbles

- David has \(2\) units and \(1\) unit represents \(7\) marbles.

Thus, the total number of marbles David has \(=2\times7=14\) marbles

- Ratio as a relationship can be represented using the double number line diagram.
- The double number line diagram is used to represent the comparison between two different units.
- We draw two number lines which are placed one below the other.
- While plotting a ratio, mark the first entry on the first number line and second entry on the second number line such that both the entries are aligned with each other.

**For example: **

Jacob earns \($27\) in \(3\) hours. At this rate, he earned \($72\) in \(8\) hours. Represent the ratio on a double number line diagram.

Here, the relationship between two different units (dollars and hours) is given. This is to be represented on a double number line diagram.

Now, we draw a double number line diagram in which one number line will represent the dollars and other the hours.

Jacob earns \($27\) in \(3\) hours.

So, we will plot \(3\) hours on the number line representing hours and \($27\) on the number line representing the dollars such that both the entries are aligned with each other, as shown in the figure.

He earned \($72\) in \(8\) hours.

Now, we will plot \(8\) hours on the number line representing the hours and \($72\) on the number line representing the dollars.

- The ratios which have the same simplified form are called equivalent ratios.

**For example:** \(2:3\) and \(4:6\) are equivalent ratios, as \(4:6\) can be simplified as \(2:3\).

- We can also understand the equivalent ratios through tape diagrams.

**Consider the following example:**

Kara has \(5\) candies and Cody has \(3\) candies.

\(Ratio=5:3\)

We can describe this ratio through a tape diagram.

When each unit represents one candy,

the ratio \(=5:3\)

When each unit represents two candies,

the ratio \(=10:6\).

The simplest form of \(10:6\) is \(5:3\).

When each unit represents three candies,

the ratio \(=15:9\).

The simplest form of \(15:9\) is \(5:3\).

Thus, \(5:3\), \(10:6\) and \(15:9\) are all equivalent ratios because their simplest forms represent the same value.

In an equivalent ratio, only the value of each unit gets changed.

- Tape diagram is a method of solving problems related to ratio by visual model.
- In tape diagram, we use rectangles to represent the part of ratio.

**Example:**

The ratio of students in Mr. Kemp and Ms. Wendy's class is \(5:6\). There are \(40\) students in Mr. Kemp's class.

- Suppose we want to calculate the total number of students in Mr. Kemp and Ms. Wendy's class altogether.
- We will use the tape diagram to solve this problem.
- First, we represent the given ratio through tape diagram, as shown in the figure.

- It is clear from the figure that Mr. Kemp's class has \(5\) units.

Also, the number of students in Mr. Kemp's class \(=40\)

Thus, \(5\) units represent \(40\) students.

\(\therefore\) \(1\) unit represents \(\dfrac{40}{5}=8\) students

Now, since Ms. Wendy's class has \(6\) units and \(1\) unit represents \(8\) students,

\(\therefore\) \(6\) units represent \(6\times8=48\) students

- It means there are \(48\) students in Ms. Wendy's class.
- Thus, the total students in Mr. Kemp and Ms. Wendy's class altogether

\(=\text{Students in Mr. Kemp's class + Students in Ms. Wendy's class}\)

\(=40+48\)

\(=88\) students

A \($275\)

B \($175\)

C \($75\)

D \($200\)

- Ratio as a relationship can be represented using the double number line diagram.
- The double number line diagram is used to represent the comparison between two different units.
- We draw two number lines which are placed one below the other.
- While plotting a ratio, mark the first entry on the first number line and second entry on the second number line such that both the entries are aligned with each other.

**For example: **

Kara is going to Kentucky by car. Her car can travel \(12\) miles on \(4\) liters of gas. How many liters of gas is needed to travel \(21\) miles?

- Here, we draw a double number line diagram in which one number line will represent the miles and another will represent the liters.

The car can travel \(12\) miles on \(4\) liters of gas.

- So, we will plot \(12\) on the number line representing the miles and \(4\) just below the \(12\) on the number line representing the liters, as shown in the figure.

The diagram represents that to travel \(12\) miles, the car needs \(4\) liters of gas.

- We will find the simplest form of \(12:4\).

The GCF of \(12\) and \(4\) is \(4\).

Thus, \(12\div4=3\) miles

and \(4\div 4=1\) liter

This means that the car can travel \(3\) miles on \(1\) liter of gas.

- Now, we will plot \(3\) and \(1\) on the corresponding number lines.

- Now, we need to determine the liters of gas needed to travel \(21\) miles.
- We know when \(3\) is multiplied by \(7\), gives \(21\).

- Since \(3\) is multiplied by \(7\), so \(1\) must also be multiplied by \(7\), which gives \(7\) liters of gas.

A \(120\)

B \(160\)

C \(40\)

D \(20\)

- The table of equivalent ratios consists of a list of equivalent ratios.
- It is used to solve problems related to ratios.

**For example:** The ratio of boys to girls in Mrs. Scott's class is \(3:4\). If there are \(15\) boys in her class, how many girls are there?

- To solve this problem, we will use the table of equivalent ratios.

In the table, we write down the multiples of the ratio.

Here, the ratio is \(3:4\).

When we multiply the ratio by \(1\),

\(3\times1=3\;,\;4\times1=4\)

the ratio becomes \(3:4\).

When we multiply by \(2\),

\(3\times2=6\; , \;4\times2=8\)

the ratio becomes \(6:8\).

\(=3:4\)

When we multiply by \(3\),

\(3\times3=9\; ,\; 4\times3=12\)

the ratio becomes \(9:12\).

\(=3:4\)

When we multiply by \(4\),

\( 3\times4=12\; ,\;4\times4=16\)

the ratio becomes \(12:16\).

\(=3:4\)

When we multiply by \(5\),

\( 3\times5=15\; ,\; 4\times5=20\)

the ratio becomes \(15:20\).

\(=3:4\)

That means on \(15\) boys, there are \(20\) girls in Mrs. Scott's class.

Boys |
Girls |

3 | 4 |

6 | 8 |

9 | 12 |

12 | 16 |

15 | 20 |