Informative line

# Solving Problems of Ratios using Tape Diagrams

• 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

#### The ratio of the lengths of Tim's rope to Jana's rope is $$3:1$$. If together they both have $$80$$ feet long rope, how long is Tim's rope? Solve using tape diagram.

A $$60$$

B $$40$$

C $$20$$

D $$80$$

×

We will represent the ratio through rectangles, as shown in the figure.

It is clear from the figure that Tim has $$3$$ units and Jana has $$1$$ unit.

Thus, the total number of units they both have is $$4$$ and the total length of the rope is $$80$$ feet.

Hence, $$1$$ unit represents $$\dfrac{80}{4}=20$$ feet

Tim has $$3$$ units and $$1$$ unit represents  $$20$$ feet.

Thus, the total length of Tim's rope

$$=3\times20$$

$$=60$$ feet

Hence, option (A) is correct.

### The ratio of the lengths of Tim's rope to Jana's rope is $$3:1$$. If together they both have $$80$$ feet long rope, how long is Tim's rope? Solve using tape diagram.

A

$$60$$

.

B

$$40$$

C

$$20$$

D

$$80$$

Option A is Correct

# Plotting Ratios on Double Number Line Diagrams

• 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.

#### To make yummy fruit punch, use $$2$$ cups of grape juice for every $$3$$ cups of apple juice and $$8$$ cups of grape juice for every $$12$$ cups of apple juice. Which one of the following options shows the double number line diagram for the given situation?

A

B

C

D

×

Here, the relationship between the quantities of apple juice and grape juice is given.

This is to be represented on the double number line diagram.

Now, we draw a double number line diagram in which one number line will represent the quantity (cups) of apple juice and other will represent the corresponding quantity (cups) of grape juice.

$$2$$ cups of grape juice is for every $$3$$ cups of apple juice.

So, we will plot $$3$$  on the number line representing the apple juice and $$2$$ on the number line representing the grape juice, as shown in the figure.

$$8$$ cups of grape juice is for every $$12$$ cups of apple juice.

So, we will plot $$12$$  on the number line representing the apple juice and  $$8$$ on the number line representing the grape juice.

Hence, option (B) is correct.

### To make yummy fruit punch, use $$2$$ cups of grape juice for every $$3$$ cups of apple juice and $$8$$ cups of grape juice for every $$12$$ cups of apple juice. Which one of the following options shows the double number line diagram for the given situation?

A
B
C
D

Option B is Correct

# Equivalent Ratios through Tape Diagram

• 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.

#### Jessica gets $$2$$ medals and Julie gets $$3$$ medals. Which one of the following options correctly represents the equivalent ratio of the given tape diagrams?

A

B

C

D

×

Given:

Jessica gets $$2$$ medals and Julie gets $$3$$ medals.

$$\therefore$$ The ratio is $$2:3$$.

We can describe the ratio through a tape diagram as;

When each unit represents one medal,

the ratio is $$2:3$$.

When each unit represents two medals,

the ratio is $$4:6$$.

The simplest form is $$2:3$$.

When each unit represents three medals,

the ratio is $$6:9$$.

The simplest form is $$2:3$$.

Thus, $$2:3$$$$4:6$$ and $$6:9$$ are all equivalent ratios because their simplest forms represent the same value.

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

Hence, option (D) is correct.

### Jessica gets $$2$$ medals and Julie gets $$3$$ medals. Which one of the following options correctly represents the equivalent ratio of the given tape diagrams?

A
B
C
D

Option D is Correct

# Finding Whole Through Its Share and Ratio

• 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

#### Jacob has $$100$$ in his pocket. If the ratio of Jacob's money to Carl's money is $$4:7$$, how much money do they both have altogether? Solve using tape diagram.

A $$275$$

B $$175$$

C $$75$$

D $$200$$

×

Given:

Money that Jacob has $$=100$$

Ratio of Jacob's money to Carl's money $$=4:7$$

Representing the given ratio through tape diagram as shown in the figure.

It is clear from the figure that Jacob has $$4$$ units.

Thus, $$4$$ units represent $$100$$.

$$\therefore$$ $$1$$ unit represents $$\dfrac{100}{4}=25$$

Now, since Carl has $$7$$ units and $$1$$ unit represents $$25$$,

$$\therefore$$ $$7$$ units represent $$7\times25=175$$

Thus, Carl has $$175$$.

Total money that Jacob and Carl have altogether

$$=\text{Jacob's Money + Carl's Money}$$

$$=100+175$$

$$=275$$

Hence, option (A) is correct.

### Jacob has $$100$$ in his pocket. If the ratio of Jacob's money to Carl's money is $$4:7$$, how much money do they both have altogether? Solve using tape diagram.

A

$$275$$

.

B

$$175$$

C

$$75$$

D

$$200$$

Option A is Correct

# Solving Problems of Ratios using Double Number Line Diagrams

• 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.

#### Alia reads $$5$$ pages in $$20$$ minutes. If she continues to read at the same pace, how many minutes will she take to read $$40$$ pages?

A $$120$$

B $$160$$

C $$40$$

D $$20$$

×

Alia reads $$5$$ pages in $$20$$ minutes.

We know when $$5$$ is multiplied by $$8$$, gives $$40$$.

Thus, $$20$$ must also be multiplied by $$8$$, which gives $$160$$ minutes.

Hence, option (B) is correct.

### Alia reads $$5$$ pages in $$20$$ minutes. If she continues to read at the same pace, how many minutes will she take to read $$40$$ pages?

A

$$120$$

.

B

$$160$$

C

$$40$$

D

$$20$$

Option B is Correct

# Table of Equivalent Ratios

• 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

#### The ratio of chocolates to muffins prepared at a bakery is $$1:2$$. If there are $$3$$ chocolates made on every $$6$$ muffins, how many muffins are made on $$12$$ chocolates?   Chocolates Muffins 3 6 6 12 9 18 12 ? 15 30 18 36

A $$12$$

B $$14$$

C $$20$$

D $$24$$

×

First, we check the pattern of the given table.

We find that the entries are multiples of $$1$$ and $$2$$.

They represent the ratio of $$1:2$$.

For the 1st row of the table, we multiply by $$3$$,

$$\Rightarrow1\times3=3$$

$$\Rightarrow2\times3=6$$

It gives the ratio $$=3:6$$

$$=1:2$$

For the 2nd row of the table, we multiply by $$6$$,

$$\Rightarrow1\times6=6$$

$$\Rightarrow2\times6=12$$

It gives the ratio $$=6:12$$

$$=1:2$$

and so on.

Similarly, for the 4th row,  we multiply by $$12$$  to find the number of muffins on $$12$$ chocolates.

$$\Rightarrow1\times12=12$$

$$\Rightarrow2\times12=24$$

It gives the ratio $$=12:24$$

$$=1:2$$

This means, on every $$12$$ chocolates there are $$24$$ muffins.

Hence, option (D) is correct.

### The ratio of chocolates to muffins prepared at a bakery is $$1:2$$. If there are $$3$$ chocolates made on every $$6$$ muffins, how many muffins are made on $$12$$ chocolates?   Chocolates Muffins 3 6 6 12 9 18 12 ? 15 30 18 36

A

$$12$$

.

B

$$14$$

C

$$20$$

D

$$24$$

Option D is Correct