Informative line

Methods Of Solving Problems Involving Ratios

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

Illustration Questions

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.

 

image

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

image

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.

Illustration Questions

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.

image

\(2\) 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.

image

\(8\) 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.

image

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 image
B image
C image
D image

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.

Illustration Questions

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 \(2:3\).

image

When each unit represents two medals,

the ratio is \(4:6\).

The simplest form is \(2:3\).

image

When each unit represents three medals,

the ratio is \(6:9\).

The simplest form is \(2:3\).

image

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?

image
A image
B image
C image
D image

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

Illustration Questions

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.

image

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.

Illustration Questions

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.

image

We know when \(5\) is multiplied by \(8\), gives \(40\).

Thus, \(20\) must also be multiplied by \(8\), which gives \(160\) minutes.

image

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?

image
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

Illustration Questions

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

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