Ratio Strategy: Repeated Identity

In this article, we are going to learn about how to solve repeated identity questions.  To solve questions on repeated identity, 

Step 1: 

Write down the two sets of ratios. 

Step 2: 

Identify the item that is repeated. 

Step 3: 

Make the units for the repeated item the same in the 2 sets of ratio by finding the common multiple. 

Step 4: 

Read the question carefully and assign a value to the units if necessary. 

Step 5:

Solve for 1 unit if necessary and answer the question. 
 

Question 1:

The ratio of the number of red pens to the number of blue pens in a box is 3 : 4. The ratio of the number of red pens to the number of black pens in the same box is 2 : 3. What is the ratio of the number of red pens to the number of blue pens to the number of black pens?

Solution:

The number of red pens is the repeated identity.

We make the units representing the red pens the same by finding the common multiple of 3 and 2. 

Answer:
6 : 8 : 9

Question 2:

Mrs Koh bought some apples, oranges, and pears. The ratio of the number of apples to the number of oranges was 3 : 2. The ratio of the number of pears to the number of apples was 5 : 2. Mrs Koh bought 18 apples. How many pears did she buy? 

Solution:

The number of apples is the repeated identity.

We make the units representing the apples the same by finding the common multiple of 3 and 2. 

6 units = 18

1 unit   = 18 ÷ 6

            = 3

Number of pears = 15 units 

    = 15 × 3

    = 45 

Answer:
45 pears 

Question 3:

The ratio of the amount of money Aini has to the amount of money Siti has is 3 : 1. The ratio of the amount of money Aini has to the amount of money Winnie has is 1 : 2. If the three girls have a total of $60, how much money does Winnie have?

Solution: 

Aini is the repeated identity.

We make the units representing Aini the same by finding the common multiple of 3 and 1. 

Total number of units the three girls have = 3 units + 1 unit + 6 units

 = 10 units

10 units = $60

1 unit     = $60 ÷ 10

              = $6

Amount of money Winnie has = 6 units 

 = 6 × $6

 = $36

Answer:
$36 

Question 4: 

The diagram shows four points A, B, C, and D in a straight line. AB is $\begin{align*} \frac { 5 } { 9 } \end{align*}$ as long as AC and CD is $\begin{align*} \frac { 2 } { 5 } \end{align*}$ as long as BD.

A) What is the ratio of the length of AB to the length of BC to the length of CD?

Solution:

BC is the repeated identity.

We make the units representing BC the same by finding the common multiple of 4 and 3. 

Answer:
15 : 12 : 8

B) Given that CD = 48 cm, what is the length of AD? 

Solution:

8 units = 48 cm

1 unit   = 48 cm ÷ 8

 = 6 cm

Length of AD = 15 units + 12 units + 8 units

  = 35 units

  = 35 × 6 cm

  = 210 cm

Answer:
210 cm

Question 5:

$\begin{align*} \frac { 9 } { 20 } \end{align*}$ of the passengers on a plane are adults and the rest are children. The ratio of the number of boys to the number of girls is 2 : 3.

A) What is the ratio of the number of adults to the number of boys to the number of girls on the plane?

Solution:

The number of children is the repeated identity. 

We make the units representing the number of children the same by finding the common multiple of 11 and 5. 

Answer:
45 : 22 : 33

B) There are 36 more adults than girls. How many passengers are there on the plane

Solution:

Difference in number of units between the number of adults and the number of girls

= 45 units – 33 units

= 12 units

12 units = 36

1 unit     = 36 ÷ 12 

              = 3

Total number of passengers = 45 units + 22 units + 33 units

      = 100 units

      = 100 × 3

      = 300

Answer:

300 passengers

Question 5:

$\begin{align*} \frac { 7 } { 11 } \end{align*}$ of the pupils in a school are boys. $\begin{align*} \frac { 2 } { 5 } \end{align*}$ of the girls can swim. The ratio of the number of boys who can swim to the number of boys who cannot swim is 2 : 3.

A) What is the ratio of the number of pupils who can swim to the number of pupils who cannot swim in the school?

Solution:

The total number of girls is repeated.

We make the units representing the girls the same by finding the common multiple of 5 and 20.

The total number of boys is also repeated.

We make the units representing the boys the same by finding the common multiple of 5 and 35.

Number of units of pupils who can swim = 8 units + 14 units

      = 22 units 

Number of units of pupils who cannot swim = 12 units + 21 units 

 = 33 units 

Answer:
2 : 3

B) 330 pupils in the school cannot swim. How many girls are there in the school?

Solution:

33 units = 330

1 unit     = 330 ÷ 33

              = 10

Number of girls = 20 units

      = 20 × 10

      = 200

Answer:

200 girls