Find Missing Values in Ratio

Worked Example 10.2.1:

Find the missing numbers in the following:

a. 12 : x = 20 : 35
b. 4 : 9 = x : 99

Solution

a. 12 : x = 20 : 35

\( \frac{12}{x} = \frac{20}{35}\; \; \; \; \; \scriptsize \; (cross \; multiply) \\ \scriptsize 20x\: = \: 12 \: \times \:35 \\ \scriptsize x = \normalsize \frac{12 \; \times \; 35}{20} \\ \scriptsize x = 21\)

b. 4 : 9 = x : 99

\( \frac{4}{9} = \frac{x}{99}\; \; \; \; \; \scriptsize \; (cross \; multiply) \\ \scriptsize 9x = 4 \: \times \: 99 \\ \scriptsize x = \normalsize \frac{4 \: \times \: 99}{9} \\ \scriptsize x = \normalsize \frac{4 \: \times \: 11}{1} \\ \scriptsize x = 44\)

Worked Example 10.2.2:

Find the missing numbers in the following equivalent ratio:

a. \( \scriptsize 12 : \square = 20 : 35 \)
b. \( \scriptsize 4 : 9 = \square : 99 \)
c. \( \scriptsize 3 : 4 =21 : \square \)

Solution

a. \( \scriptsize 12 : \square = 20 : 35 \)

\( \frac{12}{\square} = \frac{20}{35} \\ \frac{12}{\square} = \frac{4}{7}\scriptsize \; \; \; \; \; (cross \: multiply) \\ \scriptsize 4 \: \times \: \square = 12 \: \times \: 7 \\ \scriptsize \square = \normalsize \frac{12 \: \times \: 7}{4} \\ \scriptsize \square = \normalsize \frac{84}{4} \\ \scriptsize \square = 21\)

Answer: \( \scriptsize 12 : 21 = 20 : 35 \)

 b. \( \scriptsize 4 : 9 = \square : 99 \)

\( \frac{4}{9} = \frac{\square}{99} \; \; \; \; \; \scriptsize \; (cross \; multiply) \\ \scriptsize 9 \: \times \: \square = 99 \: \times \: 4 \\ \scriptsize \square = \normalsize \frac{99 \: \times \: 4}{9} \\ \scriptsize \square = 11 \: \times \: 4 \\ \scriptsize \square = 44\)

Answer: \( \scriptsize 4 : 9 = 44 : 99 \)

c. \( \scriptsize 3 : 4 = 21 : \square \)

\( \frac{3}{4} = \frac{21}{\square} \; \; \; \; \; \scriptsize \; (cross \: multiply) \\ \scriptsize 3\: \times \: \square = 21 \: \times \: 4 \\ \scriptsize \square = \normalsize \frac{21 \: \times \: 4}{3} \\ \scriptsize \square =7 \: \times \: 4 \\ \scriptsize \square = 28\)

Answer: \( \scriptsize 3 : 4 = 21 : 28\)

Worked Example 10.2.3:

In a car park, there are 30 white cars and 20 blue cars. 

a. What is the ratio of the white car to the blue car?
b. What is the ratio of the blue car to the white car?

Solution

a. Number of white cars =  30

Number of blue cars    =  20

Ratio of white cars to blue cars: 

= White car   :   blue car

= 30 : 20  

=  3 : 2

b. Ratio of blue cars to white cars:

= Number of blue cars : Number of white cars

=  20 : 30

= 2 : 3

Evaluation Question: