Finding the ratio of three quantities

Finding the ratio of three quantities

Ratio can be of three quantities. The amount of each quantity in the ratio is obtained using the total share of the quantities. The ratio of three quantities can be calculated using the following steps:

1. Write the ratio of the three quantities A, B and C as A : B : C.
2. Add the given quantities, that is, A + B + C.
3. Write the ratio of each quantity to the sum of all quantities in fraction, that is,
   • $\frac{A}{A + B + C}$
   • $\frac{B}{A + B + C}$
   • $\frac{C}{A + B + C}$
4. If you are given a total share, example, D, multiply each fraction by the total share D as follows.
   • $\frac{A}{A + B + C} \times D$
   • $\frac{B}{A + B + C} \times D$
   • $\frac{C}{A + B + C} \times D$

The answers in step 4 represent the value of each quantity in the ratio of quantity A, B and C to the given share D.

Example 1

Write the ratio of 160 to 240 to 360 using a colon and simplify.

Solution

Arrange the numbers in the question, and then change the ratio from words to a colon as follows:

160 to 240 to 360 = 160 : 240 : 360

Divide each number involved in the ratio by the G.C.F of the numbers. In this case, G.C.F of 160, 240, 360 is 40. Thus, the ratio is:

$160 \div 40 : 240 \div 40 : 360 \div 40$

This is the same as 4 : 6 : 9.

Therefore, the ratio of 160 to 240 to 360 is 4 : 6 : 9.

Example 2

Mr Samwel gave money to his three children: Mage, Elia and Imani, in the ratio of 3 : 5 : 7. If the total amount of money given was 1 950 000 shillings, how much money did each child get?

Solution

The ratio of money of each child to the total number of children is as follows:

Mage : Elia : Imani = 3 : 5 : 7

Add: 3 + 5 + 7 = 15

Write each number involved as a fraction of the total of all numbers.

The total amount of money given to the children is 1 950 000 shillings.

Multiply each fraction of the ratio of each child by the total amount of money as follows:

Mage: $\frac{3}{15} \times \text{sh } 1 950 000 = \text{sh } 390 000$

Elia: $\frac{5}{15} \times \text{sh } 1 950 000 = \text{sh } 650 000$

Imani: $\frac{7}{15} \times \text{sh } 1 950 000 = \text{sh } 910 000$

Therefore, Mage got 390 000 shillings, Elia got 650 000 shillings, and Imani got 910 000 shillings.

Example 3

If A : B : C = 3 : 15 : 7, find the value of C if A = 12 and B = 60.

Solution

Arrange the numbers in the ratio as follows:

12 : 60 : C = 3 : 15 : 7

Add all numbers in the ratio in both sides:

Left side gives: 12 + 60 + C = 72 + C

Right side gives: 3 + 15 + 7 = 25

Formulate an equation as follows:

$\frac{C}{72 + C} = \frac{7}{25}$

Cross-multiply to get a simpler equation:

$7(72 + C) = 25C$

$504 + 7C = 25C$

Collect like terms:

$18C = 504$

Divide both sides by 18:

$\frac{18C}{18} = \frac{504}{18}$

$C = 28$

Therefore, C = 28.