The incomes of P, Q, and R are in the ratio 10:12:9 and their expenditures are in the ratio 12:15:8. If Q saves 25% of his income, then what is the ratio of the savings of P, Q, and R?

Income ratio of P, Q, and R = 10:12:9

Expenditure ratio of P, Q, and R = 12:15:8

Q saves 25% of his income.

Savings = Income - Expenditure

Let the incomes of P, Q, and R be 10x, 12x, and 9x respectively.
Let the expenditures of P, Q, and R be 12y, 15y, and 8y respectively.

Q's Savings:

Since Q saves 25% of his income:

Savings of Q = 0.25 \times 12x = 3x

Also, we know that savings = income - expenditure, so:

Savings of Q = $$\text{Income of Q} - \text{Expenditure of Q}$$​​

Substituting values:

3x = 12x - 15y

Solving for y:

​$$15y = 9x \quad \Rightarrow \quad y = \frac{9x}{15} = \frac{3x}{5}$$​​

P's Savings:

The income of P is 10x and the expenditure of P is 12y. So, the savings of P:

Savings of P = 10x - 12y

Substituting y = $$\frac{3x}{5}:$$​​

Savings of P = $$10x - 12 \times \frac{3x}{5} = 10x - \frac{36x}{5} = \frac{50x}{5} - \frac{36x}{5} = \frac{14x}{5}$$​​

R's Savings:

The income of R is 9x and the expenditure of R is 8y. So, the savings of R:

Savings of R= 9x - 8y

Substituting y = $$\frac{3x}{5}$$​:

Savings of R = $$9x - 8 \times \frac{3x}{5} = 9x - \frac{24x}{5} = \frac{45x}{5} - \frac{24x}{5} = \frac{21x}{5}$$​

Now, the savings of P, Q, and R are $$\frac{14x}{5}, 3x, \frac{21x}{5}.$$​​

Savings ratio = 14x : 15x : 21x = 14:15:21