The annual incomes of A and B are in the ratio 3 : 4 and their annual expenditure are in the ratio 5 : 7. If each saves ₹ 5,000; find their annual incomes.

Given,

Annual incomes of A and B are in the ratio 3 : 4.

Let annual income of A be 3x and B be 4x.

Annual expenditure are in the ratio 5 : 7.

Let annual expenditure of A be 5y and B be 7y.

Given,

Each save ₹ 5,000.

For A,

⇒ 3x - 5y = 5000 ………..(1)

For B,

⇒ 4x - 7y = 5000 ……….(2)

Multiplying equation (1) by 4, we get :

⇒ 4(3x - 5y) = 4 × 5000

⇒ 12x - 20y = 20000 ………(3)

Multiplying equation (2) by 3, we get :

⇒ 3(4x - 7y) = 3 × 5000

⇒ 12x - 21y = 15000 ………(4)

Subtracting equation (4) from (3), we get :

⇒ 12x - 20y - (12x - 21y) = 20000 - 15000

⇒ 12x - 12x - 20y + 21y = 5000

⇒ y = 5000.

Substituting value of y in (1), we get :

⇒ 3x - 5(5000) = 5000

⇒ 3x - 25000 = 5000

⇒ 3x = 25000 + 5000

⇒ 3x = 30000

⇒ x = $\dfrac{30000}{3}$ = 10000.

⇒ 3x = 3(10000) = 30000 and 4x = 4(10000) = 40000.

Hence, annual income of A = ₹ 30,000 and B = ₹ 40,000.