Question

A man lent a part of his money at 10% p.a. and the rest at 15% p.a. His income at the end of the year is ₹ 1,900. If he had interchanged the rate of interest on the two sums, he would have earned ₹ 200 more. Find the amount lent in both cases.

Answer 1

Given:

A man lends two sums: let A be the amount lent at 10% p.a. and B be the amount lent at 15% p.a.

His total interest in one year is ₹ 1,900. 

Step-wise calculation:

1. Write the interest equations for one year:

Original lending: 0.10A + 0.15B = 1900   ...(1)

If rates are interchanged, interest becomes 0.15A + 0.10B = 1900 + 200 = 2100   ...(2)

2. Subtract (1) from (2):

(0.15A + 0.10B) – (0.10A + 0.15B) = 2100 – 1900 

⇒ 0.05A – 0.05B = 200

⇒ 0.05(A – B) = 200

⇒ A – B = `200/0.05`

⇒ A – B = 4000   ...(3)

3. Substitute A = B + 4000 into (1):

0.10(B + 4000) + 0.15B = 1900

⇒ 0.10B + 400 + 0.15B = 1900

⇒ 0.25B + 400 = 1900

⇒ 0.25B = 1500 

⇒ B = `1500/0.25`

⇒ B = 6000

4. From (3),

A = B + 4000 

= 6000 + 4000

= 10000

Check: 10% of 10,000 = 1,000; 15% of 6,000 = 900; total = 1,900.

If interchanged: 15% of 10,000 = 1,500; 10% of 6,000 = 600; total = 2,100, which is ₹ 200 more.

Amount lent at 10% = ₹ 10,000.

Amount lent at 15% = ₹ 6,000.