A person invested some amount at the rate of 12% simple interest and a certain…

Concept: For a sum P invested at a simple interest rate r% per annum, the yearly interest is P × r / 100. When two sums are invested at two different rates and are then interchanged, the change in total yearly interest equals (difference between the sums) × (difference between the rates) / 100. This relationship lets two such interest conditions be converted into two linear equations in the unknown principal amounts, which can then be solved simultaneously.

Application:

1. Let the amount invested at 12% simple interest be Rs. x and the amount invested at 10% simple interest be Rs. y.

2. From the original interest of Rs. 130: 12% of x + 10% of y = 130, that is 12x/100 + 10y/100 = 130, which simplifies to 6x + 5y = 6500 — equation (i).

3. From the interchanged interest of Rs. 134 (Rs. 4 more than 130): 10% of x + 12% of y = 134, that is 10x/100 + 12y/100 = 134, which simplifies to 5x + 6y = 6700 — equation (ii).

4. Multiply equation (i) by 6 and equation (ii) by 5: 36x + 30y = 39000 and 25x + 30y = 33500.

5. Subtracting the second from the first eliminates y: 11x = 5500, so x = 500.

6. Substituting x = 500 back into equation (i): 6(500) + 5y = 6500, so 5y = 3500 and y = 700.

Cross-check:

With x = 500 and y = 700: original interest = 12% of 500 + 10% of 700 = 60 + 70 = 130, matching the given data; interchanged interest = 10% of 500 + 12% of 700 = 50 + 84 = 134, which is exactly Rs. 4 more than 130, confirming both conditions.

Hence, the amount invested at 12% simple interest is Rs. 500.