A sum of money invested at 8% per annum simple interest grows to Rs 180 in a certain number of years. The same sum of money invested for the same number of years at 4% per annum simple interest grows to Rs 120. For how many years is the money invested?

Introduction / Context:
This question compares two simple interest scenarios for the same principal and the same time period but with different interest rates, 8% and 4% per annum. The final amounts are given for each rate. By using the simple interest formula in both cases and exploiting the fact that the principal and time are the same, we can solve for both the principal and the time, and extract the number of years for which the money is invested.

Given Data / Assumptions:
- The principal is P (same in both cases). - At 8% per annum simple interest, the amount after T years is Rs 180. - At 4% per annum simple interest, the amount after the same T years is Rs 120. - Simple interest formula applies in each scenario. - We must find T, the number of years.

Concept / Approach:
Under simple interest, Amount A = P + P * R * T / 100 = P * (1 + R * T / 100). So we can write two equations: P * (1 + 8T / 100) = 180 and P * (1 + 4T / 100) = 120. These two equations involve the two unknowns P and T. Subtracting one equation from the other eliminates P temporarily and gives a relationship in terms of P * T. Then we can substitute back into one equation to solve for P and finally solve for T. This is a system of equations approach.

Step-by-Step Solution:
Step 1: At 8% per annum, write A1 = P * (1 + 8T / 100) = 180. Step 2: At 4% per annum, write A2 = P * (1 + 4T / 100) = 120. Step 3: Subtract the second equation from the first: P * (1 + 8T / 100) - P * (1 + 4T / 100) = 180 - 120. Step 4: Factor P: P * [(1 + 8T / 100) - (1 + 4T / 100)] = 60. Step 5: Simplify inside the brackets: (1 + 8T / 100) - (1 + 4T / 100) = 4T / 100. Step 6: So we have P * (4T / 100) = 60. Step 7: Therefore P * T = 60 * 100 / 4 = 1,500. Step 8: Now use the second equation: P * (1 + 4T / 100) = 120. Step 9: Expand: P + P * 4T / 100 = 120. Step 10: Note that P * 4T / 100 = 4 * (P * T) / 100. Step 11: Substitute P * T = 1,500 to get 4 * 1,500 / 100 = 60. Step 12: So the equation becomes P + 60 = 120. Step 13: Solve for P: P = 120 - 60 = 60. Step 14: With P * T = 1,500 and P = 60, compute T = 1,500 / 60 = 25. Step 15: Thus, the money is invested for T = 25 years.

Verification / Alternative check:
Check the amounts: With P = 60 and T = 25 years, at 4% per annum the interest is I2 = 60 * 4 * 25 / 100 = 60 * 1 = 60. So amount A2 = 60 + 60 = 120, matching the given 120. At 8% per annum, interest is I1 = 60 * 8 * 25 / 100 = 60 * 2 = 120. So amount A1 = 60 + 120 = 180, matching the given 180. Both checks confirm that T = 25 years is consistent.

Why Other Options Are Wrong:
A time such as 15, 18, 20, or 22 years would not simultaneously satisfy both amount equations when the principal is the same. Substituting those into the system either breaks the equality for 8% or for 4%. Only T = 25 years keeps both A1 and A2 exactly at 180 and 120 respectively, as required.

Common Pitfalls:
Some students attempt to guess the principal and time without setting up equations, which can be time-consuming and error-prone. Others may forget that the same principal is used in both scenarios or incorrectly subtract the equations. Careful algebra and stepwise simplification ensure you correctly eliminate one variable and solve for the other.

Final Answer:
The sum of money is invested for 25 years.