Raghu invests Rs. 1000 at a simple interest rate of 6% per annum and receives a total amount of Rs. 1300 after X years. Using the simple interest formula, find the value of X, that is, the number of years for which the money was invested.

Introduction / Context:
This question checks understanding of the basic simple interest relationship between principal, rate, time, and total amount received. When a person invests a fixed sum at a known rate and later knows the final amount, the time period can be found by working with the simple interest formula. Here Raghu invests Rs. 1000 at 6 percent per annum and ends up with Rs. 1300, so we must determine for how many years the amount was left invested under simple interest.

Given Data / Assumptions:
Principal invested P is Rs. 1000.
Rate of simple interest r is 6 percent per annum.
Total amount after X years is Rs. 1300.
Simple interest is earned at a constant rate every year.
There is no compounding, only simple interest is applied.

Concept / Approach:
For simple interest, interest I is given by I = P * r * t / 100, where P is principal, r is rate per annum, and t is time in years. Also, total amount A is A = P + I. If three of the four values among P, r, t, and A are known, we can solve for the remaining one. In this case P, r, and A are known, so we find I first and then solve for t. This is a classic single step simple interest application.

Step-by-Step Solution:
Compute the simple interest earned by Raghu as I = A - P.Substitute A = 1300 and P = 1000 to get I = 1300 - 1000 = 300 rupees.Use the simple interest formula I = P * r * t / 100.So 300 = 1000 * 6 * t / 100.Simplify the right side: 1000 * 6 / 100 = 60, so 300 = 60 * t.Solve for t by dividing both sides by 60 to obtain t = 300 / 60 = 5 years.Thus Raghu must have invested the money for 5 years to receive Rs. 1300.

Verification / Alternative check:
We can check the result by computing interest for 5 years directly. For t = 5, simple interest I is 1000 * 6 * 5 / 100 = 1000 * 30 / 100 = 300 rupees. The total amount is then 1000 + 300 = 1300 rupees, which matches the information in the question. Since the calculation is consistent, the time value X equal to 5 years is verified.

Why Other Options Are Wrong:
For 4 years, interest would be 1000 * 6 * 4 / 100 = 240 rupees, giving an amount of only Rs. 1240, which is less than Rs. 1300.
For 3 years, interest is 180 rupees, total Rs. 1180, again too low.
For 2 years, interest is 120 rupees, leading to Rs. 1120, not matching Rs. 1300.
For 6 years, interest would be 360 rupees, giving Rs. 1360, which is more than the given total. Therefore those options do not fit the data.

Common Pitfalls:
A common mistake is to confuse the total amount with the interest and directly apply the formula to the wrong quantity. Another error is to forget to subtract principal from the final amount before solving for time. Some learners also incorrectly use compound interest formulas when the question clearly states simple interest. Careful reading of the phrase simple interest and careful substitution into the formula avoids these issues.

Final Answer:
The investment period X for which Raghu kept his money under simple interest is 5 years.