Introduction / Context:
This problem uses a ratio to relate the original principal to the interest earned under simple interest. Alipta invests money at a fixed rate of 6% per annum, and we are told that after some time the ratio of principal to interest becomes 10 : 3. Using this ratio and the simple interest formula, we can determine the number of years required to reach that interest amount.
Given Data / Assumptions:
• Principal P is some unknown sum received from her father.
• Rate of simple interest R = 6% per annum.
• After T years, interest I and principal P satisfy P : I = 10 : 3.
• Simple interest formula: I = (P * R * T) / 100.
Concept / Approach:
From the ratio P : I = 10 : 3, we can write P = 10k and I = 3k for some positive constant k. Using the simple interest formula, we substitute I and P in terms of k and solve for T. The principal P cancels out, so we do not need its actual value to find the time.
Step-by-Step Solution:
Given ratio P : I = 10 : 3, let P = 10k and I = 3k.
From simple interest formula: I = (P * R * T) / 100.
Substitute P and R: 3k = (10k * 6 * T) / 100.
Simplify the right side: (10k * 6 * T) / 100 = (60kT) / 100 = 0.6kT.
So 3k = 0.6kT.
Divide both sides by k (k ≠ 0): 3 = 0.6T.
Therefore, T = 3 / 0.6 = 5 years.
Verification / Alternative check:
We can pick a convenient value, for example P = Rs. 1000. Then P corresponds to 10 parts, so each part is 100, making I = Rs. 300 when ratio is 10 : 3. At 6% per year, simple interest per year is 6% of 1000 = Rs. 60. To accumulate Rs. 300 interest, time = 300 / 60 = 5 years, confirming our result.
Why Other Options Are Wrong:
Times of 3, 4, 6, or 7 years would produce interest values that correspond to different ratios than 10 : 3. For example, at 6% per year on Rs. 1000, 3 years gives Rs. 180 interest (ratio 1000 : 180 = 50 : 9), not 10 : 3. Similar mismatches occur for 4, 6, and 7 years.
Common Pitfalls:
Some students mistakenly interpret the ratio as interest to principal instead of principal to interest, which inverts the relationship. Others forget that P cancels when using the ratio, and they attempt to assume awkward principal values. Working systematically with proportional expressions keeps the algebra simple.
Final Answer:
The required time is
5 years.
