Simple interest on a certain sum is equal to 16/25 of the sum. Find the rate percent and the time in years, given that the rate percent and the time are numerically equal.

Introduction / Context:
This simple interest problem specifies the total interest as a fraction of the principal and states that the numerical values of rate and time are equal. Such questions are common in aptitude tests and require forming a simple algebraic equation from the given relationship. The focus is on translating the verbal condition into a formula and solving for the unknown rate and time.

Given Data / Assumptions:

Simple interest earned on a certain sum is equal to 16/25 of the principal sum.
Rate of interest r percent per annum and time t years are numerically equal, so r = t.
Interest is calculated on a simple interest basis.
We need to find r and t satisfying these conditions.

Concept / Approach:
The simple interest formula is I = (P * r * t) / 100, where I is the interest and P is the principal. Here I is given as (16/25) * P. Since r = t, we can write t = r and substitute into the formula, giving a single equation in r. After cancelling P, we solve the resulting quadratic equation for r. Then t is equal to r by the given condition. We choose the positive solution because rate and time must be positive.

Step-by-Step Solution:
Let principal be P rupees, rate be r percent per annum and time be t years. Given that simple interest I = (16/25) * P. By the simple interest formula, I = (P * r * t) / 100. We are also told that r and t are numerically equal, so let t = r. Then I = (P * r * r) / 100 = (P * r^2) / 100. Set this equal to (16/25) * P: (P * r^2) / 100 = (16/25) * P. Cancel P from both sides (P is not zero): r^2 / 100 = 16 / 25. Multiply both sides by 100: r^2 = 100 * (16 / 25) = 1600 / 25. Compute 1600 / 25 = 64. So r^2 = 64, implying r = 8 or r = -8. Negative rate is not meaningful here, so r = 8. Since t = r, time t = 8 years.

Verification / Alternative check:
Assume principal P = 100 for simplicity. With r = 8 percent and t = 8 years, simple interest I = (100 * 8 * 8) / 100 = 64. Then I / P = 64 / 100 = 16 / 25, which matches the given condition. This confirms that r = 8 percent per annum and t = 8 years satisfy both the fractional interest requirement and the equality of rate and time.

Why Other Options Are Wrong:
Option 7% and 7 years gives interest (P * 7 * 7) / 100 = 49P / 100, so I / P = 49 / 100, which does not equal 16 / 25.
Option 6% and 6 years yields I / P = 36 / 100, again different from 16 / 25.
Option 5% and 5 years leads to I / P = 25 / 100, which simplifies to 1 / 4, not 16 / 25.

Common Pitfalls:
Some learners forget to cancel the principal P and incorrectly think they need its value to continue, which is not required.
Others equate r and t but then incorrectly square the fraction 16/25 rather than solving r^2 / 100 = 16 / 25 correctly.
A common algebraic mistake is to forget that rate is in percent, which should always be handled with a division by 100 in the simple interest formula.

Final Answer:
The correct values are Rate = 8% per annum and Time = 8 years.