Simple Interest – Rate equals time; interest is 1/9 of principal: The simple interest on a sum is one-ninth of the principal, and the number of years equals the rate percent per annum. Find the rate percent per annum.

Introduction / Context:
Here the interest is a fixed fraction of principal (1/9), and the problem states an equality between two parameters: the number of years equals the rate percent. Such symmetry invites using the simple-interest formula with r = t to isolate the unknown rate.

Given Data / Assumptions:

• I = P / 9
• r = t (numerically equal)
• Simple interest I = P * r * t / 100

Concept / Approach:
Substitute I and r = t into the formula to get P / 9 = P * r^2 / 100, cancel P, and solve for r. Keep r in percent units when interpreting the final numeric value.

Step-by-Step Solution:
P / 9 = P * r^2 / 100Cancel P (P > 0): 1 / 9 = r^2 / 100r^2 = 100 / 9 ⇒ r = 10 / 3 ≈ 3.333... percent

Verification / Alternative check:
Let r = t = 10 / 3. Then I / P = r * t / 100 = (10 / 3) * (10 / 3) / 100 = 100 / 9 / 100 = 1 / 9, matching the premise.

Why Other Options Are Wrong:
3% and 4% yield different interest fractions; 3 1/10% (≈ 3.1%) is too small; 3.5% also does not satisfy r^2 = 100 / 9.

Common Pitfalls:
Confusing r (percent) with decimal rate, or failing to square r when r = t, results in incorrect values. Keep r in percent throughout.

Final Answer:
10/3%