the simple interest on. sum of money is 4/9 of the principal and the number of years is equal to the rate percent per annum the rate per annum is  a) 5% b) 62/3% c) 6% d) 71/5%

Let's go through the problem again.

### Given:
- Simple Interest (SI) = \(\frac{4}{9}\) of the principal \(P\).
- The number of years (\(n\)) is equal to the rate of interest (\(r\)) per annum.

### Formula for Simple Interest:
\[
\text{SI} = \frac{P \times r \times n}{100}
\]

### Since \(n = r\):
\[
\text{SI} = \frac{P \times r \times r}{100} = \frac{P \times r^2}{100}
\]

We know:
\[
\frac{P \times r^2}{100} = \frac{4}{9}P
\]

Cancel \(P\) from both sides:
\[
\frac{r^2}{100} = \frac{4}{9}
\]

Multiply both sides by 100:
\[
r^2 = \frac{400}{9}
\]

Take the square root of both sides:
\[
r = \frac{\sqrt{400}}{3} = \frac{20}{3} = 6\frac{2}{3}\%
\]

### Conclusion:
The rate per annum is \(6\frac{2}{3}\%\), which corresponds to option **b) \(6 \frac{2}{3}\%\)**.

Would you like further details or have any questions? Here are some related problems you might find interesting:

1. How does the principal amount affect the simple interest?
2. What happens to the simple interest if the rate is doubled?
3. If the number of years were different from the rate, how would that change the calculation?
4. How does compound interest differ from simple interest in similar problems?
5. Can the rate per annum be a fraction? What would that mean practically?
6. How would you calculate the time required for the interest to equal the principal?
7. What would the rate be if the simple interest were half of the principal?
8. How would you set up a similar problem for compound interest?

**Tip:** Always double-check the relationship between the variables in interest problems, especially when the time period equals the interest rate.

Learn how to calculate simple interest when the rate per annum is equal to the number of years, with a step-by-step solution. This problem involves understanding the relationship between principal, rate, time, and simple interest. Suitable for high school students.

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