Mary invested R40 000 in order to have R56 000 available in 30 months’ time. The yearly rate, compounded semi-annually, is

To find the yearly interest rate, compounded semi-annually, that will grow an investment of R40,000 to R56,000 in 30 months, we can use the compound interest formula:

\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]

Where:
- \( A \) is the amount of money accumulated after \( n \) periods, including interest (R56,000).
- \( P \) is the principal amount (the initial amount of money) (R40,000).
- \( r \) is the annual nominal interest rate (we need to find this).
- \( n \) is the number of times that interest is compounded per year (since it's semi-annually, \( n = 2 \)).
- \( t \) is the time the money is invested for, in years. (30 months is \( 2.5 \) years).

Plugging in the values:

\[
56,000 = 40,000 \left(1 + \frac{r}{2}\right)^{2 \times 2.5}
\]

Now, we need to solve for \( r \):

\[
\frac{56,000}{40,000} = \left(1 + \frac{r}{2}\right)^{5}
\]

\[
1.4 = \left(1 + \frac{r}{2}\right)^{5}
\]

Taking the fifth root of both sides:

\[
1 + \frac{r}{2} = 1.4^{\frac{1}{5}}
\]

\[
1 + \frac{r}{2} = 1.06905
\]

Subtract 1 from both sides:

\[
\frac{r}{2} = 0.06905
\]

Multiply by 2:

\[
r = 0.1381
\]

Convert to a percentage:

\[
r = 13.81\%
\]

Thus, the yearly interest rate, compounded semi-annually, is **13.81%**.

Would you like more details or have any questions?

Here are some related questions you might be interested in:
1. How would the result change if the interest were compounded quarterly?
2. What is the difference between nominal and effective interest rates?
3. How can we convert an interest rate from semi-annual compounding to monthly compounding?
4. What would the interest rate be if the desired amount was R60,000 instead of R56,000?
5. What is the formula for continuous compounding, and how would it apply here?

**Tip:** Always check whether the time period and compounding frequency are consistent (e.g., if time is in years, compounding should also reflect that period).

Learn how to calculate the yearly interest rate compounded semi-annually using the compound interest formula. This problem involves finding the interest rate that grows an investment from R40,000 to R56,000 in 30 months. Suitable for high school students studying compound interest.

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