A parent wants to invest money for her child in an account earning interest at a rate of 13% p.a. compounded half- yearly.
a) What amount should she invest on her child's 10th birthday so that the child can receive an amount of R20000 on her 21st birthday?
b) What amount should the parent invest if the child should receive R10000 on each of her 21st and 25th birthdays?

We can solve these problems using the compound interest formula, which is given by:

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

Where:
- \( A \) is the future amount (the amount the child will receive).
- \( P \) is the initial investment (the amount the parent needs to invest).
- \( r \) is the annual interest rate (13% in this case, or 0.13 as a decimal).
- \( n \) is the number of times the interest is compounded per year (since it's compounded half-yearly, \( n = 2 \)).
- \( t \) is the time in years.

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### Part (a): Investment for the 21st Birthday

We want to find the amount \( P \) the parent needs to invest so that the child receives R20000 on their 21st birthday. The time from the 10th birthday to the 21st birthday is \( t = 11 \) years.

The future value \( A = 20000 \), \( r = 0.13 \), and \( n = 2 \).

Using the compound interest formula:

\[
20000 = P \left( 1 + \frac{0.13}{2} \right)^{2 \times 11}
\]

Now, let's calculate the value of \( P \).

### Part (b): Investment for Two Payments (21st and 25th Birthdays)

For this part, the parent needs to invest such that the child receives R10000 on their 21st and 25th birthdays. This involves two steps:

1. **Investment for the 21st birthday:**
   The time period here is still \( t = 11 \) years, so we use the same formula as in part (a), but now with \( A = 10000 \).

2. **Investment for the 25th birthday:**
   The time from the 10th birthday to the 25th birthday is \( t = 15 \) years, and the child will receive R10000. We calculate the required investment for this separately with \( t = 15 \) and \( A = 10000 \).

Finally, we add the two investments required for both payments.

Let me calculate these values.### Solution:

#### Part (a):
To receive R20,000 on the child's 21st birthday, the parent should invest approximately **R5004.25** on the child's 10th birthday.

#### Part (b):
For the child to receive R10,000 on both the 21st and 25th birthdays, the parent should invest:
- **R2502.12** for the 21st birthday.
- **R1511.86** for the 25th birthday.

Thus, the total amount the parent needs to invest for both payments is approximately **R4013.98**.

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Do you need further details or have any questions? Here are 5 related questions to think about:

1. How would the required investment change if the interest rate was 10% instead of 13%?
2. What would happen if the interest was compounded quarterly instead of half-yearly?
3. How would the solution differ if the parent wanted the child to receive different amounts on their 21st and 25th birthdays?
4. How much would the child receive by their 30th birthday if no further withdrawals were made?
5. How does the formula adjust if the interest is compounded continuously?

**Tip**: When planning investments, knowing the compounding frequency is crucial as it significantly affects the future value.

A parent wants to invest money for her child in an account earning interest at a rate of 13% p.a. compounded half-yearly. a) What amount should she invest on her child's 10th birthday so that the child can receive an amount of R20000 on her 21st birthday? b) What amount should the parent invest if the child should receive R10000 on each of her 21st and 25th birthdays?

Discover how to calculate the initial investment required for a child to receive R20000 on their 21st birthday, using a 13% interest rate compounded half-yearly. This example also explores investing for two payments of R10000 on the 21st and 25th birthdays. Detailed solution includes step-by-step calculations and formulas.

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