Question

Lucy won R45 000 in a lucky draw. She wants to invest the money for 7 years.
The bank offers her the following options


OPTION A: 8,6% per annum simple interest
OPTION B: 7,9% per annum compound interest, compounded quarterly
Which option would you advise Lucy to choose? Show ALL working out to support your answer.

Answer 1

To determine which investment option is better for Lucy, we'll calculate the total amount she would have after 7 years for both options.

### OPTION A: Simple Interest

The formula for calculating simple interest is:

\[
A = P(1 + rt)
\]

Where:
- \( A \) = the total amount after time \( t \)
- \( P \) = principal amount (initial investment)
- \( r \) = annual interest rate (as a decimal)
- \( t \) = time in years

For OPTION A:
- \( P = 45,000 \)
- \( r = 8.6\% = 0.086 \)
- \( t = 7 \)

Now substituting the values into the formula:

\[
A = 45000(1 + 0.086 \times 7)
\]

Calculating \( 0.086 \times 7 \):

\[
0.086 \times 7 = 0.602
\]

Now add this to 1:

\[
1 + 0.602 = 1.602
\]

Next, calculate \( A \):

\[
A = 45000 \times 1.602
\]

\[
A = 72090
\]

So, the total amount after 7 years with OPTION A is \( R72,090 \).

### OPTION B: Compound Interest

The formula for calculating compound interest is:

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

Where:
- \( A \) = the total amount after time \( t \)
- \( P \) = principal amount (initial investment)
- \( r \) = annual interest rate (as a decimal)
- \( n \) = number of times interest is compounded per year
- \( t \) = time in years

For OPTION B:
- \( P = 45,000 \)
- \( r = 7.9\% = 0.079 \)
- \( n = 4 \) (quarterly)
- \( t = 7 \)

Now substituting the values into the formula:

\[
A = 45000\left(1 + \frac{0.079}{4}\right)^{4 \times 7}
\]

Calculating \( \frac{0.079}{4} \):

\[
\frac{0.079}{4} = 0.01975
\]

Now plug this into the formula:

\[
A = 45000\left(1 + 0.01975\right)^{28}
\]

Calculating \( 1 + 0.01975 \):

\[
1 + 0.01975 = 1.01975
\]

Next, we calculate \( 1.01975^{28} \):

Using a calculator:
\[
1.01975^{28} \approx 1.634
\]

Now calculate \( A \):

\[
A = 45000 \times 1.634
\]

\[
A \approx 73440
\]

So, the total amount after 7 years with OPTION B is approximately \( R73,440 \).

### Conclusion

Comparing both options:
- OPTION A (Simple Interest): \( R72,090 \)
- OPTION B (Compound Interest): \( R73,440 \)

**The better option for Lucy is OPTION B, with a total of approximately R73,440 after 7 years.**