Alright, let's solve this step-by-step.
1. Identify the known values:
- Initial investment (Principal, [tex]\( P \)[/tex]) = R1,500
- Annual simple interest rate ( [tex]\( r \)[/tex]) = 6.57% = 0.0657
- Time period ( [tex]\( t \)[/tex]) = 16 months
2. Convert the time period into years:
Since the interest rate is annual, we need to convert 16 months into a fraction of a year.
[tex]\[
t = \frac{16}{12} \text{ years}
\][/tex]
3. Calculate the interest earned:
Using the simple interest formula [tex]\( I = P \cdot r \cdot t \)[/tex], substitute the known values:
[tex]\[
I = 1500 \cdot 0.0657 \cdot \frac{16}{12}
\][/tex]
Calculate the interest [tex]\( I \)[/tex]:
[tex]\[
I = 1500 \cdot 0.0657 \cdot 1.3333
\][/tex]
[tex]\[
I = 131.4
\][/tex]
4. Calculate the total balance in the account after 16 months:
Add the interest earned to the initial investment:
[tex]\[
\text{Balance} = P + I
\][/tex]
[tex]\[
\text{Balance} = 1500 + 131.4
\][/tex]
[tex]\[
\text{Balance} = 1631.4
\][/tex]
Therefore, the balance in the account after 16 months is R1,631.40, which corresponds to option [tex]\( a \)[/tex]:
[tex]\[ \boxed{a. \, R1,631.40} \][/tex]
