Sure! Let's go through the solution to determine how much Moses' investment will be after four years with a compound interest rate of 3% compounded quarterly.
To find the accumulated amount (A) after a certain period using compound interest, we use the compound interest formula:
[tex]\[ A = P \left( 1 + \frac{r}{n} \right)^{nt} \][/tex]
where:
- [tex]\( A \)[/tex] is the amount of money accumulated after n periods, including interest.
- [tex]\( P \)[/tex] is the principal amount (the initial amount of money).
- [tex]\( r \)[/tex] is the annual interest rate (decimal).
- [tex]\( n \)[/tex] is the number of times interest is compounded per year.
- [tex]\( t \)[/tex] is the time the money is invested for, in years.
Given:
- The principal amount ([tex]\( P \)[/tex]) is R25,000.
- The annual interest rate ([tex]\( r \)[/tex]) is 3%, or in decimal form, 0.03.
- The interest is compounded quarterly, so [tex]\( n \)[/tex] is 4.
- The time ([tex]\( t \)[/tex]) is 4 years.
Now let's plug these values into the formula:
[tex]\[ A = 25000 \left( 1 + \frac{0.03}{4} \right)^{4 \cdot 4} \][/tex]
First, calculate the quarterly interest rate ([tex]\( r/n \)[/tex]):
[tex]\[ \frac{0.03}{4} = 0.0075 \][/tex]
Then, add 1 to the quarterly interest rate:
[tex]\[ 1 + 0.0075 = 1.0075 \][/tex]
Next, calculate the exponent part ([tex]\( nt \)[/tex]):
[tex]\[ 4 \cdot 4 = 16 \][/tex]
Raise [tex]\( 1.0075 \)[/tex] to the power of 16:
[tex]\[ 1.0075^{16} \approx 1.12699203495 \][/tex]
Now, multiply this result by the principal amount:
[tex]\[ 25000 \times 1.12699203495 \approx 28174.80284222727 \][/tex]
So, after four years, Moses' investment will be approximately R28,174.80.
Thus, the accumulated amount (A) after 4 years is:
[tex]\[ \boxed{28174.80} \][/tex]
