To determine how long it will take for Peo to save R30,835.42 by depositing R25,000 into a savings account with an annual interest rate of 10.5% compounded weekly, we use the formula for compound interest:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
where:
- [tex]\( A \)[/tex] is the future value of the investment (R30,835.42),
- [tex]\( P \)[/tex] is the principal amount (R25,000),
- [tex]\( r \)[/tex] is the annual interest rate (10.5%, or 0.105 as a decimal),
- [tex]\( n \)[/tex] is the number of times the interest is compounded per year (52 for weekly),
- [tex]\( t \)[/tex] is the number of years the money is invested.
We need to solve for [tex]\( t \)[/tex]. Starting with the compound interest formula:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
We rearrange the formula to solve for [tex]\( t \)[/tex]:
1. Divide both sides by [tex]\( P \)[/tex]:
[tex]\[ \frac{A}{P} = \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
2. Take the natural logarithm (logarithm base e) of both sides:
[tex]\[ \ln\left(\frac{A}{P}\right) = \ln\left(\left(1 + \frac{r}{n}\right)^{nt}\right) \][/tex]
3. Apply the logarithm power rule [tex]\( \ln(x^y) = y \ln(x) \)[/tex]:
[tex]\[ \ln\left(\frac{A}{P}\right) = nt \cdot \ln\left(1 + \frac{r}{n}\right) \][/tex]
4. Solve for [tex]\( t \)[/tex]:
[tex]\[ t = \frac{\ln\left(\frac{A}{P}\right)}{n \cdot \ln\left(1 + \frac{r}{n}\right)} \][/tex]
Now we plug in the values:
- [tex]\( A = 30,835.42 \)[/tex]
- [tex]\( P = 25,000 \)[/tex]
- [tex]\( r = 0.105 \)[/tex]
- [tex]\( n = 52 \)[/tex]
[tex]\[ t = \frac{\ln\left(\frac{30,835.42}{25,000}\right)}{52 \cdot \ln\left(1 + \frac{0.105}{52}\right)} \][/tex]
Using the values provided, we calculate:
[tex]\[ \frac{30,835.42}{25,000} = 1.2334168 \][/tex]
[tex]\[ \ln(1.2334168) \approx 0.210721 \][/tex]
[tex]\[ 1 + \frac{0.105}{52} = 1.002019 \][/tex]
[tex]\[ \ln(1.002019) \approx 0.002016 \][/tex]
[tex]\[ 52 \cdot 0.002016 = 0.104832 \][/tex]
Finally:
[tex]\[ t = \frac{0.210721}{0.104832} \approx 2.009603 \][/tex]
Rounding to a more reasonable precision, Peo will need approximately 2 years to reach his goal of R30,835.42 with an initial deposit of R25,000 in a savings account earning 10.5% annual interest compounded weekly.
