Question
Person X invests R 150 at the end of each month with bank X who adds interest at 6% per annum, compounded monthly. Person Y invests R 380 at the end of each year with bank Y who adds interest at 9% per annum compounded annually. Who will have the most money available after exactly 17 years?
Y with R 14 049.85
X with R 52 986
X with R 4 231.93
Y with R 1 890.70
None of the above
Y with R 14 049.85
X with R 52 986
X with R 4 231.93
Y with R 1 890.70
None of the above
Answers
GPT-4o mini
To compare the future values of Person X's and Person Y's investments, we'll calculate the future value of each investment using the appropriate formulas for compound interest.
### Person X: Investment with Bank X
- Monthly investment (PMT) = R 150
- Annual interest rate (r) = 6% = 0.06
- Monthly interest rate (i) = r / 12 = 0.06 / 12 = 0.005
- Total number of months (n) = 17 years * 12 months/year = 204 months
The future value of an annuity (FV) can be calculated using the formula:
\[
FV = PMT \times \frac{(1 + i)^n - 1}{i}
\]
Now substituting the values:
\[
FV_X = 150 \times \frac{(1 + 0.005)^{204} - 1}{0.005}
\]
\[
FV_X = 150 \times \frac{(1.005)^{204} - 1}{0.005}
\]
Calculating \((1.005)^{204}\):
\[
(1.005)^{204} \approx 2.30039
\]
Then:
\[
FV_X = 150 \times \frac{2.30039 - 1}{0.005}
\]
\[
FV_X = 150 \times \frac{1.30039}{0.005}
\]
\[
FV_X = 150 \times 260.078
\]
\[
FV_X \approx 39011.7
\]
### Person Y: Investment with Bank Y
- Yearly investment (PMT) = R 380
- Annual interest rate (r) = 9% = 0.09
- Total number of years (t) = 17 years
The future value of an annuity (FV) can also be calculated using the formula:
\[
FV = PMT \times \frac{(1 + r)^t - 1}{r}
\]
Substituting for Person Y:
\[
FV_Y = 380 \times \frac{(1 + 0.09)^{17} - 1}{0.09}
\]
Calculating \((1.09)^{17}\):
\[
(1.09)^{17} \approx 4.3604
\]
Then:
\[
FV_Y = 380 \times \frac{4.3604 - 1}{0.09}
\]
\[
FV_Y = 380 \times \frac{3.3604}{0.09}
\]
\[
FV_Y = 380 \times 37.0044
\]
\[
FV_Y \approx 14081.672
\]
### Conclusion:
- Future Value of X after 17 years: R 39,011.7
- Future Value of Y after 17 years: R 14,081.67
Therefore, the one with the most money available after exactly 17 years is **Person X** with **approximately R 39,011.7**.
The correct answer from the options provided is:
**None of the above**.
### Person X: Investment with Bank X
- Monthly investment (PMT) = R 150
- Annual interest rate (r) = 6% = 0.06
- Monthly interest rate (i) = r / 12 = 0.06 / 12 = 0.005
- Total number of months (n) = 17 years * 12 months/year = 204 months
The future value of an annuity (FV) can be calculated using the formula:
\[
FV = PMT \times \frac{(1 + i)^n - 1}{i}
\]
Now substituting the values:
\[
FV_X = 150 \times \frac{(1 + 0.005)^{204} - 1}{0.005}
\]
\[
FV_X = 150 \times \frac{(1.005)^{204} - 1}{0.005}
\]
Calculating \((1.005)^{204}\):
\[
(1.005)^{204} \approx 2.30039
\]
Then:
\[
FV_X = 150 \times \frac{2.30039 - 1}{0.005}
\]
\[
FV_X = 150 \times \frac{1.30039}{0.005}
\]
\[
FV_X = 150 \times 260.078
\]
\[
FV_X \approx 39011.7
\]
### Person Y: Investment with Bank Y
- Yearly investment (PMT) = R 380
- Annual interest rate (r) = 9% = 0.09
- Total number of years (t) = 17 years
The future value of an annuity (FV) can also be calculated using the formula:
\[
FV = PMT \times \frac{(1 + r)^t - 1}{r}
\]
Substituting for Person Y:
\[
FV_Y = 380 \times \frac{(1 + 0.09)^{17} - 1}{0.09}
\]
Calculating \((1.09)^{17}\):
\[
(1.09)^{17} \approx 4.3604
\]
Then:
\[
FV_Y = 380 \times \frac{4.3604 - 1}{0.09}
\]
\[
FV_Y = 380 \times \frac{3.3604}{0.09}
\]
\[
FV_Y = 380 \times 37.0044
\]
\[
FV_Y \approx 14081.672
\]
### Conclusion:
- Future Value of X after 17 years: R 39,011.7
- Future Value of Y after 17 years: R 14,081.67
Therefore, the one with the most money available after exactly 17 years is **Person X** with **approximately R 39,011.7**.
The correct answer from the options provided is:
**None of the above**.