To find the present value of the annuity where Sadeeq plans to deposit $700 every month for the next 7 years at an interest rate of 5% per year, we can use the formula for the present value of an ordinary annuity:
\[ PV = P \times \left(1 - (1 + r)^{-n}\right) / r \]
where:
- \(PV\) is the present value of the annuity.
- \(P\) is the payment amount per period.
- \(r\) is the interest rate per period.
- \(n\) is the total number of payments.
Given:
- Monthly payment \(P = 700\)
- Annual interest rate = 5% → Monthly interest rate \(r = \frac{0.05}{12} = \frac{0.05}{12} \approx 0.00416667\)
- Total number of payments \(n = 7 \times 12 = 84\)
Substitute into the formula:
-
Calculate \(r\) and \(n\):
\[ r = \frac{0.05}{12} \approx 0.00416667 \] \[ n = 7 \times 12 = 84 \]
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Plug in the values into the present value formula:
\[ PV = 700 \times \left(1 - (1 + 0.00416667)^{-84}\right) / 0.00416667 \]
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Calculate \( (1 + 0.00416667)^{-84} \):
\[ (1 + 0.00416667)^{-84} \approx (1.00416667)^{-84} \approx 0.680583 \]
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Continue with the calculation of \(PV\):
\[ PV = 700 \times \left(1 - 0.680583\right) / 0.00416667 \] \[ = 700 \times \left(0.319417\right) / 0.00416667 \] \[ = 700 \times 76.54604 \] \[ \approx 53582.23 \]
Final Answer
The present value of the annuity is approximately $53,582.23 when rounded to the nearest hundredth.