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Math  /  Algebra

Questionshley wants to save money to buy a motorcycle. She invests in an ordinary annuity that earns 4.8%4.8 \% interest, compounded monthly. Payments will be made at ie end of each month. ow much money will she need to pay into the annuity each month for the annuity to have a total value of $7000\$ 7000 after 5 years? ot nound intermediate computations, and round your final answer to the nearest cent. If necessary, refer to the list of financial formulas.

Studdy Solution

STEP 1

1. Ashley wants to save a total of $7000 in an annuity.
2. The annuity earns an interest rate of 4.8% per annum, compounded monthly.
3. Payments are made at the end of each month.
4. The time period for the annuity is 5 years.
5. We need to find the monthly payment amount.

STEP 2

1. Identify the formula for the future value of an ordinary annuity.
2. Define the variables in the formula.
3. Substitute the known values into the formula.
4. Solve for the monthly payment.

STEP 3

Identify the formula for the future value of an ordinary annuity. The formula is:
FV=P×(1+r)n−1r FV = P \times \frac{(1 + r)^n - 1}{r}
where: - FV FV is the future value of the annuity. - P P is the monthly payment. - r r is the monthly interest rate. - n n is the total number of payments.

STEP 4

Define the variables in the formula.
- FV=7000 FV = 7000 - r=4.8%12=0.04812 r = \frac{4.8\%}{12} = \frac{0.048}{12} - n=5×12=60 n = 5 \times 12 = 60

STEP 5

Substitute the known values into the formula.
7000=P×(1+0.04812)60−10.04812 7000 = P \times \frac{(1 + \frac{0.048}{12})^{60} - 1}{\frac{0.048}{12}}

STEP 6

Solve for the monthly payment P P .
First, calculate the monthly interest rate:
r=0.04812=0.004 r = \frac{0.048}{12} = 0.004
Next, calculate the expression:
(1+0.004)60−1 (1 + 0.004)^{60} - 1
=1.00460−1 = 1.004^{60} - 1
Calculate 1.00460 1.004^{60} :
1.00460≈1.27048 1.004^{60} \approx 1.27048
Subtract 1:
1.27048−1=0.27048 1.27048 - 1 = 0.27048
Now, substitute back into the equation:
7000=P×0.270480.004 7000 = P \times \frac{0.27048}{0.004}
7000=P×67.62 7000 = P \times 67.62
Solve for P P :
P=700067.62 P = \frac{7000}{67.62}
P≈103.49 P \approx 103.49
Therefore, Ashley needs to pay approximately 103.49 \boxed{103.49} dollars each month.

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