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Math  /  Algebra

QuestionA company will need $25,000\$ 25,000 in 8 years for a new addition. To meet this goal, the company deposits money in an account today that pays 9%9 \% annual interest compounded quarterly. Find the amount that should be invested to total $25,000\$ 25,000 in 8 years.
The company should invest \ \square$ (Do not round until the final answer. Then round to the nearest dollar as needed.)

Studdy Solution

STEP 1

1. The future value needed is \$25,000.
2. The interest rate is \(9\%\) per year, compounded quarterly.
3. The investment period is 8 years.
4. We need to find the present value (the amount to be invested today) to achieve the future value after 8 years.

STEP 2

1. Determine the quarterly interest rate from the annual interest rate.
2. Calculate the total number of compounding periods.
3. Use the formula for compound interest to solve for the present value.
4. Round the final answer to the nearest dollar.

STEP 3

Determine the quarterly interest rate from the annual interest rate.
The annual interest rate is 9%9\%, so the quarterly interest rate is: Quarterly interest rate=9%4=94%=2.25%=0.0225\text{Quarterly interest rate} = \frac{9\%}{4} = \frac{9}{4} \% = 2.25\% = 0.0225

STEP 4

Calculate the total number of compounding periods.
Since interest is compounded quarterly over 8 years: Total number of periods=8 years×4 quarters/year=32 quarters\text{Total number of periods} = 8 \text{ years} \times 4 \text{ quarters/year} = 32 \text{ quarters}

STEP 5

Use the formula for compound interest to solve for the present value.
The formula for compound interest is: FV=PV(1+rn)ntFV = PV \left(1 + \frac{r}{n}\right)^{nt} where: - FVFV is the future value (\$25,000) - \(PV\) is the present value (the amount to be invested) - \(r\) is the annual interest rate (0.09) - \(n\) is the number of times the interest is compounded per year (4) - \(t\) is the number of years (8)
Rearranging the formula to solve for PVPV: PV=FV(1+rn)ntPV = \frac{FV}{\left(1 + \frac{r}{n}\right)^{nt}}
Substitute the known values: PV=25000(1+0.094)32PV = \frac{25000}{\left(1 + \frac{0.09}{4}\right)^{32}}

STEP 6

Calculate the present value using the simplified expression.
First, calculate the base of the exponent: 1+0.094=1+0.0225=1.02251 + \frac{0.09}{4} = 1 + 0.0225 = 1.0225
Then raise this base to the power of the total number of periods: (1.0225)32≈2.0388(1.0225)^{32} \approx 2.0388
Now divide the future value by this result: PV=250002.0388≈12257.64PV = \frac{25000}{2.0388} \approx 12257.64

STEP 7

Round the final answer to the nearest dollar.
PV≈12258PV \approx 12258
Thus, the company should invest \$12,258 today.

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