Math  /  Algebra

QuestionHow much would you have in 7 years if you purchased a $1,0007\$ 1,0007-year savings certificate that paid 2%2 \% compounded quarterly? (Round your answer to the nearest cent.) \ \square$ Need Help? Read It

Studdy Solution

STEP 1

1. The principal amount (initial investment) is \$1,000.
2. The interest rate is 2% per annum.
3. The interest is compounded quarterly.
4. The investment period is 7 years.
5. We need to calculate the future value of the investment and round it to the nearest cent.

STEP 2

1. Identify the formula for compound interest.
2. Determine the values for each variable in the formula.
3. Substitute the values into the formula.
4. Calculate the future value.
5. Round the result to the nearest cent.

STEP 3

Identify the formula for compound interest. The formula to calculate the future value A A of an investment compounded periodically is:
A=P(1+rn)nt A = P \left(1 + \frac{r}{n}\right)^{nt}
where: - P P is the principal amount (initial investment), - r r is the annual interest rate (as a decimal), - n n is the number of times the interest is compounded per year, - t t is the time in years.

STEP 4

Determine the values for each variable in the formula.
- P=1000 P = 1000 - r=0.02 r = 0.02 (2% as a decimal) - n=4 n = 4 (since the interest is compounded quarterly) - t=7 t = 7

STEP 5

Substitute the values into the formula:
A=1000(1+0.024)4×7 A = 1000 \left(1 + \frac{0.02}{4}\right)^{4 \times 7}

STEP 6

Calculate the future value.
First, calculate the quarterly interest rate:
0.024=0.005 \frac{0.02}{4} = 0.005
Then, calculate the number of compounding periods:
4×7=28 4 \times 7 = 28
Now substitute these into the equation:
A=1000(1+0.005)28 A = 1000 \left(1 + 0.005\right)^{28} A=1000(1.005)28 A = 1000 \left(1.005\right)^{28}
Calculate (1.005)28 (1.005)^{28} :
(1.005)281.148882 (1.005)^{28} \approx 1.148882
Finally, calculate A A :
A=1000×1.148882 A = 1000 \times 1.148882 A1148.88 A \approx 1148.88

STEP 7

Round the result to the nearest cent.
The future value of the investment is approximately:
\$ \boxed{1148.88}

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