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Math  /  Algebra

QuestionQ. 3 If $350,000\$ 350,000 is to grow to $700,000\$ 700,000 over an 8 -year period, at what annual interest rate must it be invested, given that interest is computed quarterly?

Studdy Solution

STEP 1

1. The initial investment (principal) is \$350,000.
2. The future value of the investment is \$700,000.
3. The investment period is 8 years.
4. Interest is compounded quarterly.
5. We need to find the annual interest rate.

STEP 2

1. Identify the formula for compound interest.
2. Define the variables in the formula.
3. Substitute the known values into the formula.
4. Solve for the unknown variable, the annual interest rate.

STEP 3

Identify the formula for compound interest. The formula for compound interest when compounded quarterly is:
A=P(1+rn)nt A = P \left(1 + \frac{r}{n}\right)^{nt}
where: - A A is the future value of the investment/loan, including interest. - P P is the principal investment amount (initial deposit or loan amount). - r r is the annual interest rate (decimal). - n n is the number of times that interest is compounded per year. - t t is the time the money is invested for in years.

STEP 4

Define the variables in the formula based on the problem statement:
- A=700,000 A = 700,000 - P=350,000 P = 350,000 - n=4 n = 4 (since interest is compounded quarterly) - t=8 t = 8
We need to find r r .

STEP 5

Substitute the known values into the formula:
700,000=350,000(1+r4)4×8 700,000 = 350,000 \left(1 + \frac{r}{4}\right)^{4 \times 8}
Simplify the equation:
700,000=350,000(1+r4)32 700,000 = 350,000 \left(1 + \frac{r}{4}\right)^{32}

STEP 6

Solve for the unknown variable, the annual interest rate r r .
First, divide both sides by 350,000:
2=(1+r4)32 2 = \left(1 + \frac{r}{4}\right)^{32}
Take the 32nd root of both sides to solve for (1+r4) \left(1 + \frac{r}{4}\right) :
(1+r4)=2132 \left(1 + \frac{r}{4}\right) = 2^{\frac{1}{32}}
Subtract 1 from both sides:
r4=2132−1 \frac{r}{4} = 2^{\frac{1}{32}} - 1
Multiply both sides by 4 to solve for r r :
r=4(2132−1) r = 4 \left(2^{\frac{1}{32}} - 1\right)
Calculate the value:
r≈4×(1.0227−1) r \approx 4 \times (1.0227 - 1) r≈4×0.0227 r \approx 4 \times 0.0227 r≈0.0908 r \approx 0.0908
Convert the decimal to a percentage:
The annual interest rate is approximately 9.08% 9.08\% .

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