Suppose you want to have $$ \$ 400,000 $$ for retirement in 35 years. Your account earns $$ 10 \% $$ interest. Round answers to the nearest cent. a) How much would you need to deposit in the account each month? b) How much interest will you earn?

Question

UpStudy AI · Accepted Answer

Let the desired future amount be 
$$
FV = \$400,\!000.
$$
The account earns an annual interest rate of $$10\%$$. Since deposits are made monthly, the monthly interest rate is

$$
i = \frac{0.10}{12} \approx 0.00833333.
$$

There are

$$
n = 35 \times 12 = 420 \text{ months}.
$$

The future value of a series of monthly deposits (an annuity) is given by

$$
FV = PMT \cdot \frac{(1+i)^n - 1}{i},
$$

where $$PMT$$ is the monthly payment.

Solve for $$PMT$$:

$$
PMT = \frac{FV \cdot i}{(1+i)^n - 1}.
$$

Substitute the known values:

$$
PMT = \frac{400,\!000 \cdot \frac{0.10}{12}}{(1+\frac{0.10}{12})^{420} - 1}.
$$

Thus,

$$
PMT = \frac{400,\!000 \cdot 0.00833333}{(1.00833333)^{420} - 1}.
$$

Step 1. Compute $$(1.00833333)^{420}$$:

Using the properties of logarithms,

$$
(1.00833333)^{420} = \exp\Big(420 \cdot \ln(1.00833333)\Big).
$$

A good approximation is

$$
\ln(1.00833333) \approx 0.00829876,
$$

so

$$
420 \cdot 0.00829876 \approx 3.48550,
$$

and

$$
(1.00833333)^{420} \approx e^{3.48550} \approx 32.67.
$$

Step 2. Compute the denominator:

$$
(1.00833333)^{420} - 1 \approx 32.67 - 1 = 31.67.
$$

Step 3. Now compute $$PMT$$:

$$
PMT \approx \frac{400,\!000 \cdot 0.00833333}{31.67} \approx \frac{3333.33}{31.67} \approx 105.26.
$$

Thus, the monthly deposit is approximately

$$
\boxed{\$105.26}.
$$

For part (b), we need to determine the total interest earned.

Step 4. Total amount contributed by the deposits is

$$
\text{Total Contributions} = PMT \times n \approx 105.26 \times 420 \approx 44,\!209.20.
$$

Step 5. The interest earned is the difference between the final account balance and the total contributions:

$$
\text{Interest Earned} = FV - \text{Total Contributions} \approx 400,\!000 - 44,\!209.20 \approx 355,\!790.80.
$$

Thus, the interest earned is approximately

$$
\boxed{\$355,\!790.80}.
$$
a) You need to deposit approximately \$105.26 each month. b) You will earn about \$355,790.80 in interest.

Suppose you want to have \( \$ 400,000 \) for retirement in 35 years. Your account earns \( 10 \% \) interest. Round answers to the nearest cent. a) How much would you need to deposit in the account each month? b) How much interest will you earn?

Solución de inteligencia artificial de Upstudy

Responder

Solución

Extra Insights

preguntas relacionadas

Latest Economics Questions