Math question

Question

Imagine investing $200 monthly into a 401(k) retirement account that averages an APR of 3.00% compounded monthly. You do this consistently for 45 years until retirement. Use the savings plan formula to calculate your final balance (future value). After this calculation, find how much of this future value is principal and how much is interest.

1). Final Balance:

???$ (Round to the nearest dollar and do not use commas.)


2). Principal:

????$ (Round to the nearest dollar and do not use commas.)


3). Interest:

???$ (Round to the nearest dollar and do not use commas.)

Answer 1

Hi Carolina,

To solve this problem, we need to calculate the future value of a series of monthly investments using the future value of an annuity formula. The key details are:

- Monthly investment: $200

- Annual interest rate (APR): 3.00%

- Investment period: 45 years

**Step 1: Calculate the future value of the annuity**

The formula for the future value of an annuity is:

FV = P × ( ( (1 + r)^n − 1 ) / r )

Where:

- P is the monthly investment ($200).

- r is the monthly interest rate (APR divided by 12 months). So, r = 3% / 12 = 0.0025.

- n is the total number of payments (45 years × 12 months/year = 540 months).

Plugging in the values:

FV = 200 × ( ( (1 + 0.0025)^540 − 1 ) / 0.0025 )

Calculate:

(1 + 0.0025)^540 ≈ 3.875

Then:

FV = 200 × ( (3.875 − 1) / 0.0025 )

FV = 200 × 1150

FV ≈ $230,000

**Step 2: Calculate the principal**

Principal is simply the total amount invested:

Principal = Monthly investment × Total number of payments = 200 × 540 = $108,000

**Step 3: Calculate the interest earned**

Interest is the difference between the future value and the principal:

Interest = FV − Principal = 230,000 − 108,000 = $122,000

**Summary**

1. Final Balance (Future Value): $230000

2. Principal: $108000

3. Interest: $122000

All amounts are rounded to the nearest dollar without commas as required.