Math question

Question

Nicole deposits $150 per month in an investment account that earns 5.50% compounded monthly for 6 years.

Gabe, on the other hand, deposits $1800 at the end of each year into a similar account that also earns 5.50% compounded annually for 6 years.

Even though they each contribute the same amount of principal with the same APR and length of investment time, who comes out ahead and by how much?

1) Who comes out ahead?

Nicole
Gabe

2) How much more does he/she have? 

???$ (Round to the nearest dollar.)

Answer 1

Let's calculate both cases separately:

1) Nicole — Monthly Deposit with Monthly Compounding

1. Deposit per month = $150
2. Annual interest rate = 5.50% = 0.055
3. Monthly interest rate = 0.055 / 12 = 0.0045833
4. Number of months = 6 × 12 = 72

Future Value Formula for Monthly Payments (Ordinary Annuity):

FV = P \times \frac{(1 + r)^{n} - 1}{r}

FV = 150 \times \frac{(1 + 0.0045833)^{72} - 1}{0.0045833}

FV = 150 \times \frac{(1.0045833)^{72} - 1}{0.0045833} = 150 \times 84.486

FV \approx 12,673

2) Gabe — Annual Deposit with Annual Compounding

1. Deposit per year = $1800
2. Annual interest rate = 5.50% = 0.055

Number of years = 6

Future Value Formula for Annual Payments (Ordinary Annuity):

FV = P \times \frac{(1 + r)^{n} - 1}{r}

FV = 1800 \times \frac{(1 + 0.055)^{6} - 1}{0.055} = 1800 \times 6.995

FV \approx 12,591

3) Comparison:

1. Nicole's total = $12,673
2. Gabe's total = $12,591

Nicole comes out ahead by:

12,673 - 12,591 = 82

Final Answer:

1. Nicole comes out ahead.
2. $82 more.