Question 25 Town Bank offers a $$ 2.25 \% $$ interest rate, while Charter One offers $$ 2.8 \% $$. Both banks compound interest annually. If Rob wants to set up a new account with $$ \$ 5000 $$, how much more money will he earn at Charter One over Town Bank after 25 years? $$ \$ 1,209.79 $$ $$ \$ 1,324.10 $$ $$ \$ 1,183.41 $$ $$ \$ 1.251 .63 $$

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UpStudy AI · Accepted Answer

To determine how much more Rob will earn at Charter One compared to Town Bank after 25 years, we'll calculate the future value of his investment with both banks and then find the difference.

### Given:
- **Principal (P):** \$5,000
- **Town Bank Interest Rate (r₁):** 2.25% per annum
- **Charter One Interest Rate (r₂):** 2.8% per annum
- **Time (t):** 25 years

### Compound Interest Formula:
$$ A = P \times (1 + r)^t $$

### Calculations:

1. **Future Value with Town Bank (A₁):**
   $$
   A₁ = 5000 \times (1 + 0.0225)^{25}
   $$
   
2. **Future Value with Charter One (A₂):**
   $$
   A₂ = 5000 \times (1 + 0.028)^{25}
   $$

3. **Difference (ΔA):**
   $$
   ΔA = A₂ - A₁
   $$

### Approximate Calculations:

Using logarithmic approximations:

- **For Town Bank:**
  $$
  A₁ \approx 5000 \times e^{0.0225 \times 25} \approx 5000 \times e^{0.5625} \approx 5000 \times 1.754 \approx \$8,770
  $$

- **For Charter One:**
  $$
  A₂ \approx 5000 \times e^{0.028 \times 25} \approx 5000 \times e^{0.7} \approx 5000 \times 2.014 \approx \$10,070
  $$

- **Difference:**
  $$
  ΔA \approx 10,070 - 8,770 = \$1,300
  $$

### Closest Answer:
The closest value to our approximation is **\$1,251.63**.

**Answer:** \$1,251.63
Rob will earn approximately \$1,251.63 more at Charter One than at Town Bank after 25 years.

Question 25
Town Bank offers a interest rate, while Charter One offers . Both banks compound interest
annually. If Rob wants to set up a new account with , how much more money will he earn at
Charter One over Town Bank after 25 years?

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Given:

Compound Interest Formula:

Calculations:

Approximate Calculations:

Closest Answer:

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Question 25 Town Bank offers a interest rate, while Charter One offers . Both banks compound interest annually. If Rob wants to set up a new account with , how much more money will he earn at Charter One over Town Bank after 25 years?