Reasoning Tommy earned $$ \$ 15.60 $$ in interest after 4 years on a principal of $$ \$ 100 $$. His simple interest rate is $$ 3.9 \% $$. Jane earned $$ \$ 91.20 $$ in interest after 2 years on a principal of $$ \$ 1,200 $$. Her simple interest rate is $$ 3.8 \% $$. Which bank would you rather use, Tommy's or Jane's? Explain your reasoning. Which bank would you rather use? A. Jane's because her investment took less time. B. Tommy's because his bank accepted a smaller principal. C. Jane's because her bank paid more interest. D. Tommy's because his bank offers a better interest rate.

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To determine which bank is preferable based on the information provided, we need to analyze the interest rates and the amount of interest earned relative to the principal for both Tommy and Jane.

### Tommy's Investment:
- Principal ($$ P_T $$): $$ 100 $$
- Interest earned ($$ I_T $$): $$ 15.60 $$
- Time ($$ t_T $$): $$ 4 $$ years
- Interest rate ($$ r_T $$): $$ 3.9\% $$

The formula for simple interest is given by:
$$
I = P \times r \times t
$$
We can verify Tommy's interest rate using the formula:
$$
15.60 = 100 \times r_T \times 4
$$
Solving for $$ r_T $$:
$$
r_T = \frac{15.60}{100 \times 4} = \frac{15.60}{400} = 0.039 = 3.9\%
$$
This confirms that Tommy's interest rate is correct.

### Jane's Investment:
- Principal ($$ P_J $$): $$ 1200 $$
- Interest earned ($$ I_J $$): $$ 91.20 $$
- Time ($$ t_J $$): $$ 2 $$ years
- Interest rate ($$ r_J $$): $$ 3.8\% $$

Using the same formula for Jane:
$$
91.20 = 1200 \times r_J \times 2
$$
Solving for $$ r_J $$:
$$
r_J = \frac{91.20}{1200 \times 2} = \frac{91.20}{2400} = 0.038 = 3.8\%
$$
This confirms that Jane's interest rate is also correct.

### Comparison:
1. **Interest Earned**:
   - Tommy earned $$ 15.60 $$ over 4 years.
   - Jane earned $$ 91.20 $$ over 2 years.
   - Jane earned more interest in a shorter time.

2. **Interest Rate**:
   - Tommy's rate: $$ 3.9\% $$
   - Jane's rate: $$ 3.8\% $$
   - Tommy has a slightly better interest rate.

3. **Principal**:
   - Tommy's principal: $$ 100 $$
   - Jane's principal: $$ 1200 $$
   - Tommy's bank accepted a smaller principal.

### Conclusion:
Based on the analysis:
- **Option A**: Jane's because her investment took less time. (True)
- **Option B**: Tommy's because his bank accepted a smaller principal. (True)
- **Option C**: Jane's because her bank paid more interest. (True)
- **Option D**: Tommy's because his bank offers a better interest rate. (True)

However, the most compelling reason to choose Jane's bank is that she earned more interest in a shorter time frame, which is a significant factor for most investors.

Thus, the best choice is:
**C. Jane's because her bank paid more interest.**
You should choose Jane's bank because she earned more interest in a shorter time.

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