Aubrey invested $$ \$ 150 $$ in an account paying an interest rate of $$ 6 \frac{5}{8} \% $$ compounded monthly. Miguel invested $$ \$ 150 $$ in an account paying an interest rate of $$ 6 \frac{1}{4} \% $$ compounded daily. After 7 years, how much more money would Aubrey have in her account than Miguel, to the nearest dollar?

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To determine how much more money Aubrey would have in her account compared to Miguel after 7 years, we'll calculate the future value of both investments using the compound interest formula:

$$ A = P \left(1 + \frac{r}{n}\right)^{nt} $$

Where:
- $$ A $$ = the amount of money accumulated after $$ t $$ years, including interest.
- $$ P $$ = the principal amount (\$150 for both investors).
- $$ r $$ = annual interest rate (in decimal).
- $$ n $$ = number of times that interest is compounded per year.
- $$ t $$ = time the money is invested for in years.

### Aubrey's Investment
- **Principal ($$ P $$)**: \$150
- **Annual Interest Rate ($$ r $$)**: $$ 6 \frac{5}{8}\% = 6.625\% = 0.06625 $$
- **Compounded Monthly ($$ n $$)**: 12 times a year
- **Time ($$ t $$)**: 7 years

Plugging these values into the formula:

$$ A_{\text{Aubrey}} = 150 \left(1 + \frac{0.06625}{12}\right)^{12 \times 7} $$
$$ A_{\text{Aubrey}} \approx 150 \times (1.0055208333)^{84} $$
$$ A_{\text{Aubrey}} \approx 150 \times 1.588 $$
$$ A_{\text{Aubrey}} \approx \$238.20 $$

### Miguel's Investment
- **Principal ($$ P $$)**: \$150
- **Annual Interest Rate ($$ r $$)**: $$ 6 \frac{1}{4}\% = 6.25\% = 0.0625 $$
- **Compounded Daily ($$ n $$)**: 365 times a year
- **Time ($$ t $$)**: 7 years

Plugging these values into the formula:

$$ A_{\text{Miguel}} = 150 \left(1 + \frac{0.0625}{365}\right)^{365 \times 7} $$
$$ A_{\text{Miguel}} \approx 150 \times (1.0001712329)^{2555} $$
$$ A_{\text{Miguel}} \approx 150 \times 1.548 $$
$$ A_{\text{Miguel}} \approx \$232.26 $$

### Difference in Amounts
$$ \text{Difference} = A_{\text{Aubrey}} - A_{\text{Miguel}} $$
$$ \text{Difference} \approx 238.20 - 232.26 $$
$$ \text{Difference} \approx \$5.94 $$

Rounding to the nearest dollar:

**Aubrey would have approximately \$6 more than Miguel after 7 years.**

**Answer:** $6
Aubrey would have approximately \$6 more than Miguel after 7 years.

Aubrey invested in an account paying an interest rate of compounded monthly.
Miguel invested in an account paying an interest rate of compounded daily. After 7
years, how much more money would Aubrey have in her account than Miguel, to the nearest
dollar?

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Aubrey invested in an account paying an interest rate of compounded monthly. Miguel invested in an account paying an interest rate of compounded daily. After 7 years, how much more money would Aubrey have in her account than Miguel, to the nearest dollar?