Aaron and Shawna deposit $$ \$ 9,832.00 $$ into a savings account which earns $$ 7 \% $$ interest compounded monthly. They want to use the money in the account to go on a trip in 2 years. How much will they be able to spend? Use the formula $$ A=P\left(1+\frac{r}{n}\right)^{n t} $$, where $$ A $$ is the balance (final amount), $$ P $$ is the principal (starting amount), $$ r $$ is the interest rate expressed as a decimal, $$ n $$ is the number of times per year that the interest is compounded, and $$ t $$ is the time in years. Round your answer to the nearest cent.

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To find out how much Aaron and Shawna will be able to spend after 2 years, we need to calculate the balance in their savings account after 2 years using the formula for compound interest:

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

Given:
- Principal (starting amount) $$ P = \$9,832.00 $$
- Interest rate $$ r = 7\% = 0.07 $$ (as a decimal)
- Compounded monthly, so $$ n = 12 $$ times per year
- Time in years $$ t = 2 $$ years

Substitute the given values into the formula:

$$ A = 9832\left(1 + \frac{0.07}{12}\right)^{12 \times 2} $$

Now, we can calculate the balance in the savings account after 2 years.
Calculate the value by following steps:
- step0: Calculate:
  $$9832\left(1+\frac{0.07}{12}\right)^{12\times 2}$$
- step1: Divide the terms:
  $$9832\left(1+\frac{7}{1200}\right)^{12\times 2}$$
- step2: Add the numbers:
  $$9832\left(\frac{1207}{1200}\right)^{12\times 2}$$
- step3: Multiply the numbers:
  $$9832\left(\frac{1207}{1200}\right)^{24}$$
- step4: Simplify:
  $$9832\times \frac{1207^{24}}{1200^{24}}$$
- step5: Rewrite the expression:
  $$8\times 1229\times \frac{1207^{24}}{1200^{24}}$$
- step6: Rewrite the expression:
  $$8\times 1229\times \frac{1207^{24}}{16^{24}\times 75^{24}}$$
- step7: Rewrite the expression:
  $$2^{3}\times 1229\times \frac{1207^{24}}{2^{96}\times 75^{24}}$$
- step8: Reduce the numbers:
  $$1229\times \frac{1207^{24}}{2^{93}\times 75^{24}}$$
- step9: Multiply the fractions:
  $$\frac{1229\times 1207^{24}}{2^{93}\times 75^{24}}$$
The balance in the savings account after 2 years will be approximately $11,304.89. Therefore, Aaron and Shawna will be able to spend $11,304.89 on their trip.
After 2 years, Aaron and Shawna will have approximately \$11,304.89 in their savings account.

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