Question Allison accumulated $$ \$ 7,000 $$ in credit card debt. If the interest rate is $$ 15 \% $$ per year and she does not make any paym 3 years, how much will she owe on this debt in 3 years for quarterly compounding? Round your answer to the nearest cent. Do NOT round until you calculate the final answer

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To calculate the amount Allison will owe on her credit card debt in 3 years with quarterly compounding, we can use the formula for compound interest:

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

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

Given:
- $$ P = \$7,000 $$
- $$ r = 15\% = 0.15 $$
- $$ n = 4 $$ (quarterly compounding)
- $$ t = 3 $$ years

Substitute the given values into the formula:

$$ A = 7000 \left( 1 + \frac{0.15}{4} \right)^{4 \times 3} $$

Now, we can calculate the amount Allison will owe on her credit card debt in 3 years with quarterly compounding.
Calculate the value by following steps:
- step0: Calculate:
  $$7000\left(1+\frac{0.15}{4}\right)^{4\times 3}$$
- step1: Divide the terms:
  $$7000\left(1+\frac{3}{80}\right)^{4\times 3}$$
- step2: Add the numbers:
  $$7000\left(\frac{83}{80}\right)^{4\times 3}$$
- step3: Multiply the numbers:
  $$7000\left(\frac{83}{80}\right)^{12}$$
- step4: Simplify:
  $$7000\times \frac{83^{12}}{80^{12}}$$
- step5: Rewrite the expression:
  $$40\times 175\times \frac{83^{12}}{80^{12}}$$
- step6: Rewrite the expression:
  $$40\times 175\times \frac{83^{12}}{40^{12}\times 2^{12}}$$
- step7: Reduce the numbers:
  $$175\times \frac{83^{12}}{40^{11}\times 2^{12}}$$
- step8: Rewrite the expression:
  $$25\times 7\times \frac{83^{12}}{40^{11}\times 2^{12}}$$
- step9: Rewrite the expression:
  $$25\times 7\times \frac{83^{12}}{5^{11}\times 8^{11}\times 2^{12}}$$
- step10: Rewrite the expression:
  $$5^{2}\times 7\times \frac{83^{12}}{5^{11}\times 8^{11}\times 2^{12}}$$
- step11: Reduce the numbers:
  $$7\times \frac{83^{12}}{5^{9}\times 8^{11}\times 2^{12}}$$
- step12: Multiply the fractions:
  $$\frac{7\times 83^{12}}{5^{9}\times 8^{15}}$$
After calculating, Allison will owe approximately $10,888.18 on her credit card debt in 3 years with quarterly compounding.
Allison will owe approximately \$10,888.18 on her credit card debt in 3 years with quarterly compounding.

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