Question Edgar accumulated $$ \$ 5,000 $$ in credit card debt. If the interest rate is $$ 20 \% $$ per year and he does not make any payments for 2 years, how much will he owe on this debt in 2 years with monthly compounding? Round your answer to the nearest cent. Do NOT round until you calculate the final answer Provide your answer below:

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UpStudy AI · Accepted Answer

To calculate the amount Edgar will owe on his credit card debt after 2 years with monthly 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 (the initial amount of money).
- $$ r $$ is the annual interest rate (decimal).
- $$ n $$ is the number of times that interest is compounded per year.
- $$ t $$ is the number of years the money is invested or borrowed.

Given:
- $$ P = 5000 $$
- $$ r = 0.20 $$ (20% as a decimal)
- $$ n = 12 $$ (monthly compounding)
- $$ t = 2 $$

Now, we can substitute these values into the formula:

$$
A = 5000 \left(1 + \frac{0.20}{12}\right)^{12 \times 2}
$$

Let's calculate this step by step.

1. Calculate $$ \frac{r}{n} $$:
   $$
   \frac{0.20}{12} = 0.01666667
   $$

2. Calculate $$ nt $$:
   $$
   12 \times 2 = 24
   $$

3. Calculate $$ A $$:
   $$
   A = 5000 \left(1 + 0.01666667\right)^{24}
   $$

Now, let's perform the calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$5000\left(1+\frac{0.2}{12}\right)^{12\times 2}$$
- step1: Divide the terms:
  $$5000\left(1+\frac{1}{60}\right)^{12\times 2}$$
- step2: Add the numbers:
  $$5000\left(\frac{61}{60}\right)^{12\times 2}$$
- step3: Multiply the numbers:
  $$5000\left(\frac{61}{60}\right)^{24}$$
- step4: Simplify:
  $$5000\times \frac{61^{24}}{60^{24}}$$
- step5: Rewrite the expression:
  $$20\times 250\times \frac{61^{24}}{60^{24}}$$
- step6: Rewrite the expression:
  $$20\times 250\times \frac{61^{24}}{20^{24}\times 3^{24}}$$
- step7: Reduce the numbers:
  $$250\times \frac{61^{24}}{20^{23}\times 3^{24}}$$
- step8: Rewrite the expression:
  $$10\times 25\times \frac{61^{24}}{20^{23}\times 3^{24}}$$
- step9: Rewrite the expression:
  $$10\times 25\times \frac{61^{24}}{10^{23}\times 2^{23}\times 3^{24}}$$
- step10: Reduce the numbers:
  $$25\times \frac{61^{24}}{10^{22}\times 2^{23}\times 3^{24}}$$
- step11: Rewrite the expression:
  $$25\times \frac{61^{24}}{5^{22}\times 2^{22}\times 2^{23}\times 3^{24}}$$
- step12: Rewrite the expression:
  $$5^{2}\times \frac{61^{24}}{5^{22}\times 2^{22}\times 2^{23}\times 3^{24}}$$
- step13: Reduce the numbers:
  $$1\times \frac{61^{24}}{5^{20}\times 2^{22}\times 2^{23}\times 3^{24}}$$
- step14: Multiply the fractions:
  $$\frac{61^{24}}{5^{20}\times 2^{45}\times 3^{24}}$$
After performing the calculations, we find that the amount Edgar will owe after 2 years is approximately:

$$
A \approx 7434.57
$$

Thus, rounding to the nearest cent, Edgar will owe **\$7,434.57** on his credit card debt after 2 years with monthly compounding.
After 2 years with monthly compounding at a 20% annual interest rate, Edgar will owe \$7,434.57 on his \$5,000 credit card debt.

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