You deposit $$ \$ 500 $$ each month into an account earning $$ 7 \% $$ interest compounded monthly. Round to the nearest cent as needed. a) How much will you have in the account in 30 years? b) How much total money will you put into the account? c) How much total interest will you earn?

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UpStudy AI · Accepted Answer

To solve this problem, we will use the future value of an annuity formula, which is given by:

$$
FV = P \times \frac{(1 + r)^n - 1}{r}
$$

where:
- $$ FV $$ is the future value of the annuity,
- $$ P $$ is the amount deposited each period,
- $$ r $$ is the interest rate per period,
- $$ n $$ is the total number of deposits.

### Given:
- Monthly deposit $$ P = 500 $$
- Annual interest rate $$ 7\% = 0.07 $$
- Monthly interest rate $$ r = \frac{0.07}{12} $$
- Total number of deposits over 30 years $$ n = 30 \times 12 $$

Let's calculate each part step by step.

### Step 1: Calculate $$ r $$ and $$ n $$

1. Calculate the monthly interest rate $$ r $$:
   $$
   r = \frac{0.07}{12}
   $$

2. Calculate the total number of deposits $$ n $$:
   $$
   n = 30 \times 12
   $$

### Step 2: Calculate the future value $$ FV $$

Using the values of $$ P $$, $$ r $$, and $$ n $$ in the future value formula.

### Step 3: Calculate total money deposited

Total money deposited is simply:
$$
\text{Total Deposits} = P \times n
$$

### Step 4: Calculate total interest earned

Total interest earned can be calculated as:
$$
\text{Total Interest} = FV - \text{Total Deposits}
$$

Now, let's perform the calculations.

First, we will calculate $$ r $$ and $$ n $$, then use them to find $$ FV $$, total deposits, and total interest.

Let's proceed with the calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$30\times 12$$
- step1: Multiply the numbers:
  $$360$$
Calculate or simplify the expression $$ 0.07/12 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{0.07}{12}$$
- step1: Convert the expressions:
  $$\frac{\frac{7}{100}}{12}$$
- step2: Multiply by the reciprocal:
  $$\frac{7}{100}\times \frac{1}{12}$$
- step3: Multiply the fractions:
  $$\frac{7}{100\times 12}$$
- step4: Multiply:
  $$\frac{7}{1200}$$
Calculate or simplify the expression $$ 500*((1+(0.07/12))^(30*12)-1)/(0.07/12) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{500\left(\left(1+\left(\frac{0.07}{12}\right)\right)^{30\times 12}-1\right)}{\left(\frac{0.07}{12}\right)}$$
- step1: Remove the parentheses:
  $$\frac{500\left(\left(1+\left(\frac{0.07}{12}\right)\right)^{30\times 12}-1\right)}{\frac{0.07}{12}}$$
- step2: Divide the terms:
  $$\frac{500\left(\left(1+\frac{7}{1200}\right)^{30\times 12}-1\right)}{\frac{0.07}{12}}$$
- step3: Add the numbers:
  $$\frac{500\left(\left(\frac{1207}{1200}\right)^{30\times 12}-1\right)}{\frac{0.07}{12}}$$
- step4: Multiply the numbers:
  $$\frac{500\left(\left(\frac{1207}{1200}\right)^{360}-1\right)}{\frac{0.07}{12}}$$
- step5: Subtract the numbers:
  $$\frac{500\times \frac{1207^{360}-1200^{360}}{1200^{360}}}{\frac{0.07}{12}}$$
- step6: Divide the terms:
  $$\frac{500\times \frac{1207^{360}-1200^{360}}{1200^{360}}}{\frac{7}{1200}}$$
- step7: Multiply the numbers:
  $$\frac{\frac{1207^{360}-1200^{360}}{5^{717}\times 4^{359}\times 12^{360}}}{\frac{7}{1200}}$$
- step8: Multiply by the reciprocal:
  $$\frac{1207^{360}-1200^{360}}{5^{717}\times 4^{359}\times 12^{360}}\times \frac{1200}{7}$$
- step9: Rewrite the expression:
  $$\frac{1207^{360}-1200^{360}}{5^{717}\times 4^{359}\times 12^{360}}\times \frac{25\times 48}{7}$$
- step10: Rewrite the expression:
  $$\frac{1207^{360}-1200^{360}}{5^{717}\times 4^{359}\times 12^{360}}\times \frac{5^{2}\times 48}{7}$$
- step11: Reduce the numbers:
  $$\frac{1207^{360}-1200^{360}}{5^{715}\times 4^{359}\times 12^{360}}\times \frac{48}{7}$$
- step12: Rewrite the expression:
  $$\frac{1207^{360}-1200^{360}}{5^{715}\times 4^{359}\times 12^{360}}\times \frac{16\times 3}{7}$$
- step13: Rewrite the expression:
  $$\frac{1207^{360}-1200^{360}}{5^{715}\times 4^{359}\times 12^{360}}\times \frac{4^{2}\times 3}{7}$$
- step14: Reduce the numbers:
  $$\frac{1207^{360}-1200^{360}}{5^{715}\times 4^{357}\times 12^{360}}\times \frac{3}{7}$$
- step15: Rewrite the expression:
  $$\frac{1207^{360}-1200^{360}}{5^{715}\times 4^{357}\times 3^{360}\times 4^{360}}\times \frac{3}{7}$$
- step16: Reduce the numbers:
  $$\frac{1207^{360}-1200^{360}}{5^{715}\times 4^{357}\times 3^{359}\times 4^{360}}\times \frac{1}{7}$$
- step17: Multiply the fractions:
  $$\frac{1207^{360}-1200^{360}}{5^{715}\times 4^{717}\times 3^{359}\times 7}$$
- step18: Multiply:
  $$\frac{1207^{360}-1200^{360}}{7\times 5^{715}\times 4^{717}\times 3^{359}}$$
Calculate or simplify the expression $$ 500*360 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$500\times 360$$
- step1: Multiply the numbers:
  $$180000$$
Calculate or simplify the expression $$ 609985.497888 - 500*360 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$609985.497888-500\times 360$$
- step1: Multiply the numbers:
  $$609985.497888-180000$$
- step2: Subtract the numbers:
  $$429985.497888$$
Now that we have completed the calculations, let's summarize the results:

### a) Future Value in the Account After 30 Years
The future value $$ FV $$ of the account after 30 years is approximately:
$$
FV \approx 609985.50
$$

### b) Total Money Deposited
The total amount of money deposited into the account over 30 years is:
$$
\text{Total Deposits} = 500 \times 360 = 180000
$$

### c) Total Interest Earned
The total interest earned can be calculated as:
$$
\text{Total Interest} = FV - \text{Total Deposits} \approx 609985.50 - 180000 = 429985.50
$$

### Final Answers
- a) The amount in the account in 30 years: **\$609,985.50**
- b) Total money deposited: **\$180,000.00**
- c) Total interest earned: **\$429,985.50**
After 30 years, you will have \$609,985.50 in the account. You will have deposited a total of \$180,000, and the total interest earned will be \$429,985.50.

You deposit \( \$ 500 \) each month into an account earning \( 7 \% \) interest compounded monthly. Round to the nearest cent as needed. a) How much will you have in the account in 30 years? b) How much total money will you put into the account? c) How much total interest will you earn?

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