Homework Cam saved $$ \$ 195 $$ each month for the last six years while he was working. Since he has now gone back to school, his income is lower 15 points and he cannot continue to save this amount during the time he is studying. He plans to continue with his studies for three years and not withdraw any money from his savings account. Money is worth $$ 5.4 \% $$ compounded monthly. (a) How much will Cam have in total in his savings account when he finishes his studies? (b) How much did he contribute? (c) How much will be interest? (a) The future value is $$ \$ $$.

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UpStudy AI · Accepted Answer

To solve the problem, we need to calculate the future value of Cam's savings after he has saved for six years and then let that amount grow for three more years without any additional contributions.

### Step 1: Calculate the total contributions

Cam saved $$ \$195 $$ each month for six years.

1. **Total months of saving**: 
   $$
   6 \text{ years} \times 12 \text{ months/year} = 72 \text{ months}
   $$

2. **Total contributions**:
   $$
   \text{Total contributions} = 195 \text{ dollars/month} \times 72 \text{ months} = 14,040 \text{ dollars}
   $$

### Step 2: Calculate the future value of the savings

The future value (FV) of a series of monthly contributions can be calculated using the future value of an annuity formula:
$$
FV = P \times \frac{(1 + r)^n - 1}{r}
$$
where:
- $$ P $$ is the monthly contribution,
- $$ r $$ is the monthly interest rate,
- $$ n $$ is the total number of contributions.

1. **Monthly interest rate**:
   $$
   r = \frac{5.4\%}{12} = \frac{0.054}{12} = 0.0045
   $$

2. **Total number of contributions**:
   $$
   n = 72 \text{ months}
   $$

3. **Future value of contributions after 6 years**:
   $$
   FV = 195 \times \frac{(1 + 0.0045)^{72} - 1}{0.0045}
   $$

Calculating $$ (1 + 0.0045)^{72} $$:
$$
(1 + 0.0045)^{72} \approx 1.34885
$$

Now substituting back into the FV formula:
$$
FV = 195 \times \frac{1.34885 - 1}{0.0045} \approx 195 \times \frac{0.34885}{0.0045} \approx 195 \times 77.5222 \approx 15,116.83
$$

### Step 3: Calculate the future value after 3 more years

Now, we need to let this amount grow for another 3 years (or 36 months) without any additional contributions.

1. **Future value after 3 years**:
   $$
   FV = PV \times (1 + r)^n
   $$
   where:
   - $$ PV = 15,116.83 $$
   - $$ r = 0.0045 $$
   - $$ n = 36 $$

Calculating:
$$
FV = 15,116.83 \times (1 + 0.0045)^{36}
$$
Calculating $$ (1 + 0.0045)^{36} $$:
$$
(1 + 0.0045)^{36} \approx 1.1733
$$

Now substituting back into the FV formula:
$$
FV = 15,116.83 \times 1.1733 \approx 17,724.57
$$

### Summary of Results

(a) The future value is approximately $$ \$17,724.57 $$.

(b) The total contributions made by Cam is $$ \$14,040 $$.

(c) The interest earned can be calculated as:
$$
\text{Interest} = \text{Future Value} - \text{Total Contributions} = 17,724.57 - 14,040 = 3,684.57
$$

### Final Answers
- (a) The future value is $$ \$17,724.57 $$.
- (b) The total contributions are $$ \$14,040 $$.
- (c) The interest earned is $$ \$3,684.57 $$.
(a) The future value is approximately $$ \$17,724.57 $$. (b) The total contributions are $$ \$14,040 $$. (c) The interest earned is $$ \$3,684.57 $$.

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