You can afford monthly deposits of $240 into an account that pays 3.0% compounded monthly. How long will it be until you have $5200 to buy a boat? 

To determine how long it will take to accumulate $5,200 with monthly deposits of $240 into an account paying 3.0% compounded monthly, we can use the future value of a series formula:

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

Where:
- $FV$ is the future value (the amount you want to save, $5,200),
- $P$ is the monthly deposit ($240),
- $r$ is the monthly interest rate (annual rate divided by 12 months),
- $n$ is the number of months.

Let's break it down:
1. The annual interest rate is 3.0%, so the monthly interest rate is:

\[
r = \frac{3.0\%}{12} = 0.0025
\]

2. The formula now becomes:

\[
5200 = 240 \times \frac{(1 + 0.0025)^n - 1}{0.0025}
\]

Now, solve for $n$ (the number of months).

### Step 1: Rearrange the formula to isolate the term with $n$.

\[
\frac{5200}{240} = \frac{(1 + 0.0025)^n - 1}{0.0025}
\]

\[
21.6667 = \frac{(1 + 0.0025)^n - 1}{0.0025}
\]

### Step 2: Multiply both sides by 0.0025.

\[
21.6667 \times 0.0025 = (1 + 0.0025)^n - 1
\]

\[
0.054167 = (1 + 0.0025)^n - 1
\]

### Step 3: Add 1 to both sides.

\[
1.054167 = (1 + 0.0025)^n
\]

### Step 4: Take the natural logarithm (ln) of both sides to solve for $n$.

\[
\ln(1.054167) = n \times \ln(1.0025)
\]

### Step 5: Solve for $n$.

\[
n = \frac{\ln(1.054167)}{\ln(1.0025)}
\]

Let me calculate the value of $n$.The result shows that it will take approximately 21 months to accumulate $5,200 with monthly deposits of $240 in an account paying 3.0% interest compounded monthly.

Would you like further details or have any questions? 

Here are 5 related questions to explore:
1. What if the monthly deposit was increased to $300? How long would it take?
2. How would the time change if the interest rate were 4.0% instead of 3.0%?
3. How much total interest will you have earned after 21 months?
4. What is the formula for continuous compounding, and how does it differ from monthly compounding?
5. How would inflation impact the purchasing power of $5,200 over 21 months?

**Tip:** To minimize the time needed to save for a goal, consider increasing the deposit amount or looking for accounts with higher interest rates.

You can afford monthly deposits of $240 into an account that pays 3.0% compounded monthly. How long will it be until you have $5200 to buy a boat?

Learn how long it will take to save $5,200 with monthly deposits of $240 in an account paying 3.0% compounded monthly. This detailed step-by-step solution covers compound interest calculations, logarithms, and the future value of a series formula. Suitable for high school students in Grades 10-12.

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