Sophie invested $420 in an account paying an interest rate of 1.6% compounded monthly. Assuming no deposits or withdrawals are made, how long would it take, *to the nearest year*, for the value of the account to reach $570?

To find out how long it will take for the value of Sophie's account to reach $570 with monthly compounding, we use the formula for compound interest:

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

Where:
- $ A $ is the amount of money in the account after time $ t $ (which is $570 in this case).
- $ P $ is the principal amount (initial investment), which is $420.
- $ r $ is the annual interest rate (as a decimal), which is 1.6%, or 0.016.
- $ n $ is the number of times interest is compounded per year, which is 12 (since it's compounded monthly).
- $ t $ is the time in years.

We need to solve for $ t $:

\[
570 = 420 \left(1 + \frac{0.016}{12}\right)^{12t}
\]

First, divide both sides by 420:

\[
\frac{570}{420} = \left(1 + \frac{0.016}{12}\right)^{12t}
\]

Now, take the natural logarithm (ln) of both sides to solve for $ t $:

\[
\ln\left(\frac{570}{420}\right) = 12t \times \ln\left(1 + \frac{0.016}{12}\right)
\]

Finally, solve for $ t $:

\[
t = \frac{\ln\left(\frac{570}{420}\right)}{12 \times \ln\left(1 + \frac{0.016}{12}\right)}
\]

Let's calculate this.It would take approximately **19 years** for the value of Sophie's account to reach $570, with the interest compounded monthly.

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:
1. What would the account value be after 10 years with the same interest rate?
2. How much interest would Sophie earn over 19 years?
3. How would the time change if the interest rate were increased to 2%?
4. What would the time be if the compounding was done quarterly instead of monthly?
5. How does monthly compounding compare with continuous compounding for the same interest rate?

**Tip:** When calculating compound interest over many years, small differences in the interest rate or compounding frequency can significantly impact the total amount earned.

Discover how long it takes for an initial investment of $420, with a 1.6% annual interest rate compounded monthly, to grow to $570. Use our compound interest formula to find the solution step-by-step.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation