Jane plans to save $6,500 in a savings account that earns 6% interest annually. She wants to know how long she needs to keep her $6,500 investment in the account to grow it to $19,000. How many years does she need to keep her $6,500 investment in the account at 6% annual interest to grow it to $19,000? Round to the nearest integer.



To solve this problem, we will use the formula for compound interest:

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

Where:
- $ A $ is the amount of money accumulated after interest, including the principal.
- $ P $ is the principal amount (initial investment).
- $ r $ is the annual interest rate (in decimal form).
- $ n $ is the number of times that interest is compounded per year.
- $ t $ is the time the money is invested for (in years).

In this case:
- $ P = 6,500 $ (initial investment),
- $ A = 19,000 $ (target amount),
- $ r = 0.06 $ (6% annual interest rate),
- $ n = 1 $ (compounded annually).

We need to solve for $ t $.

The formula simplifies to:

\[
19,000 = 6,500(1 + 0.06)^t
\]

We will now solve this equation for $ t $. 

First, divide both sides by 6,500:

\[
\frac{19,000}{6,500} = (1.06)^t
\]

\[
2.923 = 1.06^t
\]

Next, take the natural logarithm (ln) of both sides:

\[
\ln(2.923) = \ln(1.06^t)
\]

\[
\ln(2.923) = t \cdot \ln(1.06)
\]

Now solve for $ t $:

\[
t = \frac{\ln(2.923)}{\ln(1.06)}
\]

Let me calculate that for you.Jane will need to keep her $6,500 investment in the account for approximately 18 years to grow it to $19,000, assuming a 6% annual interest rate.

Would you like further details on the calculation, or do you have any other questions?

Here are 5 related questions you might find useful:
1. How is compound interest different from simple interest?
2. What if the interest rate was compounded monthly? How would it change the result?
3. How can Jane calculate the amount she will have in 10 years?
4. What if Jane wanted to reach $19,000 in 15 years? What interest rate would she need?
5. How much interest will Jane earn after 18 years?

**Tip:** The power of compound interest increases significantly with time. The earlier you start investing, the more your money grows!

Jane plans to save $6,500 in a savings account that earns 6% interest annually. She wants to know how long she needs to keep her $6,500 investment in the account to grow it to $19,000. How many years does she need to keep her $6,500 investment in the account at 6% annual interest to grow it to $19,000? Round to the nearest integer.

Discover how long Jane needs to invest $6,500 at 6% annual interest to reach $19,000 using compound interest formulas. This detailed solution includes calculations using logarithms and exponential growth, perfect for high school students in Grades 10-12.

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