how many years (to two decimal places) will it take $1,000 to grow to $1,800 if it is invested at 6% compounded quarterly? compounded daily?

To calculate how long it will take for $1,000 to grow to $1,800 under different compounding frequencies, we'll use the compound interest formula:

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

Where:
- $A$ is the future value (in this case, $1,800),
- $P$ is the initial principal (in this case, $1,000),
- $r$ is the annual interest rate (6% = 0.06),
- $n$ is the number of times interest is compounded per year,
- $t$ is the number of years.

We will solve for $t$ in two cases: compounded quarterly and compounded daily.

### 1. **Compounded Quarterly**
For quarterly compounding, $n = 4$.
The equation becomes:
\[
1800 = 1000 \left(1 + \frac{0.06}{4}\right)^{4t}
\]
We can now solve for $t$.

### 2. **Compounded Daily**
For daily compounding, $n = 365$.
The equation becomes:
\[
1800 = 1000 \left(1 + \frac{0.06}{365}\right)^{365t}
\]
We will solve for $t$ in this case as well.

Let me calculate both values of $t$.It will take approximately **9.87 years** for $1,000 to grow to $1,800 if it is compounded quarterly at 6%, and approximately **9.80 years** if it is compounded daily at the same rate.

Would you like more details on how these values were calculated, or do you have any other questions?

Here are 5 related questions you might consider:
1. How does the compounding frequency affect the growth of an investment?
2. What would be the difference in time if the interest rate were 5% instead of 6%?
3. How long would it take for the investment to double at the same interest rate?
4. What if the interest was compounded monthly instead of quarterly or daily?
5. How does simple interest compare to compound interest in terms of growth over time?

**Tip**: The more frequently interest is compounded, the faster the investment grows, though the effect diminishes with very frequent compounding.

How many years (to two decimal places) will it take $1,000 to grow to $1,800 if it is invested at 6% compounded quarterly? Compounded daily?

Learn how to calculate the time required for $1,000 to grow to $1,800 when compounded quarterly and daily at a 6% interest rate. Explore the compound interest formula and exponential growth, with solutions to two decimal places. Suitable for Grades 9-12.

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