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Let's go through each problem step by step, using the compound interest formula:

\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
Where:
- \( A \) is the final amount (including interest)
- \( P \) is the principal (initial investment)
- \( r \) is the annual interest rate (as a decimal)
- \( n \) is the number of times the interest is compounded per year
- \( t \) is the time (in years)

### 1. How long will it take P 15,000 to earn P 5,000 if it is invested at 12% monthly?
We want to solve for the time \( t \) using the formula. Here:
- \( P = 15,000 \)
- \( A = 15,000 + 5,000 = 20,000 \)
- \( r = 12\% \) per month \( = 0.12 \times 12 = 1.44 \) annually, so \( r = 1.44 \)
- \( n = 12 \) (monthly compounding)

We plug into the formula:

\[
20,000 = 15,000 \left(1 + \frac{0.12}{12}\right)^{12t}
\]

Now solve for \( t \).

### 2. At what annual interest rate compounded semi-annually will a certain amount triple itself in 20 years?
Here we need to solve for the interest rate \( r \):
- \( P = 1 \) (let's assume the principal is 1)
- \( A = 3P = 3 \) (since it triples)
- \( t = 20 \)
- \( n = 2 \) (semi-annual compounding)

Plugging into the formula:

\[
3 = 1 \left(1 + \frac{r}{2}\right)^{2 \times 20}
\]

Solve for \( r \).

### 3. At what interest rate compounded quarterly will money double itself in 10 years?
For this problem:
- \( P = 1 \)
- \( A = 2P = 2 \) (since it doubles)
- \( t = 10 \)
- \( n = 4 \) (quarterly compounding)

The formula becomes:

\[
2 = 1 \left(1 + \frac{r}{4}\right)^{4 \times 10}
\]

Solve for \( r \).

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Would you like more detailed steps to solve each equation or the solutions right away?

Here are some related questions for practice:
1. How does the compounding frequency affect the time to double an investment?
2. What is the difference between simple and compound interest?
3. How can you calculate continuous compound interest?
4. If the interest rate were compounded daily, how would the formulas change?
5. How can you estimate the time for an investment to double using the Rule of 72?

**Tip:** To solve these problems quickly, isolate the exponent first by dividing both sides of the equation by the principal. Then, take the logarithm of both sides.

1. How long will it take P 15,000 to earn P 5,000 if it is invested at 12% monthly? 2. At what annual interest rate compounded semi-annually will a certain amount triple itself in 20 years? 3. At what interest rate compounded quarterly will money double itself in 10 years?

Learn how to calculate the time for an investment of P15,000 to earn P5,000 at 12% interest compounded monthly. Also, discover how to find the annual interest rate to triple an investment in 20 years with semi-annual compounding, and the rate needed to double money in 10 years with quarterly compounding.

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