To find the interest rate for this investment, we can use the formula for compound interest:

$A = P \left( 1 + r \right)^t$

Where:

• $A$ is the final amount (₱41,070),
• $P$ is the principal or initial amount (₱27,750),
• $r$ is the annual interest rate (what we are solving for),
• $t$ is the time in years (6 years).

We will solve for $r$.

Step 1: Substitute the known values into the formula

$41,070 = 27,750 \left( 1 + r \right)^6$

Step 2: Solve for $(1 + r)^6$

$\frac{41,070}{27,750} = \left( 1 + r \right)^6$

$1.47913 \approx \left( 1 + r \right)^6$

Step 3: Take the 6th root of both sides

$1 + r = \left( 1.47913 \right)^{\frac{1}{6}}$

$1 + r \approx 1.06605$

Step 4: Solve for $r$

$r \approx 1.06605 - 1$

$r \approx 0.06605$

Thus, the annual interest rate is approximately 6.61%.

Would you like more details, or do you have any questions?

Here are 5 related questions for further exploration:

1. What would the balance be after 10 years at the same interest rate?
2. How would the interest rate change if the final amount was ₱50,000?
3. What is the difference between compound interest and simple interest?
4. How can you calculate the time needed to reach a certain amount with a fixed interest rate?
5. What impact does compounding frequency (annual, semi-annual, etc.) have on the interest rate?

Tip: When calculating interest over several years, compound interest grows faster than simple interest, as it includes interest on previously earned interest!