17. Find the compound amount at the end of ten years if ₱50,000 is invested at 8 % compounded monthly for the first three years and at 6 ½ % compounded bimonthly for the remaining years.

To solve this, we need to calculate the compound amount for two distinct periods with different compounding rates and frequencies.

### Step 1: Calculate the Compound Amount for the First Three Years
1. **Principal (initial amount), \( P \):** 50,000
2. **Rate for the first 3 years, \( r \):** 8% per annum, compounded monthly
3. **Time for this period, \( t \):** 3 years

Using the compound interest formula:
\[
A = P \left(1 + \frac{r}{n}\right)^{n \cdot t}
\]
where:
- \( n \) is the number of compounding periods per year (monthly means \( n = 12 \)),
- \( r \) is the rate in decimal form (8% = 0.08).

For the first three years:
\[
A_1 = 50000 \left(1 + \frac{0.08}{12}\right)^{12 \cdot 3}
\]

Calculating this value:
\[
A_1 \approx 50000 \times 1.26824 \approx 63412
\]

### Step 2: Use the Result as Principal for the Remaining Seven Years
Now, \( A_1 = 63412 \) becomes the principal for the remaining seven years.

1. **Principal for the second period:** \( 63412 \)
2. **Rate for the remaining 7 years, \( r \):** 6.5% per annum, compounded bimonthly
3. **Time for this period, \( t \):** 7 years

Since it’s compounded bimonthly, \( n = 6 \) (as there are 6 two-month periods in a year).

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

\[
A_2 = 63412 \left(1 + \frac{0.065}{6}\right)^{6 \cdot 7}
\]

Calculating this:
\[
A_2 \approx 63412 \times 1.56314 \approx 99134
\]

### Final Answer
The compound amount at the end of ten years is approximately **₱99,134**.

Would you like further breakdown on each calculation step?

---

1. How does compounding frequency affect compound interest?
2. What happens if the rates change more frequently in the investment?
3. Why does the amount grow faster with monthly compounding?
4. How can we adapt this to other compounding periods?
5. What is continuous compounding, and how does it differ?

**Tip:** The compounding frequency greatly influences the final amount—higher frequencies yield more compound interest.

Find the compound amount at the end of ten years if ₱50,000 is invested at 8% compounded monthly for the first three years and at 6 ½% compounded bimonthly for the remaining years.

Learn how to calculate the compound amount for a 10-year investment with an initial ₱50,000 at 8% interest compounded monthly for 3 years, followed by 6 ½% compounded bimonthly for the remaining 7 years. This detailed solution walks through each step for calculating the final compound amount, covering key financial mathematics concepts suitable for high school students.

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