ms lim deposits php3,500 every end of the month for 8 years and 9 months in a trust fund that pays 19 2/3% interest compounded monthly. how much would she have after 45th deposit? how much should she have after 5 years and 8 months? how much interest will the fund earn after 6 years? how much should she have after 7 yers and 10 months

To solve these questions, we'll apply the formula for the future value of an ordinary annuity, which is used when regular deposits are made at the end of each period. The future value formula is:

\[
FV = P \times \frac{(1 + r)^n - 1}{r}
\]

where:
- \( FV \) = future value of the annuity
- \( P \) = regular deposit amount
- \( r \) = monthly interest rate (annual interest rate divided by 12)
- \( n \) = total number of deposits

Given:
- \( P = 3,500 \) PHP
- Annual interest rate = 19 2/3% = \( \frac{59}{3}\% = 19.6667\% \)
- Monthly interest rate \( r = \frac{19.6667\%}{12} = \frac{0.196667}{12} \approx 0.016389 \)
- Number of months in different scenarios:
  - After 45 deposits: \( n = 45 \)
  - After 5 years and 8 months: \( n = 5 \times 12 + 8 = 60 + 8 = 68 \)
  - After 6 years: \( n = 6 \times 12 = 72 \)
  - After 7 years and 10 months: \( n = 7 \times 12 + 10 = 84 + 10 = 94 \)

Let's calculate each value:

### 1. **Amount after 45 deposits**

\[
FV_{45} = 3500 \times \frac{(1 + 0.016389)^{45} - 1}{0.016389}
\]

### 2. **Amount after 5 years and 8 months (68 deposits)**

\[
FV_{68} = 3500 \times \frac{(1 + 0.016389)^{68} - 1}{0.016389}
\]

### 3. **Interest earned after 6 years (72 deposits)**

First, calculate the future value after 6 years:

\[
FV_{72} = 3500 \times \frac{(1 + 0.016389)^{72} - 1}{0.016389}
\]

Then, the total interest earned after 6 years would be:

\[
\text{Interest}_{72} = FV_{72} - (3500 \times 72)
\]

### 4. **Amount after 7 years and 10 months (94 deposits)**

\[
FV_{94} = 3500 \times \frac{(1 + 0.016389)^{94} - 1}{0.016389}
\]

Let's compute these values.Here are the results based on the calculations:

1. **Amount after 45 deposits**: PHP 230,268.56
2. **Amount after 5 years and 8 months (68 deposits)**: PHP 431,489.58
3. **Interest earned after 6 years (72 deposits)**: PHP 222,827.12
4. **Amount after 7 years and 10 months (94 deposits)**: PHP 770,792.82

Would you like further details or have any additional questions? Here are some related questions:

1. How would the total amount change if the deposit amount was increased to PHP 4,000?
2. What if the interest rate was lower, say 15% per annum?
3. How would the amount be affected if the deposits were made at the beginning of each month instead?
4. What if Ms. Lim stopped depositing after 5 years? How much would she have after 8 years and 9 months?
5. How much would she have after 10 years if she continues the same deposit plan?

**Tip:** When working with compound interest, small changes in the interest rate or the frequency of deposits can significantly affect the final amount due to the compounding effect.

Learn how to calculate the future value of regular deposits with compound interest using the formula for the future value of an annuity. Solve specific scenarios including after 45 deposits, after 5 years and 8 months, and after 7 years and 10 months.

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