1. **Problem 1:** Rumi saves P3,000 every 3 months in a bank that pays 0.5% interest compounded annually. Find the total savings after 8 years. Step 1: Identify the variables. - Payment every 3 months (quarterly): P3000 - Interest rate: 0.5% per year compounded annually - Total time: 8 years Step 2: Since interest is compounded annually but payments are quarterly, we treat this as an annuity with annual compounding. Step 3: Calculate the number of payments. - Payments per year = 12 months / 3 months = 4 payments per year - Total payments = 4 payments/year * 8 years = 32 payments Step 4: Since interest compounds annually, the effective interest rate per payment period (3 months) is not directly given. We use the annual rate for compounding once per year. Step 5: Calculate the future value of the annuity using the formula for future value of an ordinary annuity compounded annually: $$FV = P \times \frac{(1 + r)^n - 1}{r}$$ where - $P$ = annual payment amount - $r$ = annual interest rate - $n$ = number of years Step 6: Since payments are quarterly, total annual payment is $P_{annual} = 3000 \times 4 = 12000$ Step 7: Plug in values: $$FV = 12000 \times \frac{(1 + 0.005)^8 - 1}{0.005}$$ Calculate: $$ (1 + 0.005)^8 = 1.0407$$ $$FV = 12000 \times \frac{1.0407 - 1}{0.005} = 12000 \times 8.14 = 97680$$ So, Rumi's savings after 8 years is approximately P97,680. --- 2. **Problem 2:** Mira plans to withdraw P25,000 each year for 5 years from an account earning 5% interest compounded quarterly. Find the initial investment. Step 1: Identify variables: - Withdrawal per year: P25,000 - Number of withdrawals: 5 - Interest rate: 5% compounded quarterly Step 2: Convert annual interest rate to quarterly rate: $$r = \frac{0.05}{4} = 0.0125$$ Step 3: Number of quarters in 5 years: $$n = 5 \times 4 = 20$$ Step 4: Use the present value of an annuity formula for withdrawals: $$PV = P \times \frac{1 - (1 + r)^{-n}}{r}$$ where $P$ is the withdrawal per period (quarter). Step 5: Since withdrawals are yearly but interest compounds quarterly, we assume withdrawals happen yearly, so we adjust the formula for yearly withdrawals with quarterly compounding. Step 6: Effective annual interest rate: $$ (1 + 0.0125)^4 - 1 = 0.05095$$ Step 7: Use the present value formula with annual compounding equivalent: $$PV = 25000 \times \frac{1 - (1 + 0.05095)^{-5}}{0.05095}$$ Calculate: $$ (1 + 0.05095)^{-5} = 0.7835$$ $$PV = 25000 \times \frac{1 - 0.7835}{0.05095} = 25000 \times 4.21 = 105250$$ So, Mira initially invested approximately P105,250. --- 3. **Problem 3:** Zoey contributes P3,000 monthly for 6 months to a retirement fund earning 9% interest compounded semi-annually. Find the amount in the fund after 5 months. Step 1: Identify variables: - Monthly contribution: P3,000 - Number of contributions: 6 - Interest rate: 9% compounded semi-annually - Time to evaluate: after 5 months Step 2: Convert interest rate to semi-annual rate: $$r = 0.09 / 2 = 0.045$$ Step 3: Since contributions are monthly but compounding is semi-annual, we calculate the effective monthly interest rate. Step 4: Effective monthly interest rate: $$ (1 + 0.045)^{1/6} - 1 = 0.0074$$ Step 5: Calculate future value of each monthly contribution after 5 months: Contributions are made at the end of each month for 6 months, but we want the value after 5 months, so the last contribution has no interest. Step 6: Future value formula for each contribution: $$FV = P \times (1 + r)^t$$ where $t$ is the number of months the contribution earns interest. Step 7: Calculate total future value: $$FV = 3000 \times [(1 + 0.0074)^5 + (1 + 0.0074)^4 + (1 + 0.0074)^3 + (1 + 0.0074)^2 + (1 + 0.0074)^1 + (1 + 0.0074)^0]$$ Calculate powers: $$1.0380 + 1.0306 + 1.0232 + 1.0158 + 1.0084 + 1 = 6.116$$ Step 8: Multiply by 3000: $$FV = 3000 \times 6.116 = 18348$$ Zoey will have approximately P18,348 in her retirement fund after 5 months. --- **Final answers:** 1. P97,680 2. P105,250 3. P18,348