1. Problem 1: Find the initial price of the car given monthly payments of $315, for 3 years, with an annual interest rate of 12.4% compounded monthly. Step 1: Identify variables: - Payment per month $P = 315$ - Number of payments $n = 3 \times 12 = 36$ - Monthly interest rate $i = \frac{12.4}{100} \div 12 = 0.0103333$ Step 2: Use the present value of annuity formula: $$PV = P \times \frac{1-(1+i)^{-n}}{i}$$ Step 3: Calculate: $$PV = 315 \times \frac{1-(1+0.0103333)^{-36}}{0.0103333}$$ Step 4: Compute the terms: Calculate $(1+0.0103333)^{-36} = (1.0103333)^{-36} \approx 0.6942$ So: $$PV = 315 \times \frac{1-0.6942}{0.0103333} = 315 \times \frac{0.3058}{0.0103333} \approx 315 \times 29.58 = 9311.7$$ The initial price of the car is approximately $9311.7$ 2. Problem 2: Calculate the lump sum payout equivalent to $10,000 received annually for 20 years at 6% interest. Step 1: Identify variables: - Annual payment $P = 10000$ - Number of payments $n = 20$ - Interest rate $i = 0.06$ Step 2: Use the present value of annuity formula: $$PV = P \times \frac{1-(1+i)^{-n}}{i}$$ Step 3: Calculate: $$PV = 10000 \times \frac{1-(1+0.06)^{-20}}{0.06}$$ Step 4: Compute the term: $(1+0.06)^{-20} = (1.06)^{-20} \approx 0.3118$ So: $$PV = 10000 \times \frac{1-0.3118}{0.06} = 10000 \times \frac{0.6882}{0.06} \approx 10000 \times 11.47=114700$$ The lump sum payout is approximately $114700$ 3. Problem 3: Find future value of saving PHP 5,000 annually for 18 years at 6.5% interest. Step 1: Identify variables: - Annual payment $P = 5000$ - Number of payments $n = 18$ - Interest rate $i = 0.065$ Step 2: Use future value of annuity formula: $$FV = P \times \frac{(1+i)^n -1}{i}$$ Step 3: Calculate: $$FV= 5000 \times \frac{(1.065)^{18} -1}{0.065}$$ Step 4: Compute terms: $(1.065)^{18} \approx 3.099$ So: $$FV = 5000 \times \frac{3.099 -1}{0.065} = 5000 \times \frac{2.099}{0.065} \approx 5000 \times 32.293 =161465$$ The amount saved after 18 years is approximately PHP 161465 4. Problem 4: Find future worth of saving PHP 15,000 every 6 months for 10 years at 5% compounded semi-annually. Step 1: Identify variables: - Payment $P = 15000$ - Number of payments $n = 10 \times 2 = 20$ - Semi-annual interest rate $i = 0.05 \div 2 = 0.025$ Step 2: Use future value of annuity formula: $$FV = P \times \frac{(1+i)^n -1}{i}$$ Step 3: Calculate: $$FV= 15000 \times \frac{(1.025)^{20} -1}{0.025}$$ Step 4: Compute terms: $(1.025)^{20} \approx 1.6386$ So: $$FV = 15000 \times \frac{1.6386 -1}{0.025} = 15000 \times \frac{0.6386}{0.025} \approx 15000 \times 25.544 = 383160$$ The future worth after 10 years is approximately PHP 383160