PV=PMT×(1−(1+r) −n )/r where: 𝑃 𝑉 PV is the present value of the annuity 𝑃 𝑀 𝑇 PMT is the payment amount 𝑟 r is the interest rate per period 𝑛 n is the total number of payments Part (a) For the first annuity: 𝑃 𝑀 𝑇 = $ 325 PMT=$325 The annual interest rate is 3.5%, so the monthly interest rate 𝑟 = 3.5 % 12 = 0.035 12 ≈ 0.0029167 r= 12 3.5% ​ = 12 0.035 ​ ≈0.0029167 The number of monthly payments 𝑛 = 36 n=36 Using the formula: 𝑃 𝑉 = 325 × ( 1 − ( 1 + 0.0029167 ) − 36 ) / 0.0029167 PV=325×(1−(1+0.0029167) −36 )/0.0029167 First, calculate ( 1 + 0.0029167 ) − 36 (1+0.0029167) −36 : ( 1 + 0.0029167 ) − 36 ≈ 1 / ( 1 + 0.0029167 ) 36 ≈ 1 / 1.107073 ≈ 0.90334 (1+0.0029167) −36 ≈1/(1+0.0029167) 36 ≈1/1.107073≈0.90334 Then, calculate 1 − 0.90334 = 0.09666 1−0.90334=0.09666. Now, compute the present value: 𝑃 𝑉 = 325 × 0.09666 0.0029167 ≈ 325 × 33.135 ≈ 10769.88 PV=325× 0.0029167 0.09666 ​ ≈325×33.135≈10769.88 So, the present value of the annuity is approximately $10,769.88. Part (b) For the second annuity: 𝑃 𝑀 𝑇 = $ 450 PMT=$450 The annual interest rate is 5%, so the weekly interest rate 𝑟 = 5 % 52 = 0.05 52 ≈ 0.0009615 r= 52 5% ​ = 52 0.05 ​ ≈0.0009615 The number of weekly payments 𝑛 = 60 n=60 Using the formula: 𝑃 𝑉 = 450 × ( 1 − ( 1 + 0.0009615 ) − 60 ) / 0.0009615 PV=450×(1−(1+0.0009615) −60 )/0.0009615 First, calculate ( 1 + 0.0009615 ) − 60 (1+0.0009615) −60 : ( 1 + 0.0009615 ) − 60 ≈ 1 / ( 1 + 0.0009615 ) 60 ≈ 1 / 1.059808 ≈ 0.9436 (1+0.0009615) −60 ≈1/(1+0.0009615) 60 ≈1/1.059808≈0.9436 Then, calculate 1 − 0.9436 = 0.0564 1−0.9436=0.0564. Now, compute the present value: 𝑃 𝑉 = 450 × 0.0564 0.0009615 ≈ 450 × 58.69 ≈ 26410.5 PV=450× 0.0009615 0.0564 ​ ≈450×58.69≈26410.5