A universal welder has a cash value of $585 and can be purchased financed with the following plan: down payment of 30% of the cash value and the rest in 18 equal monthly installments; the first installment must be paid within 8 months and a final payment of $80 three months after the last monthly installment. If the interest rate is 2.58% per month during the first seven months and 2.91% per month thereafter, find the value of the uniform monthly installments

To find the value of the uniform monthly installments, we need to calculate the total amount financed and divide it by the number of monthly installments. Let's break down the problem step by step: Calculate the down payment: The cash value of the welder is $585, and the down payment is 30% of that: Down payment = 30% of $585 = 0.30 * $585 = $175.50 Calculate the amount financed: The amount financed is the cash value minus the down payment: Amount financed = Cash value - Down payment = $585 - $175.50 = $409.50 Calculate the monthly interest rate for the first seven months: The interest rate is given as 2.58% per month during the first seven months. Monthly interest rate = 2.58% = 2.58 / 100 = 0.0258 Calculate the monthly interest rate for the remaining months: The interest rate is given as 2.91% per month after the first seven months. Monthly interest rate = 2.91% = 2.91 / 100 = 0.0291 Calculate the number of monthly installments: The first installment must be paid within 8 months, and there is a final payment of $80 three months after the last monthly installment. Number of monthly installments = 8 + 18 + 3 = 29 Calculate the monthly installment amount: We'll use the formula for calculating the monthly installment for a loan with monthly compounding interest: Monthly installment = Amount financed / Present value factor The present value factor can be calculated using the monthly interest rate and the number of monthly installments. For the first seven months: Present value factor = (1 - (1 + r)^(-n)) / r For the remaining months: Present value factor = (1 - (1 + r)^(-n)) / (r * (1 + r)^m) Where: r = Monthly interest rate n = Number of monthly installments m = Number of installments without the final payment Let's calculate the monthly installment amount: For the first seven months: r = 0.0258 (monthly interest rate) n = 7 (number of monthly installments) m = 7 (number of installments without the final payment) Present value factor = (1 - (1 + 0.0258)^(-7)) / 0.0258 For the remaining months: r = 0.0291 (monthly interest rate) n = 29 - 7 - 1 = 21 (number of monthly installments after the first seven months, excluding the final payment) m = 21 (number of installments without the final payment) Present value factor = (1 - (1 + 0.0291)^(-21)) / (0.0291 * (1 + 0.0291)^21) Now we can calculate the monthly installment amount: Monthly installment = Amount financed / Present value factor For the first seven months: Monthly installment = $409.50 / Present value factor for the first seven months For the remaining months: Monthly installment = $409.50 / Present value factor for the remaining months By plugging in the values and performing the calculations, you will obtain the value of the uniform monthly installments