To purchase a specialty guitar for his band, for the last year JJ Morrison has made payments of $86 at the end of each month into a savings account earning interest at 5.45%compounded monthly. If he leaves the accumulated money in the savings account for another three years at 6.29%compounded quarterly, how much will he have saved to buy the guitar?

Step 1 ：First, we need to calculate the total amount of money JJ Morrison has saved in the first year. Since he made payments of $86 at the end of each month, and the interest rate is 5.45% compounded monthly, we can use the formula for the future value of an ordinary annuity to calculate this. The formula is: \(FV = P * [(1 + r/n)^(nt) - 1] / (r/n)\) where: - FV is the future value of the annuity - P is the payment made each period - r is the annual interest rate (in decimal form) - n is the number of times that interest is compounded per year - t is the time the money is invested for, in years

Step 2 ：After we have calculated the total amount of money saved in the first year, we need to calculate how much this amount will grow in the next three years, with an interest rate of 6.29% compounded quarterly. We can use the formula for the future value of a lump sum to calculate this. The formula is: \(FV = PV * (1 + r/n)^(nt)\) where: - FV is the future value of the lump sum - PV is the present value of the lump sum (the amount of money saved in the first year) - r is the annual interest rate (in decimal form) - n is the number of times that interest is compounded per year - t is the time the money is invested for, in years

Step 3 ：Given that the monthly payment P = $86, the annual interest rate for the first year r1 = 5.45% or 0.0545, the number of times that interest is compounded per year for the first year n1 = 12, the time the money is invested for the first year t1 = 1 year, the annual interest rate for the next three years r2 = 6.29% or 0.0629, and the number of times that interest is compounded per year for the next three years n2 = 4, we can calculate the future value of the annuity after the first year FV1 and the future value of the lump sum after the next three years FV2.

Step 4 ：The future value of the annuity after the first year is approximately \(FV1 = $1058.17\)

Step 5 ：The future value of the lump sum after the next three years is approximately \(FV2 = $1276.06\)

Step 6 ：Final Answer: The total amount of money JJ Morrison will have saved to buy the guitar after three more years is approximately \(\boxed{1276.06}\)