B owes $3000 due 2 years from today without interest and $2000 with interest at 4% compounded quarterly due in 6 years from today.if money worth is 5% compounded semiannually what single payment made 4 years from today will discharge his debts?
To find the single payment that would discharge B's debts, we need to calculate the present value of the two debts.
First, let's calculate the present value of the $3000 debt due in 2 years without interest. We'll use the formula for calculating present value:
PV = FV / (1 + r)^n
Where:
PV = Present Value
FV = Future Value
r = interest rate per period
n = number of periods
In this case, the future value (FV) is $3000, the interest rate (r) is 5%, and the number of periods (n) is 2 years.
PV1 = 3000 / (1 + 0.05)^2
= 3000 / (1.05)^2
= 3000 / 1.1025
= 2721.61
The present value of the $3000 debt due in 2 years is approximately $2721.61.
Now let's calculate the present value of the $2000 debt due in 6 years with 4% interest compounded quarterly. Using the same present value formula:
PV2 = 2000 / (1 + r)^n
In this case, the interest rate (r) is 4% compounded quarterly, which means we need to adjust the rate and the number of periods.
The interest rate per period becomes 4% divided by 4 (because it's compounded quarterly), which is 1% or 0.01. The number of periods (n) becomes 6 years multiplied by 2 (because it's compounded semiannually), which is 12 periods.
PV2 = 2000 / (1 + 0.01)^12
= 2000 / (1.01)^12
= 2000 / 1.12682503
= 1774.02
The present value of the $2000 debt due in 6 years is approximately $1774.02.
Finally, to find the single payment that would discharge B's debts, we need to add the present values of both debts:
Total Present Value = PV1 + PV2
= 2721.61 + 1774.02
= 4495.63
Therefore, a single payment of approximately $4495.63 made 4 years from today will discharge B's debts.
To calculate the single payment that will discharge B's debts, we need to find the present value (PV) of both debts and then add them together.
Let's start with the debt of $3000 due in 2 years without interest. Since there is no interest involved, the present value of this debt is the same as the future value (FV) of the amount. We can use the formula for the future value of a lump sum:
FV = PV * (1 + r)^n
Where:
FV = Future Value
PV = Present Value (to be calculated)
r = interest rate (as a decimal)
n = number of periods
Using the given information, we can calculate the present value:
PV = FV / (1 + r)^n
PV = $3000 / (1 + 0.05)^2
PV = $3000 / (1.05)^2
PV = $3000 / 1.1025
PV = $2720.18 (rounded to the nearest cent)
Now, let's move on to the debt of $2000 due in 6 years with an interest rate of 4% compounded quarterly. We can use the formula for calculating the future value of a lump sum with compound interest:
FV = PV * (1 + r/n)^(n*t)
Where:
FV = Future Value ($2000)
PV = Present Value (to be calculated)
r = interest rate (as a decimal, 4% = 0.04)
n = number of times the interest is compounded per period (quarterly, so 4)
t = number of periods (6 years)
Using the given information, we can calculate the present value:
PV = FV / (1 + r/n)^(n*t)
PV = $2000 / (1 + 0.04/4)^(4*6)
PV = $2000 / (1.01)^(24)
PV = $2000 / 1.2697
PV = $1575.64 (rounded to the nearest cent)
To find the single payment that will discharge B's debts, we add the present values of both debts:
Single Payment = $2720.18 + $1575.64
Single Payment = $4295.82 (rounded to the nearest cent)
Therefore, a single payment made 4 years from today of $4295.82 will discharge B's debts.