A owes $3000 now and $2000 in 4 months. After re-arrangement of payments, he
agreed to pay $X in 6 months and $2X in 8 months from now. Determine the value
of $X with j12 = 12% per annum.
Ans please
owes $3000 now and $2000 in 4 months. After re-arrangement of payments, he
agreed to pay $X in 6 months and $2X in 8 months from now. Determine the value
of $X with j12 = 12% per annum.
$1759.63
Assuming that the interest rate is 12% per annum, the value of $X can be determined using the formula for present value of an annuity:
PV = A * ((1 - (1 / (1 + i)^n)) / i)
where PV is the present value, A is the annuity payment, i is the interest rate, and n is the number of payments.
In this case, the annuity payment is $X, the interest rate is 12% per annum, and the number of payments is 2. Plugging these values into the formula, we get:
PV = X * ((1 - (1 / (1 + 0.12)^2)) / 0.12)
PV = X * ((1 - (1 / 1.2496)) / 0.12)
PV = X * (0.1992 / 0.12)
PV = X * 1.66
Therefore, X = PV / 1.66, and PV = $3000.
X = $3000 / 1.66
X = $1759.63
Paying $3000 now and $2000 in 4 months, the interest rate is $12\%$ p.a.. In this case $j=12\%$ p.a. and $n=4$.
Now you owe $3000$ now and $2000$ in $6$ months time.
So, $X(1+\frac jn)^n=3000$
$X(1+\frac j{12})^{12}=3000$
$X(1+j)^n=3000$
$X(1+j)^4=3000$
$X(1.12)^4=3000$
$X(\frac{1.12^{12}}{1.12^4})=3000$
$X=3000(\frac{1.12^{12}}{1.12^4})$
$X=\frac{3000}{1.12^8}$
$X=\frac{3000}{1.2^8}$
$X\approx 1719.02$
0.12)^6 = $1759.63
To determine the value of $X, we can use the concept of present value. The present value is the current value of a future cash flow, taking into account the time value of money.
Let's calculate the present value of A's payments using the formula:
PV = FV / (1 + r)^n
Where PV is the present value, FV is the future value, r is the interest rate per period, and n is the number of periods.
First, let's calculate the present value of $2000 in 4 months:
PV1 = 2000 / (1 + 0.12/12)^(4/12)
PV1 = 2000 / (1 + 0.01)^0.33
PV1 = 2000 / 1.01033
PV1 ≈ 1976.14
Next, let's calculate the present value of $X in 6 months:
PV2 = X / (1 + 0.12/12)^(6/12)
PV2 = X / (1 + 0.01)^0.5
PV2 = X / 1.005
PV2 ≈ X/1.005
Finally, let's calculate the present value of $2X in 8 months:
PV3 = (2X) / (1 + 0.12/12)^(8/12)
PV3 = (2X) / (1 + 0.01)^(2/3)
PV3 = (2X) / 1.01066
PV3 ≈ (2X)/1.01066
Now, let's express the total present value as a sum of the individual present values and set it equal to the total amount owed:
PV1 + PV2 + PV3 = 3000
Substituting the derived values:
1976.14 + X/1.005 + (2X)/1.01066 = 3000
Simplifying the equation:
1976.14 + 0.99503X + 1.98934X = 3000
3.98437X = 1023.86
X ≈ 257.00
Therefore, the value of $X is approximately $257.00.
To determine the value of X, we can start by calculating the present value of the debts A owes.
Let's calculate the present value of the $2000 debt after 4 months. To do this, we'll use the formula for present value:
PV = FV / (1 + r)^n
Where:
PV is the present value
FV is the future value
r is the interest rate per period
n is the number of periods
In this case, FV = $2000, r = 12% per annum, but since we need to calculate the present value after 4 months, we'll use r/12 as the monthly interest rate, and n = 4 months.
PV = 2000 / (1 + 12%/12)^4
PV = 2000 / (1 + 1%)^4
PV = 2000 / (1 + 0.01)^4
PV ≈ 1830.08
So the present value of the $2000 debt after 4 months is approximately $1830.08.
Next, let's calculate the present value of the remaining debt, which is $3000. Since this amount is already the present value, we don't need to calculate it further.
Now, let's set up an equation using the present values of the debts and the agreed payment amounts in 6 and 8 months:
1830.08 + 3000 = X / (1 + 12%/12)^6 + 2X / (1 + 12%/12)^8
Simplifying the equation:
1830.08 + 3000 = X / (1 + 1%)^6 + 2X / (1 + 1%)^8
4830.08 = X / (1.01)^6 + 2X / (1.01)^8
To find the value of X, we need to solve this equation. Let's rearrange it:
4830.08 = X / 1.06171739433 + 2X / 1.08243244985
4830.08 * 1.06171739433 * 1.08243244985 = X + 2.03174387425X
5956.4281 = 3.03174387425X
5956.4281 / 3.03174387425 = X
1963.15 ≈ X
So, the value of X is approximately $1963.15.