Salote owes $3000 now and $2000 in 4 months. After re-arrangement of payments,
she agreed to pay $X in 6 months and $2X in 8 months from now.
Determine the value of $X with j12 = 12% per annum.
A. $1,759.63
B. $1,719.02
C. $1,722.32
D. $1,764.77
Make a time graph from "now" to 8 months
Above the line, place 3000 at now and 2000 at month 4
Below the line place x at month 6 and 2x at month 8
Pick month 8 as your focal point and "move" all monies to that time
i = .01 , and
x(1.01)^2 + 2x = 3000(1.01)^8 + 2000(1.01)^4
x[ 1.01^2 + 2] = 5329.778
x = 5329.778/3.0201 = 1764.77
To determine the value of $X, we can set up an equation based on the information given.
Let's break down the payments and their corresponding time periods:
1. The first payment of $3000 is due immediately (0 months).
2. The second payment of $2000 is due in 4 months.
3. The third payment of $X is due in 6 months.
4. The fourth payment of $2X is due in 8 months.
To find the present value of the future payments, we need to discount them back to the present using the formula for present value of a future sum:
PV = FV / (1 + r/n)^(n*t)
Where:
PV = Present Value
FV = Future Value
r = interest rate
n = number of compounding periods per year
t = time period in years
We'll start by finding the present value of the second payment of $2000 due in 4 months:
PV2 = 2000 / (1 + 0.12/12)^(12/3)
PV2 = 2000 / (1 + 0.01)^4
PV2 = 2000 / (1.01)^4
PV2 = 2000 / 1.04060401
PV2 = $1,923.21
Next, let's find the present value of the third payment of $X due in 6 months:
PV3 = X / (1 + 0.12/12)^(12/2)
PV3 = X / (1 + 0.01)^6
PV3 = X / (1.01)^6
PV3 = X / 1.06152015
PV3 = $0.9434X
Finally, let's find the present value of the fourth payment of $2X due in 8 months:
PV4 = 2X / (1 + 0.12/12)^(12/4)
PV4 = 2X / (1 + 0.01)^8
PV4 = 2X / (1.01)^8
PV4 = 2X / 1.08243285
PV4 = $1.8454X
Now, we can set up an equation to find the value of X:
$3000 + $1923.21 + $0.9434X + $1.8454X = $0
Combining like terms:
$4923.21 + $2.7888X = $0
Subtracting $4923.21 from both sides:
$2.7888X = -$4923.21
Dividing by $2.7888:
X = -$4923.21 / 2.7888
X = -$1763.83
Since a negative value doesn't make sense in this context, we'll disregard it.
Therefore, the correct answer is not listed among the options provided.
To solve this problem, we can use the concept of present value. We need to find the values of $X and 2X that will make the total present value of the payments equal to the total amount owed by Salote.
First, let's calculate the present value of the $2000 payment in 4 months. We are given the interest rate as j12 = 12% per annum, which means the monthly interest rate is 12%/12 = 1% or 0.01. Using the formula for present value (PV):
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.
PV1 = $2000 / (1 + 0.01)^4
= $2000 / (1.01)^4
= $2000 / 1.04060401
≈ $1922.23
Now, let's calculate the present value of the $X payment in 6 months:
PV2 = X / (1 + 0.01)^6
= X / (1.01)^6
Finally, let's calculate the present value of the $2X payment in 8 months:
PV3 = 2X / (1 + 0.01)^8
= 2X / (1.01)^8
The total present value of these payments should be equal to the total amount owed by Salote:
PV1 + PV2 + PV3 = $3000 + $1922.23
Now, we can form the equation and solve for X:
1922.23 + X / (1.01)^6 + (2X / (1.01)^8) = 3000 + 1922.23
Simplifying the equation:
1922.23 + X / 1.0616782 + (2X / 1.08516504) = 4922.23
Multiplying through by 1.08516504 to clear the denominator:
2079.51 + 1.08516504X + 2X = 5345.01
Collecting like terms:
3.08516504X = 5345.01 - 2079.51
3.08516504X = 3265.50
Dividing by 3.08516504:
X ≈ 3265.50 / 3.08516504
X ≈ 1058.61
Therefore, the value of $X is approximately $1058.61.
Now let's check which option matches this value of X. Subtract X from $3000, we get $1941.39. Substitute X = $1058.61 and 2X = $2117.22 into equation (1941.39/1.01^6) + 1941.39/1.01^8) + X + 2X = $3000 + 1941.39, we get $1941.39 ≈ $1941.39+ $1941.39 + X + 2X = $3000 + $1941.39. The sum of the left-hand side is 4215.99, and the sum of the right-hand side is 4941.39. So the equation is incorrect, the value of $X should be different. Let's continue checking other options.
Similar steps can be done with the other options:
For option A. $1,759.63:
X ≈ 1759.63 / 3.08516504
X ≈ $570.06
Again, let's check this option. Subtract X from $3000, we get $2429.94. Substitute X = $570.06 and 2X = $1,140.12 into equation (2429.94/1.01^6) + (2429.94/1.01^8) + X + 2X = $3000 + $2429.94, we get $2429.94 ≈ $2429.94 + $2429.94 + X + 2X = $3000 + $2429.94. The sum of the left-hand side is 5370.1, and the sum of the right-hand side is 5429.94. So the equation is incorrect.
For option B. $1,719.02:
X ≈ 1719.02 / 3.08516504
X ≈ $557.15
Let's check this option. Subtract X from $3000, we get $2442.85. Substitute X = $557.15 and 2X = $1114.30 into equation (2442.85/1.01^6) + (2442.85/1.01^8) + X + 2X = $3000 + $2442.85, we get $2442.86 ≈ $2442.85 + $2442.85 + x + 2x = $3000 + $2442.85. The sum of the left-hand side is 5370.1, and the sum of the right-hand side is 5442.85. So the equation is incorrect.
For option C. $1,722.32:
X ≈ 1722.32 / 3.08516504
X ≈ $558.79
Let's check this option. Subtract X from $3000, we get $2441.21. Substitute X = $558.79 and 2X = $1117.58 into equation (2441.21/1.01^6) + (2441.21/1.01^8) + X + 2X = $3000 + $2441.21, we get $2441.20 ≈ $2441.21 + $2441.21 + X + 2X = $3000 + $2441.21. The sum of the left-hand side is 5356.79, and the sum of the right-hand side is 5441.21. So the equation is incorrect.
For option D. $1,764.77:
X ≈ 1764.77 / 3.08516504
X ≈ $571.77
Again, let's check this option. Subtract X from $3000, we get $2428.23. Substitute X = $571.77 and 2X = $1143.54 into equation (2428.23/1.01^6) + (2428.23/1.01^8) + X + 2X = $3000 + $2428.23, we get $2428.22 ≈ $2428.23 + $2428.23 + X + 2X = $3000 + $2428.23. The sum of the left-hand side is 5367.46 and the sum of the right-hand side is 5428.23. So the equation is incorrect.
Therefore, option D. $1,764.77 is not the correct value for $X.