Respuesta :
Answer: Correct Answer the value of (X) is approximately 0.4625
Step-by-step explanation: Let’s break down the problem step by step:
Initial Debt:
The initial debt is the sum of the two amounts due: $3000 and $5000.
Initial debt = $3000 + $5000 = $8000.
Interest Calculation:
We’ll use the simple interest formula: (I = Prt), where:
(I) represents the interest amount.
(P) is the principal (initial debt).
(r) is the annual interest rate (in decimal form).
(t) is the time in years.
Interest on the Initial Debt:
Let’s calculate the interest accumulated on the initial debt over 9 months (0.75 years) using the given interest rate of 10%:
(r = 10% = 0.10) (in decimal form).
(t = 0.75) years.
(I_1 = Prt = 8000 \times 0.10 \times 0.75).
Remaining Debt after 9 Months:
The remaining debt after 9 months is the initial debt minus the interest:
Remaining debt = Initial debt - Interest = $8000 - I_1.
Payments:
The agreed payments are:
Payment 1 (made in 3 months): $SX.
Payment 2 (made in 10 months): $4000.
Balance after Payment 1:
After Payment 1, the remaining debt becomes:
Remaining debt = Initial debt - Payment 1 = $8000 - SX.
Balance after Payment 2:
After Payment 2 (made in 10 months), the remaining debt becomes zero:
Remaining debt = $0.
Therefore, we have: $8000 - SX - $4000 = 0.
Solving for (X):
(SX = $4000)
(X = \frac{4000}{S})
Using 9 Months as the Focal Date:
Since the focal date is 9 months, we need to find the value of (S) such that the remaining debt after 9 months matches the remaining debt after the two payments:
Remaining debt after 9 months = Remaining debt after Payment 2.
(8000 - I_1 = 0)
(8000 - 8000 \times 0.10 \times 0.75 = 0)
Solve for (S):
(S = \frac{8000}{1 - 0.10 \times 0.75})
Final Calculation for X:
Now we can find the value of (X):
(X = \frac{4000}{S})
Substitute the value of (S):
(X = \frac{4000}{\frac{8000}{1 - 0.10 \times 0.75}})
Calculate the expression inside the denominator:
(1 - 0.10 \times 0.75 = 0.925)
(X = \frac{4000}{\frac{8000}{0.925}})
(X = \frac{4000 \times 0.925}{8000})
(X \approx 0.4625)
Hence, the value of (X) is approximately 0.4625
