Let's solve the problem step by step:

Problem:

John borrowed money from Mary as follows:

• a) $100 due in 1 year.
• b) $300 due in 2 years.
• c) $400 due in 2.5 years.

John will pay $800 as a single payment, and we need to determine when he can make this payment. The interest rate is 4% compounded semiannually.

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Key Steps:

1. Find the present value (PV) of each debt at time $t$ (unknown).

   • Formula for present value with compounding: $PV = FV \cdot \left(1 + \frac{r}{n}\right)^{-n \cdot t}$ where:
     • $FV$: Future value (debt amount),
     • $r = 0.04$: Annual interest rate,
     • $n = 2$: Semiannual compounding periods per year,
     • $t$: Time in years.

2. Find the equivalent single payment of $800 by discounting it back to the same time, $t$, using the same formula.

3. Equate the total PV of all debts to the PV of $800 to find $t$.

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Step-by-Step Solution:

1. Debts' Present Values at time $t$:

   • For $100$ due in 1 year: $PV_1 = 100 \cdot \left(1 + \frac{0.04}{2}\right)^{-2(1 - t)}$

   • For $300$ due in 2 years: $PV_2 = 300 \cdot \left(1 + \frac{0.04}{2}\right)^{-2(2 - t)}$

   • For $400$ due in 2.5 years: $PV_3 = 400 \cdot \left(1 + \frac{0.04}{2}\right)^{-2(2.5 - t)}$

2. Present Value of $800 at time $t$: $PV_{800} = 800$

3. Equate total PVs: $PV_1 + PV_2 + PV_3 = PV_{800}$

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Let’s calculate $t$:

We will calculate this value using numerical approximation or algebraic manipulation. Let me compute it step by step. Given constants FV1 = 100 # Debt due in 1 year FV2 = 300 # Debt due in 2 years FV3 = 400 # Debt due in 2.5 years FV_payment = 800 # Single payment by John r = 0.04 # Annual interest rate n = 2 # Compounding semiannually

Import numerical solver

from sympy import symbols, Eq, solve

Symbol for time t (unknown)

t = symbols('t')

Compounded discount formula for each debt

def PV(FV, years_to_due, t, r, n): return FV * (1 + r/n)**(-n * (years_to_due - t))

Present value of each debt

PV1 = PV(FV1, 1, t, r, n) PV2 = PV(FV2, 2, t, r, n) PV3 = PV(FV3, 2.5, t, r, n)

Present value of the payment

PV_payment_at_t = FV_payment

Equation: Total debts' PV = Payment's PV

equation = Eq(PV1 + PV2 + PV3, PV_payment_at_t)

Solve for t

t_solution = solve(equation, t) t_solutionThe solution for $t$ is approximately 2.12 years. This means John can discharge all his debts with a single payment of $800 in 2.12 years (around 2 years and 1.5 months).

Would you like further clarification or a detailed breakdown of the computations? Here are some additional related questions:

1. How does compounding frequency affect the present value calculations?
2. Can you explain the formula for the present value in more detail?
3. What would happen if the interest rate were compounded quarterly instead of semiannually?
4. How would changing the timing of the debts (e.g., due dates) alter the solution?
5. Can you derive the result using continuous compounding instead of semiannual?

Tip: Always ensure consistency in time units (years, months, etc.) when working with compounded interest formulas!