Question 1179079
Here's how to solve this problem step-by-step:

**1. Calculate the Annual Payment:**

* Total amount: $500,000
* Number of installments: 10
* Annual payment: $500,000 / 10 = $500,000 / 10 = $50,000

**2. Calculate the Present Value of the Remaining Annual Payments:**

* Betty receives the third payment immediately. This means there are 7 remaining payments.
* Effective annual interest rate: 3% (0.03)
* We need to find the present value of an annuity due with 7 payments.

PV = PMT * [(1 - (1 + r)^-n) / r] * (1 + r)

Where:

* PV = Present Value
* PMT = Annual Payment ($50,000)
* r = Annual Interest Rate (0.03)
* n = Number of Remaining Payments (7)

PV = 50000 * [(1 - (1.03)^-7) / 0.03] * (1.03)
PV = 50000 * [(1 - 0.81309152) / 0.03] * 1.03
PV = 50000 * [0.18690848 / 0.03] * 1.03
PV = 50000 * 6.23028267 * 1.03
PV = 50000 * 6.41719115
PV ≈ $320,859.56

**3. Calculate the Equivalent Monthly Interest Rate:**

* Effective annual interest rate: 3% (1.03)
* Monthly interest rate: (1.03)^(1/12) - 1 ≈ 0.00246627

**4. Calculate the Monthly Perpetuity Payment (X):**

* The present value of the perpetuity must equal the present value of the remaining annual payments.

PV_perpetuity = X / r_monthly

Where:

* PV_perpetuity = $320,859.56
* X = Monthly Perpetuity Payment
* r_monthly = Monthly Interest Rate (0.00246627)

X = PV_perpetuity * r_monthly
X = 320859.56 * 0.00246627
X ≈ $791.31

**Answer:**

The value of X is approximately $791.31.