Question 1172543
Absolutely, let's break down this problem step-by-step.

**1. Calculate the Maturity Value of the Note**

* Principal (P): $800,000
* Interest Rate (R): 8.6% or 0.086
* Time (T): 90/360 (ordinary interest)

* Interest = P * R * T = $800,000 * 0.086 * (90/360) = $17,200
* Maturity Value (MV) = Principal + Interest = $800,000 + $17,200 = $817,200

**2. Calculate the Bank Discount for Different Days**

* Discount Rate (d): 13% or 0.13
* Time remaining (t) = (90-days)/360
* Bank Discount (BD) = MV * d * t
* Proceeds (PR) = MV - BD

* **Day 20:**
    * Time remaining = (90 - 20) / 360 = 70 / 360
    * BD = $817,200 * 0.13 * (70 / 360) = $20,601.33
    * PR = $817,200 - $20,601.33 = $796,598.67
    * Too high. Proceeds are less than $800,000.

* **Day 30:**
    * Time remaining = (90 - 30) / 360 = 60 / 360
    * BD = $817,200 * 0.13 * (60 / 360) = $17,762.67
    * PR = $817,200 - $17,762.67 = $799,437.33
    * Too high. Proceeds are less than $800,000.

* **Day 40:**
    * Time remaining = (90 - 40) / 360 = 50 / 360
    * BD = $817,200 * 0.13 * (50 / 360) = $14,802.22
    * PR = $817,200 - $14,802.22 = $802,397.78
    * Low enough. Proceeds are greater than $800,000.

**3. Calculate the Exact Day**

* We need to find the day when the proceeds are exactly $800,000.
* $800,000 = $817,200 - BD
* BD = $17,200
* $17,200 = $817,200 * 0.13 * t
* t = $17,200 / ($817,200 * 0.13)
* t = 0.1618333 (approximately)
* t * 360 = 58.26 days remaining.
* 90 - 58.26 = 31.74 days into the note.

Therefore, the bank discount on:

* Day 20: $20,601.33 (Too high)
* Day 30: $17,762.67 (Too high)
* Day 40: $14,802.22 (Low enough)

And Richie should discount the note on the 32nd day (rounding 31.74 up) to meet his goal.