# Module 06: Thinking Out Loud
## Demo Problems for Chain-of-Thought

Three problems at escalating complexity. Show each one with and without chain-of-thought prompting.

---

## Problem 1: The Vendor Decision (Simple)

```
We're choosing between two shipping vendors for our East Coast distribution.

Vendor A: $4.20 per package, guaranteed 2-day delivery, 99.1% on-time rate, $800/month platform fee, minimum 500 packages/month.

Vendor B: $3.85 per package, guaranteed 3-day delivery, 97.4% on-time rate, no platform fee, no minimum.

We currently ship about 1,800 packages per month to the East Coast. About 15% of our customers have indicated that 2-day delivery influences their purchase decision.

Which vendor should we choose?
```

**Without chain-of-thought:** Claude will likely pick one vendor and give a brief justification.

**With chain-of-thought prompt:** Add "Think through this step by step, showing your calculations and reasoning before making a recommendation."

**What changes:** Claude will calculate total monthly costs for both ($8,360 vs. $6,930), then quantify the value of the on-time rate difference, then estimate the revenue impact of the 2-day vs. 3-day delivery difference on that 15% of customers. The recommendation becomes grounded in numbers, not instinct.

---

## Problem 2: The Hiring Plan (Medium)

```
Our engineering team has asked for 4 new hires this year. Here's the situation:

Current team: 12 engineers
Current quarterly output: 18 features shipped
Target quarterly output for next year: 28 features shipped
Average ramp time for new engineers: 3 months to 50% productivity, 6 months to full productivity

New hire fully loaded cost: $165,000/year (salary + benefits + equipment + overhead)
Current revenue per feature: approximately $45,000 in first-year value
Our annual engineering budget is being increased from $2.4M to $2.9M

The VP of Product says we need 28 features per quarter by Q4 2026. The CFO wants to know if 4 hires gets us there and whether the math works within the new budget.

Analyze this and give me a recommendation.
```

**With extended thinking prompt:** Add "Use extended thinking to work through this. Consider: can 4 new hires achieve the target given ramp times? What's the realistic quarter-by-quarter output trajectory? Does the budget support this? Are there alternatives if the numbers don't work?"

**What to highlight for the room:**
- Claude should calculate current per-engineer output (1.5 features/quarter)
- Model the ramp: Q1 hires at 50%, Q2 at 75%, Q3-Q4 at 100%
- Show that 4 new hires at full productivity = 6 additional features/quarter, getting to 24, not 28
- Flag the gap and suggest alternatives (hire 6 instead, or improve per-engineer velocity, or adjust the target)
- Verify the budget math: 16 engineers x $165K = $2.64M, fits the $2.9M budget with room

---

## Problem 3: The Pricing Strategy (Complex)

```
We sell industrial sensors. Current pricing and volume data for our three product tiers:

Basic Sensor (Model S-100):
  Price: $285/unit
  Monthly volume: 2,400 units
  COGS: $112/unit
  Gross margin: 60.7%
  Customer retention rate: 72% annually

Pro Sensor (Model S-300):
  Price: $640/unit
  Monthly volume: 890 units
  COGS: $195/unit
  Gross margin: 69.5%
  Customer retention rate: 88% annually

Enterprise Sensor (Model S-500):
  Price: $1,450/unit
  Monthly volume: 210 units
  COGS: $380/unit
  Gross margin: 73.8%
  Customer retention rate: 94% annually

Our main competitor just dropped their mid-tier price by 15%. Our sales team is seeing more competitive bids at the Pro level. They want us to match the price cut.

Before we react, analyze:
1. If we drop the S-300 price by 15%, what volume increase do we need to maintain the same gross profit dollars?
2. What's the risk of a price cut on the S-300 cannibalizing S-500 sales?
3. Is there a better strategic response than matching the price cut?

Think through this step by step. Show all calculations. Consider second-order effects.
```

**What to highlight for the room:**
- This is where extended thinking earns its keep. The problem has interacting variables.
- Claude should calculate: current S-300 gross profit = $890 x (640-195) = $396,050/month. At 15% discount ($544), new margin = $349/unit, needed volume = 1,135 units (27.5% increase) to match current GP.
- Second-order: will S-500 customers trade down? Even a 10% cannibalization (21 units) at the margin difference is significant.
- Alternative strategies: bundle, add features to justify price, target different segments.

---

## When NOT to Use Chain-of-Thought

Demo these quick tasks to show the contrast:

```
Rewrite this email to be more professional: "hey dude we need the report by friday or we're screwed"
```

```
Translate "The meeting has been moved to 3 PM Thursday" into Spanish.
```

```
What's the capital of Uruguay?
```

**Teaching point:** Chain-of-thought adds value when the problem has multiple variables, requires calculation, or involves trade-offs. For simple generation, translation, or factual recall, it adds latency without improving quality. Know when to ask for the work and when to ask for the answer.