Weighted Average Problem: Quality Control

What This Problem Teaches

• Setting up weighted average equations where quantities and rates both vary
• Understanding that percentages must be weighted by their corresponding quantities
• Recognizing mixture problems disguised as manufacturing scenarios
• Using algebraic manipulation to isolate unknown quantities in percentage contexts
• Verification strategies for percentage mixture problems

Visualizing the Problem

The visual shows how the overall 5% rate sits between the two plant rates. Since 5% is closer to Plant B's 2% rate than Plant A's 6% rate, we might expect Plant A to have produced more panels — which we'll verify algebraically.

Solution: Method 1 — The Weighted Average Approach

Step 1 — Define the unknown

Let x = number of panels produced by Plant B.

Step 2 — Understand what "overall percentage" means

The overall 5% defective rate means that out of all panels produced by both plants combined, exactly 5% are defective. This gives us our fundamental equation:

Step 3 — Express each component

We need to count defective panels from each plant separately:

• Plant A: 0.06 × 12000 = 720 defective panels
• Plant B: 0.02 × x defective panels
• Total panels: 12000 + x
• Total defective: 0.05 × (12000 + x)

Step 4 — Set up the equation

Step 5 — Solve for x

Expand the right side:

Subtract 0.02x from both sides:

Subtract 600 from both sides:

Divide by 0.03:

Solution: Method 2 — The Alligation Method

Step 1 — Find the "distances" from target

In mixture problems, we can use the alligation shortcut. Find how far each rate is from the target rate:

• Plant A: 6% - 5% = 1% above target
• Plant B: 5% - 2% = 3% below target

Step 2 — Apply the inverse ratio rule

The quantities must be in the inverse ratio of their distances. Since Plant A is 1% away and Plant B is 3% away:

This means for every 3 units Plant A produces, Plant B produces 1 unit.

Step 3 — Calculate Plant B's production

Since Plant A produced 12000 panels, and the ratio is 3:1:

Therefore, Plant B produced 4000 panels.

Verification

Let's check our answer by computing the total defective rate:

What Trips Students Up

The Pattern Behind This

This is a classic weighted average problem. The general formula for any two-component mixture is:

Where w represents weights (quantities) and r represents rates (percentages). In our case:

• w₁ = 12000 (Plant A panels), r₁ = 0.06
• w₂ = x (Plant B panels), r₂ = 0.02
• r_target = 0.05

How to Spot This Problem Type

Look for these telltale phrases:

• "Overall percentage" or "combined rate"
• Two or more sources with different rates/percentages
• One quantity known, another unknown
• Context involving mixing, manufacturing, or combining groups
• Words like "defective," "success rate," "concentration," or "composition"

These problems appear in many disguises: drug concentrations in pharmacy, pollution levels in environmental science, quality control in manufacturing, and portfolio returns in finance.

Real Applications

• Manufacturing quality control: Exactly this scenario — combining output from multiple production lines with different defect rates to meet overall quality targets.
• Financial portfolio management: Mixing investments with different return rates to achieve a target portfolio return.
• Pharmaceutical compounding: Combining solutions of different concentrations to create a medication with the prescribed strength.