Study guide: Weighted Averages of Points

📋 Course Outline

1. Averages, Midpoints, and Ratios
2. Calculating a Simple Average
3. Weighted Averages of Two Points
4. How Changing Weights Moves Points
5. Weighted Averages of Multiple Points
6. Composite Scores and Grade Examples

📖 1. Averages, Midpoints, and Ratios

🔑 Key Concepts & Definitions

• Midpoint: The exact middle of a line segment.
• Weighted average: A type of mean calculated by assigning different levels of importance, or weights, to a set of values.
• Average: The total of all values divided by the number of values.
• Ratio: A mathematical comparison of two quantities.

💡 Memory Hook

An average combines values; a midpoint marks the exact center of a segment.

📖 2. Calculating a Simple Average

★ Must-know

📐 Formula - The average of two numbers is $\frac{\text{first number} + \text{second number}}{2}$.

Further detail

• The average of 80 and 90 is $\frac{80+90}{2}=85$, which is also the midpoint between 80 and 90 on a number line.

📖 3. Weighted Averages of Two Points

★ Must-know

📐 Formula - A weighted average is the sum of each value multiplied by its weight: $\text{Weighted average}=w_1x_1+w_2x_2$.

• With a daily grade of 80 weighted at 80% and an exam grade of 90 weighted at 20%, the weighted average is $0.80(80)+0.20(90)=82$.

📌 A weighted average lies proportionally closer to the value with greater weight, rather than necessarily at the midpoint between the values.

Further detail

• For points 5 and 13 with weights in the ratio 2:3, the weighted average is $\frac{2}{5}(5)+\frac{3}{5}(13)=9.8$.

💡 Memory Hook

A heavier weight acts like a magnet, pulling the average toward its point.

📖 4. How Changing Weights Moves Points

📝 Essential Points

• For points 80 and 90 with weights 80% and 20%, the weighted average is 82, which is 2 units from 80 and 8 units from 90; those distances have ratio 1:4.

📌 In the number-line example with points 80 and 90, the distances from the weighted average 82 are inversely proportional to their weights: the heavier weight at 80 corresponds to the shorter distance.

• For points 3 and 9 with weights 2:1, the weighted average is 5; changing the ratio to 1:2 gives a weighted average of 7.

• For points 3 and 9, changing the weight ratio from 1:2 to 1:3 changes the weighted average from 7 to 7.5, moving it closer to 9.

💡 Memory Hook

Increase a point’s relative weight → the weighted average moves toward that point.

📖 5. Weighted Averages of Multiple Points

★ Must-know

• 🔄 To calculate a weighted average for multiple values:
  1. Determine the weight of each value.
  2. Find the total weight.
  3. Multiply each value by its weight.
  4. Add the weighted values.

Further detail

• For a ratio of 1:3:2:4, the total is 10 parts, giving normalized weights of 1/10, 3/10, 2/10, and 4/10.

💡 Memory Hook

Weight, multiply, then add the contributions.

📖 6. Composite Scores and Grade Examples

📝 Essential Points

• A coffee blend using Bean A at $10 per pound and Bean B at $20 per pound in a 1:3 ratio has a weighted average cost of $17.50 per pound.

• A gaming score combining Game A at 1000 points with 50% weight, Game B at 1500 points with 30% weight, and Game C at 800 points with 20% weight is 1110.

• In a grade example, homework and quiz averages are each 92.5 with weights 10% and 20%, the test grade is 90 with a 30% weight, and the final exam grade is 80 with a 40% weight, producing a weighted average of 86.75%.