21. Weighted averages

21. Weighted averages

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Below are 10 weighted averages problems, each with a step-by-step solution procedure and detailed explanation. Weighted averages are used when different groups or items contribute unequally to the overall average, with weights reflecting their relative contributions (e.g., quantities, frequencies, or proportions). The key formula is:
Weighted Average = (Sum of (Value × Weight)) / (Sum of Weights)
Problems may involve mixing quantities (e.g., prices, concentrations), averaging speeds, grades, or other metrics with weights like quantities, time, or counts. Units vary (e.g., dollars/kg, km/h, percentages), and each problem explores different scenarios for a comprehensive understanding. I’ll ensure clarity with step-by-step solutions and detailed explanations, addressing potential pitfalls.
Problem 1: Mixing Two Quantities
A shopkeeper mixes 20 kg of rice at $5/kg with 30 kg of rice at $6/kg. Find the average price per kg of the mixture.
Step-by-Step Solution:
  1. Identify given values:
    • Rice 1: 20 kg, $5/kg
    • Rice 2: 30 kg, $6/kg
  2. Calculate weighted sum: (20 × 5) + (30 × 6) = 100 + 180 = 280 dollars
  3. Calculate total weight: 20 + 30 = 50 kg
  4. Calculate weighted average: 280 / 50 = $5.6/kg
  5. State result: The average price is $5.6/kg.
Detailed Explanation: The weighted average accounts for the different quantities. Multiplying each price by its quantity gives the total cost, and dividing by the total weight gives the average price per kg. This is not the simple average of prices ($5.5), as the larger quantity of the costlier rice increases the average.
Answer: $5.6/kg
Problem 2: Average Speed
A car travels 100 km at 40 km/h and 150 km at 60 km/h. Find the average speed for the entire journey.
Step-by-Step Solution:
  1. Identify given values:
    • Distance 1: 100 km, Speed 1: 40 km/h
    • Distance 2: 150 km, Speed 2: 60 km/h
  2. Calculate times:
    • Time 1 = 100 / 40 = 2.5 hours
    • Time 2 = 150 / 60 = 2.5 hours
  3. Calculate total distance: 100 + 150 = 250 km
  4. Calculate total time: 2.5 + 2.5 = 5 hours
  5. Calculate weighted average speed: 250 / 5 = 50 km/h
  6. State result: The average speed is 50 km/h.
Detailed Explanation: Average speed is total distance divided by total time, not the arithmetic mean of speeds. Time is the weight (distance/speed), and the equal times for each segment balance the average between the speeds, weighted by distance. Pitfall: Averaging speeds directly (40 + 60)/2 = 50 is coincidental here but often incorrect.
Answer: 50 km/h
Problem 3: Weighted Grades
A student scores 80% in 3 tests (weight 20% each) and 90% in a final exam (weight 40%). Find the overall percentage.
Step-by-Step Solution:
  1. Identify given values:
    • Tests: 80%, weight = 20% × 3 = 60%
    • Final: 90%, weight = 40%
  2. Calculate weighted sum: (80 × 0.6) + (90 × 0.4) = 48 + 36 = 84%
  3. Verify total weight: 60% + 40% = 100%
  4. State result: The overall percentage is 84%.
Detailed Explanation: Weights are percentages of the total grade (100%). The weighted sum of scores gives the overall percentage, with the final exam’s higher weight increasing the average above the test scores. This reflects real-world grading systems where components have different importance.
Answer: 84%
Problem 4: Mixing Solutions
40 liters of a 20% sugar solution is mixed with 60 liters of a 30% sugar solution. Find the percentage of sugar in the mixture.
Step-by-Step Solution:
  1. Identify given values:
    • Solution 1: 40 liters, 20% sugar
    • Solution 2: 60 liters, 30% sugar
  2. Calculate sugar amounts:
    • Sugar 1 = 40 × 0.2 = 8 liters
    • Sugar 2 = 60 × 0.3 = 18 liters
  3. Calculate total sugar: 8 + 18 = 26 liters
  4. Calculate total volume: 40 + 60 = 100 liters
  5. Calculate weighted average: (26 / 100) × 100 = 26%
  6. State result: The sugar percentage is 26%.
Detailed Explanation: The weight is the volume of each solution. The total sugar divided by the total volume gives the concentration. This is equivalent to (40 × 20 + 60 × 30) / 100 = 26%, showing the weighted contribution of each solution. Pitfall: Averaging percentages (25%) ignores volume differences.
Answer: 26%
Problem 5: Average of Multiple Groups
A class has 3 groups: 10 students averaging 70 marks, 15 students averaging 80 marks, and 5 students averaging 90 marks. Find the class average.
Step-by-Step Solution:
  1. Identify given values:
    • Group 1: 10 students, 70 marks
    • Group 2: 15 students, 80 marks
    • Group 3: 5 students, 90 marks
  2. Calculate weighted sum: (10 × 70) + (15 × 80) + (5 × 90) = 700 + 1200 + 450 = 2350 marks
  3. Calculate total students: 10 + 15 + 5 = 30
  4. Calculate weighted average: 2350 / 30 ≈ 78.333 marks
  5. State result: The class average is approximately 78.33 marks.
Detailed Explanation: The number of students in each group is the weight. The weighted sum of marks divided by the total students gives the average, reflecting each group’s contribution. This extends the concept to multiple groups with different sizes.
Answer: ≈ 78.33 marks
Problem 6: Cost of Mixed Goods
A grocer mixes 25 kg of apples at $4/kg with 35 kg of apples at $5/kg and 40 kg at $6/kg. Find the average cost per kg.
Step-by-Step Solution:
  1. Identify given values:
    • Apples 1: 25 kg, $4/kg
    • Apples 2: 35 kg, $5/kg
    • Apples 3: 40 kg, $6/kg
  2. Calculate weighted sum: (25 × 4) + (35 × 5) + (40 × 6) = 100 + 175 + 240 = 515 dollars
  3. Calculate total weight: 25 + 35 + 40 = 100 kg
  4. Calculate weighted average: 515 / 100 = $5.15/kg
  5. State result: The average cost is $5.15/kg.
Detailed Explanation: Each type’s cost is weighted by its quantity. The total cost divided by total weight gives the average cost per kg. This is practical for businesses mixing goods, showing how quantities skew the average toward higher-cost items.
Answer: $5.15/kg
Problem 7: Average Temperature
A city records temperatures of 20°C for 10 days, 25°C for 12 days, and 30°C for 8 days. Find the average temperature over 30 days.
Step-by-Step Solution:
  1. Identify given values:
    • Temp 1: 20°C, 10 days
    • Temp 2: 25°C, 12 days
    • Temp 3: 30°C, 8 days
  2. Calculate weighted sum: (20 × 10) + (25 × 12) + (30 × 8) = 200 + 300 + 240 = 740°C
  3. Calculate total days: 10 + 12 + 8 = 30 days
  4. Calculate weighted average: 740 / 30 ≈ 24.667°C
  5. State result: The average temperature is approximately 24.67°C.
Detailed Explanation: Days are the weights, reflecting the duration of each temperature. The weighted sum divided by total days gives the average, slightly above the middle temperature due to more days at 25°C. This applies weighted averages to time-based data.
Answer: ≈ 24.67°C
Problem 8: Weighted Average with Ratios
Two types of tea are mixed in the ratio 2:3. Type A costs $10/kg, and Type B costs $12/kg. Find the average cost per kg.
Step-by-Step Solution:
  1. Identify given values:
    • Ratio A:B = 2:3
    • Type A: $10/kg
    • Type B: $12/kg
  2. Assign weights: A = 2 parts, B = 3 parts
  3. Calculate weighted sum: (2 × 10) + (3 × 12) = 20 + 36 = 56 dollars
  4. Calculate total parts: 2 + 3 = 5 parts
  5. Calculate weighted average: 56 / 5 = $11.2/kg
  6. State result: The average cost is $11.2/kg.
Detailed Explanation: The ratio provides the weights (parts of each type). The weighted sum divided by total parts gives the average cost, skewed toward Type B due to its higher weight and cost. This is common in mixing problems with proportional quantities.
Answer: $11.2/kg
Problem 9: Average with Partial Replacement
A tank contains 50 liters of a 40% alcohol solution. 10 liters are removed and replaced with a 60% alcohol solution. Find the new alcohol percentage.
Step-by-Step Solution:
  1. Identify given values:
    • Initial: 50 liters, 40% alcohol
    • Removed: 10 liters
    • Added: 10 liters, 60% alcohol
  2. Calculate initial alcohol: 50 × 0.4 = 20 liters
  3. Calculate alcohol removed: 10 × 0.4 = 4 liters
  4. Calculate alcohol after removal: 20 - 4 = 16 liters
  5. Calculate alcohol added: 10 × 0.6 = 6 liters
  6. Calculate final alcohol: 16 + 6 = 22 liters
  7. Calculate final volume: 50 liters (unchanged)
  8. Calculate weighted average: (22 / 50) × 100 = 44%
  9. State result: The new alcohol percentage is 44%.
Detailed Explanation: This is a weighted average after replacement. The removed solution carries the original concentration, and the added solution contributes its concentration. The final concentration is the total alcohol divided by the unchanged volume, showing how replacement alters the average.
Answer: 44%
Problem 10: Average Wage
A company has 20 employees earning $30/hour, 30 employees earning $40/hour, and 10 employees earning $50/hour. Find the average hourly wage.
Step-by-Step Solution:
  1. Identify given values:
    • Group 1: 20 employees, $30/hour
    • Group 2: 30 employees, $40/hour
    • Group 3: 10 employees, $50/hour
  2. Calculate weighted sum: (20 × 30) + (30 × 40) + (10 × 50) = 600 + 1200 + 500 = 2300 dollars/hour
  3. Calculate total employees: 20 + 30 + 10 = 60
  4. Calculate weighted average: 2300 / 60 ≈ $38.333/hour
  5. State result: The average wage is approximately $38.33/hour.
Detailed Explanation: The number of employees is the weight for each wage. The weighted sum divided by total employees gives the average wage, reflecting the larger group at $40/hour. This applies weighted averages to labor economics, showing group size impacts.
Answer: ≈ $38.33/hour
Summary
These 10 weighted averages problems cover mixing quantities (rice, solutions, tea), average speed, weighted grades, multiple groups (students, temperatures, employees), ratios, and partial replacement. Each uses the formula (Sum of (Value × Weight)) / (Sum of Weights), with weights like quantities, time, or counts. Step-by-step solutions ensure clarity, and detailed explanations highlight why weighted averages differ from simple averages, how weights affect results, and pitfalls (e.g., averaging speeds directly). These problems build a strong understanding of weighted averages across practical contexts.

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