Averages

Master average calculations for RRB exam preparation with comprehensive concepts and practice problems.

Basic Concepts

Simple Average (Arithmetic Mean)

Definition

• Average = (Sum of all values) ÷ (Number of values)
• Formula: Average = Σx/n

Properties

• Sum = Average × Number of items
• New Average: Changes when new items are added
• Weighted Average: Different weights for different items

Types of Averages

Arithmetic Mean

• Most common type
• Sum divided by count
• Used in most RRB questions

Weighted Average

• Formula: (Σ(weights × values)) ÷ Σ(weights)
• Used when items have different importance
• Common in marks calculations

Geometric Mean

• Formula: (Product of values)^(1/n)
• Used for growth rates
• Less common in RRB

Important Formulas

Basic Average Formulas

• Average = (Sum of observations) ÷ (Number of observations)
• Sum = Average × Number of observations
• Number of observations = (Sum of observations) ÷ Average

Weighted Average

• WA = (w₁x₁ + w₂x₂ + … + wₙxₙ) ÷ (w₁ + w₂ + … + wₙ)

Average Speed

• Average Speed = (Total Distance) ÷ (Total Time)
• For equal distances: 2xy/(x+y) where x, y are speeds

Average Age Problems

• Current Average: (Sum of current ages) ÷ (Number of people)
• After n years: Current Average + n
• n years ago: Current Average - n

Practice Problems

Question 1

The average of 5 numbers is 20. If one number is excluded, the average becomes 18. Find the excluded number.

Question 2

A batsman scores 80, 90, and 110 runs in three innings. What should be his score in the fourth innings to have an average of 100?

Question 3

The average age of 30 students in a class is 15 years. If the age of the teacher is included, the average becomes 16 years. Find the age of the teacher.

Question 4

A person travels at 40 km/h for 3 hours and 60 km/h for 2 hours. Find the average speed.

Question 5

The average of 10 numbers is 25. If each number is multiplied by 2, what will be the new average?

Question 6

A student scored 85, 90, 78, and 92 marks in four subjects. What should be his score in the fifth subject to have an average of 85?

Question 7

The average weight of 8 persons increases by 2 kg when one person weighing 60 kg is replaced by a new person. Find the weight of the new person.

Question 8

A shopkeeper sells 3 items at ₹20 each, 5 items at ₹30 each, and 2 items at ₹50 each. Find the average selling price.

Question 9

The average of first 5 prime numbers is?

Question 10

The average of 11 numbers is 50. If the average of first 6 numbers is 45 and the average of last 6 numbers is 55, find the 6th number.

Special Cases

Age Problems

• General Rule: When average changes by ‘a’ due to adding/removing one person:
  • Added person’s age = New average + (Number of persons × change in average)
  • Removed person’s age = Old average + (Number of persons × change in average)

Temperature Problems

• Average Temperature: (Sum of temperatures) ÷ (Number of days)
• Conversion: Remember to convert Celsius to Fahrenheit if needed

Cricket Average

• Batting Average: Total runs ÷ Number of innings (not-outs considered)
• Bowling Average: Runs conceded ÷ Wickets taken

Quick Methods

Mental Calculations

• Sum Shortcut: If average is ‘a’ for ’n’ items, sum = a × n
• Quick Check: Average should be between minimum and maximum values
• Even Distribution: If values are consecutive integers, average = middle value

Common Patterns

• Arithmetic Series: Average = (First + Last) ÷ 2
• Symmetric Values: Average = middle value
• Equal Contribution: If all values are same, average = that value

Important Notes

Points to Remember

1. Units: Keep track of units (km/h, years, kg, etc.)
2. Zero Values: Include zeros in calculations
3. Negative Values: Can have negative averages
4. Decimal Averages: Average can be fractional

Common Mistakes

1. Counting Errors: Ensure correct count of items
2. Division Errors: Check division calculations
3. Unit Mistakes: Maintain consistent units
4. Formula Confusion: Use correct formula for the problem type

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