Average

Average

Overview

Average is one of the most fundamental concepts in quantitative aptitude and appears frequently in MPESB Group 1/2/3 exams. Questions on average test your ability to quickly calculate mean values, handle additions or removals from groups, and work with weighted data sets. This topic forms the foundation for understanding more complex concepts like statistics, data interpretation, and mixture problems.

In MPESB exams, average questions typically range from direct calculations to word problems involving age, marks, income, or speed. Mastering this topic gives you quick wins in the exam since most problems can be solved in under a minute with the right approach. The key is understanding when to use simple arithmetic mean versus weighted average and recognizing the patterns in how questions are framed.

Key Concepts

• Arithmetic Mean is the sum of all observations divided by the total number of observations. It represents the central value of a data set.
• Weighted Average assigns different weights (importance) to different values. Use this when items contribute unequally to the total.
• Average of a series: For consecutive numbers, the average equals the middle term. For an AP, average = (first term + last term) ÷ 2.
• Effect of adding/removing items: When a new item is added, the change in average depends on how far the new item is from the old average.
• Replacement concept: When one item replaces another, the change in sum equals (new item − old item), and this affects the average proportionally.
• Average speed is NOT the simple average of speeds. It equals total distance ÷ total time.
• Combined average: When two groups merge, the combined average lies between the two individual averages, closer to the group with more members.

Formulas / Key Facts

Basic Average Formula Average = Sum of observations ÷ Number of observations Sum = Average × Number of observations

Weighted Average Formula Weighted Average = (w₁×x₁ + w₂×x₂ + ... + wₙ×xₙ) ÷ (w₁ + w₂ + ... + wₙ) where w = weight, x = value

Average of first n natural numbers Average = (n + 1) ÷ 2

Average of first n even numbers Average = n + 1

Average of first n odd numbers Average = n

Average of consecutive numbers from a to b Average = (a + b) ÷ 2

New average when a member joins New sum = Old average × Old count + New member's value New average = New sum ÷ New count

Average speed for equal distances Average speed = 2×S₁×S₂ ÷ (S₁ + S₂) where S₁ and S₂ are speeds for two equal distances

Change in average when one value changes Change in average = Change in value ÷ Number of items

Worked Examples

Example 1: Basic Average Calculation The average of 5 numbers is 27. If one number is excluded, the average becomes 25. Find the excluded number.

Solution: Sum of 5 numbers = 27 × 5 = 135 Sum of remaining 4 numbers = 25 × 4 = 100 Excluded number = 135 − 100 = 35

Example 2: New Member Joining The average age of 30 students is 15 years. When the teacher's age is included, the average increases by 1 year. Find the teacher's age.

Solution: Sum of students' ages = 30 × 15 = 450 New average = 15 + 1 = 16 years New sum (31 people) = 16 × 31 = 496 Teacher's age = 496 − 450 = 46 years

Example 3: Weighted Average A student scores 70 in Physics (weight 3), 80 in Chemistry (weight 2), and 90 in Biology (weight 1). Find the weighted average.

Solution: Weighted sum = (70×3) + (80×2) + (90×1) = 210 + 160 + 90 = 460 Total weight = 3 + 2 + 1 = 6 Weighted average = 460 ÷ 6 = 76.67

Example 4: Average Speed A person travels from A to B at 40 km/hr and returns at 60 km/hr. Find the average speed for the entire journey.

Solution: For equal distances, average speed = 2×40×60 ÷ (40+60) = 4800 ÷ 100 = 48 km/hr

Note: Simple average would give 50, but that's incorrect here.

Example 5: Replacement Problem The average weight of 8 persons increases by 2.5 kg when a new person replaces one weighing 65 kg. Find the weight of the new person.

Solution: Total increase in sum = 2.5 × 8 = 20 kg Weight of new person = 65 + 20 = 85 kg

Common Mistakes

• Using simple average for speed problems → For equal distances traveled at different speeds, always use the harmonic mean formula: 2×S₁×S₂÷(S₁+S₂). Simple average only works when equal time is spent at each speed.
• Forgetting to update the count → When a person joins or leaves, students often calculate the new sum but divide by the old count. Always track whether the number of items has changed.
• Confusing weighted average conditions → Use weighted average only when different items have different importance or quantities. If all items are equally weighted, simple average works.
• Sign errors in replacement problems → When average increases, the new value is greater than the replaced value. When average decreases, the new value is smaller. Keep track of whether the difference should be added or subtracted.
• Assuming average of averages → The average of two group averages is NOT the combined average unless both groups have equal sizes. Always calculate total sum ÷ total count.

Quick Reference

• Average = Sum ÷ Count; Sum = Average × Count
• First n natural numbers: Average = (n+1)/2
• Average speed for equal distances = 2ab/(a+b)
• New member's value = (New avg × New count) − (Old avg × Old count)
• When one value changes by x, average changes by x/n
• Combined average lies closer to the larger group's average