Average — UPSSSC PET Study Notes

Overview

Average is a foundational topic in Elementary Arithmetic that appears in every UPSSSC PET examination with 2–4 direct questions. It tests your ability to find central tendencies in numerical data and solve word problems involving groups, ages, weights, marks, speeds, and incomes. Mastery of average is essential not just for direct questions but also for time-speed-distance problems, age-based puzzles, and data interpretation sections.

The PET exam focuses on three core areas: simple average (arithmetic mean), weighted average where different quantities have different importance, and special formulas for consecutive numbers. Most questions involve straightforward computation, but tricky variants test whether you understand the underlying logic—such as when new members join a group or when items are replaced. Students who grasp the "deviation method" and pattern recognition for consecutive series gain significant speed advantages.

Strong performance in average questions requires comfort with fractions, decimals, and quick mental arithmetic. The problems are designed to be solvable within 30–45 seconds each, so formula clarity and calculation accuracy determine success.

Key Concepts

• **Average (Arithmetic Mean)** is the sum of all observations divided by the number of observations: Average = Total Sum ÷ Number of Items. It represents the central value that would result if all quantities were equal.

• **Total Sum Relationship**: If you know the average and count, you can find the total: Total = Average × Number of Items. This reverse calculation is crucial for multi-step problems.

• **Effect of Adding/Removing Items**: When a new member joins or leaves, the average changes. The change depends on whether the new value is above or below the current average. Calculate new total, then divide by new count.

• **Replacement Effect**: When one item replaces another, the change in total equals (new value – old value). The average change equals this difference divided by the total number of items.

• **Weighted Average**: When different groups have different counts, you cannot simply average the averages. You must find total sum of all groups combined, then divide by total count: Weighted Average = (Sum₁ + Sum₂ + ... + Sumₙ) ÷ (Count₁ + Count₂ + ... + Countₙ).

• **Consecutive Numbers Formula**: For any set of consecutive integers, the average always equals the middle term (if odd count) or the average of two middle terms (if even count). Shortcut: Average = (First Term + Last Term) ÷ 2.

• **Average Speed**: When equal distances are covered at different speeds, average speed ≠ average of speeds. Use: Average Speed = Total Distance ÷ Total Time. For two equal distances at speeds s₁ and s₂: Average Speed = 2s₁s₂ ÷ (s₁ + s₂).

• **Deviation Method**: When numbers are close to a convenient assumed average A, calculate deviations from A, find average of deviations d, then Actual Average = A + d. This speeds up calculation for numbers like 497, 503, 511.

Formulas / Key Facts

**1. Basic Average Formula**: Average = Sum of all observations ÷ Number of observations

**2. Sum from Average**: Sum = Average × Number of observations

**3. New Member Effect**: New Average = (Old Total + New Value) ÷ (Old Count + 1)

**4. Replacement Formula**: New Average = Old Average + (New Value – Replaced Value) ÷ Total Count

**5. Consecutive Integers**: Average = (First Number + Last Number) ÷ 2, or simply the middle number

**6. First n Natural Numbers**: Average = (n + 1) ÷ 2. Sum = n(n + 1) ÷ 2

**7. First n Odd Numbers**: Average = n. Sum = n²

**8. First n Even Numbers**: Average = n + 1. Sum = n(n + 1)

**9. Weighted Average (2 groups)**: Combined Average = (n₁A₁ + n₂A₂) ÷ (n₁ + n₂)

**10. Average Speed (equal distances)**: Average Speed = 2s₁s₂ ÷ (s₁ + s₂) for two speeds

Worked Examples

**Example 1: Simple Average Calculation**

*Problem*: The marks of 5 students are 68, 72, 75, 80, and 85. Find their average marks.

*Solution*:

• Sum = 68 + 72 + 75 + 80 + 85 = 380
• Number of students = 5
• Average = 380 ÷ 5 = 76 marks

**Example 2: New Member Joining**

*Problem*: The average age of 8 persons is 30 years. If a new person aged 38 years joins them, what is the new average age?

*Solution*:

• Old total age = 30 × 8 = 240 years
• New total age = 240 + 38 = 278 years
• New count = 8 + 1 = 9 persons
• New average = 278 ÷ 9 = 30.89 years (approximately 31 years)

**Example 3: Consecutive Numbers**

*Problem*: Find the average of all numbers from 25 to 45 (inclusive).

*Solution*:

• For consecutive numbers, average = (First + Last) ÷ 2
• Average = (25 + 45) ÷ 2 = 70 ÷ 2 = 35

*Verification*: Count = 45 – 25 + 1 = 21 numbers. Sum = 21 × 35 = 735. This equals (25 + 26 + ... + 45).

**Example 4: Weighted Average**

*Problem*: In a class, 20 students scored an average of 60 marks and 30 students scored an average of 75 marks. Find the average marks of the entire class.

*Solution*:

• Total marks of first group = 20 × 60 = 1200
• Total marks of second group = 30 × 75 = 2250
• Combined total = 1200 + 2250 = 3450
• Total students = 20 + 30 = 50
• Class average = 3450 ÷ 50 = 69 marks

*Note*: Simply averaging 60 and 75 gives 67.5, which is incorrect because it ignores the different group sizes.

**Example 5: Replacement Problem**

*Problem*: The average weight of 10 boys is 50 kg. If one boy weighing 45 kg is replaced by another boy, the new average becomes 51 kg. Find the weight of the new boy.

*Solution*:

• Increase in average = 51 – 50 = 1 kg
• Total increase for 10 boys = 10 × 1 = 10 kg
• Weight of new boy = 45 + 10 = 55 kg

*Shortcut*: New boy's weight = Replaced weight + (Average increase × Total count) = 45 + (1 × 10) = 55 kg

Common Mistakes

**Mistake 1**: Averaging averages without considering group sizes → **Fix**: Always find total sums first, then divide by total count. Never average two averages directly unless groups are equal in size.

**Mistake 2**: Forgetting to adjust count when members are added or removed → **Fix**: Carefully track whether count increases, decreases, or stays same. New count = Old count ± change.

**Mistake 3**: Confusing average speed with average of speeds → **Fix**: For distance problems, always use Average Speed = Total Distance ÷ Total Time. The formula 2s₁s₂ ÷ (s₁ + s₂) applies only when equal distances are covered at each speed.

**Mistake 4**: In consecutive number problems, forgetting the "+1" when counting terms → **Fix**: From a to b inclusive, count = b – a + 1. For example, numbers from 10 to 20 are 11 numbers, not 10.

**Mistake 5**: Calculation errors in multi-digit multiplication and division → **Fix**: Practice mental math shortcuts. For division, check if numerator is divisible. Use approximation to verify reasonableness—if averaging numbers around 70, answer cannot be 35 or 140.

Quick Reference

• **Core Formula**: Average = Sum ÷ Count; conversely Sum = Average × Count
• **Consecutive integers**: Average = (First + Last) ÷ 2 = Middle term
• **n natural numbers**: Average = (n + 1) ÷ 2; n odd numbers: Average = n
• **Weighted average requires totals**: Never average averages of unequal groups
• **New member impact**: New Avg = (Old Total + New Value) ÷ (Count + 1)
• **Replacement**: Change in average = (New – Old) ÷ Total count