Average — Study Notes for UP Police Constable

Overview

Average (or arithmetic mean) is one of the most frequently tested topics in the Numerical & Mental Ability section of the UP Police Constable exam. Questions on average appear in nearly every paper, often disguised in word problems involving ages, marks, income, temperature, or cricket scores. The topic tests your ability to quickly compute the central value of a data set and solve real-world problems where quantities are added or removed from a group.

Mastering average requires solid understanding of three core relationships: the formula itself, weighted averages when groups combine, and the effect of adding/removing elements. Questions typically involve 2–4 calculation steps and reward speed and accuracy. Since the exam is time-bound, recognizing problem patterns and applying shortcut techniques is essential. Strong performance on average questions builds confidence and frees time for tougher reasoning problems.

The key to success is practicing varied problem types — simple averages, weighted averages, replacement scenarios, and combined group problems — until the patterns become automatic.

Key Concepts

• **Average** is the sum of all observations divided by the number of observations. It represents the central or typical value of a dataset.

• **Formula relationship**: Average = Total Sum / Number of Items, which rearranges to Total Sum = Average × Number of Items. This second form is critical for solving most exam problems.

• **Weighted Average** applies when different groups have different counts. The combined average is NOT the simple average of the two averages; you must account for the weight (count) of each group.

• **Effect of Adding/Removing**: When a new element is added or an old one removed, the average changes. Calculate the new total sum, then divide by the new count.

• **Replacement Impact**: When one element replaces another, the change in average reveals the difference between the new and old elements.

• **Age Problems**: Average age questions often involve time shifts (e.g., "5 years ago" or "after 3 years"). Remember that when n people age by t years, the total age increases by n×t.

• **Shortcut for Consecutive Numbers**: The average of consecutive integers is the middle number (for odd count) or the average of the two middle numbers (for even count).

• **Zero-Sum Thinking**: Deviations from the average cancel out. If some values are above average, others must be below by an equal total amount.

Formulas / Key Facts

1. **Basic Average Formula**: Average = Sum of Observations / Number of Observations

2. **Reverse Formula**: Sum = Average × Number of Items (use this to find totals quickly)

3. **Weighted Average of Two Groups**: Combined Average = (n₁A₁ + n₂A₂) / (n₁ + n₂), where n₁, n₂ are counts and A₁, A₂ are respective averages.

4. **Change in Average**: If a new item x is added to n items with average A, new average = (nA + x) / (n + 1)

5. **Replacement Formula**: If replacing one item changes average from A to B in n items, then Difference = New Item − Old Item = n(B − A)

6. **Age Increase**: When all members of a group age by t years, total age increases by n×t, but average age increases by t (not n×t)

7. **Consecutive Numbers**: Average of first n natural numbers = (n + 1) / 2; Average of first n odd numbers = n; Average of first n even numbers = n + 1

8. **Speed-Distance Average**: Average speed ≠ average of speeds. Use Total Distance / Total Time instead.

Worked Examples

**Example 1: Basic Average with Replacement**

The average of 5 numbers is 27. If one number 25 is replaced by 40, what is the new average?

*Solution*:

• Original sum = 5 × 27 = 135
• After replacement: sum increases by (40 − 25) = 15
• New sum = 135 + 15 = 150
• New average = 150 / 5 = 30

**Shortcut**: Change in average = 15 / 5 = 3, so new average = 27 + 3 = 30 ✓

**Example 2: Weighted Average**

The average marks of 40 students in Section A is 75, and the average marks of 60 students in Section B is 80. Find the combined average of all 100 students.

*Solution*:

• Total marks of Section A = 40 × 75 = 3000
• Total marks of Section B = 60 × 80 = 4800
• Combined total = 3000 + 4800 = 7800
• Combined average = 7800 / 100 = 78

**Common Error**: (75 + 80) / 2 = 77.5 ✗ (ignores weights)

**Example 3: Age Problem with Time Shift**

The average age of a family of 5 members is 24 years. If the youngest member is 4 years old, what was the average age of the family at the time of the youngest member's birth?

*Solution*:

• Current total age = 5 × 24 = 120 years
• 4 years ago (at birth), each of 4 older members was 4 years younger
• Total age 4 years ago = 120 − 4 (youngest's age) − 4 × 4 (age decrease of 4 members) = 120 − 4 − 16 = 100 years
• Number of members then = 4
• Average age then = 100 / 4 = 25 years

Common Mistakes

1. **Averaging the averages without weights** → When combining groups, don't just add averages and divide by 2. Always multiply each average by its count, sum the totals, then divide by the combined count.

2. **Confusing individual change with total change** → If 5 people each gain 3 kg, total weight increases by 15 kg, not 3 kg. Always multiply individual change by the number of items.

3. **Wrong formula for speed average** → Average speed is NOT (speed₁ + speed₂) / 2. It's total distance / total time. For equal distances: Average speed = 2xy / (x + y) (harmonic mean).

4. **Forgetting to adjust the count** → When adding or removing items, remember to change the denominator. If 10 items become 11, divide by 11, not 10.

5. **Sign errors in replacement** → When an old value is replaced by a new value, the change is (new − old), not (old − new). A larger replacement increases the average; a smaller replacement decreases it.

Quick Reference

• Average = Sum / Count; Sum = Average × Count (use the second form to find totals)
• Weighted average needs counts: (n₁A₁ + n₂A₂) / (n₁ + n₂)
• Replacement impact: n(New Avg − Old Avg) = (New Item − Old Item)
• When n people age by t years: total age increases by n×t; average age increases by t
• Average of 1 to n = (n+1)/2; Average of n consecutive numbers = (First + Last)/2
• Always adjust both numerator AND denominator when adding/removing items