SSC CGL Preparation – Day 5

Quantitative Aptitude – AVERAGE

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1. What is Average?

Average = (Sum of Observations) / (Number of Observations)

It is the arithmetic mean of a given set of numbers.

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2. Basic Formula:

$$\text{Average} = \frac{\text{Sum of all values}}{\text{Number of values}}$$

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3. Important Concepts:

a. Average of Natural Numbers:

• 1 to n: $$\text{Average} = \frac{n + 1}{2}$$
• n consecutive numbers starting from a: $$\text{Average} = a + \frac{n – 1}{2}$$

b. Average of Even/Odd Numbers:

• First n even numbers: $$\text{Average} = n + 1$$
• First n odd numbers: $$\text{Average} = n$$

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4. Weighted Average:

If two groups have different sizes and averages: $$\text{Overall Average} = \frac{(A_1 \times N_1) + (A_2 \times N_2)}{N_1 + N_2}$$

Where:

• A₁, A₂ = Averages of groups
• N₁, N₂ = Number of elements in groups

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5. Change in Average When an Element is Added or Removed:

➕ When a new number is added:

$$\text{New Average} = \frac{(\text{Old Average} \times n) + \text{New Value}}{n + 1}$$

➖ When a number is removed:

$$\text{New Average} = \frac{(\text{Old Average} \times n) – \text{Removed Value}}{n – 1}$$

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6. Shortcut Tricks:

🔁 Equal Increase in Values:

If each number increases by x, then:

New average = Old average + x

💡 Equal Decrease in Values:

If each number decreases by x, then:

New average = Old average − x

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7. Common SSC CGL Patterns:

• Missing value based on given average
• Finding total sum when average and number of items are given
• Average of consecutive numbers (like days, years, marks, etc.)
• Questions involving addition or removal of a person from a group
• Cricket score average (batsman’s performance)

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8. Examples:

📍 Example 1:

The average of 5 numbers is 20. Find their total sum.

Solution:
Sum = 20 × 5 = 100

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📍 Example 2:

The average of 4 numbers is 25. A fifth number 35 is added. What is the new average?

Solution:
Old Sum = 25 × 4 = 100
New Sum = 100 + 35 = 135
New Average = 135 / 5 = 27

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📍 Example 3:

The average age of 10 people is 40 years. One person leaves and the new average becomes 39 years. What is the age of the person who left?

Solution:
Old total = 10 × 40 = 400
New total = 9 × 39 = 351
Age of person who left = 400 − 351 = 49 years

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📍 Example 4:

The average of 7 consecutive numbers is 17. Find the smallest number.

Solution:
Middle number = 17 (since average of odd number of consecutive terms = middle term)
So numbers are: 14, 15, 16, 17, 18, 19, 20
Smallest = 14

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📍 Example 5:

Average marks of 30 students is 60. If the average marks of boys (18 students) is 55, find the average marks of the girls.

Solution:
Total = 30 × 60 = 1800
Boys’ total = 18 × 55 = 990
Girls’ total = 1800 − 990 = 810
Girls = 12
Average = 810 / 12 = 67.5

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9. Important Points for SSC CGL:

• Never forget to use weighted average when mixing groups.
• If values are in arithmetic progression, average is the middle term.
• When someone joins/leaves a group, always track the change in sum.
• Practice questions with multiple group averages (school class, salaries, etc.)