SSC CGL Preparation – Day 15

Quantitative Aptitude: AVERAGES

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🔍 What is an Average?

Average is the value that represents the central or typical value in a set of data.
It is found by dividing the sum of quantities by the number of quantities.

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📘 Basic Formula

$$\text{Average} = \frac{\text{Sum of Observations}}{\text{Number of Observations}}$$

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📘 Extended Formulas and Shortcuts

1. Average Speed (2 equal distances):

$$\text{Average Speed} = \frac{2ab}{a + b}$$

Where a and b are two speeds.

2. Effect of Replacing a Value:
   If one value is removed and another is added:

$$\text{Change in Total} = (\text{New value} – \text{Old value}) \Rightarrow \text{Change in Average} = \frac{\text{Change in Total}}{\text{Total Elements}}$$

3. Adding a New Value to the Group:

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

4. When Average of Group Changes:
   If average increases/decreases by a value due to addition of new value:

$$\text{New Value} = \text{Old Average} \pm (\text{Change in Average} \times \text{Total No. of People})$$

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📌 Types of Problems in SSC CGL

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✅ 1. Simple Average Problems

Example: Find the average of 12, 15, 18, 10, 20
Solution:
Sum = 12 + 15 + 18 + 10 + 20 = 75
Count = 5
Average = 75/5 = 15

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✅ 2. Finding a Missing Number

Example: The average of 5 numbers is 26. If four numbers are 24, 30, 28, 20, find the fifth number.
Solution:
Total = 26 × 5 = 130
Known sum = 24 + 30 + 28 + 20 = 102
Missing = 130 − 102 = 28

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✅ 3. Weighted Average

Used when groups have different sizes.

$$\text{Weighted Average} = \frac{n_1a_1 + n_2a_2 + \ldots}{n_1 + n_2 + \ldots}$$

Example: The average marks of 30 boys is 60 and that of 20 girls is 70. Find average marks of all.

$$\text{Avg} = \frac{30×60 + 20×70}{30 + 20} = \frac{1800 + 1400}{50} = \frac{3200}{50} = 64$$

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✅ 4. Average of Consecutive Numbers

• n consecutive numbers:

$$\text{Average} = \frac{\text{First Number} + \text{Last Number}}{2}$$​

• Odd consecutive numbers:
  Average is the middle number.

Example: Find average of first 7 odd numbers: 1, 3, 5, 7, 9, 11, 13
Average = Middle number = 7

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✅ 5. Effect of Replacing a Value

Example: Average of 10 numbers is 48. If one number 60 is replaced by 20, find new average.
Change in total = 20 − 60 = −40
New Total = (48×10) − 40 = 440
New Average = 440/10 = 44

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✅ 6. Average Speed

If a man travels with two different speeds for equal distances: $$\text{Average Speed} = \frac{2ab}{a + b}$$​

Example: A man travels to office at 60 km/hr and returns at 40 km/hr. Find average speed.

$$\text{Avg Speed} = \frac{2×60×40}{60+40} = \frac{4800}{100} = \textbf{48 km/hr}$$

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🧠 Tips for SSC CGL

• If values are equally spaced, the average is the middle value.
• Alligation method can be used when two groups with different averages are combined.
• Focus on shortcuts for replacing, adding values, or computing weighted averages.