📘 Average

Definition

Average means the fair value obtained after dividing the total into equal parts.

Simple Life Example

If 3 friends have 3, 5 and 7 chocolates
Total = \(3+5+7=15\)
Average = \(15÷3=5\)

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Average Formulas

\[ \text{Average of first } n \text{ natural numbers}=\frac{n+1}{2} \]

\[ \text{Average of squares till } n=\frac{(n+1)(2n+1)}{6} \]

\[ \text{Average of cubes till } n=\frac{n(n+1)^2}{4} \]

\[ \text{Average of odd numbers}=\frac{\text{Last odd}+1}{2} \]

\[ \text{Average of even numbers}=\frac{\text{Last even}+2}{2} \]

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Examples

Average of odd numbers from 1 to 40 \[ =\frac{39+1}{2}=20 \]

Average of even numbers from 1 to 81 \[ =\frac{80+2}{2}=41 \]

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🔥 Exam Shortcuts & Tricks

Shortcut 1: Average of consecutive numbers = Middle number
Example: 11,12,13,14,15 → Average = 13

Shortcut 2: Average of odd numbers = middle odd number

Shortcut 3: Average of even numbers = middle even number

Shortcut 4: If average of \(n\) numbers is \(A\) \[ \text{Sum}=nA \]

Shortcut 5 (Replacement): \[ \text{New Avg}=A+\frac{\text{New}-\text{Old}}{n} \]

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⏱️ Time & Work Based Average

\[ \text{Work}=\text{Efficiency} \times \text{Time} \]

If A takes \(a\) days and B takes \(b\) days: \[ \text{Combined Time}=\frac{ab}{a+b} \]

Efficiency ratio of A and B: \[ b:a \]

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🔥 Time & Work Exam Shortcuts

Shortcut 1: Never find LCM — use \[ \frac{ab}{a+b} \]

Shortcut 2: Average time \(=\frac{a+b}{2}\) ❌ (not together time)

Shortcut 3: Slower person → less efficiency

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📘 Solved Examples (Time & Work)

A can do work in 10 days, B in 20 days \[ \text{Time together}=\frac{10×20}{30}=\frac{20}{3}\text{ days} \]

A takes 12 days, B takes 18 days \[ \text{Efficiency ratio}=18:12=3:2 \]

Average efficiency = \(\frac{5}{24}\) work/day \[ \text{Time}=\frac{24}{5}\text{ days} \]

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