Chapter 06: Average (औसत)
Chapter Outline & Learning Goals
By the end of this chapter, you will be able to:
- Explain what an average really means (equal sharing).
- Recall and apply the three golden forms of the average formula instantly.
- Avoid the most dangerous trap in weighted average.
- Solve “new entry” (inclusion), “removal” (exclusion), and “wrong average” (error correction) questions with confidence.
- Use a hilarious memory trick to never mix up the steps again.
1. The Shocking Story of Rahul’s Batting
Rahul is a cricket lover. After 10 innings, his batting average is 45. He decides:
> *“I want my average to become 50 after the next innings. How many runs must I score?”*
If your mind immediately says 50, you are not alone — most students think the same at first.
The real answer is 100 runs.
Yes, Rahul needs 100, not 50! This one idea, if you master it today, will set you apart from the crowd in every competitive exam.
🧠 Before reading further, guess why the answer is 100?
*(Hint: 50 is not enough because the new innings increases the divisor as well!)*
2. Let’s Build Empathy — Why Does Average Feel Tricky?
Most learners confuse average with the “middle number.” They think adding a new value will move the average by an equal amount.
The truth is, average is a balancing act — and every new value must lift the total weight of the entire group, not just itself. This chapter will gently erase that confusion forever.
3. The Golden Formula (Smallest Possible Chunk)
Think of average as equal sharing:
If 5 friends have ₹100 and share equally, each gets ₹20. That ₹20 is the average.
The formula in words:
$$\text{Average} = \frac{\text{Total Sum}}{\text{Number of Items}}$$
In symbols:
$$A = \frac{S}{N}$$
From this one simple formula, three superhero versions emerge:
- $S = A \times N$ (Find total sum if you know average and count)
- $N = \frac{S}{A}$ (Find how many items if you know sum and average)
- $A = \frac{S}{N}$ (The foundational formula)
If you memorize only these three, you can solve most basic average questions.
🧠 Cover the screen. Say the three forms out loud like a chant:
> *“A = S/N, S = A × N, N = S/A.”*
4. Crack the Batting Puzzle (Step-by-Step Chunk)
- Old situation: 10 innings, average 45.
$$\text{Old total runs} = 10 \times 45 = 450$$ - New situation after 11 innings: average wanted 50.
$$\text{New total runs needed} = 11 \times 50 = 550$$ - Runs required in 11th innings:
$$\text{Runs} = 550 - 450 = 100 \text{ runs}$$
This is called an inclusion problem. Always use:
$$\text{New value} = (\text{New Avg} \times \text{New Count}) - (\text{Old Avg} \times \text{Old Count})$$
No guesswork, no panic.
🧠 Mental Drill: A batsman has an average of 30 after 8 innings. How many runs in the 9th innings to make his average 35?
*(Answer: $9 \times 35 - 8 \times 30 = 315 - 240 = 75$ runs)*
5. The Weird and Funny Memory Trick (Mnemonic)
Meet the SAD Panda 🐼.
He sits under a bamboo tree and repeats:
> *“Sum all the values. Amount (number) count karo. Divide karo.”*
> SAD = Sum, Amount, Divide.
Why is the panda sad? Because whenever a student forgets to divide at the end, the panda cries!
To make the panda happy, always do the D step last.
Reverse Magic: If someone gives you Average and Amount, multiply them to get the Sum (panda hugs his bamboo!).
6. Weighted Average — The Trap That Hurts
Imagine a class: 4 boys weigh 50 kg on average, 6 girls weigh 40 kg on average.
Many students do:
$$\frac{50 + 40}{2} = 45 \quad \text{❌ WRONG!}$$
That would be true only if boys and girls were equal in number. Here the weights (counts) are different!
Correct Weighted Average:
$$A_w = \frac{(4 \times 50) + (6 \times 40)}{4 + 6} = \frac{200 + 240}{10} = \frac{440}{10} = 44 \text{ kg}$$
Always keep the total weighted sum in the numerator and the sum of weights in the denominator. Never fall for the 50–50 trap.
🧠 Practice: In a factory, 10 workers earn ₹300 on average, 5 earn ₹500 on average. Find overall average.
*(Answer: $\frac{10 \times 300 + 5 \times 500}{15} = \frac{3000 + 2500}{15} = 366.67$)*
7. Consecutive Numbers Shortcut (No Addition Needed)
For consecutive numbers like $1, 2, 3, \dots, 100$:
$$\text{Average} = \frac{\text{First} + \text{Last}}{2} = \frac{1 + 100}{2} = 50.5$$
- For an odd count of terms: e.g., $11, 12, 13, 14, 15$ $\implies$ average is the exact middle term = $13$.
- For an even count of terms: e.g., $11 \text{ to } 20$ $\implies$ average = $\frac{11 + 20}{2} = 15.5$.
This turns a long manual addition into a 5-second answer!
🧠 Quick Quiz: Average of numbers from 101 to 200?
*(Answer: $\frac{101 + 200}{2} = 150.5$)*
8. Correcting Mistakes (Error Correction Questions)
If someone mistakenly recorded 56 instead of 65 while summing, and the average of 20 items came out as 40:
- Wrong sum: $40 \times 20 = 800$.
- Correct sum: $800 - 56 + 65 = 809$.
- Correct average: $\frac{809}{20} = 40.45$.
The Golden Rule: Fix the sum first, then divide again!
9. Summary Table — Your Quick Reference
| Concept | Formula / Shortcut |
|---|---|
| Basic Average | $A = \text{Sum} \div N$ |
| Total Sum | $\text{Sum} = A \times N$ |
| New Entry (Inclusion) | $\text{New value} = (\text{New } A \times \text{New } N) - (\text{Old } A \times \text{Old } N)$ |
| Effect of One Value | $\text{New } A = \text{Old } A + \frac{\text{New value} - \text{Old } A}{\text{New } N}$ |
| Weighted Average | $A_w = \frac{w_1 x_1 + w_2 x_2 + \dots}{w_1 + w_2 + \dots}$ |
| Consecutive Numbers (AP) | $\text{Average} = \frac{\text{First} + \text{Last}}{2}$ |
| Error Correction | $\text{Correct Sum} = \text{Wrong Sum} - \text{Wrong Value} + \text{Right Value}$ |
📅 Review This After 2 Days (Spaced Repetition)
- Rewrite the SAD panda story without looking.
- Solve the Rahul batting question again — this time for an average rise from 45 to 52 after 12 innings.
- Attempt at least 5 mixed average questions using only the summary table.
- Teach the concept to a study partner — if you can explain it simply, you own it!