📚 Average – Quantitative Aptitude for Patwari / Lekhpal Exam
(Madhya Pradesh Government – 2024)
Why it matters
Averages are the backbone of many aptitude questions – from calculating the mean of scores to finding the combined average of two groups. Mastering averages means you can solve a wide array of problems quickly and accurately.
1. Simple Explanation
-
Average (Arithmetic Mean) – The sum of all values divided by the number of values.
[ \text{Average} = \frac{\text{Sum of all values}}{\text{Number of values}} ] -
Weighted Average – When different values carry different importance (weights).
[ \text{Weighted Avg.} = \frac{w_1x_1 + w_2x_2 + \dots + w_nx_n}{w_1 + w_2 + \dots + w_n} ] -
Average of Averages – When you know the averages of sub‑groups and their sizes, you can find the overall average.
[ \text{Overall Avg.} = \frac{n_1A_1 + n_2A_2 + \dots + n_kA_k}{n_1 + n_2 + \dots + n_k} ] -
Average of Two Numbers –
[ \text{Avg.} = \frac{x + y}{2} ] -
Average of Two Percentages – Often used to find a combined percentage when the total number of items is equal.
[ \text{Avg. %} = \frac{p_1 + p_2}{2} ]
2. Key Points & Cheat‑Sheet
| Concept | Formula | Quick Tip |
|---|---|---|
| Arithmetic Mean | (\displaystyle \frac{\sum x_i}{n}) | For quick mental math, add the numbers and divide by the count. |
| Weighted Mean | (\displaystyle \frac{\sum w_i x_i}{\sum w_i}) | Think of each value as repeated (w_i) times. |
| Average of Two Numbers | (\displaystyle \frac{x+y}{2}) | If you know the difference (d) and one number (x), the other is (x+d). |
| Average of Two Percentages | (\displaystyle \frac{a+b}{2}) | Use only when the base (total items) is the same for both percentages. |
| Average of Averages | (\displaystyle \frac{n_1A_1 + n_2A_2 + \dots}{n_1+n_2+\dots}) | Multiply each average by its group size. |
| Average of a Number and its Reciprocal | (\displaystyle \frac{n + 1/n}{2}) | Useful in ratio problems. |
| Finding Missing Value from Average | If average (= A) and sum of knowns (= S), missing (x = nA - S). | Great for “missing number” problems. |
| Average of Two Numbers Given One’s Average and Ratio | If (x/y = r) and ((x+y)/2 = a), then (x = a(1+r)), (y = a(1+1/r)). | Avoid heavy algebra. |
| Average of Two Sets with Different Sizes | (\displaystyle \frac{n_1A_1 + n_2A_2}{n_1+n_2}) | Same as “Average of Averages”. |
Common Patterns in Exam Questions
- “Find the average of 5 numbers if 4 of them are known.”
- “Weighted average of scores where each subject has a different credit.”
- “Combined average after adding a new item.”
- “Average of two percentages when the total number of items is equal.”
- “Average of two numbers given their sum and product.”
3. Worked Example
Problem (Typical Patwari Question)
A land record officer has the following data for two villages:
• Village‑A: 80 ha, average land price ₹12,000/ha.
• Village‑B: 120 ha, average land price ₹15,000/ha.
Find the overall average land price per hectare for the combined area.
Step‑by‑Step Solution
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Identify group sizes (weights)
- Village‑A weight (w_1 = 80) ha
- Village‑B weight (w_2 = 120) ha
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**Identify averages (