📊 Study Notes – Average
Program: Telangana Patwari / Lekhpal
Subject: Quantitative Aptitude
Topic: Average
“Average” is the most frequently asked concept in the Patwari/ Lekhpal exam. Mastery of it will help you solve problems related to mean, weighted mean, rates, ratios, and comparisons quickly and accurately.
1. Simple Explanation
| Term | Definition | Symbol |
|---|---|---|
| Average (Mean) | Sum of all values divided by the number of values. | (\bar{x}) |
| Weighted Average | Average where each value has a different importance (weight). | (\bar{x}_w) |
Formulae
-
Simple Average
[ \bar{x} = \frac{S}{n} ] where (S) = sum of all values, (n) = number of values. -
Weighted Average
[ \bar{x}_w = \frac{\sum w_i x_i}{\sum w_i} ] where (w_i) = weight of the (i^{th}) value (x_i).
2. Key Points
| Key Point | Explanation | Example |
|---|---|---|
| 1. Use in “Make-up” Problems | When a group’s overall average changes after adding new members, use the formula to find the missing data. | “The average of 10 students is 80. After adding 5 more with an average of 70, overall average becomes 78.” |
| 2. Weighted Average for Rates | For combined rates (e.g., speed, cost, density), treat each rate as a value and the corresponding time/quantity as the weight. | “Two workers finish a job in 4 hrs and 6 hrs respectively. What is the combined rate?” |
| 3. “Average of Averages” | Do NOT simply average the averages; convert each average to total sum first. | “Class A avg = 70 (30 students), Class B avg = 80 (40 students). Overall avg?” |
| 4. “Average of a Set & One Element” | If you know the average of (n) numbers and the average after adding one more number, you can find that number. | “Avg of 5 numbers is 20. After adding 30, new avg = 22.” |
| 5. “Rate of Change” | Average can help compute average rate (e.g., speed = total distance / total time). | “A car travels 120 km in 2 hrs and 90 km in 1.5 hrs. Avg speed?” |
| 6. “Percentage Increase/Decrease” | Use average to compare two sets or to find the required percentage to reach a target average. | “Average score of 6 exams is 65. To achieve 70 average, what score needed on 7th?” |
| 7. Rounding & Approximation | Keep track of rounding errors; always work with fractions or decimals until the final step. | |
| 8. Common Mistakes | • Averaging averages without weights. • Treating sum of averages as total sum. • Ignoring units (e.g., km/h vs km). |
3. Worked Example
Problem
A Patwari records the following monthly sales (in ₹): 12, 15, 10, 20, 18. The average sales for these 5 months is calculated. Later, a new month’s sales of ₹25 is added. What is the new average sales?
Step‑by‑Step
-
Compute the initial sum
[ S_{\text{old}} = 12+15+10+20+18 = 75 ] -
Initial average
[ \bar{x}_{\text{old}} = \frac{75}{5} = 15 ] (This confirms the given average.) -
Add new month’s sales
[ S_{\text{new}} = S_{\text{old}} + 25 = 75 + 25 = 100 ] -
New number of months
[ n_{\text{new}} = 5 + 1 = 6 ] -
New average
[ \bar{x}{\text{new}} = \frac{S{\text{new}}}{n_{\text{new}}} = \frac{100}{6} \approx 16.67 ] → ₹16.67 (rounded to 2 decimal places).
Quick Formulaic Shortcut
[
\bar{x}{\text{new}} = \frac{n{\text{old}}\bar{x}{\text{old}} + x{\text{new}}}{n_{\text{old}}+1}
]
Plugging in:
[
\bar{x}_{\text{new}} = \frac{5\times15 + 25}{6} = \frac{100}{6} = 16.67
]
4. Quick Revision Summary
| Topic | Quick Tip |
|---|---|
| Simple Average | Sum ÷ Count |
| Weighted Average | Weighted sum ÷ Total weight |
| Adding a New Value | ((n\bar{x} + x_{\text{new}})/(n+1)) |
| Removing a Value | ((n\bar{x} - x_{\text{removed}})/(n-1)) |
| Average of Averages | Convert each avg to total sum first |
| Rate Problems | Treat rate = value ÷ time; combine using weighted average |
| Percentage Increase | ((\text{Target Avg} - \bar{x}) \times n) gives required sum for remaining values |
| Common Pitfall | Never average averages directly; always use totals |
Final Note
- Practice: Solve at least 5 “average” problems daily, covering all sub‑types (add/remove, weighted, rate, comparison).
- Check Work