📚 Study Notes – Average
(Quantitative Aptitude – J&K Forest Guard Examination)
1. Simple Explanation
Average (or mean) is a measure of central tendency that represents the typical value of a set of numbers.
Average = (Sum of all values) ÷ (Number of values)
It gives a single number that summarizes a collection of data, making it easier to compare groups, solve problems, or check consistency.
2. Key Points & Tips
| Concept | Formula | When to Use | Quick Tip |
|---|---|---|---|
| Simple (Arithmetic) Mean | (\displaystyle \bar{x} = \frac{\sum x_i}{n}) | Single data set | Compute sum first, then divide. |
| Combined (Weighted) Average | (\displaystyle \frac{w_1x_1 + w_2x_2 + ... + w_kx_k}{w_1 + w_2 + ... + w_k}) | Two or more groups with different sizes | Treat each group as a weight. |
| Average (Rate) – Speed | (\displaystyle \text{Avg Speed} = \frac{\text{Total Distance}}{\text{Total Time}}) | Travel problems | Use distance and time in same units. |
| Average (Time) – Work | (\displaystyle \frac{1}{\text{Avg Rate}} = \frac{1}{r_1} + \frac{1}{r_2} + ...) | Two or more workers/vehicles | Inverse of rate gives time. |
| Average (Cost/Price) | (\displaystyle \frac{\text{Total Cost}}{\text{Total Units}}) | Cost‑price problems | Useful for “average price per unit”. |
| Percentage Change in Average | (\displaystyle \frac{\text{New Avg} - \text{Old Avg}}{\text{Old Avg}} \times 100) | Compare two averages | Avoid negative sign confusion. |
Quick Tips for the Exam
- Read Carefully – Identify if the question asks for a simple average, weighted average, or average rate.
- Units Matter – Convert all quantities to the same unit before computing the average.
- Check for “Average Speed” – If the question gives “average speed” and you need distance or time, use ( \text{Speed} = \frac{\text{Distance}}{\text{Time}}).
- Use Inverse for Time – When dealing with “average time” for a job done by multiple workers, use the reciprocal of their rates.
- Avoid Redundant Calculations – If the problem gives total distance and total time, the average speed is simply the ratio; no need to add segments unless specified.
3. Worked Example
Problem:
A forest guard drives a jeep from City A to City B and back.
- Distance from A to B = 120 km.
- Speed from A to B = 60 km/h.
- Speed from B to A = 80 km/h.
Find the average speed for the whole journey.
Step‑by‑Step Solution
-
Compute time for each leg
[ \text{Time}{AB} = \frac{120\text{ km}}{60\text{ km/h}} = 2\text{ h} ] [ \text{Time}{BA} = \frac{120\text{ km}}{80\text{ km/h}} = 1.5\text{ h} ] -
Total distance
[ D_{\text{total}} = 120\text{ km} + 120\text{ km} = 240\text{ km} ] -
Total time
[ T_{\text{total}} = 2\text{ h} + 1.5\text{ h} = 3.5\text{ h} ] -
Average speed
[ \text{Avg Speed} = \frac{D_{\text{total}}}{T_{\text{total}}} = \frac{240}{3.5} \approx 68.57\text{ km/h} ]
Answer: The average speed for the round trip is ≈ 68.6 km/h.
4. Quick Revision Summary
| Topic | Formula | Typical Question |
|---|---|---|
| Simple Average | (\displaystyle \bar{x} = \frac{\sum x_i}{n}) | “What is the average of 5, 8, 12, 15, 20?” |
| Weighted Average | (\displaystyle \frac{\sum w_i x_i}{\sum w_i}) | “Two groups with 30 and 50 members have averages 70 and 80. What is the overall average?” |
| Average Speed | (\displaystyle \frac{\text{Total Distance}}{\text{Total Time}}) | “A train travels 200 km at 100 km/h and back at 120 km/h. Find average speed.” |
| Average Time (Work) | (\displaystyle \frac{1}{\text{Avg Rate}} = \frac{1}{r_1} + \frac{1}{r_2}) | “Two workers complete a task in 6 h and 9 h respectively. Work together, how long?” |
| Average Cost/Price | (\displaystyle \frac{\text{Total Cost}}{\text{Total Units}}) | “Buy 5 pens at ₹10 each and 3 pens at ₹15 each. Find average cost per pen.” |
Remember: In average speed problems, total distance ÷ total time gives the correct answer. For average time problems involving rates, use reciprocals.
Happy studying! 🚀