📚 Goa Forest Guard – Quantitative Aptitude
Topic: Average
Average (or mean) is a central‑tendency measure that gives the “average” value of a set of numbers.
In exam questions it is used to compare quantities, check consistency, or simplify calculations.
1️⃣ Simple Explanation
- Average = Sum of all items ÷ Number of items
- It represents a typical value of a data set.
- In many questions we need to find the average of a group of numbers, or to determine a missing number when the average is known.
2️⃣ Key Points to Remember
| Point | Detail |
|---|---|
| Formula | Average = (Sum of all values) / (Count of values) |
| Average of 2 numbers | (a + b) / 2 |
| Weighted Average | Avg = (w1·x1 + w2·x2 + … + wn·xn) / (w1 + w2 + … + wn) |
| Combined Average | If two groups have averages A1 (size n1) and A2 (size n2):Combined Avg = (A1·n1 + A2·n2) / (n1 + n2) |
| Finding a missing value | Use the average formula rearranged: Missing = Avg·Count – Sum_of_known_values |
| Quick Tricks | 1. Adding & Subtracting – If you know the average and one element, you can find the sum of the rest quickly. 2. Cross‑Multiplication – For weighted averages, multiply the average by the total weight to get the weighted sum. |
| Common Mistakes | - Forgetting to divide by the correct count. - Mixing up weighted vs unweighted averages. - Not converting units (e.g., hours to minutes) before averaging. |
3️⃣ Worked Example
Problem (Typical Goa Forest Guard style)
The average speed of a forest guard’s patrol car over 3 days was 60 km/h.
On the first day the speed was 55 km/h, on the second day 65 km/h.
What was the speed on the third day?
Solution
- Total average
[ \text{Avg} = 60 \text{ km/h} ] - Number of days
[ n = 3 ] - Total distance (or total speed sum)
[ \text{Total sum} = \text{Avg} \times n = 60 \times 3 = 180 ] - Sum of known speeds
[ 55 + 65 = 120 ] - Speed on third day
[ \text{Missing} = \text{Total sum} - \text{Sum of known} = 180 - 120 = 60 \text{ km/h} ]
Answer: 60 km/h
Check –
[
\frac{55 + 60 + 65}{3} = \frac{180}{3} = 60 \text{ km/h}
]
4️⃣ Quick Revision Summary
| Concept | Formula | Quick Tip |
|---|---|---|
| Average of n numbers | Avg = Σxi / n |
Remember “Sum ÷ Count”. |
| Average of two numbers | (a + b)/2 |
Same as “Average of one pair”. |
| Weighted average | (w1·x1 + ... + wn·xn)/(w1 + ... + wn) |
Multiply each value by its weight, then divide by total weight. |
| Combined average of two groups | (A1·n1 + A2·n2)/(n1 + n2) |
Think “(Total sum of group 1 + Total sum of group 2) ÷ (Total items)”. |
| Finding missing value | Missing = Avg·Count – ΣKnown |
Rearrange average formula; use “Total sum” shortcut. |
| Common trap | – | Always check the units and the exact count. |
Quick Practice Question
The average height of 5 students is 160 cm. If the heights of 4 students are 158, 162, 150, and 170 cm, find the height of the 5th student.
- Total sum = 160 × 5 = 800
- Sum of known = 158 + 162 + 150 + 170 = 640
- Missing = 800 – 640 = 160 cm
Stay sharp!
Practice converting between sums and averages, and always double‑check the number of items in the set. Good luck with your preparation!