Quantitative Aptitude – Average
Targeted Exam: Puducherry Police Constable (PPC)
Topic: Average
Level: Beginner – Intermediate
📌 1. Simple Explanation
- Average (Mean) is the sum of all values divided by the number of values.
- It gives a “central” value that represents the whole data set.
- Formula
[ \text{Average} = \frac{\text{Sum of all values}}{\text{Number of values}} ]
Quick mental rule:
Add all numbers → divide by how many you added.
📌 2. Key Points
| Feature | Description |
|---|---|
| Simple Average | (\displaystyle \bar{x} = \frac{S}{n}) where (S) = sum, (n) = count |
| Weighted Average | (\displaystyle \bar{x}_w = \frac{\sum w_i x_i}{\sum w_i}) where (w_i) = weight for value (x_i) |
| Average of Averages | If groups have different sizes: (\displaystyle \bar{x} = \frac{\sum (n_i \bar{x}_i)}{\sum n_i}) |
| Adding/Removing Data | Adding a new value (x_{new}) → new sum (S' = S + x_{new}); new average computed accordingly. |
| Common Mistakes | • Forgetting to divide by correct count. • Using wrong weights. • Mixing up sum and average in “average of averages”. |
| Useful in Problems | • “What is the average of the remaining group?” • “If the average increases by X, find the new value added.” • “Average of scores before and after a test.” |
📌 3. Worked Example
Problem
A police officer’s monthly salary for 3 consecutive months is ₹18,000, ₹20,000, and ₹22,000.
- Find the average monthly salary.
- After the next month, the salary rises to ₹24,000. What is the new average?
- If the officer wants the average to be ₹21,000, how much should the next month’s salary be?
1. Initial Average
[ \bar{x}_1 = \frac{18,000 + 20,000 + 22,000}{3} = \frac{60,000}{3} = ₹20,000 ]
2. New Average (after 4th month)
- New sum (S' = 60,000 + 24,000 = 84,000)
- New count (n' = 4)
[ \bar{x}_2 = \frac{84,000}{4} = ₹21,000 ]
3. Desired Average ₹21,000
Let the 5th month salary be (X).
- Current total after 4 months: (84,000)
- Desired total after 5 months: [ \text{Desired total} = 21,000 \times 5 = 105,000 ]
- Solve for (X): [ X = 105,000 - 84,000 = ₹21,000 ]
Answer
- ₹20,000
- ₹21,000
- ₹21,000
📌 4. Practice Problem (Try it)
A group of 6 patrol officers have the following number of cases solved in a week: 12, 15, 20, 10, 18, 14.
- What is the average number of cases solved per officer?
- Two more officers join and solve 16 and 19 cases respectively. What becomes the new average?
- If the average needs to increase to 17 cases, how many cases must the next officer solve?
(Use the formulas above to calculate. Write your steps clearly.)
📌 5. Quick Revision Summary
| What to Remember | How to Apply |
|---|---|
| Average = Sum / Count | Add all numbers, divide by how many you added. |
| Weighted average | Multiply each value by its weight, sum, divide by total weight. |
| Average of averages | Multiply each group’s average by its size, sum, divide by total size. |
| Updating average | Keep running sum and count; update after each new value. |
| Reverse problem | Set up equation: New average × New count = Old total + New value. Solve for new value. |
| Common pitfalls | Wrong divisor, wrong weights, forgetting to include all values. |
📌 Tips for the Exam
- Read the question carefully – look for phrases like “average of the remaining”, “average after addition”, “weighted average”.
- Write down the formula before starting – it keeps you organized.
- Check units – sometimes the problem involves percentages or ratios; convert them before using the formula.
- Double‑check the divisor – number of items, not the sum.
- Practice quick mental addition – it speeds up your calculations.
Good luck, Constables! Master the average, and you’ll ace many of the quantitative problems that come your way.