Average, Mensuration and Data Interpretation
Average, Mensuration and Data Interpretation
This topic closes the maths block and is the most formula-heavy part of it. Expect 2 to 3 questions — typically one average sum, one mensuration formula and one small table or bar-chart reading. None of it is hard, but the formulas must be recalled instantly.
Average and Weighted Average
Average = Sum of observations ÷ Number of observations
Two situations get asked repeatedly. First, when a new member joins or leaves and the average shifts. Second, when two groups merge.
Weighted average = (n₁a₁ + n₂a₂) ÷ (n₁ + n₂)
A class of 30 students averages 60 marks and another of 20 averages 70. The combined average is (30 × 60 + 20 × 70) ÷ 50 = 64, not 65.
For a replacement, the change in total equals the number of members multiplied by the change in average. If the average age of 10 teachers rises by 2 years when one leaves and a new one joins, the new teacher is 20 years older than the one replaced.
Mensuration Formula Sheet
| Shape | Area / Volume | Perimeter / Surface Area |
|---|---|---|
| Square | side² | 4 × side |
| Rectangle | length × breadth | 2(l + b) |
| Triangle | ½ × base × height | Sum of three sides |
| Equilateral triangle | (√3/4) × side² | 3 × side |
| Circle | πr² | 2πr |
| Cube | side³ | 6 × side² |
| Cuboid | l × b × h | 2(lb + bh + hl) |
| Cylinder | πr²h | 2πr(r + h) |
✗ Circumference of a circle = πr² | ✓ Circumference = 2πr; πr² is the area
Use π = 22/7 when the radius is a multiple of 7, and 3.14 otherwise. The examiner picks radii like 7, 14 or 21 precisely so that 22/7 cancels cleanly.
Reading Tables, Bar Charts and Pie Charts
Data interpretation in Part-II stays simple: read a value, find a total, compute a percentage or a ratio. Three habits protect your marks.
- Read the heading and the unit first — thousands and lakhs are mixed up deliberately.
- Note whether the question asks for a difference or a percentage difference.
- In a pie chart, remember that 360 degrees equals the whole, so 1% is 3.6 degrees.
Pie chart: Degrees = (Value ÷ Total) × 360
वृत्त की परिधि = 2πr और क्षेत्रफल = πr² — दोनों को मिलाना सबसे आम गलती है।
TRE pointer: Because you are being selected to teach classes 6 to 8, mensuration here is asked exactly at NCERT level — area, perimeter, surface area and volume of standard shapes, with no derivations. The favourite trap is swapping circumference with area. In data interpretation, the wrapper is often a district-wise or school-wise table, so read the unit row carefully. And keep the marking scheme in mind: with −1/3 per wrong answer and a compulsory option E for skipping, it is better to leave a doubtful DI set with E than to guess three of them.
60-Second Recap
- Average = sum ÷ number; combined average is weighted, not the plain mean.
- Replacement change = number of members × change in average.
- Circle: area πr², circumference 2πr. Use π = 22/7 when the radius is a multiple of 7.
- Cuboid volume lbh, surface area 2(lb + bh + hl); cylinder volume πr²h.
- Pie chart: degrees = (value ÷ total) × 360, so 1% = 3.6 degrees.
- Read units and headings before calculating — that is where DI marks leak.
Frequently Asked Questions
Which mensuration formulas are needed for BPSC TRE Part-II?
Stick to the NCERT class 6-8 set: area and perimeter of square, rectangle, triangle and circle, plus volume and surface area of cube, cuboid and cylinder. Derivations are not asked. Knowing when to use 22/7 for pi saves more time than learning extra formulas.
How is combined average different from simple average?
A simple average divides the total by the count. A combined average weights each group by its size, so it is (n₁a₁ + n₂a₂) divided by (n₁ + n₂). Taking the plain mean of two group averages is correct only when both groups have exactly the same number of members.
How do I convert a pie chart value into degrees?
Divide the value by the total, then multiply by 360. A quick check is that one percent of the whole equals 3.6 degrees, so 25 percent is 90 degrees. Pie chart questions in Part-II rarely go beyond this single conversion plus a ratio comparison.
Should I guess a data interpretation question in TRE 4.0?
Be careful. Every paper has five options and each wrong answer costs one-third of a mark, so three wrong guesses cancel a correct answer. If you cannot read the set confidently, mark option E, which is the compulsory way to leave a question unattempted. A blank row is penalised the same as a wrong one.
Next step: you have the concepts — now test them under exam conditions. Practise this chapter's questions in the BPSC TRE 4.0 mock test on ExamAtlas, with the same five-option pattern and −1/3 negative marking you will face on exam day.