Class 8 Mathematics Complete Study Guide for BLE Exam 2026
Complete Class 8 Mathematics study guide for BLE exam covering algebra, geometry, arithmetic, and statistics with formulas, examples, and practice questions.
Table of Contents
Table of Contents
Introduction to Class 8 Mathematics (BLE)#
The Basic Level Examination (BLE) Mathematics tests fundamental concepts that bridge primary and secondary mathematics. This guide covers all topics in the CDC Class 8 curriculum with BLE-focused preparation.
Unit 1: Number System#
Rational and Irrational Numbers#
Number expressible as p/q where p,q ∈ Z, q ≠ 0
Number not expressible as p/q. Non-terminating, non-recurring decimals.
| Rational | Irrational |
|---|---|
| 1/2, 3/4, 0.75 | √2, √3, π, e |
| Terminating decimals | Non-terminating, non-recurring |
| Recurring decimals | 0.1010010001... |
| Integers | √5, √7 |
- Sum/product of rationals = rational
- Sum/product of rational + irrational = irrational
- Product of rational (≠0) and irrational = irrational
- Sum of two irrationals can be rational (e.g., √2 + (2-√2) = 2)
Real Numbers#
- R = Q ∪ Q' (Rationals ∪ Irrationals)
- Represented on number line
- Every point on number line represents a real number
1/(√3 + √2) = (√3 - √2)/((√3+√2)(√3-√2)) = (√3 - √2)/(3-2) = √3 - √2
Surds#
| Type | Form | Example |
|---|---|---|
| Pure surd | √a | √2, √3, √5 |
| Mixed surd | a√b | 2√3, 5√2 |
| Like surds | Same radicand | 3√2, 5√2 |
| Unlike surds | Different radicands | √2, √3 |
- a√b ± c√b = (a±c)√b (like surds)
- √a × √b = √(ab)
- √a / √b = √(a/b)
- (√a)² = a
Unit 2: Algebra#
Polynomials#
Expression of form aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀ where aᵢ are constants
| Degree | Name | General Form |
|---|---|---|
| 0 | Constant | a₀ |
| 1 | Linear | ax + b |
| 2 | Quadratic | ax² + bx + c |
| 3 | Cubic | ax³ + bx² + cx + d |
| 4 | Quartic | ax⁴ + bx³ + cx² + dx + e |
- Addition: Add like terms
- Multiplication: Use distributive law
- Division: Long division or synthetic division
Factorization#
| Method | Formula/Rule |
|---|---|
| Common factor | ax + ay = a(x + y) |
| Grouping | ax + ay + bx + by = (a+b)(x+y) |
| a² - b² | (a+b)(a-b) |
| a² + 2ab + b² | (a+b)² |
| a² - 2ab + b² | (a-b)² |
| a² + b² + c² + 2ab + 2bc + 2ca | (a+b+c)² |
| a³ + b³ | (a+b)(a² - ab + b²) |
| a³ - b³ | (a-b)(a² + ab + b²) |
x² + 5x + 6 = (x+2)(x+3)
4x² - 9 = (2x+3)(2x-3)
x³ + 8 = (x+2)(x² - 2x + 4)
x³ - 27 = (x-3)(x² + 3x + 9)
Algebraic Fractions#
- Addition/Subtraction: LCM of denominators
- Multiplication: Numerator × Numerator / Denominator × Denominator
- Division: Multiply by reciprocal
- Simplification: Factorize, cancel common factors
Linear Equations in Two Variables#
Form: ax + by + c = 0
| Method | Steps |
|---|---|
| Substitution | Solve one for x, substitute in other |
| Elimination | Multiply to make coefficients equal, add/subtract |
| Cross-multiplication | x/(b₁c₂-b₂c₁) = y/(c₁a₂-c₂a₁) = 1/(a₁b₂-a₂b₁) |
2x + 3y = 13
3x - 2y = 1
Multiply first by 2, second by 3:
4x + 6y = 26
9x - 6y = 3
Add: 13x = 29 → x = 29/13 = 2.23...
y = (13 - 2x)/3
Quadratic Equations#
Form: ax² + bx + c = 0
| Method | Formula/Steps |
|---|---|
| Factorization | Find factors of ac summing to b |
| Quadratic Formula | x = [-b ± √(b²-4ac)]/2a |
| Completing Square | x² + bx = (x+b/2)² - (b/2)² |
D = b² - 4ac
- D > 0: Two distinct real roots
- D = 0: Equal real roots
- D < 0: No real roots (complex)
x² - 5x + 6 = 0 → (x-2)(x-3) = 0 → x = 2, 3
2x² - 4x - 6 = 0 → x = [4 ± √(16+48)]/4 = [4 ± 8]/4 → x = 3, -1
Find k for equal roots: x² + 4x + k = 0 → D = 16 - 4k = 0 → k = 4
Unit 3: Geometry#
Lines and Angles#
| Angle Pair | Sum/Relation |
|---|---|
| Complementary | 90° |
| Supplementary | 180° |
| Linear pair | 180° |
| Vertically opposite | Equal |
| Corresponding (parallel lines) | Equal |
| Alternate interior (parallel lines) | Equal |
| Interior on same side | Supplementary |
Triangles#
| Congruence Rule | Condition |
|---|---|
| SSS | Three sides equal |
| SAS | Two sides and included angle |
| ASA | Two angles and included side |
| AAS | Two angles and non-included side |
| RHS | Right angle, hypotenuse, side |
- Sum of angles = 180°
- Exterior angle = Sum of opposite interior angles
- Larger side opposite larger angle
- Sum of any two sides > third side
- Difference of any two sides < third side
| Similarity Criterion | Condition |
|---|---|
| AAA | All angles equal |
| SSS | Sides proportional |
| SAS | Two sides proportional, included angle equal |
Quadrilaterals#
| Quadrilateral | Properties |
|---|---|
| Parallelogram | Opposite sides parallel & equal, diagonals bisect |
| Rectangle | All angles 90°, diagonals equal |
| Square | All sides equal, all angles 90°, diagonals equal & perpendicular |
| Rhombus | All sides equal, diagonals perpendicular bisectors |
| Trapezium | One pair of opposite sides parallel |
| Kite | Adjacent sides equal, one diagonal bisects other |
Circles#
- Angle at center = 2 × Angle at circumference (same arc)
- Angles in same segment = Equal
- Angle in semicircle = 90°
- Opposite angles of cyclic quadrilateral = 180°
- Tangent ⊥ Radius at point of contact
- Tangents from external point = Equal length
| Term | Definition |
|---|---|
| Chord | Line segment joining two points on circle |
| Secant | Line intersecting circle at two points |
| Tangent | Line touching circle at one point |
| Sector | Region between two radii and arc |
| Segment | Region between chord and arc |
Constructions#
- Perpendicular bisector of line segment
- Angle bisector
- Perpendicular from point to line
- Parallel line through given point
- Triangle given SSS, SAS, ASA, RHS
- Quadrilateral with given measurements
- Tangent to circle from external point
- Circumcircle and incircle of triangle
Unit 4: Arithmetic#
Ratio and Proportion#
| Concept | Formula/Rule |
|---|---|
| Ratio | a = a/b (a, b same units) |
| Proportion | a = c ↔ a/b = c/d ↔ ad = bc |
| Continued proportion | a = b → b² = ac |
| Mean proportional | b = √(ac) |
| Third proportional | a = b → c = b²/a |
| Fourth proportional | a = c → d = bc/a |
Percentage#
| Type | Formula |
|---|---|
| Percentage | % = (Part/Whole) × 100 |
| Increase | New = Original × (1 + %/100) |
| Decrease | New = Original × (1 - %/100) |
| Original from increased | Original = New / (1 + %/100) |
| Original from decreased | Original = New / (1 - %/100) |
| Profit % | (Profit/CP) × 100 |
| Loss % | (Loss/CP) × 100 |
| Discount % | (Discount/MP) × 100 |
Simple and Compound Interest#
| Type | Formula |
|---|---|
| Simple Interest | SI = PRT/100, Amount = P + SI |
| Compound Interest (annual) | A = P(1 + R/100)ᵀ, CI = A - P |
| Compound Interest (half-yearly) | A = P(1 + R/200)²ᵀ |
| Compound Interest (quarterly) | A = P(1 + R/400)⁴ᵀ |
P = 10000, R = 10%, T = 2 years
SI = 10000×10×2/100 = 2000
CI (annual) = 10000(1.1)² - 10000 = 2100
CI (half-yearly) = 10000(1.05)⁴ - 10000 = 2155.06
Profit, Loss, and Discount#
| Term | Formula |
|---|---|
| Profit | SP - CP (if SP > CP) |
| Loss | CP - SP (if CP > SP) |
| Profit % | (Profit/CP) × 100 |
| Loss % | (Loss/CP) × 100 |
| SP with profit | CP × (100 + P%)/100 |
| SP with loss | CP × (100 - L%)/100 |
| CP from SP & profit | SP × 100/(100 + P%) |
| CP from SP & loss | SP × 100/(100 - L%) |
| Discount | MP - SP |
| Discount % | (Discount/MP) × 100 |
Unit 5: Statistics#
Measures of Central Tendency#
| Measure | Formula |
|---|---|
| Mean (Ungrouped) | Σx/n |
| Mean (Grouped) | Σfx/Σf |
| Median (Odd n) | (n+1)/2 th value |
| Median (Even n) | Average of n/2 and n/2+1 th |
| Median (Grouped) | L + ((N/2 - cf)/f) × h |
| Mode (Ungrouped) | Most frequent value |
| Mode (Grouped) | L + ((f₁-f₀)/(2f₁-f₀-f₂)) × h |
Measures of Dispersion#
| Measure | Formula |
|---|---|
| Range | Max - Min |
| Quartile Deviation | (Q₃ - Q₁)/2 |
| Mean Deviation | Σf |
| Standard Deviation (Ungrouped) | √[Σ(x - x̄)²/n] |
| Standard Deviation (Grouped) | √[Σf(x - x̄)²/Σf] |
| Variance | (SD)² |
| Coefficient of Variation | (SD/Mean) × 100% |
Data: 5, 8, 10, 12, 15, 15, 18, 20
Mean = 103/8 = 12.875
Median = (12+15)/2 = 13.5
Mode = 15 (appears twice)
Range = 20 - 5 = 15
Practice Questions#
Algebra:
- Simplify: (2x+3)² - (2x-3)²
- Factorize: x² - 7x + 12
- Solve: 3x + 4y = 10, 2x - y = 1
- Roots of x² - 5x + 6 = 0
- If x + 1/x = 5, find x² + 1/x²
Geometry:
- Prove: Sum of angles of triangle = 180°
- Construct triangle ABC: AB=5cm, BC=6cm, ∠B=60°
- Prove: Angle in semicircle = 90°
- Construct tangent from point 5cm from center of circle radius 3cm
Arithmetic:
- Find 15% of 2400
- CP = 800, Profit = 15%. Find SP
- MP = 1200, Discount = 10%. Find SP
- CI on 5000 at 10% for 2 years
- Ratio 3 = x. Find x
Statistics:
- Mean of 5, 8, 10, 12, 15
- Median of 3, 7, 9, 12, 15, 18
- Mode of 2, 4, 4, 6, 7, 4, 8
- Range of 10, 15, 20, 25, 30
BLE Exam Tips#
| Tip | Description |
|---|---|
| Time Management | 2.5 hrs theory, 0.5 hr practical. Allocate ~2 min/mark |
| Show Working | Always write steps. Partial marks for correct method |
| Diagrams | Draw neat, labeled diagrams for geometry |
| Constructions | Practice with compass, ruler, protractor |
| Formula Sheet | Create one-page formula sheet for revision |
| Mental Math | Practice basic calculations without calculator |
| Past Papers | Solve at least 5 past BLE papers |
Conclusion#
Class 8 Mathematics is about building a strong foundation. Focus on understanding concepts rather than memorizing. Practice daily: 5 algebra, 3 geometry, 2 arithmetic problems. Draw diagrams for every geometry problem. With consistent practice, BLE Mathematics is very scoring.
"Mathematics is not about numbers, equations, or algorithms. It is about understanding."
Good luck with your BLE 2026 preparation!
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