BLE Math Formula Sheet
Grade 8 CDC Nepal Curriculum
Sets & Number Systems
1.1 Subsets & Types of Sets
• Subsets: Total possible subsets = $2^n$
• Proper Subsets: Subsets excluding the set itself = $2^n – 1$
• Improper Subset: The set itself = Always 1
• Overlapping Sets: Sets having at least one common element ($A \cap B \neq \emptyset$).
• Disjoint Sets: Sets having no common elements ($A \cap B = \emptyset$).
1.2 Binary & Quinary Number Systems
• Decimal $\to$ Binary: Divide by 2 continuously.
• Decimal $\to$ Quinary: Divide by 5 continuously.
• Binary $\to$ Decimal: Multiply by powers of 2.
• Quinary $\to$ Decimal: Multiply by powers of 5.
• Binary $\to$ Quinary: First convert Binary $\to$ Decimal, then Decimal $\to$ Quinary.
Rational & Irrational Numbers
Cannot be written in the form $\frac{p}{q}$
e.g. $\sqrt{2}, \sqrt{3}, \pi, 0.101001\dots$
Scientific Notation (Standard Form)
$N = a \times 10^k$ where $1 \le a < 10$ and $k \in \mathbb{Z}$.
Ratio and Proportion
Ratio & Compound Ratio
• Ratio of $a$ and $b = a : b = \frac{a}{b}$
• Compound Ratio of $(a:b)$ and $(c:d) = ac : bd$
Proportion Relations (Step-by-step)
If $a : b = c : d$
$\implies \frac{a}{b} = \frac{c}{d}$
$\implies a \times d = b \times c$
(Product of Extremes = Product of Means)
Continued Proportion ($a, b, c$):
$\implies \frac{a}{b} = \frac{b}{c}$
$\implies b^2 = a \times c$
Profit, Loss, Discount & VAT
1. Profit Case
• $\text{Profit Amount} = SP – CP$
• $\text{Profit Amount} = \frac{\text{Profit}\% \times CP}{100}$
• $SP = \frac{(100 + \text{Profit})\% \times CP}{100}$
• $CP = \frac{SP \times 100}{(100 + \text{Profit})\%}$
• $\text{Profit}\% = \frac{\text{Profit}}{CP} \times 100\%$
2. Loss Case
• $\text{Loss Amount} = CP – SP$
• $\text{Loss Amount} = \frac{\text{Loss}\% \times CP}{100}$
• $SP = \frac{(100 – \text{Loss})\% \times CP}{100}$
• $CP = \frac{SP \times 100}{(100 – \text{Loss})\%}$
• $\text{Loss}\% = \frac{\text{Loss}}{CP} \times 100\%$
3. Discount Case
• $\text{Discount Amount} = MP – SP$
• $\text{Discount Amount} = \frac{\text{Discount}\% \times MP}{100}$
• $SP = \frac{(100 – \text{Discount})\% \times MP}{100}$
• $\text{Discount}\% = \frac{\text{Discount}}{MP} \times 100\%$
4. VAT (Value Added Tax) Case
• $\text{VAT Amount} = \frac{\text{VAT}\% \times SP}{100}$
• $\text{Price with VAT} = SP + \text{VAT Amount}$
• $\text{Price with VAT} = \frac{(100 + \text{VAT})\% \times SP}{100}$
Unitary Method
1. Direct Variation
• To find unit cost $\to$ Divide
• To find more cost $\to$ Multiply
$\frac{\text{Quantity}_1}{\text{Cost}_1} = \frac{\text{Quantity}_2}{\text{Cost}_2}$
2. Indirect Variation
• To find 1 person/day work $\to$ Multiply
• To find more than 1 person work $\to$ Divide
$M_1 \times D_1 \times H_1 = M_2 \times D_2 \times H_2$
Simple Interest
1. Interest ($I$)
$I = \frac{P \times T \times R}{100}$
$I = A – P$
2. Principal ($P$)
$P = \frac{I \times 100}{T \times R}$
$P = \frac{A \times 100}{100 + T \times R}$
$P = A – I$
3. Time ($T$ in yrs)
$T = \frac{I \times 100}{P \times R}$
4. Rate ($R\%$)
$R = \frac{I \times 100}{P \times T}$
Total Amount ($A$): $A = P + I = P\left(1 + \frac{T \times R}{100}\right)$
Area and Perimeter (Plane Figures)
1. Rectangle
• $\text{Area} = l \times b$
• $\text{Perimeter} = 2(l+b)$
2. Square
• $\text{Area} = l^2$ or $\frac{d^2}{2}$
• $\text{Perimeter} = 4l$
3. Right-Angled Triangle
• $\text{Area} = \frac{1}{2} b \times h$
• $\text{Perimeter} = a+b+c$
4. Circle
• $\text{Area} = \pi r^2$
• $\text{Circumference} = 2\pi r$
5. Trapezium
Parallel sides $p_1, p_2$, Height $h$
$\text{Area} = \frac{1}{2} \times h \times (p_1 + p_2)$
6. General Quadrilateral
Diagonal $d$, Heights $h_1, h_2$
$\text{Area} = \frac{1}{2} \times d \times (h_1 + h_2)$
7. Rhombus
Diagonals $d_1, d_2$
$\text{Area} = \frac{1}{2} \times d_1 \times d_2$
8. Equilateral Triangle
All sides equal ($a$)
$\text{Area} = \frac{\sqrt{3}}{4} a^2$
9. Isosceles Triangle
Two equal sides ($a$), Base ($b$)
$\text{Area} = \frac{b}{4} \sqrt{4a^2 – b^2}$
10. Scalene Triangle
All sides different ($a,b,c$)
$\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$
where $s = \frac{a+b+c}{2}$
Indices (Laws of Exponents)
Product Law
$a^m \cdot a^n = a^{m+n}$
Quotient Law
$\frac{a^m}{a^n} = a^{m-n}$
Power Law
$(a^m)^n = a^{mn}$
Zero Index
$a^0 = 1$ ($a \neq 0$)
Negative Index
$a^{-n} = \frac{1}{a^n}$
Root Index
$a^{m/n} = \sqrt[n]{a^m}$
Distributive Power
$(ab)^n = a^n b^n, \quad \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$
Algebraic Expressions & Identities
1. Square Identities
• $(a+b)^2 = a^2 + 2ab + b^2$
• $(a-b)^2 = a^2 – 2ab + b^2$
• $a^2 – b^2 = (a+b)(a-b)$
• $a^2 + b^2 = (a+b)^2 – 2ab$
$= (a-b)^2 + 2ab$
2. Cube Identities (Stepwise)
• $(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$
$= a^3 + b^3 + 3ab(a+b)$
• $(a-b)^3 = a^3 – 3a^2b + 3ab^2 – b^3$
$= a^3 – b^3 – 3ab(a-b)$
• $a^3 + b^3 = (a+b)(a^2 – ab + b^2)$
• $a^3 – b^3 = (a-b)(a^2 + ab + b^2)$
Algebraic Fractions, HCF & LCM
1. Fraction Operations
• Addition/Subtraction: $\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}$
• Multiplication: $\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$
• Division: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$
2. HCF & LCM Rules
• HCF: Product of common factors with lowest powers.
• LCM: Product of common factors and remaining factors.
• $\text{HCF} \times \text{LCM} = \text{Expression}_1 \times \text{Expression}_2$
Equations and Graphs
1. Linear Equations & Value Table
• In any equation: $y = mx + c$ ($m = \text{slope}, c = \text{y-intercept}$)
• Example table calculation for graph ($y = 2x + 1$):
| x | -1 | 0 | 1 | 2 |
|---|---|---|---|---|
| y | -1 | 1 | 3 | 5 |
2. Graph Quadrants Rules
• Origin Point: $(0,0)$
• Quadrant I: $(+,+)$, Quadrant II: $(-,+)$
• Quadrant III: $(-,-)$, Quadrant IV: $(+,-)$
Lines and Angles
1. Basic Angle Types
• Complementary Angles: $a + b = 90^\circ$
• Supplementary Angles: $a + b = 180^\circ$
• Vertically Opposite Angles: Equal ($a = b$)
• Angles on a Straight Line: Sum $= 180^\circ$
• Angles around a point: Sum $= 360^\circ$
2. Parallel Lines & Transversal
• Corresponding Angles: Equal ($F$-shape)
• Alternate Angles: Equal ($Z$-shape)
• Co-interior Angles: Sum $= 180^\circ$ ($C$-shape)
Plane Figures & Geometrical Theorems
1. Triangles Theorems
• Sum of angles of a triangle $= 180^\circ$
• Exterior angle $=$ Sum of two opposite interior angles.
• Pythagoras Theorem (Right $\Delta$): $h^2 = p^2 + b^2$
2. Quadrilaterals & Polygons
• Sum of angles of a quadrilateral $= 360^\circ$
• Parallelogram Area $= b \times h$
• Rhombus Area $= \frac{1}{2} d_1 d_2$
• Trapezium Area $= \frac{1}{2} (a + b) h$
• Sum of interior angles of $n$-sided polygon $= (n-2) \times 180^\circ$
Congruency and Similarity
1. Congruent Triangles ($\cong$)
Same shape and same size.
• S.S.S. (Side-Side-Side)
• S.A.S. (Side-Angle-Side)
• A.S.A. / A.A.S. (Angle-Side-Angle)
• R.H.S. (Right Angle-Hypotenuse-Side)
2. Similar Triangles ($\sim$)
Same shape, proportional size.
• Conditions: A.A.A., S.S.S., S.A.S.
• Ratio of corresponding sides are equal:
$\frac{AB}{PQ} = \frac{BC}{QR} = \frac{AC}{PR}$
Solid Objects (3D Geometry)
1. Cube
• Volume $= l^3$
• Total Surface Area $= 6l^2$
• Lateral Surface Area $= 4l^2$
2. Cuboid
• Volume $= l \times b \times h$
• TSA $= 2(lb + bh + lh)$
• LSA $= 2h(l + b)$
3. Cylinder
• Volume $= \pi r^2 h$
• Curved Area $= 2\pi r h$
• TSA $= 2\pi r (r + h)$
4. Prism
• Volume $= \text{Base Area} \times h$
• LSA $= \text{Base Perimeter} \times h$
Coordinate Geometry
1. Distance Formula
Distance between $P(x_1, y_1)$ and $Q(x_2, y_2)$:
$d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}$
2. Midpoint Formula
Midpoint $M(x, y)$ between $(x_1, y_1)$ and $(x_2, y_2)$:
$M(x, y) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$
Tessellation
1. Regular Tessellation
Made of only one type of regular polygon (e.g. Squares, Triangles, Hexagons).
2. Semi-Regular Tessellation
Made of two or more types of regular polygons at each vertex.
3. Irregular Tessellation
Made of non-regular polygons or asymmetrical shapes without gaps.
Geometric Transformations
1. Reflection Rules $P(x,y) \to P’$
• In X-axis ($y=0$): $(x, -y)$
• In Y-axis ($x=0$): $(-x, y)$
2. Rotation about Origin $(0,0)$
• $+90^\circ$ (Quarter turn anticlockwise / $270^\circ$ clockwise): $(-y, x)$
• $-90^\circ$ (Quarter turn clockwise / $270^\circ$ anticlockwise): $(y, -x)$
• $\pm 180^\circ$ (Half turn): $(-x, -y)$
3. Translation
• Vector $T = \begin{pmatrix} a \\ b \end{pmatrix}$: $P(x,y) \to P'(x+a, y+b)$
Bearing and Scale Drawing
1. Three-Figure Bearing Rules
1. Always measured from True North Line.
2. Always measured in Clockwise direction.
3. Always written in 3 Digits (e.g. $045^\circ, 090^\circ, 225^\circ$).
• Compass Bearing: Written using N/S direction e.g., $N 45^\circ E$.
• Back Bearing Calculation:
– If angle $\theta < 180^\circ \implies \text{Back Bearing} = \theta + 180^\circ$
– If angle $\theta \ge 180^\circ \implies \text{Back Bearing} = \theta – 180^\circ$
2. Scale Drawing Formulas
• $\text{Scale Factor} = \frac{\text{Map / Drawing Distance}}{\text{Actual Ground Distance}}$
• $\text{Actual Distance} = \text{Drawing Distance} \times \text{Scale Factor}$
• Example: Scale $1 : 100,000 \implies 1\text{ cm on map} = 1\text{ km in ground}$.
Statistics & Pie Chart
1. Mean ($\bar{x}$)
• Individual Data: $\bar{x} = \frac{\sum x}{N}$
• Discrete Frequency: $\bar{x} = \frac{\sum fx}{N}$ ($N = \sum f$)
2. Median ($M_d$) Position
• Individual/Discrete Data: $\text{Position} = \left(\frac{N + 1}{2}\right)^{\text{th}} \text{ item}$
3. Mode & Range
• Mode: Value with highest frequency.
• Range: $H – L$ (Highest $-$ Lowest value)
4. Pie Chart Calculations & Drawing Steps
$\text{Central Angle} = \frac{\text{Given Value}}{\text{Total Value}} \times 360^\circ$
2. To find Value from Central Angle:$\text{Value} = \frac{\text{Central Angle}}{360^\circ} \times \text{Total Value}$
- Find the sum of all components ($\text{Total}$).
- Calculate the central angle for each component.
- Draw a circle of suitable radius with a baseline.
- Use a protractor to draw angles successively.