Subjects
Grades
| Property | Addition | Multiplication |
|---|---|---|
| ||a+b=b+a|| | ||a\times b=b\times a|| |
| ||(a+b)+c=a+(b+c)|| | ||(a\times b)\times c=a\times(b\times c)|| |
| ||a+0=0+a=a|| | ||a\times1=1\times a=a|| |
| ||a\times0=0\times a=0|| | |
| ||a+-a=-a+a=0|| | ||a\times\dfrac{1}{a}=1|| |
| ||a\times(b\pm c)=a\times b\pm a\times c|| | |
| Functions | Basic rule | Transformed rule | ||
|---|---|---|---|---|
| 0 Degree | ||y=b|| | |||
| 1st degree | ||y=x|| | Functional form | Symmetrical form | General form |
||y=ax+b|||a|: rate of change (slope) |b|: y-intercept||a=\dfrac{y_2-y_1}{x_2-x_1}|| | ||\dfrac{x}{a}+\dfrac{y}{b}=1|||a|: x-intercept |b|: y-intercept | ||Ax+By+C=0|| | ||
| |\Rightarrow| Symmetrical||\begin{align}a_s&=\dfrac{-b_f}{a_f}\\b_s&=b_f\end{align}|| | |\Rightarrow| Functional||\begin{align}a_f&=\dfrac{-b_s}{a_s}\\b_f&=b_s\end{align}|| | |\Rightarrow| Functional||\begin{align}a_f&=\dfrac{-A}{B}\\b_f&=\dfrac{-C}{B}\end{align}|| | ||
|\Rightarrow| General Find the common denominator and bring everything to the same side of the equation. | |\Rightarrow| General Find the common denominator and bring everything to the same side of the equation. | |\Rightarrow| Symmetrical||\begin{align}a_s&=\dfrac{-C}{A}\\\\b_s&=\dfrac{-C}{B}\end{align}|| | ||
| |\text{km}| | |\text{hm}| | |\text{dam}| | |\text{m}| | |\text{dm}| | |\text{cm}| | |\text{mm}| |
| In this direction |\Rightarrow \times 10\qquad \qquad\qquad| In this direction |\Leftarrow \div 10| | ||||||
| |\text{km}^2| | |\text{hm}^2| | |\text{dam}^2| | |\text{m}^2| | |\text{dm}^2| | |\text{cm}^2| | |\text{mm}^2| |
| In this direction |\Rightarrow \times 100\qquad \qquad\qquad| In this direction |\Leftarrow \div 100| | ||||||
| |\text{km}^3| | |\text{hm}^3| | |\text{dam}^3| | |\text{m}^3| | |\text{dm}^3| | |\text{cm}^3| | |\text{mm}^3| |
| In this direction |\Rightarrow \times 1000\qquad \qquad\qquad| In this direction |\Leftarrow \div 1000| | ||||||
| Figure | Perimeter | Area | |
|---|---|---|---|
| Triangle | The sum of all sides | |A =\dfrac{b\times h}{2}| |A = \sqrt{p(p-a)(p-b)(p-c)}| |A=\dfrac{ab\sin C}{2}| | |
| Square | |P=4 \times s| | |\begin{align} A &= s \times s\\ A &= s^2 \end{align}| | |
| Rectangle | |\begin{align} P &= b+h+b+h\\ P &= 2(b+h) \end{align}| | |A=bh| | |
| Rhombus | P=|4 \times s| | |A=\dfrac{D\times d}{2}| | |
| Parallelogram | The sum of all sides | |A=bh| | |
| Trapezoid | The sum of all sides | |A=\dfrac{(B+b)\times h}{2}| | |
| Regular polygon | |P=n \times s| | |A=\dfrac{san}{2}| | |
| Any polygon | The sum of all sides | The sum of the areas of all the triangles that make up the polygon | |
| Circle | |\begin{align} d &= 2r\\\\ r &= \frac{d}{2} \end{align}| | ||\begin{align} C &= \pi d\\\\ C &= 2 \pi r \end{align}|| | |A=\pi r^2| |
| Circular arc and sector of a circle | |\displaystyle \frac{\text{Central angle}}{360^o}=\frac{\text{Arc length}}{2\pi r}| | |\displaystyle \frac{\text{Central angle}}{360^o}=\frac{\text{Area of sector}}{\pi r^2}| | |
| Solids | Lateral area | Total area | Volume |
|---|---|---|---|
| Prism and cylinder | Sum of the areas of the lateral faces of the solid |A_L=P_b\times h| | Sum of the areas of all faces of the solid |A_T = A_L+2A_b| | |V=A_b\times h| |
| Pyramid and cone | Sum of the areas of the lateral faces of the solid |A_L=\displaystyle \frac{P_b\times a}{2}| | Sum of the areas of all faces of the solid |A_T = A_L+A_b| | |V=\displaystyle \frac{A_b\times h}{3}| |
| Sphere | |A=4\pi r^2| | |V=\displaystyle \frac{4\pi r^3}{3}| | |
| Theorems in a right triangle |
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| Similarity ratio (Scale factor) | Area ratio | Volume ratio |
|---|---|---|
| ||k=\dfrac{\text{Length of image figure}}{\text{Length of initial figure}}|| | ||k^2=\dfrac{\text{Area of image figure}}{\text{Area of initial figure}}|| | ||k^3=\dfrac{\text{Volume of image solid}}{\text{Volume of initial solid}}|| |
| Concept | Formulas | ||
|---|---|---|---|
| Displacements | ||\begin{align}\Delta x&=x_2-x_1\\[3pt]\Delta y&=y_2-y_1\end{align}|| | ||
| Slope (rate of change) of a line | ||a=\dfrac{\Delta y}{\Delta x}=\dfrac{y_2-y_1}{x_2-x_1}|| | ||
| Relative position of two lines with equations of the form |y=ax+b| | Coinciding parallel lines | Disjoint parallel lines | Perpendicular lines |
| ||\begin{align}a_1&=a_2\\[3pt]b_1&=b_2\end{align}|| | ||\begin{align}a_1&=a_2\\[3pt]b_1&\neq b_2\end{align}|| | ||a_1=-\dfrac{1}{a_2}|| | |
| Concept | Formulas |
|---|---|
| Probability | ||\text{Probability}=\dfrac{\text{No. of favourable outcomes}}{\text{No. of possible outcomes}}|| |
| Complementary probability | ||\mathbb{P}(A')=1-P(A)|| |
| Probability of incompatible events | ||\mathbb{P}(A\cup B)=\mathbb{P}(A)+\mathbb{P}(B)|| |
| Probability of compatible events | ||\mathbb{P}(A\cup B)=\mathbb{P}(A)+\mathbb{P}(B)-\mathbb{P}(A\cap B)|| |
| Measure | Listed data | Condensed data | Grouped data |
|---|---|---|---|
| Mean | ||\overline{x}=\dfrac{\sum x_i}{n}|| | ||\overline{x}=\dfrac{\sum x_i n_i}{n}|| | Approximate mean: ||\overline{x}=\dfrac{\sum m_i n_i}{n}|| |
| Median | ||\text{Rank}_\text{median}=\left(\dfrac{n+1}{2}\right)|| If |n| is odd, the median is obtained directly. If |n| is even, the median is obtained by calculating the mean of the two central data values. | ||\text{Rank}_\text{median}=\left(\dfrac{n+1}{2}\right)|| If |n| is odd, the median is obtained directly. If |n| is even, the median is obtained by calculating the mean of the two central data values. | Medial class: The class that contains the median. The median of a grouped-data distribution is often estimated by calculating the middle of the medial class. |
| Mode | The most frequent data value | The most frequent data value | Modal class: The class with the largest frequency |
| Measure | Listed data | Condensed data | Grouped data |
|---|---|---|---|
| Range | ||R=x_\text{max}-x_\text{min}|| | ||R=\text{Value}_\text{Max}-\text{Value}_\text{Min}|| | ||R=\text{Boundary}_\text{upper}-\text{Boundary}_\text{lower}|| |
| Interquartile range | ||IR=Q_3-Q_1|| | ||IR=Q_3-Q_1|| | ||IR=Q_3-Q_1|| |
| Quarter range | ||Q=\dfrac{EI}{2}=\dfrac{Q_3-Q_1}{2}|| | ||Q=\dfrac{EI}{2}=\dfrac{Q_3-Q_1}{2}|| | ||Q=\dfrac{EI}{2}=\dfrac{Q_3-Q_1}{2}|| |