Maths.md

Maths

E2.2

Use language, notation and Venn diagrams to describe sets and represent relationshops between sets.

E1.12

Calculate percentage increase or decrease. Carry out calculations involving reverse percentages.

If the original price is £96 and the sale price is £84, find the percentage decrease.

$$96-\frac{r}{100}\times96=84$$

$$96 - 84=\frac{r}{100}\times 96$$

$$\frac{96-84}{96}=\frac{r}{100}$$

$$\frac{96-84}{96}\times 100 = r$$

$$r=12.5$$

If the original price is £120 and it is reduced by 15%, find the sale price.

$$120-\frac{15}{100}\times 120=y$$

$$120(1-\frac{15}{100})=y$$

$$\pounds 120 = y$$

If the original price is reduced by %18 and the sale price is £14.60, find the original price.

$$x-\frac{18}{100}\times x = 24.6$$

$$0.82x = 24.6$$

$$x=\frac{24.6}{0.82}$$

$$x=\pounds 30$$

E1.16

Use given data to solve problems on personal and small business finance involving earnings, simple interest and compound interest.

Let **I** = amount of**Interest** earned

Let **P** =**Principal**, the amount initially invested

Let **R** = interest**Rate** the prinicipal earns

Let **T** = amount of **Time** principal is invested

To determine the amount of Simple Interest:

$$I=PRT$$

To determine to total amount earned with Simple Interest:

$$P+I=P+PRT$$

To determine the total amount earned with Compound Interest:

$$\text Amount\scriptstyle total = P(1+\frac{R}{100})^T$$

E2.5

Solve quadratic eqations by use of the formula.

The general quadratic equation is

$$ax^2 + bx + c = 0$$

Here x represents an unknown, while a, b, and c are constants with a not equal to 0. One can verify that the quadratic formula satisfies the quadratic equation by inserting the former into the latter. With the above parameterization, the quadratic formula is:

$$x = \frac{ -b \pm \sqrt{ b^2 - 4ac } }{ 2a }$$

E7.4

Calculate the determinant |A| and inverse |A^-1| of a non-singular matrix A.

$$A = \begin{bmatrix} a & b \ c & d \end{bmatrix}$$

$$Det. = \frac { 1 }{ ad - bc }$$

$$Inverse = \frac { 1 }{ ad - bc } \begin{bmatrix} d & -b \ -c & a \end{bmatrix}$$

E6.3

Solve problems using the sine and cosine rules for any triangle and the formula area of a triangle.

Sin rule

$$\frac{a}{\sin a} = \frac{b}{\sin B} = \frac{c}{\sin C}$$

Cosine rule

$$a^2 = b^2 + c^2 - 2bc\cos A$$

Area of triangle

$$\text{Area of} \space \triangle = \frac{1}{2}bh = \frac{1}{2}bc\sin A$$