# Approximation & Formula Sheets

- Created by: Kathryn
- Created on: 10-05-14 06:35

## Rounding

**Decimal Places** - count from the decimal point

**1.** Count the number of decimal places you need.

**2.** Look at the next digit. If it's 4 or below just write down the answer with the right amount of decimal places. If it's 5 or above write down the number but put your last decimal place up by one.

*Significant Figures*

- Count from the first digit.
- Don't count zeros unless they're in-between non-zero digits.

e.g. 12 736 to **three **significant figures is 12 700

e.g. 6530 to **one** significant figure is 7000

e.g. 0.576 to **two** significant figures is 0.58

## Estimating - Upper & Lower Bounds

- Measurement is only approximate.

- The real answer can be half the rounded unit either way.

We call them **upper** and **lower** bounds.

The real value can be as much as **half the rounded unit** above or below the value given.

So, if you are given 5.4cm the **upper bound** is 5.45cm and the **lower bound** is 5.35cm.

For 6.0kg you need to go 0.05kg either way so the **upper bound** is 6.05kg and the **lower bound** is 5.95kg.

- Addition and Multiplication - use all upper bounds.

- Subtraction and Division - first number upper bound, second number lower bound.

And *vice versa* for mimimun values.

## Trial & Improvement

- Use tables to display guesses and answers. Say whether too high or too low.

- Be methodical.
- If you know which two numbers the answer is between try the middle and that will tell you which one it is closer to.

*For example:*

A rectangle has an area of 100cm^{2 }and its base is 1cm more than it's height. Find its height to 2 decimal places.

Answer: 9.51cm.

## Rules of Indices

*Rule 1:* When you multiply indices of the same number you **add** the powers.*Rule 2:* When you divide indices of the same number you **subtract** the powers.*Rule 3:* Indices outside a bracket **multiply.**

*Rule 4:*

**Negative indices**mean

**reciprocal**, i.e. 'one over....' or 'put on the bottom of a fraction'.

*Rule 5:*When the power is a fraction the top of the fraction (numerator) is a power and the bottom of the fraction is a root.

*Rule 6:* Anything to a power of 1 is just itself and we normally don't bother putting the 1 there.

## Angle Formulae

*Rule 1:* Angles around a single point add up to 360^{°}.

*Rule 2:* Angles on a straight line add up to 180°.

*Rule 3:* Vertically opposite angles are equal. (This is when two straight lines cross!).

*Rule 4:* Angles in a triangle add up to 180°.

*Rule 5:* Angles in a quadrilateral add up to 360°.

#### The nth Term

*n ^{th} term = dn + (a - d)*

*d = difference between the two terms. a = the actual term.*

*For example: *6, 11, 16, 21, ... for this sequence d = 5, a = 6

## Parallel Lines & Triganometry

*Rule 1:* Corresponding angles are equal - these are angles in a letter 'F'.

*Rule 2: *Alternate angles are equal - these are angles in a letter 'Z'.

*Rule 3:* Supplementary angles add up to 180° - these are angles in a letter 'U' or 'C' (when the 'U' and the 'C' are made of three straight sides, of course).

#### Sin, Cos, Tan

*[SOH CAH TOA]*

*Rule 1:* **Sine** is** Opposite** over **Hypotenuse**

*Rule 2:* **Cos** is **Adjacent** over **Hypotenuse**

*Rule 3:* **Tan** is **Opposite** over **Adjacent**

## Pythagoras & Area

*Rule: ***The square on the hypotenuse is equal to the sum of the squares on the other two sides **or,* a ^{2 }+ b^{2} = c^{2}*

#### Area

*Square:* Area = Length^{2}

*Rectangle:* Area = Length x Width

*Right-angled Triangle: *Area = ½ x Base x Height

*Other Triangle:* Area = ½ x Base x Perpendicular Height

*Circle:* Area = π r^{2}

*Trapezium:* Area = Average of Parallel sides x Distance between them

## Surface Area & Volume

*(Where n = pi )*

*Curved Surface of a Cylinder:* Area = 2π rh Volume = π r^{2}h

*Surface of a Sphere: *Area = 4π r^{2} Volume = ^{4}/_{3}π r^{3}

*Curved Surface of a Cone: *Area = π rl

*Cube: *Volume = Length^{3}

*Cuboid: *Volume = Length x Width x Height

*Prism:* Volume = Area of Cross-section x Length Volume = ^{1}/_{3}π r2h

## Polygons and Circles

For a regular polygon with 'n' sides, *External angle:*

For a regular polygon with 'n' sides, *Internal angle:*

#### Circles

Circumference = 2π r **or**, Circumference = πd

Area = π r^{2}

## Similarity & Probability

If we call a particular event 'A' then the probability of A happening is:

*The 'and' rule:*

**p (A and B) = p (A) x p (B)**

*The 'or' rule:*

**p (A or B) = p (A) + p (B)**

## Graphs

The equation of a straight line is *y = mx + c*

The gradient, m:

Quadratic functions are written in the form *y = ax ^{2} + bx + c*

Cubics are in the form *y = ax ^{3} + bx^{2} + cx + d*

In a pie chart, to find out the frequency that each section represents measure the angle for the section then:

## Standard Form

Any number can be written in the form **a x 10 ^{n}** where

**a**is a number between

**1 and 10**and

**b**is an

**integer**.As numbers written in Standard Form are still numbers you can multiply them, divide them, etc. The rules of indices can provide some shortcuts!

*For example:*

(2 x 10^{7}) x (4 x 10^{9}) can be rearranged to (2 x 4)x (10^{7} x 10^{9}) giving 8 x 10^{16}

*Important:* The number at the front must be between 1 and 10 so be careful! You may need to adjust it a bit by moving the point and changing the power of 10 to make up for your change.

## Indices in Algebra

Algebraic fractions can be **simplified** by following Rule 2 of Indices and taking the powers on the bottom away from the powers on the top or, in other words, **cancelling the powers**.

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