Approximation & Formula Sheets

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

1. Count from the first digit.
2. 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

1. Measurement is only approximate.
   
   
2. 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.

• 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 100cm² and its base is 1cm more than it's height. Find its height to 2 decimal places.

Answer: 9.51cm.

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.

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

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

Rule: The square on the hypotenuse is equal to the sum of the squares on the other two sides or, a² + b² = c²

Area

Square: Area = Length²

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 r²

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

(Where n = pi )

Curved Surface of a Cylinder: Area = 2π rh Volume = π r²h

Surface of a Sphere: Area = 4π r² Volume = ⁴/₃π r³

Curved Surface of a Cone: Area = π rl

Cube: Volume = Length³

Cuboid: Volume = Length x Width x Height

Prism: Volume = Area of Cross-section x Length Volume = ¹/₃π r2h

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 r²

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)

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

The gradient, m:

Quadratic functions are written in the form y = ax² + bx + c

Cubics are in the form y = ax³ + bx² + cx + d

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

Any number can be written in the form a x 10ⁿ 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 10⁷) x (4 x 10⁹) can be rearranged to (2 x 4)x (10⁷ x 10⁹) giving 8 x 10¹⁶

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.