# Maths Rules and Equations

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• Created by: faholly21
• Created on: 27-05-15 14:52

## Standard Form Rules

Multiplying numbers in standard form

- Rearrange so powers of 10 are together

-Multiply the number parts

-Rewrite in standard form if neccessary

Dividing numbers in standard form

– Rearrange so powers of 10 are together

- Divide the number parts

- Subtract the powers

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## Surd Rules

√ab = √a x √b

√a/b = √a / √b

Rationalising the denominator

Multiply the top and bottom of the fraction by the surd part in the denominator.

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## Working out the nth term

Write the difference between each term e.g 4

Find the 0th term of the sequence

nth term= difference x n + zero term.

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## Factorising

Look for the largest factor you can take out of every term in the expression if it is x(y -or+ z)

x^2 + bx + c - find tow numbers which add up to b that are the same as two numbers that multiply to c

Both numbers are positive if b and c are positive

Bigger number is positive and the small number is negative if b is positive and c is negative

Smaller number is positive and the big number is positive if b is negative and c is positive

Both numbers are negative if b and c are negative

if ax^2 + bx + c means on of the brackets must have one ax or  two numbers in front of x that multiply to a.

x^2-c means the equation must be (x-a) (x+a) Difference of two squares.

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## Linear equations

If letters are on both sides of the equation - have to get it on one side - collect the like terms on one side.

If there are brackets within equations- multiply them out.

Equations w/ fractions - Get rid of the fractions before solving – multiply all the terms by the lowest common multiple of the denominators ( Look over in book)

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## Straight Line Graphs

Gradient - Change in y / change in x

Y intercept-  use equation y=mx + c and gradient value to rearrange to find the intercept.

Parralel lines have a gradient of 1; Perpendicular lines gradients multiply to -1

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## 3D co-ordinates

They have a z axis (x, y, z)

To find missing co-ordinates look at x co-ordinate at face and the y and z co-ordinates of the line adjecent to the point.

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## Rearranging Formulae

Letter appearing twice

- Group all terms with that letter on one side of the formula and all other terms on the other side

- Factorise so the leeter only appears once

- Divide everuthing in the bracket to get the letter on its own.

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## Inequalities

Open circle for < and >

Closed circle for ≤ and ≥

On graphs  show x ≥2 by doing a solid line on x=2 and do an arrow in the right direction

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## Graphs of k/a and a^x

y= k/x are reiprocal graphs y=a^ x or y=a^-x are  Exponential graphs 10 of 61

## Simultaneous Equations

Algebraic solution

IF NECCESSARY, multiply equations so co-effecients of one unknown are the same

Once one unknown found use subsititution

Check the answer by substituting both values in the orignal equations.

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ax^2 +bx+c = 0 - if not in this form, should rearrange before starting the calculating.

Factorise the left hand side to find the answers w/o a calculator

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## Completing the Square

when written as (x+p)^2 + q in completed square form.

Use the formulae x^2 + 2bx + c = (x+b)^2 - b^2 + c

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## Simultaneous Equations 2

If there is x^2 or y^2 in a pair of simultaneous quations need to use Substitution of one equation into another to make one letter the subject. Then factorise to find solutions. Then subsitute to find unknown value.

There would also be 2 solutions

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## Proportionality formulae

A statement of proportionailty y is directly proportional to x  =   y=kx

k= constant of proportonality

y is directly proportional to the square of x =    y= kx^2

y is directly proportional to the cube of x =   y= kx^3

y is directly proportional to the square root of x =  y= k√x

y is inversley proportional to x =  y= k/x

y is inversley proportional to the square of x =     y= k/ x^2

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## Transformations

y=f(x) + a     MOVE UP A UNITS

y= f(x) - a   MOVE DOWN A UNITS

y=f (x+a)    MOVE LEFT A UNITS

y=f (x-a)   MOVE RIGHT A UNITS

y= af(x)     STRETCH IN THE VERTICAL DIRECTION, SCALE FACTOR A (X VALUE STAYS THE

SAME)

y= f(ax)    STRETCH IN HOROZONTIAL DIRECTION, SCALE FACTOR 1/A (Y VALUE STAYS THE

SAME)

y= -f(x)  REFLECTION IN THE X AXIS

y= f(–x)  REFLECTION IN THE Y AXIS

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## Graphs of sin and cos

Graphs of  y= cos x intercept at 1 and pass through x=0 at 90 and 270

Graphs of y= sin x intercept at 0 and pass through x= 0 at 180 and 360 Sine graph Cosine graph

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## Algebraic Fractions

To add/ subtract algebaric fractions with diff denominators

Find a common denominator;  Add or subtract the numerators; Don't change denominator and Simplify

To Multiply A fractions

multiply numerators/ denominaters and Simplify

To Divide A Fractions

Change second fraction to be reciprocal; Change / to x; Multiply fractions and simplify.

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## Angle Properties

Corresponding Angles are equal

Alternate Angles are equal

Opposite Angles are equal

Allied Angles add up to 180

The Exterior angle of a triangle = sum of interior angles of other two vertices

Opposite angles of a parallelogram are equal

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## Reasons in Angle Problems

• Angles on a straight line add up to 180
• Angles around a point add up to 360
• Opposite angles are equal
• Corresponding angles are equal
• Alternate angles are equal
• Angles in a triangle add up to 180
• Base angles of an isoceles triangle are equal.
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## Angles in Polygons

n= number of sides

Sum of interior angles = 180 x (n–2)

Sum of exterior angles = 360 to find regular exterior angle 360/n

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## Perimeter and Area

Area of a Triangle = 1/2 * b * h

Area of Parallelogram= bh

Area of a Trapezium = 1/2 (a+b) h

To calculate areas and perimeters of more complicated shapes split them into parts.

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## Prisms

Volume of a Cubiod = length x width x height

Volume of a Prisim = area of cross section x length

Surface area of shapes =  add together all the areas of faces.

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## Circles and cylinders

Circumference of a Circle- 2πr or πd

Area of a Circle- πr^2

Volume of a Cylinder- πr^2h

Surface Area of a Cylinder- 2πr^2 + 2πrh

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## Sectors of A Circle

Area of a sector= x/360 x πr^2

Arc length= x/360 x 2πr

Rearrange first equation to find angle

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## Volumes of 3D shapes

Cubiod-  length x width x height (lwd)

Pyramid 1/3 x area of base x vertical height (1/3 Ah)

Cone: 1/3 x area of base x vertical height (1/3 πr^2 h)

Sphere: 4/3 πr^3

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## Pythagoras

a^2 + b^2 = c^2   Must be a right angled triangle. c= hypotense

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## Surface Area

Curved Surface Area of Cone= π x radius x slant height (πrl)

Whole Surface Area = πr^2 (area of base)+ πrl

Surface Area of a Sphere- 4πr^2

Surface Area of a Hemisphere (1/2 x 4πr^2)+πr^2 (area of base)

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## Converting Units

1km = 1000m   1m= 100cm  1cm= 10mm

1 tonne= 1000kg  1kg= 1000g  1g= 1000mg

1 litre = 100cl  1000 litres= 1m^3  1cl= 100ml

Metric Units      Imperial Units

1kg                      2.2 pounds (lb)

1 litre                   1.75 pints

4.5 litres              1 gallon

8km                     5 miles

30cm                   1 foot

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## Area and Volume conversions

Area conversions

1cm^2= 100mm^2

1m^2= 10,000cm^2

1km^2= 1,000,000m^2

Volume Conversions

1cm^3 = 1000mm^3

1m^3 = 1,000,000cm^3

1 litre= 1000cm^3

1ml= 1cm^3

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## Speed

Speed = distance/ time

Distance = speed * time

Time= Distance/ Speed

Units = m/s (metres per second) kilometres per hour (km/h) and mph (miles per hour) DEPENDS ON WHAT YOU USE IN THE FORMULAE.

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## Density

Density = mass/ volume

Volume= mass/ density

Mass= Volume x Density

Units of Density = g/cm^3 (grams per cubic centimetre); kg/m^3( kilograms per cubic metre)

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## Congruent Triangles

Congruent if they have exactly the same size and shape. To prove, show one of these conditions are true

SSS (three sides are equal)

AAS( two angles and a corresponding side are equal )

SAS ( two sides and the included angle are equal)

RHS ( right angle, hypotenuse and a side are equal)

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## Similar Shapes

All three angles are equal;

All three pairs of sides are in the same ratio

Two sides are in the same ratio and the included angle is equal

To Calculate a length  AB/FG= AE/FJ input values and rearrange to find answer

Scale factor =k

Enlarged surface area = k^2 x original surface area

Enlarged volume = k^3 x original volume

Enlarged mass = k^3 x original mass

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## Constructions

Construct a triangle-  Draw and label one side with a ruler, use a compass to find the other vertex by extending it to the size of the other sides.

Construct a angle bisector-  Mark two equal distanced from angle points on the arms. Then use arcs to find third point which equi-distant from these points

Construct a perpindicular bisector- Use compass to draw intersecting arcs with centres A and B

Construct a perpendicular line that passes through a certain point  Use compass to mark two points on the line equidistant from the point. widen compass and draw arcs with their centres at these points

Construct a perpendicular line that passes through a point off the line  Use compass to mark two points equidistant from P. Draw two arcs using this points

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## Line Segments

Use Pythagoras to find the length of a line segment

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## Trigonometry 1

SOH CAH TOA

sin x = opposite/ hypotenuse; cos x= adjecent / hypotenuse; tan x = opposite/ adjacent

To find the size of an angle. Use tan/sin/cos ^–1 to get the angle on its own

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## Trigonometry 2

To find a length of another side in a right- angled triangle - you need to have the length of another side and the size of one of the acute angles.

Subsititute it into the equation e.g sin40 = a/10  a=10 x sin40 = 6.43 cm

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## Pythagoras in 3D

d^2 = a^2 +b^2+c^2

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## Triangles and segments

Area of triangle if you only have two sides and the angle between them = 1/2 ab sin C

Area of segments= Area of whole sector – Area of triangle

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## Sine Rule

Applies to any triangle. To find a missing side :

a/ sinA = b/ sinB = c/ sinC

To find a missing angle:

sinA/ a =  sinB/ b = sinC/ c

To use this rule, must know either two angles and a side (ASA) or two sides and a non included angle (SSA)

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## Cosine Rule

Applies to any triangle To find a missing side:

a^2 = b^2 + c^2 –2bc cos A

To find a missing angle:

cos A = b^2 + c^2 –a2/ 2bc

Use if you have two sides and the included angle to find a missing side or three sides and are looking for a missing angle.

Use sine when a problem involves two sides and two angles

Use cosine when a problem involves three sides and one angle

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## Circle facts

The angle between a radius and a tangent is 90 degress

Two tangents which meet at a point outside a circle are the same length

A triangle which has one vertex at the centre of the circle and two angles on the circumference is an ISOSCELES TRIANGLE.

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## Circle Theorems

A perpendicular line from a chord to the centre of the circle bisects the chord.

The angle at the centre of the circle is twice the angle on the circumference

Angles in the same segment are equal

The angle in a semicircle is 90 degrees

Angle between a tangent and a chord is equal to the angle in the alternate segment - ALTERNATE SEGMENT THEOREM

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## Collecting Data

Make sure response boxes are numerical and dont overlap

Don't ask biased questions- Statement and response agree, disagree neither

Data Collection sheets: Tally and Frequency columns

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## Stratified Sampling

The equation is( sample/ whole population) x amount of people from group

e.g How many Year 7 boys will be sampled out of the population of 382 and a sample of 85

Year 7

Boys       50                        85/382 * 50 = 11.12 -> 11 students

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## Mean, Median and Mode

Mean- Add up all the values; Divide by total number of values

Median- Write the values smallest to largest; Count the number of values; Odd no is middle value, Even is half way between two middle values

Mode- Look for the mose common value

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## Frequency table averages

To find the median: Find the total frequency; divide by two- that value is the median.

To calculate the mean: Multiply the frequency by the value. Add these together to find the total frequency. total value/ total frequency = mean

If it has class interval values and it asks to calculate the mean - find the midpoint of the value and multiply that by the frequency.

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## Interquartile Range

Interquartile range is between the Lower quartile(Q1)  and the Upper quartile (Q3). (Q3-Q1)

To find the Lower quartile n+1 /4  th value, n= number of data values

To find the Upper quartile 3(n+1)/4 th value

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## Frequency Polygons

Plot the midpoints of each class interval

Draw points with straight lines

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## Histograms

Frequency Density = Frequency/ Class Width

Vertical axis is labelled Frequency Density

Area of each bar proportional to frequency

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## Culmulative frequency

Plot 0 at the beginning

Plot each value at the upper end of the class interval

Join points as a SMOOTH CURVE

Find Lower and upper quartile values by using the equations and looking where the value lies on the graph.

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## Box Plots

When comparing box plots use the range, interquartile range and the largest and smallest values and the median. It is always better to compare measures of spread rather than medians or end points.

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## Scatter Graphs

Negative Correlation - Points almost on a negative straight line

Positive Correlation- Points almost on a positive straight line

No Correlation - Completely random points.

Lines of best fit- Straight line which is as close as possible to all points; X need to go thru 0,0 ; Drawn with a ruler; ignores isolated points.

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## Highest Common Factor and Lowest Common Multiple

HCF - all prime factors that are common in both numbers multiplied together

LCM multiply all prime factors from both numbers

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## Indices Rules

a^m x a^n = a^m+n

a^m/ a^n = a^m-n

(a^m)^n= a^mn

a^0= 1

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## Fractions

To divide fractions, turn the second fraction upside down and multiply the two fractions together

To convert the fraction into a decimal- divide the numerator by the denominator

To convert recurring decimals into fractions - write decimal as 'n'  and multiply by ( 10 if one recurring digit; 100 if two recurring digits; 1000 if three recurring digit). Then subtract n and divide both sides by 9, 99 or 999 to get a fraction - simplify . IF there is one unrecurring digit e.g 0.4737373etc let 99n be 46.9 and multiply by ten to get 469/990

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## Rounding and Estimation

5 or more - round up

less than 5 - round down

to estimate - round to one sig figure

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## Upper Bounds and Lower Bounds

Greatest value     UB+UB           UB–LB           UB*UB          UB/LB

Lowest Value       LB+ LB           LB–UB           LB*LB            LB/UB

LB= lower bound

UB= Upper bound

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## Fractions Percentages and Decimals

Find fraction of amount: e.g 2/5 of 90 - divide by denominater and multiply by numerator 90/5 =18*2= 36

Write one quantity as a percentage of another e.g 7 out of 20 - Divide first value by second then multiply by 100  7/20 * 100=35%

Find % of an amount  45% of 200 divide percentage by 100 then multiply by value 45/100 *200= 90 Without a calculator use multiples of 1% and 10%

% change : Find percentage of amount and subtract the decrease or find % of original amount e.g 280 is reduced by 25%   0.75 x 280= 210

Calculate increase/decrease Work out the amount of increase/decrease- write as a percentage of the original amount e.g 60 now 39,  60-39=21,   21/60= 35%

Reverse Percentage: Divide by the multipler e.g find orginal price of 132.88 before a 12% decrease 132.88/0.88

Compound Intrest: (starting amount) x (multiplier)^n  n= number of years

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## Indices 2

a^–n = 1/ a^n

a^–1 =1/a (reciprocal)

(a/b)^n= a^n / b^n

(a/b)^–n = b^n / a^n (upside down and negative to positive power )

a^1/2 = 2√a  a^1/3 = 3√a

a^m/n= (a^1/n)^m

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