Maths GCSE - EDexcel - Unit 1

The sequence of doing operations in a mathematic formula can be remembered by BODMAS

• B - Brackets
• O - Other  (example squaring or cubing or square root)
• D - Division
• M - Multiplication
• A - Addition
• S - Subtraction

Remember :-

• Minus and Minus equals Plus
• Minus and Plus equals Minus
• Plus and Minus equals Minus
• Plus and Plus equals Plus

Problem questions:-

• Read  the question carefully
• Work out the bit of maths you need
• Underline the information you need
  • remember you may not need all the information you are given
• Write out the question in mathematical terms
• Show your working
• Always include the units in your answer
• Always check for whether there are mixed units in the information you have been given

DON'T PANIC !!!!!!!!!

Finding a percentage :-

Find x% of Y     e.g. Find 15% of £46.00

Method a:-

• convert the percentage you want into a fraction
• then multiply it by the amount you want  the percentage from
• so 15/100 of £46.00 = 15/100 * 46 = £6.90

Method b:-

• convert the percentage into a decimal
• then multiply it by the amount you want  the percentage from
• 15% = 0.15
• 0.15 * 46 = £6.90

Express as number as a percentage of another number

Express x as a percentage of y

e.g give 40p as a percentage of £3.34

• remember to make both elements into the same units pence - 40  and 334
• Convert to a fraction what you want 40/344 and multiply by 100
• so (40/344) * 100 = 12%

Find a new value when a number decreases or increases by a percentage

e.g. 1 A shirt is on sale for 20% off the original price. If the original price was £30 what is the price in the sale :-

• If there is 20% off the new price will be 80% (100-20) of the original price
• 80% of £30 = £30 * 0.80 = £24

e.g. 2 The price of volvo cars has gone up by 15%. The old price of a C70 was £20,000 - what is the price now :-

• If there is 15% added to the old price the new price will be 115% (100+15) of the original price
• 115% of £20,000 = 115/100 * 20,000 or 1.15 * 20,000 = £23,000

Finding an original value when you have a percentage of it

e.g. A car is in the sale at 85% of its full price. It is now on offer at £6,000. What was its original price ?

You know 85% is £6,000 so 1% is £6000 divided by 85.

6000 / 85 = 70.5882

if 1% = 70.5882 then 100% = 70.5882 * 100 = £7058.82

If you can describe it as how many of something go into  something else you need to divide

My book shelf is 30 inches long - all my books are 1/2 " wide - how many books can I fit on my shelf - a bit like how many books will go into  my shelf - this is a division

answer 30 divided by 1/2 = 60

If you can describe it as times or of it is multiplication 

My books are 1/2 " wide - how big will my bookshelf need to be built to hold my collection of 40 books ? 

I could write this as :-

• I want 40 books times 1/2 inch - this is a multiplication or
• I want 40 books of 1/2 inch - this is a multiplication

answer 40 multiplied by 1/2 = 20

Lowest Common Multiple (LCM) :-

The smallest number that will divide by all the numbers in question

e.g. the LCM of 3 and 5

• multiples of 3 are - 3, 6, 9, 12, 15, 18, 21
• multiples of 5 are - 5, 10, 15, 20, 21, 25, 30

The smallest number that 3 and 5 will both go into is 15. This is the LCM

Highest Common Factor (HCF) :-

The biggest number that will divide into all the numbers in question

e.g. the HCF of 8 and 12

• the factors of 8 are - 1, 2, 4, 8
• the factors of 12 are - 1, 2, 3, 4, 6, 12 

The highest number that will divide into both 8 and 12 is 4. This is the HCM

Top tip !

How to get all the factors for a number easily - write them out from either end until they meet in the middle - e.g. factors of 48

• 1                                                   48 (1*48=48)
• 1, 2,                                        24, 48 (2*24=48)
• 1, 2, 3,                              16, 24, 48 (3*16=48)
• 1, 2, 3, 4,                    12, 16, 24, 48 (4*12=48) 
• 1, 2, 3, 4, 6,            8, 12, 16, 24, 48  (6*8=48)

You've finished - the factors of 48 are :-

1, 2, 3, 4, 6, 8, 12, 16, 24, 48

8/9   =   0.888888888 recurring

21/99 = 0.212121212121 recurring

451/999 = 0.451451451451 recurring

This works both ways - you can work out the fraction from the recurring decimal

0.1212121212     = 12/99

0.678678678678 = 678/999   etc

Quite often in an exam the question will be in mixed units

• Always remember to convert to a single unit to do your calculations
• then you may need to convert your answer back to the appropriate units
• Always show your units in your answer

Converting decimals to fractions is easy - just follow the rules in the following examples :-

• 0.125 - because there are 3 places after the decimal point this is in thousandths so as a fraction it is 125/1000 or in its simplest form 1/8
• 0.x - because 'x' is the first place after the decimal point 'x' is in tenths and the fraction is x/10
• 1.6  - 1 is the whole number and .6 is one place after the decimal point so is in tenths - the fraction is 1 and 6/10 as a mixed fraction or you can show this as a heavy fraction 16/10 

Converting fractions to decimals is easy - first you need to convert your fraction into tenths or hundredths or thousandths -just follow the rules in the following examples :-

• 7/2 - 2 will go into 10 so we convert the fraction into tenths by multiplying the top and the bottom by 5 - we get 35/10. we have 35 tenths = 3.5
• 3/20 - 20 will go into 100 so we convert the fraction into hundredths by multiplying the top and the bottom by 5 - we get 15/100 we have 15 hundredths = 0.15
• x/100 - x is in hundredths - one hundredth as a decimal is 0.01 so x hundredths is 0.0x

What the fraction form of the ratio actually means

• Suppose in a class the girls and boys are in the ratio 3 : 4
• this means there are 3/4 as many girls as boys
• so if there were 20 boys there would be 3/4 * 20 girls = 15 girls
• It does not mean  3/4 of  the people are girls, in fact 3/7 of the class are girls

Reducing ratios to their simplest form:-

• this is just like you do with fractions
• e.g. take the ratio 15:18
  • 15 and 18 will both divide by 3
  • that gives 5:6
  • ths simplest form is 5:6

The sneaky bit :-

• Use your calculator as if the ratio was a fraction and it will convert it to the simplest terms for you !

Special cases :-

• If the ratio is in fractions - e.g 1 1/4 : 3 / 1/2
  • multiply both of the numbers by the same number until they are both whole numbers
  • multiply both by 4 gives 5 : 14
• if the ratio is mixed units - e.g. 24mm : 7.2cm
  • convert both sides into the smaller units
  • 24mm :72mm
  • you can now simplify this to 1 : 3
• if you need the ratio in the form 1:n or n:1 (where n can be any number) - e.g take 3 : 56
  • Divide both sides by the smallest side - in this case 3
  • dividing both sides by 3 gives 1 : 18.7

Using the formula triangle in ratio questions

example - Mortar is made from sand and cement in the ratio of 7:2. If 9 buckets of sand are used - how much cement is needed

•  
  •  
    •  
      • This is the basic formula triangle for ratios but NOTE:
      • The ratio must be the right way round, with the first number of the ratio relating to the item on top in the triangle
      • You'll always need to convert the ratio into its equivalent fraction or decimal  to work out the answer

Using the formula triangle in ratio questions

example - Mortar is made from sand and cement in the ratio of 7:2. If 9 buckets of sand are used - how much cement is needed

Here is the formula for the mortar question

• the equivalent fraction is 7/2 or, as a decimal, 3.5
• so covering up cement in the triangle gives
  • cement = sand / 3.5 = 9 / 3.5
    • = 2.57 buckets of cement

Proportional Division

In a proportional division question a total amount  is to be split in a certain ratio

e.g. £9,100 is to be split into 2:4:7. Find the three amounts

• The key word is parts
  • add up the parts - the ratio 2:4:7 means there are 2+4+7 parts - a total of 13 parts
  • Find the amount for 1 part - = £9,100 divided by 13  = £700 for one part
  • Now you can find the three amounts
    • 2 * 700 = £1,400
    • 4 * 700 = £2,800
    • 7 * 700  = £4,900

Compound Interest Calculation

Examples to show how easy it is

Key facts about probabilities :-

• All probabilities are between 0 and 1
  • a probability of zero - it will never happen
  • a probability of one - it will definately happen
• Probabilities are given as either a fraction (1/4) or a decimal (0.25) or a percentage (25%)
• The notation P(x) = 1/2 means :
  • The probability that 'x' will happen is 1/2
• Probability always adds up to 1
  • if P(pass) = 1/4, then P(fail) must = 3/4

There are 2 rules you must remember:-

• The AND rule
  • the probability of Event A and Event B happening is equal to the two separate probabilities multiplied together
  • P(A and B) = P(A) * P(B)
• The OR rule
  • the probability of Event A or Event B happening is equal to the two separate probabilities added together
  • P(A orB) = P(A) + P(B)

Three simple steps to solving probability questions :-

• Allways break down a complicated looking question into a sequence of separate single events
• Find the probability of each of these separate single events
• Apply the And/Or rule

Using the three steps we can solve probability problems :-

Example :- find the probability of picking two kings from a pack of cards ( assuming you do not put the first king back)

• Split into two events - picking the first king  and picking the second king
• Find the separate probabilities  of these two separate events
  • P(1st king) = 4/52
  • P(2nd King) = 3/51
  • Apply the and/or rule - Both events must happen so its the and rule
  • So multiply the two separate probabilities
    • 4/52 * 3/51 = 1/221

Relative frequency:-

• Relative frequency is used when you are measuring things that are biased
  • e.g. a fair dice or a dice which is biased
• You repeat the experiment again and again ( the more times you repeat, the more accurate the result)
• Formula for relative probability
  • The number of times it has happened divided by the number of times you tried

Example - the wonky dice :-

    paste table here

Tree Diagrams:-

• Rules -
  • Always multiply along the branches to get the end results
  • On any set of branches that meet at a point, the numbers must add up to 1
  • Check that your diagram is right by making sure that end results add up to 1
  • To answer any question simply add up the relevant end results

Extra tips for Tree Diagrams :-

• Always break the question into a sequence of separate events
• Don't feel you have to draw complete  tree diagrams - just draw the branches you need
• With at least  questions it's always 1 - the probability of the other outcome
  • e.g. the probability of having at least one girl in four children is the same as 1 - the probability of having 4 boys
• Watch out for conditional probabilities
  • where the fraction on each branch depends on what happened on the previous branch
    • where the number on the bottom of the fractiion changes as items are  removed
    • e.g. picking two kings from a pack of cards is 4/52 followed by 3/51

Random sampling types:-

• Simple Random Sampling
  • Every item is chosen at random

• Stratified Sampling
  • The population is split into groups (strata) that have something in common. A random sample is then taken from each group in proportion to the size of each group

Non-random sampling

• Cluster sampling - when the population naturally falls into clusters, a number of clusters are selected randomly and each item in these clusters is included

• Quota Sampling - A quota of subjects of a specified type are interviewed

• Systematic Sampling - From the sampling frame, a starting point is chosen at random, and therefter items are chosen at regular intervals

Questionnaire design:-

• Make sure questions are relevant
• Questions should be clear, brief and easy to understand
  • idiot proof
• Allow all possible answers to your question
• Questions should not be leading or biased
• Questions should be unambiguous
• People may not answer truthfully
  • e.g. they may be embarrassed to give their age - it is a sensitive question - you get round this by using groups so they don't have to give their exact age
• Make sure any groups do not overlap
• Think carefully about how to distribute your questionnaires - post, ask people to pick one up e.g. restaurant, give out personally 

When to use them

Paste table here

Key points :-

• quartiles divide the data into four equal groups
• the quartiles are the lower quartile Q1, the median Q2, and the upper quartile Q3
• If you put the data in order the quartiles are 1/4 (25%), 1/2 (50%) and 3/4 (75%) through the list
• The interquartile range is the difference between the lower quartile and the upper quartile
• Values that are outside of the interquartile range are referred to as outliers
• Formulas (n is the number of values you have):-
  • Q1 = (n+1)/4
  • Q2 = 2(n+1)/4
  • Q3= 3(n+1)/4

The word frequency just means how many - so a frequency table is only a 'how many in each group' table

• Frequency tables are shown either in rows or in columns
• A completed frequency table looks like this :-
  • it contains the list of groups you are measuring (number of sisters)
  • The frequency - how many times each group occurred (frequency)
  • The number multipled by the frequency

         paste table here

Calculating the mean, median and mode

Look at the table again :-

Grouped frequency tables :-

• where the groups are in ranges as in the following table 
•  
•  
•  

• Estimating the mean using mid-interval values
• Mean = overall total / frequency total = 3220/60 = 53.7

Cumulative frequency - adding it up as you go along - example

Rules with cumulative frequency

• when plotting a graph always use the Highest Value from each group
• when plotting a graph - cumulative frequency is always the y axis

  Photocopy

A histogram is a bar chart where the bars have different widths

• There are 2 basic rules to histograms
  • It's not the height, but the area of each bar that matters
  • Divide the bars into small squares of exactly the same size - add the number of squares in each block up to work out the area
• Example question - this histogram represents the number of people arrested in a town in 1995. Given there are 36 people in the 55-65 age range find the total number of people arrested

There are three simple rules to stem and leaf diagrams :-

• Put the data in order
• Put in the groups and make a key
• Draw the diagram

Photo copies

Scatter graphs show correlation and the line of best fit

• they compare two data sets
• the line of best fit is the straight line through the data points
  • it goes roughly through the middle of all the points
• the closer the points are to the line of best fit the stronger the correlation
• Outliers are values that don't fit the general pattern

Dual bar charts can also be used to compare two data sets

Composite bar charts show proportions

• It has single bars split into sections
• its easy to reas off total frequencies
  • the height of each bar

              paste here

You can

• estimate the mean from a histogram
• see the spread from a histogram

The first histogram has a large spread - lots of values away from the mean - the second has values closer to the mean - a narrow spread

• Two way tables - plot two variables against each other
  • example
•  
•  
•  

• Line graphs - a set of points joined by straight lines

• A frequency polygon looks the same - it is used to show the information from a frequency table

• Pie Charts - the golden rule is the TOTAL of everything =360 degrees

Its as easy as abc !!!

Rules :-

              insert table here

Expressions, Equations, Formulas and identities

• Expression - A bunch of letters and/or numbers added, subtracted, multiplied or divided together
• Equation - Two expressions joined with an equals sign.
• Formula - This is a relationship or rule for working something out, written in symbols
  • e.g. speed = distance * time
• Identity - This is an equation that's true for all values of the variables  - e.g. a + b = b +a

Whenever the formula van be expressed as A = B * C, you can use a formula triangle :-

• speed = distance * time
• Area of a triangle = 1/2 base * height

Covering up any one part of the triangle will give you the calculation you need

Rules for the types of straight line graphs -

• A horizontal line means y has a constant value
• A vertical line means x has a constant value
• A 'main diagonal' (45 degrees) through the origin means
  • y = x if the diagonal goes uphill
  • y = -x if the diagonal goes downhill
• Other diagonals through the origin
  • y = ax and y = -ax -where a is a constant number
• These are the easy types -  there are other straight line graphs but all straight line equations just contain - something x, something y and a number.

Finding the gradient

• Find two accurate points
• Find the change in x
• Find the change in y
• Gradient =
•     Change in y divided by change in x
• check the signs right -
  • uphill = +
  • downhill = -

Drawing straight line graphs

• Method 1
  • Chose 3 values of x
  • calculate the y values
  • plot the co-ordinates
  • draw the straight line that passes through all the points

• Method 2
  • set x = 0 and calculate the value of y
  • set y = 0 and calculate the value of x
  • plot these two points and draw a line through them

Method one is better because it has 3 points and is an additional check that the equation is really a straight line graph

Conversion graphs -

• graphs that convert units from one unit to another
  • e.g. £ to $ or km to miles

You often get a question where you are asked to interpret  a conversion graph e.g. how many km is 50 miles

Method:-

• 1 Draw a line from the value on one axis
•    keep going until you hit the line
• 2 then change direction
  • and go straight to the other axis
• 3 Read off the new value from the axis
  • that's the answer

No matter what the graph is representing :-

• the gradient always means

(y-axis units) per (x-axis units)

Equations of graphs :-

Calculation of Stratified sample size =

• sample size / population size * strata size

Standardised Score =

• score - mean / standard deviation

Index number

• Current value / last value * 100

Weighted Mean

Sum of (weighting * x value) / Sum of weighting

Example :-

In an exam Paper 1 has a weighting of 40, paper 2 has a weighting of 40, paper 3 has a weighting of 10 and paper 4 has a weighing of 10

A candidate scores the following marks :-

paper 1 62%, paper 2 38%, paper 3 58% and paper 4 39%

Work out the candidates final mark:-

= (40*62)+(40*38)+(58*10)+(39*10) / 40+40+10+10

= 2480+1520+580+390 / 100

=4970 / 100

=49.7%

Population - everything or everybody that couls be involved in an investigation 

Sampling Frame - the list of people or items to be smapled

Sampling Unit - the people or items to be samples

Cencus - data about every meember of the population

Sample - data about part of the population

Control group - Often used to test the effewctiveness of drugs - the control group and the group to be tested are both randomly selected. The control group is given an inactive substance and the other group is given the actual drug being tested.

First calculate the Interquartile range

Then multiply this by 1.5

An outlier is any value that lies more than this value outside the upper or lower quartile

Example - A box plot shows an interquartile range of 14. The lower quartile is 38 and the upper quartile is 52

The IQR * 1.5 = 14 *1.5 = 21

An outlier is any value that is lower than 38-21 or higher than 52+21

A Probability Distribution is a list of all possible outcomes together with their probabilities

Discrete uniform distribution :- Has n discrete outcomes. Each outcome is equally likely - example

• fair sided dice
• days of week given an event is equally likely on each day

Binomial distributions :- has a fixed number of independent trials n. Each trial has only two outcomes - success or failure

Normal Distribution :-  there are 4 properties that make a distribution normal

• the distribution is symmetrical about the mean
• the mode, median and mean are all equal (becasue the distribution is symmetrical)
• 95% of observations lie within plus or minus two standard deviations from the mean
• 99.8% (virtually all) observations lie within plus or minus three standard deviations from the mean