Maths GCSE  EDexcel  Unit 1
Quick reminders for topics in edxcel unit 1
 Created by: Katy Head
 Created on: 300511 13:55
Top Tip  BODMAS
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
Top Tip  Approach to written (verbal) questions
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 !!!!!!!!!
Calculations using percentages  1
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
Calculations using percentages  2
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%
Calculations using percentages  3
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% (10020) 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
Calculations using percentages  4
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
Top Tip  for wordy questions 1
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
Top Tip  for wordy questions 2
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)
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) 1
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  Highest Common Factor (HCF)
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
Top Tip  dividing by 9 or 99 or 999 or 9999 etc
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
Top Tip  Units, Units, Units
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
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
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
Ratios 1
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
Ratios 2
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 !
Ratios 3
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
Ratio 4
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


Ratio 5
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
 cement = sand / 3.5 = 9 / 3.5
Ratio 6
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 and Depreciation calculations
Compound Interest Calculation
Compound Interest and Depreciation 2
Examples to show how easy it is
Probability 1
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
Probability 2
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)
Probability 3
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
Probability 4
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
Probability 5
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
Probability 6
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
Probability 7
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
 where the fraction on each branch depends on what happened on the previous branch
Sample types 1
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
Sample types 2
Nonrandom 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
Questionnaires
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
Mean, Median and Mode
When to use them
Paste table here
Quartiles and Interquartile range
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
Frequency Tables 1
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
Frequency tables 2
Calculating the mean, median and mode
Look at the table again :
Grouped frequency tables
Grouped frequency tables :
 where the groups are in ranges as in the following table
 Estimating the mean using midinterval values
 Mean = overall total / frequency total = 3220/60 = 53.7
Cumulative frequency 1
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
Cumulative frequency 2
Photocopy
Histograms and frequency density
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 5565 age range find the total number of people arrested
Stem and Leaf Diagrams
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 and Bar charts
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
Averages and Spread
paste here
Shapes of histogram and spread
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
Other Graphs and Charts 1
 Two way tables  plot two variables against each other
 example
 Line graphs  a set of points joined by straight lines
Other Graphs and Charts 2
 A frequency polygon looks the same  it is used to show the information from a frequency table
Other Graphs and Charts 2
 Pie Charts  the golden rule is the TOTAL of everything =360 degrees
Basic Algebra 1
Its as easy as abc !!!
Rules :
insert table here
Basic Algebra 2
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
Formula triangles
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
Straight line graphs 1
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.
Straight line graphs 2
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 = 
Straight line graphs 3
Drawing straight line graphs
 Method 1
 Chose 3 values of x
 calculate the y values
 plot the coordinates
 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
Real life Graphs
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
What the gradient of a graph means
No matter what the graph is representing :
 the gradient always means
(yaxis units) per (xaxis units)
Extra tips for Statistics  equations for line gra
Equations of graphs :
Extra tips for Statistics  Useful Equations
Calculation of Stratified sample size =
 sample size / population size * strata size
Standardised Score =
 score  mean / standard deviation
Index number
 Current value / last value * 100
Extra tips for Statistics  weighted mean
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%
Extra tips for Statistics  Geometric Mean
Extra tips for statistics  sampling definitions
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.
Extra tips for Statistics  calculating what is an
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 3821 or higher than 52+21
Extra tips for Statistics  probability distributi
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
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