Mathematics - PBworks

Mathematics Number & Algebra Revision Notes For

Higher Tier

Thomas Whitham Sixth Form

S J Cooper

Factors, Primes & Prime Factors

Approximations

Fractions: Equivalent, expressing as, fractions of a quantity.

Percentages: Expressing as, Percentage of a quantity, finding the

original amount, compound measures.

Proportion: Ratio, Direct proportion and Inverse proportion.

Standard from.

Surds

Index Notation

Algebra: Collection of like terms, Solving equations

Factorisation

Graphs: Linear & Quadratic.

Solving simultaneous equations using their graphs.

Thomas Whitham Sixth Form Page 1

Factors

A factor is any number which will divide into a given number an exact

number of times.

Example 3 is a factor of 12 since 3 divides into 12 exactly 4 times.

Example List all the factors of 18.

Factors of 18 = {1, 2, 3, 6, 9, 18}

Prime Factors

A prime number is a number who’s factors are itself and 1.

A prime factor is a factor which is a prime number. For example the prime

factors of 12 are found in the diagram below.

The prime factors of 12 are 2 x 2 x 3 = 22 x 3

Example

Express each of the following as products of their prime factors

(i) 135 (ii) 54

24

6

2

4

3 2 2


(i) (ii)

135 = 5 x 3 x 3 x 3 54 = 3 x 3 x 3 x 2

= 5 x 33 = 33 x 2

NB Here the highest common factor (HCF) = 33 = 27

Approximation

Example

Find an approximate value of 02.0

8.43.54

02.0

8.43.54

135

5 27

3 9

3 3

54

9 6

3 2 3 3

Nearer 50 than 60 Nearer 5 than 4

Nearer 0 than 1. So

leave as 0.02


02.0

250

02.0

550

02.0

8.43.54

2

25000

= 12500

Example

(a) Estimate 96.1

25.360.2 giving your answer to 2 significant figures

(b) Evaluate 96.1

25.360.2 giving your answer to 2 significant figures

(a) 5.42

9

2

33

96.1

25.360.2

(b) 3.496.1

45.8

96.1

25.360.2

Example Estimate the value of 02.0

83.7986.4 giving your answer to

one significant figure.

20002

4000

02.0

40

02.0

85

02.0

83.7986.4

Clearly here if we round 0.02 off we

have 0 which won’t do!

Simplify numerator

Get rid of decimal by multiplying

top and bottom by 100


Example

Lewis uses his calculator to calculate 54.1 x 0.036 and gets

the answer 19.476.

Use estimation to work out whether his answer is reasonable.

54.1 x 0.036 50 x 0.04 = 2

Answer is unreasonable, as he is approximately a factor of 10 out.

Fractions

1. Equivalent fractions

Example

Express each of the following fractions in their simplest form.

(a) 6

10 (b)

24

42 (c)

60

84

(a) Here we notice that we have a common factor of 2. Since 2 will divide

into both 6 and 10.

6

10

2 3

2 5

3

5

(b) 24

42

6 4

6 7

4

7


(c) 60

84

6 10

6 12

10

12

10

12

2 5

2 6

5

6

With practice you will not require the intermediate step but move from

the given fraction to the final answer.

Example

Eight pupils out of ninety two pupils failed to turn up for their Mathematics

examination.

What fraction of the group (a) failed to turn up? (b) did turn up for the

exam?

(a) Fraction who did not turn up = 8

92

2

23 {no marks are awarded for “8 out of 92”}

(b) fraction who did turn up = 21

23 {i.e. the rest}

Example

On Saturday 9000 people won ten pounds on the National Lottery draw.

However 360 people failed to claim their £10 prize. What fraction failed to

claim their prize?

Fraction failed to claim prize = 360

9000

36

900

4

100

1

25

Notice here we have not completely

cancelled down the fraction so we must

repeat the process.


Example

What fraction has been shaded in for each of the following shapes?

(a)

Here there is a total of 15

squares, of which 3 are

shaded

fraction shaded = 3

15

1

5

(b)

Fraction shaded = 15

24

5

6

2. Fraction of a calculator (without calculator)

Example Find 1

448 of

1

448 of means

1

448 {i.e. 48 divided by 4}

Answer = 12

However we have a technique for showing our working as follows:

1

448 12

1

12

Example Find 3

570 of


3

570 of means

3

570

{again we could find 1

5

of 70 by dividing 70 by 5}

{so we find 3

5 of 70 by dividing 70 by 5 and then multiplying the answer by 3}

3

570

3

570 42

1

14 of

Example Find 2

7238 of

2

7238

2

7238 68

1

34 of 7 23 8

3 42

Example

In a school with 720 pupils, 9

10 stay in school at lunch time, and

3

8 of

these pupils bring a packed lunch. How many pupils bring a packed lunch?

Number staying at school = 9

10720 648

1

72

7 2

9

2 4 3

Number with packed lunch = 3

8648 243

1

81 8 64881

More complicated divisions may require

some additional calculations at the side

of your page


3. Fraction of a quantity (with calculator)

Example Find 2

37 41 of £ .

2

37 41 of £ . means

2

37 41 £ .

Using the calculator we have 2 7 41 3 .

Answer = £4.94 [Don’t forget the units]

Example Find 3

4 of 1488km

3

4

3

4 of 1488km = 1488 = 1116km

Example

Mr Ashworth earns £295.68 in one week. After tax is deducted, he receives

only five sevenths of this amount. How much does he receive?

We require 5

7

5

7 of £295.68 = 295.68 = 211.2

He receives £211.20 [remember currency has two decimal places]

4. Fractions to percentages/decimals

Example

Express each of the following as fractions in their simplest form.

(a) 45% (b) 0.34 (c) 2.6 (d) 12%


(a) 45% = 45

100

9

20 (b) 0.34 =

34

100

17

50

(c) 2.6 = 26

102

3

5 (d) 12% =

25

3

100

12

5. Addition and subtraction of fractions

Example Work out 4

7

5

7

4

7

5

7

9

71

2

7

Which is easily done provided the denominators are the same.

Example Work out 3

4

5

12

First change the first fraction with its equivalent in twelfths.

3

4

3 3

3 4

9

12

3

4

5

12

9

12

5

12

14

121

2

121

1

6


Example Work out 51

32

2

5

51

32

2

57

5

15

6

15

711

15

Example Work out 31

71

2

3

31

71

2

32

3

21

14

21

124

21

14

21

110

21

Example

A piece of wood is 72

5 metres long. If 1

7

15 metres is cut off, what length

of wood is left?

72

51

7

156

6

15

7

15

521

15

7

15

514

15

First add the whole numbers together.

Next the common denominator is 15.

hence replace each fraction with its

equivalent in terms of fifteenths.

Subtract the whole numbers and

place each fraction with common

denominator 21.

Since we cannot subtract 14 from 3 w

must use one of the whole numbers.

hence 321 becomes 24

21 and we now

have only 1 whole one.

Length of wood left =


6. Multiplication and Division of fractions

Example Work out 2

3

5

7

2

3

5

7

2 5

3 7

10

21

Example Work out 4

9

3

7

4

9

3

7

4

213

1

Example Work out 11

22

2

5

11

22

2

9

3

2

20

9

10

3

31

3

1

1

10

3

To multiply any two fractions

together simply multiply

numerators together and then

multiply denominators together.

In the event that a number on the

numerator has a common factor to a

number on the denominator, cancel

to start with. i.e. 3 will divide exactly

into both 3 and 9.

Change fractions to top heavy fractions.

Next cancel where possible


Example Work out 10

13

5

6

10

13

5

6

10

13

6

5

12

13

2

1

Example Work out 35

91

1

2

35

91

1

2

32

9

3

2

9

32

2

3

18

96

Example

Find the exact value of ba

11 when

3

2a and

5

4b

2

3

2

31

3

2

11

a That is the reciprocal of

3

2 is

2

3

Similarly the reciprocal of 5

4 is

4

5

1. Invert the second fraction and

change the ‘‘ to a ‘x’ sign.

2. Cancel where possible

3. multiply across.

change fractions to top heavy fractions

Invert second fraction and cancel where possible.

Finally multiply across


4

32

4

11

4

5

4

6

4

5

2

311

ba

Example

Louise has 71

4m of ribbon.

She makes 9 skirts and uses 2

5 of a metre for each one. How much ribbon

does she have left?

5

33

5

18

5

29

Amount left = 5

33

4

17

= 5

3

4

14

m20

133

20

12

20

253

20

12

20

54

Place over common

denominators

Subtract whole quantities

Since we cannot subtract 12

from 5 here , use one of the

whole ones as 20

20


Example

Express each of the following as fractions in their simplest form

a) 0.237

b) 0.2373737.....

c) 0.1373737....

a) 1000

237237.0

b) Let ...237237237.0x

...237237.2371000 x

Subtract from previous expression gives 237999 x

Hence 333

79

999

237x and since ...237237237.0x

333

79...237237237.0

c) Again Let ...1373737.0x

...73737.13100 x

Subtract from previous expression gives 6.1399 x

Hence 495

68

990

136

99

6.13x


Percentages

1. Expressing as a percentage

Example Express each of the following percentages as

(a) Decimals (b) Fractions in their simplest form.

(i) 30% (ii) 46% (iii) 2% (iv) 154%

(a)(i) 30% = 3.0100

30 (ii) 46% = 46.0

100

46

(iii) 2% = 02.0100

2 (iv) 154% = 54.1

100

154

(b)(i) 30% = 10

3

100

30 (ii) 46% =

50

23

100

46

(iii) 2% = 50

1

100

2 (iv) 154% =

50

271

100

541

100

154

Example Mark scored 32 out of 40 in a recent mathematics test.

Express his score as a percentage.

Test result = %8010020

16

40

32 5

1


Example The height of a tree increased from 2.73m to 2.98m in one

year. What percentage increase is this?

Increase = 2.98 – 2.73 = 0.25

Percentage increase = %16.910073.2

25.0

2. Percentage of a quantity

a) Without a calculator

Example What is 20% of 40 kg

Method 1: Using fractions

20% = 5

1

100

20 as a fraction

hence 20% of 40kg = 5

1 of 40 = kg840

5

1 8

1

Method 2 Using unity

10% of 40kg = 4 kg {i.e. divide by 10}

20% of 40 kg = 2 x 4 = 8kg

Example What is 35% of £60?


35% = 20

7

100

35 as a fraction


hence 35% of £60 = 20

7 of 60 = 21£60

20

7 3

1


10% of £60 = £6 {i.e. divide by 10}

5% of £60 = £3 {i.e. half of 10%}

30% of £60 = 3 x 6 = £18

35% of £60 = 18 + 3 = £21

Example What is 25% of 144cm?


25% = 4

1

100

25 as a fraction

hence 25% of 144cm = 4

1 of 144 = cm36144

4

1 36

1

63

4144 2


10% of 144cm = 14.4cm {i.e. divide by 10}

5% of 144cm = 7.2cm {i.e. half of 10%}

20% of 144cm = 2 x 14.4 = 28.8cm

25% of 144cm = 28.8 + 7.2 = 36cm

63

4144 2

Division more complicated so

done at the side of our page


Example

The price of a new television is £176 plus VAT at 17½%.

(a) Work out the VAT to be added to the price of the television.

(b) What is the total cost of the television?

(a) 10% of £176 = £17.60

5% of £176 = £ 8.80 {i.e. half of 10%}

2½% of £176 = £ 4.40 {i.e. half of 5%}

17½% of £176 = £30.80 =VAT {i.e. add the previous answers together}

(b) Total cost = 176+30.80 = £206.80

Example The number of girls attending football matches is expected

to increased by 3% this year. If there were 12500 girls

attending matches last year, how many girls are expected

to attend this year?

1% of 12500girls = 125 girls {i.e. divide by 100}

3% of 12500girls = 3 x 125 = 375girls

Number of girls = 12500 + 375 = 12875

b) With a calculator

Example What is 23% of 40 kg

With a calculator there is no need to cancel down fractions or use the unity

method. Simply set out your sum as a fraction and use the calculator.

23% of 40 kg = kg2.940100

23


Keys used:

(there are alternative combinations for inputs to the calculator)

Example What is 17% of £6.50?

17% of £6.50 = 11.1£105.150.6100

17

{here we must round up the answer since money involves 2 decimal places}

Example I pay 6% of my salary into a pension fund. How much do I

pay into my fund if my salary is £12450 per annum?

Amount paid = 6% of £12450 = 747£12450100

6

Example Every year a car loses 15% of its value at the beginning of

that year. If it was originally worth £5000, what will it be

worth after 2 years?

Method 1: Find the amount lost and then subtract from the original.

Amount lost = 15% of 5000 = 750£5000100

15

Value after 1st year = 5000 – 750 = £4250

Amount lost in 2nd year = 15% of 4250 = 50.637£4250100

15

Value after 2nd year = 4250 – 637.50 = £3612.50

X

4 0 ÷ 1 2 3 = 0 0


Method 2: Using the percentage Loss we can determine the percentage

left!

If the value of the car is 15% less each year, then the new value will be

85% of the original.

(i.e. 85% + 15% = 100%)

value after 1st year = 85% of 5000 = 4250£5000100

85

Or better still 4250£500085.0

value after 2nd year = 85% of 4250 = 50.3612£4250100

85

{answer comes out quicker!}

Example

The population of Umbridge increased by 37% during the years 1960-80.

If the population in 1960 was 447, what was the population in 1980?

An increase of 37% means than the new population is 137% what it was!

{i.e. an increase implies we add on to 100%}

New population = 137% of 447 = 61239.612447100

137

Or better still 61239.61244737.1


Example

This year nurses were given a 3.5% pay rise. If Susan earned £16 540 per

annum last year how much more will she get in her pay packet this year?

What will be her new salary?

Increase of 3.5% means that Nurses now earn 103.5% what they did the

year before!

New Wage = 103.5% of £16 540 = 75.17025£16450100

5.103

3. Finding the original percentage

Here we make use of the formula:

Example The population of Villanova has increased by 63% during

the last five years and is now 124 000. What was its

population five years ago?

Here we have been told the answer. That is the new value is 124 000.

after an increase of 63%

163% of original value = 124 000

{i.e. increase means 63+100%}

1% of original value = ...73.760163

124000

{i.e divide by 163 to find 1%}

New value = Percentage of Original value

Or

New value = Percentage x Original value


100% of original value = original value = 76074...73.760100

NB original value is always 100% and we have either increased it or

decreased it to find the new value

Example The price of houses in Villanova has increased by 18%

during the last year. If the house costs $45 000 now, what

would it have cost a year ago?

Here we have been told the answer. That is the new value is $45 000.

after an increase of 18%

118% of original value = 45 000 {i.e. increase means 18+100%}


45000

{i.e divide by 118 to find 1%}

100% of original value = original value = 59.38135$...355.381100

Example The attendance of Burnley football club fell by 7% in 2001.

If 2030 fewer people went to matches in 2001, how many

went in 2000?

Here we have been told the answer. That is the new value is 2030.

which represents the 7%!

7% of original value = 2030


2030 {i.e divide by 7 to find 1%}

100% of original value = original value = 29000.290100 people


Example During a Grand Prix race, the tyres on a car are reduced in

weight by 3%. If they weigh 388 kg at the end of the race,

how much did they weigh at the start?

Here we have been told the answer. That is the new value is 388kg.

after an decrease of 3%

97% of original value = 388 {i.e. increase means 100 – 3%}



100% of original value = original value = kg4004100

Example A car, which failed its MOT test, was sold for £456, thereby

making a loss of 35% on the cost price. What was the cost

price?

Here we have been told the answer. That is the new value is £456. after

an decrease of 35%

65% of original value = 456 {i.e. increase means 100 – 35%}



100% of original value = original value = 54.701£...015.7100


Example

In 2002 the population of England was 49,561,800.

The population of England is increasing by an annual rate of 0.3 per cent.

a) Write down the single number that we must multiply 49,561,800 by

if we wish to calculate an estimate for what the population was in

2004.

b) Assuming that England’s population continues at this same rate,

calculate the population of England in 2012.

a) increase implies add on to 100%, therefore 100.3% = 1.003

This is the number which must be multiplied to 49,561,800 if we want the

population in 2003. For 2004 we must multiply by 1.003 again.

Hence number must be 1.0032 = 1.006009

b) Population in 2012 = 1.00310 x 49561800 = 4,700,516.

Example

In 2008 the State Pension was increased by 2.5 per cent to £95.25 a

week.

What was the state pension before this increase?

Increase 102.5% of original value = 95.25

1.025 x Original pension = 95.25

Pension was 93.92£025.1

25.95


Compound Measures

Example

A bank pays interest of 4% on money in deposit accounts. Mr Smith Puts

£2000 in the bank. How much has he after

a) One year b) three years?

a) After one year

Amount = 104% of 2000 = 1.04 x 2000 =£2080

b) After three years

Method 1 In 2nd year Amount = 104% of 2080 = 1.04 x 2080 = £2163.20

in 3rd year Amount = 104% of 2163.20 = 1.04 x 2163.20

= £2249.73

Method 2

After three years we have multiplied 1.04 a total of three times. i.e. 1.043

After three years Amount = 1.043 x 2000 = £2249.73

In general: For an initial amount of £P at an annual rate of interest r%, the

amount in the account after n years will be A where A is worked out using

the formula below.

However don’t learn the formula, just learn its meaning!

nr

PA

1001


Example

The population of an island increases by 8% each year. If the population in

2009 was 10 million, what is the expected population of the island in 2015?

Here an increase by 8% means 108% or 1.08

2009 to 2015 means the increase over 6 years.

Expected population = 1.086 x 10 =15.87 million (rounded to 4 significant

figures)

Example

A new car is valued at £15 000. At the end of each year its value is reduced

by 15% of its value at the start of the year. What will it be worth after 6

years?

Here a decrease by 15% means it’s worth 85% or 0.85

Value after 6 years = 0.856 x 15000 = £5657.24

Ratio & Proportion

Example Simplify each of the following ratios

a) 4 : 28

b) 18 : 27

c) £3 : £1.80

d) 300m : 5.1km

e) 1250 cm2 : 3 litres

a) 4 : 28 = 1 : 7 {Divide both sides by 4}

b) 18 : 27 = 2 : 3 {Divide both sides by 9}

c) £3 : £1.80 = 300 : 180 {Change both into pence}

= 30 : 18 { Divide by 10}

= 5 : 3 { Divide by 6}


d) 300m : 5.1km = 300 : 5100 { Change both into metres}

= 3 : 51 { Divide by 100}

= 1 : 17 { Divide by 3}

e) 1250 cm2 : 3 litres = 1250 : 3000 {Change both into cm2}

= 25 : 60 {divide both sides by 50}

= 5 : 12 {divide both sides by 5}

Example A school decides to give 20% of the proceeds of a jumble

sale to charity and the rest to the school fund. In what ratio

are the proceeds to be divided?

Charity to school fund = 20 : 80 = 1 : 4

Example One bottle of wine holds 750 cm2 whereas another holds

1.2 litres. Give the simplest ratio of their capacities.

Ratio = 750cm2 : 1.2 litres = 750 : 1200 {change units into cm2}

= 15 : 24

= 5 : 8

Example Jane took 45 minutes to do her homework, but her sister

Lucy took 1¼ hours. What is the simplest ratio of their

times taken?


Jane to Lucy = 45 : 1¼ = 45 : 75 {change units into minutes}

= 9 : 15 = 3 : 5

Example The standard gauge of railway track is 1.43m. A model is to

be made with gauge 11mm. Calculate the scale in the form

1 : n, where n>1.

Model to Standard gauge = 11mm : 1.43m

= 11 : 1430 {change into mm}

= 1 : 130

Example £420 is divided between two people in the ratio 2 : 5. Work

out what each person will receive.

There are two ways of looking at this problem.

Method 1

There are a total of 7 parts 7 parts represents £420

1 part represents 60£7

420

2 parts = £60 x 2 = £120 and 5 parts = £60 x 5 = £300

Method 2

The fraction for the first person is 2 out of 7 parts, that is 7

2 and the

fraction for the second person will be 7

5


First person = 120£4207

2 and

second person = 420 - 120 = £300

Both methods work for most cases.

For the following I will use method 1

Example A sum of money is divided in the ratio 3 : 4 and the smallest

share is equivalent to £27. What is

(a) the amount given to the largest share.

(b) the total amount of money shared?

The smallest share is worth 3 parts so here 3 parts was £27.

1 part = 9£3

27

(a) Largest share = 4 x 9 = £36

(b) Total amount = 36 + 27 = £63

Example Two lines have lengths in the ratio 5 : 2. If the longer line is

15 cm long, find the length of the other line.

Longer line = 5 parts so 5 parts = 15 cm

1 part = cm35

15

Other line = 2 parts = 2 x 3 = 6cm


Example The ratio of my gas bill to my electricity bill was 13 : 5. If my

gas bill was £182, how much was my electricity bill?

Gas bill = 13 parts so 13 parts = £182

1 part = 14£13

182

Electricity bill = 5 parts = 5 x 14 = £70

Example Three people stake £10 on the national lottery and win

£850. Peter paid £2, John paid £4.50 and Claire paid the

rest towards the stake. The winnings are shared in the ratio

of the contributions. How much does Claire receive?

Ratio of share = Peter to John to Claire = £2 : £4.50 : £3.50

= 200 : 450 : 350

= 4 : 9 : 7

Hence out of 20 parts of the money Claire will receive 7 parts.

20 parts = £850

1 part = 50.42£20

850

Calire = 7 parts = 7 x 42.50 = £297.50


Example

For every 9 teenagers who like pop music there are 2 that does not.

In a youth club of 187 members, how many do not like pop music?

Ratio = Like pop to not like pop = 9 : 2

11 parts = 187

1 part = 1711

187

Not like pop = 2 parts = 2 x 174 = 34 people

Example A lorry is loaded up with fruit and vegetables for market.

The mass of fruit to vegetables is in the ratio of 7 : 8. If the

lorry’s load is 18.6 tonnes, find the mass of fruit and the

mass of vegetables it is carrying.

15 parts = 18.6 tonnes

1 part = 24.115

6.18 tonnes

Fruit = 7 parts = 7 x 1.24 = 8.68 tonnes

Vegetables = 8 parts = 8 x 1.24 = 9.92 tonnes

Example When £195 is divided in the ratio 2 : 4 : 7, what is the

difference between the largest share and the smallest?


Difference between the largest and smallest share is 5 parts(7 -2)

13 parts = £195

1 part = 15£13

195

Difference = 5 parts = 5 x 15 = £75

Example

A man and a woman share a bingo prize of £1000 between them in the

ratio 1 : 4. The woman shares her part between herself, her mother and her

two daughters in the ratio 2 : 1 : 1

How much does the woman receive?

5 parts = £1000

1 part = 200£5

1000

Woman’s original share = 4 parts = 4 x 200 = £800

Hence 4 parts = £800

1 part = 200£4

800

Woman’s final share = 2 parts = 2 x 200 = £400


Example £400 is divided between Ann, Brian and Carol so that Ann

has twice as much as Brian and Brian has three times as

much as Carol. How much does Brian receive?

If Carol received one part Brian would have to receive three parts hence

Ann would have to receive six parts. This gives the ratio

Ann : Brian : carol = 6 : 3 : 1

Hence 10 parts = £400

1 part = 40£10

400

Brian = 3 parts = 3 x 40 = £120

Example

Mrs Simms inherits £24 000.

She divides the money between her three children, aged 9, Alice, Brenda,

aged 7 and Charles, aged 8,in the ratio of their ages.

How much does Charles receive?

Ratio = 9 : 7 : 8

24 parts = 24 000

1 part = 1 000

Charles = 8 parts = £8 000


Direct Proportion

If two quantities are directly proportional to one another then one can be

written as a constant (k) multiplied by the other.

In order to find the constant k, more information needs to be provided.

Example

W and P are both positive quantities.

W is directly proportional to the square of P.

When W = 12, P = 4.

(a) Express W in terms of P.

(b) What is the value of W when P = 6?

(c) What is the value of P when W = 75?

a) Using the definition above

W is directly proportional to the square of P means W = k x P2

Using the information W = 12, P = 4 12 = k x 42 or 12 = k x 16

k = 75.04

3

16

12

W = 4

3P2

b) P = 6 W = 27364

36

4

3 2

c) W = 75 2

4

375 P

= k x

Don’t forget once k is found to

write the equation down


2

75.0

75P

2100 P

10P

Example

Y are X are both positive quantities.

Y is directly proportional to the square root of X.

When Y = 16, X = 16

a) Express Y in terms of X

b) What is the value of Y when X = 25?

c) What is the value of X when Y = 60?

a) XkY

Y = 16, X = 16 1616 k

4

416

k

k

XY 4

b) X = 25 2054254 Y

c) Y = 60 X460

X15

225

152

X


Inverse Proportion

If two quantities are inversely proportional to one another then one can be

written as a constant (k) divided by the other.

In order to find the constant k, more information needs to be provided.

Example

Given that M varies inversely to P and that M = 12 when P = 4

a) Obtain an expression for M in terms of P

b) What is the value of M when P = 8?

c) What is the value of P when M = 0.5?

a) P

kM

M = 12, P = 4 4

12k

k412

48k

PM

48

b) P = 8 68

48M

k =


c) M = 0.5 P

485.0

965.0

48P

Example

y is inversely proportional to the square root of x.

When y = 6, x = 9.

a) What is the value of y when x = 4.

b) What is the value of x when y = 10.

y is inversely proportional to the square root of x means x

ky

When y = 6, x = 9. 9

6k

or 3

6k

k36

18k x

y18

a) X = 4 92

18

4

18y

b) Y =10 x

1810

10

18x

8.1x

24.3x

Using the algebraic knowledge that

the P and 0.5 can be “swapped”


Standard form

A number expressed in standard form is a number written between 1 and

10 multiplied by 10 to an appropriate power.

The use of standard form is to represent very small numbers or very large

numbers

For example

0.000 000 000 000 32 represented in standard form will be 13102.3

214000 000 000 represented in standard form will be 111014.2

Example Express each of the following in standard form

(i) 0.000 000 462

(ii) 0.004

(iii) 90 000

(iv) 5910 000 000

(i) 0.000 000 462 = 71062.4

(ii) 0.004 = 3104

(iii) 90 000 = 4109

(iv) 5910 000 000 = 91091.5

Example Express each of the following as ordinary numbers

(i) 8106.1

(ii) 61054.2

(iii) 12107

(iv) 210414.9

(i) 8106.1 = 160 000 000

(ii) 61054.2 = 0.000 00254

(iii) 12107 = 7000 000 000 000

(iv) 210414.9 = 0.09414


Example The surface of the earth is about 509 970 000 km2. Express

this in standard form correct to two significant figures.

509 970 000 = 8101.5 km2

Example

Given that 4106A and 7104B work out, without a calculator

(i) AB (ii) B

A

(i) (ii)

12

11

74

74

104.2

1024

10106

104106

AB

3

7

4

7

4

105.1

10

10

4

6

104

106

B

A

NB Here we used the laws of indices. That is nmnm aaa and

nm

n

mnm a

a

aaa

Example

Given that 5102.1 X and 9105 Y work out , in standard form

(i) XY (ii) Y

X

(i) (ii)

14

95

106

105102.1

XY

3

9

5

104.2

2400

105

102.1

Y

X

Here we use the button for the standard form and type for X EXP


And for Y we type

Example

A rectangle has length AB = 5103.1 km and width BC = 4105.2 km.

Giving your answer in standard form, find

(i) the area,

(ii) the perimeter of this rectangle.

(i) Area = 3250000000105.2103.1 45

{this number is not in standard form}

Area = 91025.3 km2

(ii) Perimeter = 50000260000105.22103.12 45

= 310000 = 5101.3 km

Example

The weight of 1 grain of sand is given as 91043 grams.

a) (i) Write 91043 in standard form

(ii) What is the weight of 5 billion grains of sand? [1 billion is 1000 million]

b) A piece of sandstone weighs 1kg.

How many grains of sand is this equivalent to?

a) (i) 89 103.41043

(ii) Weight = 215103.45000000000 8 grams

b) Number of grains of sand = 10

81033.2

103.4

1000

{since 1kg = 1000g}

EXP 1 . 2 – 5

EXP 5 – 9


Example 7104.5 p and 6105.3 q

Calculate the value of each of the following.

Give all your answers in standard form.

(a) qp

(b) qp 85

(c) qp

(a) 1467 1089.1105.3104.5 qp

(b)

867 1098.2298000000105.38104.5585 qp

Here the typing could be done as I have done above or you could break it

down. Simply type:

(c) 767 1005.550500000105.3104.5 qp

Example The space shuttle is covered with 4101.3 heat resistant

tiles. The total surface area of the shuttle is 15102 m2.

How many tiles per m2 does the shuttle have?

Number of tiles per m2 = 10

4

15

1045.6101.3

102

Example The mean weight of all men, women, children and babies in

the UK is 42.1kg. The population of the UK is 56 million.

Work out the total weight of the entire population giving

your answer in standard form.

Total weight = 9103576.223576000001.4256000000

EXP 5 . 4 7 + 5 X EXP 3 . 5 6 = 8 X


Example

(a) At 300 000km/s, how long, to one significant figure, does it take

light to travel from the sun to Saturn, a distance of 9104.1

km?

(b) An electron carries a charge of 19106.1 coulomb. To one

significant figure, how many electrons are required for a total

charge of 1 coulomb?

(a) Using the formula for distance, speed time

Time taken = ssec..666.4666300000

104.1 9

(b) Number of electrons = 1818

191061025.6

106.1

1

(to 1 s.f.)

Example On an antique map (on which no scale is shown) the

distance between Oakford and Stanton is 3.6 cm whereas in

fact the distance as the crow flies is 19km. It is required to

express the scale in the form 1 : n to an accuracy of 2

significant figures in standard form.

Oakford to Stanton = 3.6 cm : 19km

= 3.6 : 1900000 (measurements in cm)

= 1 : 6.3

1900000

= 1 : ...7777.527777

= 1 : 5103.5

D

S T


Example

The mass of one atom of oxygen is given as 231066.2 grams.

The mass of one atom of hydrogen is given as 241067.1 grams.

a) Find the difference in mass between one atom of oxygen and one

atom of hydrogen.

b) A molecule of water contains two atoms of hydrogen and one atom of

oxygen.

i) Calculate the mass of one molecule of water.

ii) Calculate the number of molecules in 1 gram of water.

a) difference = 232423 10493.21067.11066.2

b) i) Mass of 1 molecule of water =

232423 10994.21067.121066.2 grams

ii) Number of molecules = 22

231034.3

10994.2

1

Surds 5,3,2 are irrational numbers expressed in surd form.

A rational number is one which can be written in the form q

p where p and

q are integers. An irrational number cannot be written in this form.

2 for example is an irrational number, and so is the numbers .

Numbers written in the form a are called surds.

Laws

abba

b

a

b

a


Special case

Example Simplify (i) 12 (ii) 27 (iii) 75

Hence simplify 752712

0

353332752712

= =

= =

= =

aaa

34 32

39 33

325 35


Example

Express with rational denominator 5

20

Here we multiply by one! However we make one 5

5 as this will help us.

545

520

5

5

5

20

5

20

Example

Given that 3

91227 can be express in the form 𝑎 3. Find the

value a.

333927

323412

333

39

3

3

3

9

3

9

323332333

91227

From the fact that

555


Example simplify (i) 1898 (ii) 4875

(i) 242327292491898

(ii) 4

5

34

35

316

3254875

Index notation

Laws

Special Cases 10 a

aa 2

1

, n aa n 1

,

n

n

aa

1 , n

na

a

1 ,

nn

a

b

b

a

nmnm aaa

nmnm aaa

mnnm aa


n maa nm

mn a

Example 288 33

1

Example 3

1

9

1

9

19

2

1

2

1

Example

(a) Write down the value for each of the following

(i) 06 (ii) 32

27 (iii) 43

(i) 160 {anything to the power 0 is 1}

(ii) 93272722

332

(iii) 81

1

3

13

4

4

(b) Simplify 25.0 336

3

2

9

16

3

136336

2

25.0

Example

Simplify each of the following (i) 34 xx (ii) 75 yy (iii) 53t

(i) 734 xxx {Add the powers}

Power 0.5 means square root


(ii) 275 yyy or 2

1

y {subtract the powers}

(iii) 1553 tt {multiply the powers}

Example

Simplify

24

323

6

23

ba

baa

ba

ba

ba

ba

baa

ba

baa

5

24

39

24

363

24

323

4

6

24

6

83

6

23

Calculator calculations

Example

a) Use your calculator to find 22 6.102.37

b) express this number correct to 3 significant figures.

Working out 2 cubed

Multiplying powers for a

Multiplying powers for b

Adding powers for a on

numerator Working out

3 x 8

Subtracting powers for b Dividing 24

by 6

Subtracting powers for a


a) To ensure you get it right work out 37.22-10.62 first on the calculator.

Then square root.

37.22-10.62 = 1271.48

35.65781822

b) 3 significant figures means the first three numbers of value. Which is 35.6

However the number after the 6 is a 5 and therefore we must round up (as

we do with decimal places) Answer = 35.7

Example

Find the value of 65 107.4105.3 , giving your answer in standard

form.

Algebra

Collection of like terms

When collecting like terms remember we can collect together

equivalent letters i.e. aaa 1037 and we can collect together numerical terms i.e. 4 + 8 – 3 = 9

However we cannot collect together terms that are not alike i.e.

ba 54 cannot be simplified. Nor can 43 a Example Simplify each of the following

a) bababa 855439 {Remember to take note of the sign

in front of each letter}

b) yxyxyx 87626

c) fedfdefed 27647324

𝒙 Ans EXE


d) qpqpqp 553782

Solving simple equations

Example Solve each of the following equations

a) 1743 x

b) 7325 xx

c) 9457 xx

d) 19231 xx

e) 1214 x

f) 7523 x

g) 53

52

x

h)

95

73

x

Golden rules The equation starts balanced and must remain balanced! So whatever you do to one side you must do to the other side.

i.e. (a) 1743 x

417443 x {Add 4 to both sides of the equation}

213 x {Simplify}

3

21

3

3

x {Divide both sides by 3}

7x {Simplify giving answer}


An alternative way of thinking! When moving a number from one side of

the equal sign to the other we perform the opposite operation.

The opposite of Addition is subtraction and visa versa.

The opposite of Multiplication is Division and visa versa.

(b) 7325 xx

52

{Simplify} 52

subtract} andover 3 the{take 535

2}subtract andover 2 the{take 2735

7325

.x

x

xxx

xx

xx

(c) 9457 xx

33333.13

4

43

447

5947

9457

x

x

xx

xx

xx

(d) 19231 xx


4

{Simplify} 520

add} andover 3 the{Move 3220

}add! andover 19 the{Move 23191

19231

x

x

xxx

xx

xx

(e) 1214 x

4

164

before} as rearrangecan we{now 4124

first} brackets {remove 1244

1214

x

x

x

x

x

(f) 7523 x

333333.13

4

6

8

86

1576

7156

7523

x

x

x

x

x

(g) 53

52

x


5

102

5152

times}andover 3 the{take 1552

53

52

x

x

x

x

x

(h)

95

73

x

8

243

21453

bracket} the{Remove 45213

times}andover 5 the{Take 4573

95

73

x

x

x

x

x

x

Factorisation

Example Factorise each of the following

a) 342 xx

b) 652 xx

c) 1582 xx

d) 542 xx

e) 1522 xx

f) 62 xx

g) 2832 xx


When factorising a quadratic such as 342 xx we express as two

brackets multiplied together. i.e. bxax where a and b are numbers

that

1. Multiply to give, in this case, 3

2. add to give 4.

This means they can be 1 and 3 or –1 and –3.

Since 1 and 3 add to give 4 then

31342 xxxx

(b) 652 xx

Here the numbers could be 1 and 6, –1 and –6, 2 and 3 or –2 and –3.

Since –2 and –3 add to give –5 the answer is found.

32652 xxxx

(c) 1582 xx

Here the numbers could be 1 and 15, -1 and –15, 3 and 5 or –3 and –5.

Since 3 and 5 add to give 8

531582 xxxx

(d) 542 xx

Here the numbers could be 1 and –5 or –1 and 5 (one positive and one

negative.)

Since 1 and –5 add to give –4.

51542 xxxx


(e) 1522 xx

Here the numbers could be 1 and –15, –1 and 15, 3 and –5 or –3 and 5

Since –3 and 5 add to give 2

531522 xxxx

(f) 62 xx

Here the numbers could be 2 and –3, –2 and 3, 1 and –6 or –1 and 6

Since 2 and –3 add to give –1.

3262 xxxx

(g) 2832 xx

Here the numbers could be 1 and –28, –1and 28, 2 and –14, 14 and –2, 4

and –7 or –4 and 7.

Since –4 and 7 add to give 3.

742832 xxxx

Example Factorise each of the following:

a) xyx 32

b) qppq 22 4

c) mnmm 6104 2

d) abcbaba 223 37


Here we cannot place into two brackets since it does not follow the

pattern of 2x followed by x, followed by a constant!

However we have a common factor. So place the common factor

outside a bracket.

a) 332 xyxxyx

b) pqpqqppq 44 22

c) nmmmnmm 35226104 2

d) cabaababcbaba 3737 2223

Difference of squares

Example factorise each of the following

(i) 𝑥2 − 36 (ii) 4𝑥2 − 49

(i) Here 𝑥2 − 36 represents the difference of two squares which

are 𝑥2 − 62. When we factorise 𝑥2 − 36 we are looking for

two numbers that have a sum of zero (for the middle term) this

is +6 − 6 = 0

Hence 𝑥2 − 36 = 𝑥 − 6 𝑥 + 6

(ii) 4𝑥2 − 49 = 2𝑥 − 7 2𝑥 + 7

Since

yxxyx 2

Since

xx 33

2𝑥 2 72


Example (a) factorise the quadratic 1272 xx

(b) Hence solve the equation 01272 xx

(a) 431272 xxxx {since –3 and –4

add to give –7.

(b) 01272 xx is the same as saying 043 xx

Which means 03 x or 04 x

Since 0ba means 0a or 0b

3x or 4x

Example (a) factorise the quadratic 2762 xx

(b) Hence solve the equation 02762 xx

(a) 392762 xxxx

(b) 02762 xx

039 xx

09 x or 03x 9x or

3x

Difference of two squares

Example Factorise each of the following:


Inequalities

Example Solve each of the following inequalities

a) 932 x

b) 1973 x

c) 8314 xx

d) 7335 xx

e) 9142 x

f) 73

5

x

g) 252 x

h) 812 x

Solving inequalities can be like solving equations. However don’t

forget to write the correct symbol; and not the equal sign!

a) 932 x

6

{Simplify} 122

side}other the to3 {Add 392

932

x

x

x

x

b) 1973 x

4

123

7193

1973

x

x

x

x


c) 8314 xx

7

734

1834

8314

x

xx

xx

xx

d) 7335 xx

2

42

435

3735

7335

x

x

xx

xx

xx

e) 9142 x

8

11

118

298

928

9142

x

x

x

x

x

f) 73

5

x


16

521

215

73

5

x

x

x

x

g) 252 x

5

252

x

x

However when dealing with a quadratic we have two solutions. NB

If 6x then 362 x which is also greater than 25. Hence

5x is a second solution.

h) 812 x

9 and 9

812

xx

x

Example

Given that n is an integer, find the values of n such that −7 ≤ 2𝑛 < 6

−3.5 ≤ 𝑛 < 3 {divide throughout by 2}

Hence n can equal −3, −2, −1 ,0 ,1 ,2

Whole number


Example List the values of n such that n is an integer value and

a) 23 n

b) 51 n

c) 34 n

Here we are not asked to find the solution by solving an equation

but by listing the possible solutions.

An integer value means possible whole number answers whether

positive or negative.

a) 23 n means n can be –2 –1, 0 or 1 we write

1,0,1,2 n

b) 51 n Answer: 5,4,3,2,1,0n

c) 34 n Answer: 2,1,0,1,2,3,4 n

Graphs

1. The straight line

Example Draw the graph of 𝑦 = 3𝑥 − 4 for values of 𝑥 from −3 to 3

For any straight line we can get away with plotting three points and

then a line through these three points.

Three points are selected to make sure we have only one straight line.

X – 3 0 3

y – 13 – 4 5

13

49

433

y

4

403

y

11

415

453

y


Example Draw the graph of 𝑥 + 𝑦 = 7 for values of 𝑥 from −1 to 7

X 4 0 3

y 3 7 4

3

74

y

y 70 y

4

73

y

y

2

4

6

-2

-4

-6

-8

-10

-12

-14

2 -2 0 x

y


2. The quadratic curve

The quadratic Curve could appear on both papers but for non-

calculator papers it will be a very basic curve.

Example

(a) Complete the table of values of y given 𝑦 = 𝑥2 − 2𝑥 − 5

(b) Draw the graph of 𝑦 = 𝑥2 − 2𝑥 − 5

(c) Use the graph to state the values of x for which 𝑥2 − 2𝑥 − 5 = 0

(d) State the minimum value for y on the graph.

x –4 –3 –2 –1 0 1 2 3 4 y 19 10 –2 –5 –6 –2 3

2

4

6

8

10

-2

2 4 6 8 -2 0 x

y


a)

X –4 –3 –2 –1 0 1 2 3 4

y 19 10 3 –2 –5 –6 –5 –2 3

b)

c) the curve equals zero at the points where the curve cuts the x-axis

𝑥 = −1.4, 𝑥 = 3.4

d) Minimum value of curve is where the curves gradient changes from

going down to going upwards. i.e. 𝑦 = −6 when 𝑥 = 1

3

544

52222

y

5

544

52222

y

5

10

15

20

-5

1 2 3 4 -1 -2 -3 -4 0 x

y


Example

a) Complete the table below for the graph of 𝑦 = 2𝑥2 − 3𝑥

b) Draw the graph of 𝑦 = 2𝑥2 − 3𝑥

c) Draw on the same axis the graph of 𝑦 + 2𝑥 = 14

d) Use the graph to obtain a suitable approximation to solution of the

simultaneous equations given by 𝑦 = 2𝑥2 − 3𝑥 and 𝑦 + 2𝑥 = 14

a)

b)

x –3 –2 –1 0 1 2 3 4 y 27 14 0 –1 2 20

x –3 –2 –1 0 1 2 3 4 y 27 14 5 0 –1 2 9 20

Type in -1 and exe

Then type Ans2 – 3Ans

Type in 3 and exe

Then type Ans2 – 3Ans

5

10

15

20

25

30

-5

1 2 3 4 -1 -2 -3 0 x

y

𝑦 + 2𝑥 = 14

𝑦 = 2𝑥2 − 3𝑥


c) 𝑦 + 2𝑥 = 14

X 0 1 3

y 14 12 8

Line drawn on the graph in part (b)

d) The curve 𝑦 = 2𝑥2 − 3𝑥 and the line 𝑦 + 2𝑥 = 14 are equal at the

points of intersection. Which occur at −2.4,19 and 2.9,8.1

approximately.

14

140

y

y

12

142

y

y

8

146

y

y


Notes

Mathematics - PBworks

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