Mathematics. Number & Algebra Revision Notes For Higher Tier. Thomas Whitham Sixth Form S J Cooper

Transcription

1 Mathematics Number & Algebra Revision Notes For Higher Tier Thomas Whitham Sith Form S J Cooper Factors, Primes & Prime Factors Approimations Fractions: Equivalent, epressing as, fractions of a quantity. Percentages: Epressing as, Percentage of a quantity, finding the original amount, compound measures. Proportion: Ratio, Direct proportion and Inverse proportion. Standard from. Surds Inde Notation Algebra: Collection of like terms, Solving equations Factorisation Graphs: Linear & Quadratic. Solving simultaneous equations using their graphs.

2 Factors A factor is any number which will divide into a given number an eact number of times. is a factor of since divides into eactly times. List all the factors of 8. Prime Factors Factors of 8 = {,,, 6, 9, 8} A prime number is a number who s factors are itself and. A prime factor is a factor which is a prime number. For eample the prime factors of are found in the diagram below. 6 The prime factors of are = Epress each of the following as products of their prime factors (i) (ii) Thomas Whitham Sith Form Page

3 (i) (ii) = = = = NB Here the highest common factor (HCF) = = 7 Approimation Find an approimate value of Nearer 0 than 60 Nearer than Nearer 0 than. So leave as 0.0 Thomas Whitham Sith Form Page

4 Simplify numerator Get rid of decimal by multiplying top and bottom by 00 = 00 (a) Estimate giving your answer to significant figures (b) Evaluate giving your answer to significant figures (a) (b) Estimate the value of 0.0 one significant figure. giving your answer to Clearly here if we round 0.0 off we have 0 which won t do! Thomas Whitham Sith Form Page

5 Lewis uses his calculator to calculate and gets the answer Use estimation to work out whether his answer is reasonable = Answer is unreasonable, as he is approimately a factor of 0 out. Fractions. Equivalent fractions Epress each of the following fractions in their simplest form. (a) 6 0 (b) (c) 60 8 (a) Here we notice that we have a common factor of. Since will divide into both 6 and (b) Thomas Whitham Sith Form Page

6 (c) Notice here we have not completely cancelled down the fraction so we must repeat the process With practice you will not require the intermediate step but move from the given fraction to the final answer. Eight pupils out of ninety two pupils failed to turn up for their Mathematics eamination. What fraction of the group (a) failed to turn up? (b) did turn up for the eam? (a) Fraction who did not turn up = 8 {no marks are awarded for 8 out of 9 } 9 (b) fraction who did turn up = {i.e. the rest} On Saturday 9000 people won ten pounds on the National Lottery draw. However 60 people failed to claim their 0 prize. What fraction failed to claim their prize? Fraction failed to claim prize = Thomas Whitham Sith Form Page

7 What fraction has been shaded in for each of the following shapes? (a) (b) Here there is a total of squares, of which are shaded fraction shaded = Fraction shaded = 6. Fraction of a calculator (without calculator) Find of 8 of 8 means Answer = 8 {i.e. 8 divided by } However we have a technique for showing our working as follows: 8 Find of 70 Thomas Whitham Sith Form Page 6

8 of 70 means 70 {again we could find of 70 by dividing 70 by } {so we find of 70 by dividing 70 by and then multiplying the answer by } of Find 7 of 8 7 of More complicated divisions may require some additional calculations at the side of your page 9 In a school with 70 pupils, 0 stay in school at lunch time, and 8 of these pupils bring a packed lunch. How many pupils bring a packed lunch? Number staying at school = Number with packed lunch = Thomas Whitham Sith Form Page 7

9 . Fraction of a quantity (with calculator) Find of 7. of 7. means 7. Using the calculator we have 7. Answer =.9 [Don t forget the units] Find of 88km of 88km = 88 = 6km Mr Ashworth earns 9.68 in one week. After ta is deducted, he receives only five sevenths of this amount. How much does he receive? We require 7 of 9.68 = 9.68 =. 7 He receives.0 [remember currency has two decimal places]. Fractions to percentages/decimals Epress each of the following as fractions in their simplest form. (a) % (b) 0. (c).6 (d) % Thomas Whitham Sith Form Page 8

10 (a) % = 00 9 (b) 0. = (c).6 = 6 0 (d) % = 00. Addition and subtraction of fractions Work out Which is easily done provided the denominators are the same. Work out First change the first fraction with its equivalent in twelfths Thomas Whitham Sith Form Page 9

11 Work out First add the whole numbers together Net the common denominator is. hence replace each fraction with its equivalent in terms of fifteenths. Work out Subtract the whole numbers and place each fraction with common denominator. Since we cannot subtract from w must use one of the whole numbers. hence becomes have only whole one. and we now A piece of wood is 7 of wood is left? metres long. If 7 metres is cut off, what length Length of wood left = Thomas Whitham Sith Form Page 0

12 6. Multiplication and Division of fractions Work out To multiply any two fractions together simply multiply numerators together and then multiply denominators together. Work out 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. will divide eactly into both and 9. Work out Change fractions to top heavy fractions Net cancel where possible Thomas Whitham Sith Form Page

13 Work out Invert the second fraction and change the to a sign.. Cancel where possible Work out change fractions to top heavy fractions Invert second fraction and cancel where possible. Finally multiply across Find the eact value of when a a b and b a That is the reciprocal of is Similarly the reciprocal of is Thomas Whitham Sith Form Page

14 a b 6 Louise has 7 m of ribbon. She makes 9 skirts and uses of a metre for each one. How much ribbon does she have left? 8 9 Amount left = 7 = m 0 Subtract whole quantities Place over common denominators Since we cannot subtract from here, use one of the whole ones as 0 0 Thomas Whitham Sith Form Page

15 Epress each of the following as fractions in their simplest form a) 0.7 b) c) a) b) Let Subtract from previous epression gives Hence 7 79 and since c) Again Let Subtract from previous epression gives Hence Thomas Whitham Sith Form Page

16 Percentages. Epressing as a percentage Epress each of the following percentages as (a) Decimals (b) Fractions in their simplest form. (i) 0% (ii) 6% (iii) % (iv) % (a)(i) 0% = (ii) 6% = (iii) % = 0. 0 (iv) % = (b)(i) 0% = 0 6 (ii) 6% = (iii) % = 7 (iv) % = Mark scored out of 0 in a recent mathematics test. Epress his score as a percentage. Test result = % 0 0 Thomas Whitham Sith Form Page

17 The height of a tree increased from.7m to.98m in one year. What percentage increase is this? Increase =.98.7 = Percentage increase = %.7. Percentage of a quantity a) Without a calculator What is 0% of 0 kg Method : Using fractions 0% = 0 as a fraction 00 Method Using unity hence 0% of 0kg = of 0 = 8 0 8kg 0% of 0kg = kg {i.e. divide by 0} 0% of 0 kg = = 8kg What is % of 60? Method : Using fractions % = 7 as a fraction 00 0 Thomas Whitham Sith Form Page 6

18 7 hence % of 60 = 7 of 60 = Method Using unity 0% of 60 = 6 {i.e. divide by 0} % of 60 = {i.e. half of 0%} 0% of 60 = 6 = 8 % of 60 = 8 + = What is % of cm? Method : Using fractions % = as a fraction 00 hence % of cm = of = Division more complicated so done at the side of our page Method Using unity 6 6cm 6 0% of cm =.cm {i.e. divide by 0} % of cm = 7.cm {i.e. half of 0%} 0% of cm =. = 8.8cm % of cm = = 6cm Thomas Whitham Sith Form Page 7

19 The price of a new television is 76 plus VAT at 7½%. (a) (b) Work out the VAT to be added to the price of the television. What is the total cost of the television? (a) 0% of 76 = 7.60 % of 76 = 8.80 {i.e. half of 0%} ½% of 76 =.0 {i.e. half of %} 7½% of 76 = 0.80 =VAT {i.e. add the previous answers together} (b) Total cost = = The number of girls attending football matches is epected to increased by % this year. If there were 00 girls attending matches last year, how many girls are epected to attend this year? % of 00girls = girls {i.e. divide by 00} % of 00girls = = 7girls Number of girls = = 87 b) With a calculator What is % of 0 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. % of 0 kg = 0 9.kg 00 Thomas Whitham Sith Form Page 8

20 Keys used: X = (there are alternative combinations for inputs to the calculator) What is 7% of 6.0? 7 7% of 6.0 = {here we must round up the answer since money involves decimal places} I pay 6% of my salary into a pension fund. How much do I pay into my fund if my salary is 0 per annum? 6 Amount paid = 6% of 0 = Every year a car loses % of its value at the beginning of that year. If it was originally worth 000, what will it be worth after years? Method : Find the amount lost and then subtract from the original. Amount lost = % of 000 = Value after st year = = 0 Amount lost in nd year = % of 0 = Value after nd year = = 6.0 Thomas Whitham Sith Form Page 9

21 Method : Using the percentage Loss we can determine the percentage left! If the value of the car is % less each year, then the new value will be 8% of the original. (i.e. 8% + % = 00%) 8 value after st year = 8% of 000 = Or better still value after nd year = 8% of 0 = {answer comes out quicker!} The population of Umbridge increased by 7% during the years If the population in 960 was 7, what was the population in 980? An increase of 7% means than the new population is 7% what it was! {i.e. an increase implies we add on to 00%} 7 New population = 7% of 7 = Or better still Thomas Whitham Sith Form Page 0

22 This year nurses were given a.% pay rise. If Susan earned 6 0 per annum last year how much more will she get in her pay packet this year? What will be her new salary? Increase of.% means that Nurses now earn 0.% what they did the year before! 0. New Wage = 0.% of 6 0 = Finding the original percentage Here we make use of the formula: New value = Percentage of Original value Or New value = Percentage Original value The population of Villanova has increased by 6% during the last five years and is now 000. What was its population five years ago? Here we have been told the answer. That is the new value is 000. after an increase of 6% 6% of original value = 000 {i.e. increase means 6+00%} 000 % of original value = {i.e divide by 6 to find %} Thomas Whitham Sith Form Page

23 00% of original value = original value = NB original value is always 00% and we have either increased it or decreased it to find the new value The price of houses in Villanova has increased by 8% during the last year. If the house costs $ 000 now, what would it have cost a year ago? Here we have been told the answer. That is the new value is $ 000. after an increase of 8% 8% of original value = 000 {i.e. increase means 8+00%} 000 % of original value = {i.e divide by 8 to find %} 00% of original value = original value = $8. 9 The attendance of Burnley football club fell by 7% in 00. If 00 fewer people went to matches in 00, how many went in 000? Here we have been told the answer. That is the new value is 00. which represents the 7%! 7% of original value = % of original value = 90 {i.e divide by 7 to find %} 7 00% of original value = original value = people Thomas Whitham Sith Form Page

24 During a Grand Pri race, the tyres on a car are reduced in weight by %. If they weigh 88 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 88kg. after an decrease of % 97% of original value = 88 {i.e. increase means 00 %} 88 % of original value = {i.e divide by 97 to find %} 97 00% of original value = original value = 00 00kg A car, which failed its MOT test, was sold for 6, thereby making a loss of % on the cost price. What was the cost price? Here we have been told the answer. That is the new value is 6. after an decrease of % 6% of original value = 6 {i.e. increase means 00 %} % of original value = {i.e divide by 6 to find %} 6 00% of original value = original value = Thomas Whitham Sith Form Page

25 In 00 the population of England was 9,6,800. The population of England is increasing by an annual rate of 0. per cent. a) Write down the single number that we must multiply 9,6,800 by if we wish to calculate an estimate for what the population was in 00. b) Assuming that England s population continues at this same rate, calculate the population of England in 0. a) increase implies add on to 00%, therefore 00.% =.00 This is the number which must be multiplied to 9,6,800 if we want the population in 00. For 00 we must multiply by.00 again. Hence number must be.00 = b) Population in 0 = =,700,6. In 008 the State Pension was increased by. per cent to 9. a week. What was the state pension before this increase? Increase 0.% of original value = 9..0 Original pension = Pension was Thomas Whitham Sith Form Page

26 Compound Measures A bank pays interest of % on money in deposit accounts. Mr Smith Puts 000 in the bank. How much has he after a) One year b) three years? a) After one year Amount = 0% of 000 = = 080 b) After three years Method In nd year Amount = 0% of 080 = = 6.0 Method in rd year Amount = 0% of 6.0 = = 9.7 After three years we have multiplied.0 a total of three times. i.e..0 After three years Amount = = 9.7 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. r A P 00 n However don t learn the formula, just learn its meaning! Thomas Whitham Sith Form Page

27 The population of an island increases by 8% each year. If the population in 009 was 0 million, what is the epected population of the island in 0? Here an increase by 8% means 08% or to 0 means the increase over 6 years. Epected population = =.87 million (rounded to significant figures) A new car is valued at 000. At the end of each year its value is reduced by % of its value at the start of the year. What will it be worth after 6 years? Here a decrease by % means it s worth 8% or 0.8 Value after 6 years = = 67. Ratio & Proportion Simplify each of the following ratios a) : 8 b) 8 : 7 c) :.80 d) 00m :.km e) 0 cm : litres a) : 8 = : 7 {Divide both sides by } b) 8 : 7 = : {Divide both sides by 9} c) :.80 = 00 : 80 {Change both into pence} = 0 : 8 { Divide by 0} = : { Divide by 6} Thomas Whitham Sith Form Page 6

28 d) 00m :.km = 00 : 00 { Change both into metres} = : { Divide by 00} = : 7 { Divide by } e) 0 cm : litres = 0 : 000 {Change both into cm } = : 60 {divide both sides by 0} = : {divide both sides by } A school decides to give 0% 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 = 0 : 80 = : One bottle of wine holds 70 cm whereas another holds. litres. Give the simplest ratio of their capacities. Ratio = 70cm :. litres = 70 : 00 {change units into cm } = : = : 8 Jane took minutes to do her homework, but her sister Lucy took ¼ hours. What is the simplest ratio of their times taken? Thomas Whitham Sith Form Page 7

29 Jane to Lucy = : ¼ = : 7 {change units into minutes} = 9 : = : The standard gauge of railway track is.m. A model is to be made with gauge mm. Calculate the scale in the form : n, where n>. Model to Standard gauge = mm :.m = : 0 {change into mm} = : 0 0 is divided between two people in the ratio :. Work out what each person will receive. There are two ways of looking at this problem. Method There are a total of 7 parts 7 parts represents 0 0 part represents 60 7 parts = 60 = 0 and parts = 60 = 00 Method The fraction for the first person is out of 7 parts, that is 7 and the fraction for the second person will be 7 Thomas Whitham Sith Form Page 8

30 First person = 0 0 and 7 second person = 0-0 = 00 Both methods work for most cases. For the following I will use method A sum of money is divided in the ratio : and the smallest share is equivalent to 7. What is (a) the amount given to the largest share. (b) the total amount of money shared? The smallest share is worth parts so here parts was 7. 7 part = 9 (a) Largest share = 9 = 6 (b) Total amount = = 6 Two lines have lengths in the ratio :. If the longer line is cm long, find the length of the other line. Longer line = parts so parts = cm part = cm Other line = parts = = 6cm Thomas Whitham Sith Form Page 9

31 The ratio of my gas bill to my electricity bill was :. If my gas bill was 8, how much was my electricity bill? Gas bill = parts so parts = 8 8 part = Electricity bill = parts = = 70 Three people stake 0 on the national lottery and win 80. Peter paid, John paid.0 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 = :.0 :.0 = 00 : 0 : 0 = : 9 : 7 Hence out of 0 parts of the money Claire will receive 7 parts. 0 parts = part =. 0 0 Calire = 7 parts = 7.0 = 97.0 Thomas Whitham Sith Form Page 0

32 For every 9 teenagers who like pop music there are that does not. In a youth club of 87 members, how many do not like pop music? Ratio = Like pop to not like pop = 9 : parts = part = 7 Not like pop = parts = 7 = people 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 8.6 tonnes, find the mass of fruit and the mass of vegetables it is carrying. parts = 8.6 tonnes 8.6 part =. tonnes Fruit = 7 parts = 7. = 8.68 tonnes Vegetables = 8 parts = 8. = 9.9 tonnes When 9 is divided in the ratio : : 7, what is the difference between the largest share and the smallest? Thomas Whitham Sith Form Page

33 Difference between the largest and smallest share is parts(7 -) parts = 9 9 part = Difference = parts = = 7 A man and a woman share a bingo prize of 000 between them in the ratio :. The woman shares her part between herself, her mother and her two daughters in the ratio : : How much does the woman receive? parts = part = 00 Woman s original share = parts = 00 = 800 Hence parts = part = 00 Woman s final share = parts = 00 = 00 Thomas Whitham Sith Form Page

34 00 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 si parts. This gives the ratio Ann : Brian : carol = 6 : : Hence 0 parts = part = 0 0 Brian = parts = 0 = 0 Mrs Simms inherits 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 parts = 000 part = 000 Charles = 8 parts = Thomas Whitham Sith Form Page

35 Direct Proportion If two quantities are directly proportional to one another then one can be written as a constant (k) multiplied by the other. = k In order to find the constant k, more information needs to be provided. W and P are both positive quantities. W is directly proportional to the square of P. When W =, P =. (a) Epress W in terms of P. (b) What is the value of W when P = 6? (c) What is the value of P when W = 7? a) Using the definition above W is directly proportional to the square of P means W = k P Using the information W =, P = = k or = k 6 k = W = P b) P = 6 W = Don t forget once k is found to write the equation down c) W = 7 7 P Thomas Whitham Sith Form Page

36 7 0.7 P 00 P P 0 Y are X are both positive quantities. Y is directly proportional to the square root of X. When Y = 6, X = 6 a) Epress Y in terms of X b) What is the value of Y when X =? c) What is the value of X when Y = 60? a) Y k X Y = 6, X = 6 6 k 6 6 k k Y X b) X = Y 0 c) Y = X X X Thomas Whitham Sith Form Page

37 Inverse Proportion If two quantities are inversely proportional to one another then one can be written as a constant (k) divided by the other. = k In order to find the constant k, more information needs to be provided. Given that M varies inversely to P and that M = when P = a) Obtain an epression 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.? a) M M =, P = k P k k k 8 M 8 P 8 b) P = 8 M 6 8 Thomas Whitham Sith Form Page 6

38 c) M = P 8 P Using the algebraic knowledge that the P and 0. can be swapped y is inversely proportional to the square root of. When y = 6, = 9. a) What is the value of y when =. b) What is the value of when y = 0. y is inversely proportional to the square root of means When y = 6, = 9. 6 k k or k k 8 8 y a) X = 8 8 y 9 b) Y = y k Thomas Whitham Sith Form Page 7

39 Standard form A number epressed in standard form is a number written between and 0 multiplied by 0 to an appropriate power. The use of standard form is to represent very small numbers or very large numbers For eample represented in standard form will be represented in standard form will be. 0 Epress each of the following in standard form (i) (ii) 0.00 (iii) (iv) (i) = (ii) 0.00 = (iii) = (iv) = Epress each of the following as ordinary numbers (i) (ii) (iii) (iv) (i) (ii) (iii) (iv) = = = = 0.09 Thomas Whitham Sith Form Page 8

40 = The surface of the earth is about km. Epress this in standard form correct to two significant figures km Given that (i) AB 60 NB 60 Given that A 60 and 0 7 B 0 work out, without a calculator A B (ii) (i) AB (ii) A B Here we used the laws of indices. That is a X. 0 and (i) XY (ii) (i) XY Here we use the button 0 9 EXP a m n mn a a and m a n a a m n a mn 9 Y 0 work out, in standard form X Y (ii) X Y for the standard form and type for X Thomas Whitham Sith Form Page 9

41 . EXP And for Y we type EXP 9 A rectangle has length AB =.0 km and width BC =.0 km. Giving your answer in standard form, find (i) the area, (ii) the perimeter of this rectangle. (i) Area = {this number is not in standard form} 9 Area =. 0 km (ii) Perimeter = = 0000 =.0 km The weight of grain of sand is given as 0 9 grams. a) (i) 9 Write 0 in standard form (ii) What is the weight of billion grains of sand? [ billion is 000 million] b) A piece of sandstone weighs kg. How many grains of sand is this equivalent to? a) (i) (ii) Weight = grams b) Number of grains of sand = {since kg = 000g} 8.0 Thomas Whitham Sith Form Page 0

42 7 p. 0 and q.0 Calculate the value of each of the following. Give all your answers in standard form. (a) p q (b) (c) p 8q p q p q (a) (b) p 8q Here the typing could be done as I have done above or you could break it down. Simply type: X. EXP X. EXP 6 = p q (c) 6 The space shuttle is covered with. 0 heat resistant tiles. The total surface area of the shuttle is 0 m. How many tiles per m does the shuttle have? Number of tiles per m = The mean weight of all men, women, children and babies in the UK is.kg. The population of the UK is 6 million. Work out the total weight of the entire population giving your answer in standard form. Total weight = Thomas Whitham Sith Form Page

43 (a) At km/s, how long, to one significant figure, does it take light to travel from the sun to Saturn, a distance of. 0 km? 9 (b) An electron carries a charge of.6 0 coulomb. To one significant figure, how many electrons are required for a total charge of coulomb? (a) Using the formula for distance, speed time 9. 0 Time taken = sec s S D 9 T (b) Number of electrons = (to s.f.) On an antique map (on which no scale is shown) the distance between Oakford and Stanton is.6 cm whereas in fact the distance as the crow flies is 9km. It is required to epress the scale in the form : n to an accuracy of significant figures in standard form. Oakford to Stanton =.6 cm : 9km =.6 : (measurements in cm) = : = : = :. 0 Thomas Whitham Sith Form Page

44 The mass of one atom of oygen is given as.660 grams. The mass of one atom of hydrogen is given as.670 grams. a) Find the difference in mass between one atom of oygen and one atom of hydrogen. b) A molecule of water contains two atoms of hydrogen and one atom of oygen. i) Calculate the mass of one molecule of water. ii) Calculate the number of molecules in gram of water. a) difference = b) i) Mass of molecule of water =.660 ii) Number of molecules = Surds,, grams are irrational numbers epressed 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. for eample is an irrational number, and so is the numbers. Numbers written in the form a are called surds. Laws a b ab a b a b Thomas Whitham Sith Form Page

45 Special case a a a Simplify (i) (ii) 7 (iii) 7 Hence simplify 7 7 = = = 9 = = = Thomas Whitham Sith Form Page

46 Epress with rational denominator 0 Here we multiply by one! However we make one as this will help us From the fact that Given that value a. 9 7 can be epress in the form a. Find the Thomas Whitham Sith Form Page

47 Thomas Whitham Sith Form Page 6 simplify (i) 8 98 (ii) 8 7 (i) (ii) Inde notation Laws Special Cases 0 a a a, a n n a, n n a a, n n a a, n n a b b a n m n m a a a n m n m a a a mn n m a a

48 m a n n m a m n a (a) Write down the value for each of the following (i) 0 6 (ii) 7 (iii) (i) 6 0 {anything to the power 0 is } (ii) (iii) 8 (b) Simplify Power 0. means square root Simplify each of the following (i) (ii) 7 y y (iii) t (i) 7 {Add the powers} Thomas Whitham Sith Form Page 7

49 (ii) 7 y y y or y {subtract the powers} (iii) t t {multiply the powers} a a b Simplify 6a b Working out cubed a 6a Working out 8 Dividing by 6 b a b a 8a b 6a b a b 6a b a 9 b 6 Multiplying powers for a Multiplying powers for b Adding powers for a on numerator Subtracting powers for b Subtracting powers for a Calculator calculations a) Use your calculator to find b) epress this number correct to significant figures. Thomas Whitham Sith Form Page 8

50 a) To ensure you get it right work out first on the calculator. Then square root = 7.8 Ans EXE.6788 b) significant figures means the first three numbers of value. Which is.6 However the number after the 6 is a and therefore we must round up (as we do with decimal places) Answer =.7 6 Find the value of form. Algebra Collection of like terms, giving your answer in standard When collecting like terms remember we can collect together equivalent letters i.e. 7a a 0a and we can collect together numerical terms i.e. + 8 = 9 However we cannot collect together terms that are not alike i.e. a b cannot be simplified. Nor can a Simplify each of the following a) 9a b a b a 8b {Remember to take note of the sign in front of each letter} b) 6 y 6y 7 8y c) d e f e 7d f 6d 7e f Thomas Whitham Sith Form Page 9

51 d) p 8q 7 p q p q Solving simple equations Solve each of the following equations a) 7 b) 7 c) 7 9 d) 9 e) f) 7 g) h) 7 9 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) 7 7 {Add to both sides of the equation} {Simplify} {Divide both sides by } 7 {Simplify giving answer} Thomas Whitham Sith Form Page 0

52 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) {take the over and subtract } {take the over and subtract} {Simplify} (c) (d) 9. Thomas Whitham Sith Form Page

53 {M ovethe9 over and add!} {Movethe over and add} {Simplify} (e) 6 {remove brackets first} {now we can rearrange as before} (f) 7 (g) Thomas Whitham Sith Form Page

54 0 (h) 7 9 {take the over and times} {Take the over and times} {Remove the bracket} Factorisation Factorise each of the following a) b) 6 c) 8 d) e) f) 6 g) 8 Thomas Whitham Sith Form Page

55 When factorising a quadratic such as we epress as two a b where a and b are numbers brackets multiplied together. i.e. that. Multiply to give, in this case,. add to give. This means they can be and or and. Since and add to give then (b) 6 Here the numbers could be and 6, and 6, and or and. Since and add to give the answer is found. (c) 8 6 Here the numbers could be and, - and, and or and. Since and add to give 8 (d) 8 Here the numbers could be and or and (one positive and one negative.) Since and add to give. Thomas Whitham Sith Form Page

56 (e) Here the numbers could be and, and, and or and Since and add to give (f) 6 Here the numbers could be and, and, and 6 or and 6 Since and add to give. (g) 8 6 Here the numbers could be and 8, and 8, and, and, and 7 or and 7. Since and 7 add to give. 8 7 Factorise each of the following: a) y b) pq p q c) m 0m 6mn d) 7 a b a b abc Thomas Whitham Sith Form Page

57 Here we cannot place into two brackets since it does not follow the pattern of followed by, followed by a constant! However we have a common factor. So place the common factor outside a bracket. a) y y Since y y Since b) pq p q pqq p c) m 0m 6mn m m n d) 7a b a b abc ab7a ab c Difference of squares (i) factorise each of the following (i) 6 (ii) 9 Here 6 represents the difference of two squares which are 6. When we factorise 6 we are looking for two numbers that have a sum of zero (for the middle term) this is +6 6 = 0 Hence 6 = (ii) 9 = Thomas Whitham Sith Form Page 6

58 (a) factorise the quadratic 7 (b) Hence solve the equation 7 0 (a) 7 add to give 7. {since and is the same as saying 0 (b) 7 0 Which means 0 or 0 Since a b 0 means a 0 or b 0 or (a) factorise the quadratic 6 7 (b) Hence solve the equation (a) (b) or 0 9 or Difference of two squares Factorise each of the following: Thomas Whitham Sith Form Page 7

59 Inequalities Solve each of the following inequalities a) 9 b) 7 9 c) 8 d) 7 e) 9 f) 7 g) h) 8 Solving inequalities can be like solving equations. However don t forget to write the correct symbol; and not the equal sign! a) {Add to theother side} {Simplify} b) Thomas Whitham Sith Form Page 8

60 Thomas Whitham Sith Form Page 9 c) d) e) f) 7

61 7 6 g) However when dealing with a quadratic we have two solutions. NB If 6 then 6 which is also greater than. Hence is a second solution. h) and 9 Whole number Given that n is an integer, find the values of n such that 7 n < 6. n < {divide throughout by } Hence n can equal,,,0,, Thomas Whitham Sith Form Page 60

62 List the values of n such that n is an integer value and a) n b) n c) 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) n means n can be, 0 or we write n,,0, n Answer: n 0,,,,, b) n Answer: n,,,,0,, c) Graphs. The straight line Draw the graph of y = for values of from to 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. y 9 X 0 y y 0 Thomas Whitham Sith Form Page 6 y

63 y Draw the graph of + y = 7 for values of from to 7 X 0 y 7 y 7 y 0 y 7 y 7 y Thomas Whitham Sith Form Page 6

64 y The quadratic curve The quadratic Curve could appear on both papers but for noncalculator papers it will be a very basic curve. (a) Complete the table of values of y given y = 0 y (b) Draw the graph of y = (c) Use the graph to state the values of for which = 0 (d) State the minimum value for y on the graph. Thomas Whitham Sith Form Page 6

65 a) X 0 y y y b) y c) the curve equals zero at the points where the curve cuts the -ais =., =. d) Minimum value of curve is where the curves gradient changes from going down to going upwards. i.e. y = 6 when = Thomas Whitham Sith Form Page 6

66 a) a) Complete the table below for the graph of y = 0 y b) Draw the graph of y = c) Draw on the same ais the graph of y + = d) Use the graph to obtain a suitable approimation to solution of the simultaneous equations given by y = and y + = 0 y Type in - and ee Then type Ans Ans b) y 0 Type in and ee Then type Ans Ans 0 y = 0 y + = Thomas Whitham Sith Form Page 6

67 c) y + = X 0 y 8 y 0 y y y Line drawn on the graph in part (b) y 6 y 8 d) The curve y = and the line y + = are equal at the points of intersection. Which occur at.,9 and.9,8. approimately. Thomas Whitham Sith Form Page 66

68 Notes Thomas Whitham Sith Form Page 67