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Maths Common Entrance revision Year 8 Topic/Objective To multiply/divide any number including decimals by 10, 100, 1000 Examples of questions A) 5.8 x 1000 = B) 3.9 x 100 = C) 517.6 ÷ 1000 = D) 536 ÷ 10000 = Fill in the missing blanks: A) ______ x 1000 = 7350 B) 283 ÷ _____ = 0.0283 Harder questions: A) _____ x 300 = 69000 B) 4600 ÷ 20 = _______ To multiply/divide/add/subtract using decimal numbers A) 5.36 + 0.2 = B) 3.76 + 14 = C) 7.98 - 2.19 = D) 13.26 - 7.03 = Find the area of the following rectangle: 8.4cm 6.3cm Find the length of the missing side of the rectangle if it’s area is 42.63cm2 xcm 7cm If 268 x 32 = 8576. What is A) 26.8 x 3.2? B) 2.68 x 3.2? C) 0.268 x 32? D) 0.268 x 0.32? To order decimal numbers according to size order Order the following decimals from smallest to largest: 4.606, 4.6, 4.06, 4.66, 4.066 1|Page Maths Common Entrance revision Year 8 Topic/Objective To round numbers to the nearest whole, ten, hundred or thousand To round numbers to a given number of decimals places To round numbers to a given number of significant figures Examples of questions Round the following numbers to the nearest 10: A) 764 B) 253 C) 2874 D) 1836 E) 26.43 Round the above numbers to the nearest 1000: (harder) Round the following numbers to the stated decimal place A) 2.6843 to 2dp B) 5.927 to 2dp C) 5.962 to 1dp D) 8.0999 to 3dp Round the following numbers to the stated number of significant figures: A) 7.456 to 2sf B) 0.0604 to 2 sf C) 0.070036 to 4sf D) 6347 to 1sf E) 3982 to 2sf F) 5341 to 3sf Estimate the value of this calculation: 3.94 x 6.03 Rewrite the following calculation, rounding each number to 1 significant figure: Now work out the value of this. Using a calculator, find the exact value of the above calculation. Write down all of the digits shown on your calculator. Round this number to A) 3decimal places B) 3 significant figures 2|Page Maths Common Entrance revision Year 8 Topic/Objective Examples of questions To round numbers to a given number of significant figures (Continued) Repeat the above for the following: To use all four operations with negative numbers A) -3-5 B) 2+ -6 C) 3- - 9 D) -4-6-+4- -11 What is the difference between -11˚C and 64˚C? What is the difference between -137 and -149? A) -3 x 5 B) -2 x -9 C) 5 x -7 D) -3 x -4 A) What is (-5)2 B) There are two answers to the following: (x) 2 = 36. What could they be? Fill in the missing blanks: To be able to find equivalent fractions or cancel down a fraction into its simplest terms 24 = [ ] 36 6 4 = [] 5 40 66 = [ ] 77 7 Cancel down the following fractions: 21 34 36 35 40 13 60 70 90 63 56 52 Write 64 centimetres as a fraction of 4 metres. Give your answer in its simplest form. To use equivalent fractions to size order Order these fractions from smallest to largest: 7 5 2 1 fractions 12 6 1 3 6 15 5 8 3 5 3 4 4 6 7 10 4 10 11 20 3|Page Maths Common Entrance revision Year 8 Topic/Objective To add/subtract fractions Examples of questions 3 7 + 11 21 4 5 + 3 15 5 6 + 3 4 5 8 - 4 7 3 4/5 To find a fraction of an amount To multiply and divide using fractions + 2 1/9 2/5 of 45 3/7 of 406 3/8 of 768 5/6 of 426 Calculate ⅜ of 2 metres, giving your answer in centimetres. ⅓ of a class of 24 children are girls. How many children in the class are boys? ¾ of the girls and ½ of the boys learn a musical instrument. How many children learn a musical instrument? 2/3 x 4/5 3/4 x 7/5 2/9 x 5/3 4/3 ÷ 3/8 2/3 ÷ 6/7 2/9 ÷ 1/3 Find the area of the following rectangles: 3/7cm 4/9cm 2 ⅓ cm 5⅜ cm 4|Page Maths Common Entrance revision Year 8 Topic/Objective To multiply and divide using fractions (continued) To find a percentage of an amount To express one number as a percentage of another To find percentage increase and decrease Examples of questions It takes ⅔ of an hour to drive the red bus round its route. The red bus is driven round its route for six hours each day. How many times does the red bus complete its route each day? Sheba the cat drinks ⅔ of a pint of milk each day. For how long will 6 pints of milk last Sheba? A) 30% of £710 B) 25% of £640 C) 38% of £540 D) 33⅓% of £963 E) 5% of £340 F) 66⅔% of 1462 Work out 15% of £12 Calculate 90% of £15 Mrs Collins sold a rare book on a website for £60 She then paid a fee of 12% of the selling price. How much was the fee? During 2010, Callumʼs height increased by 12%. At the start of the year, he was 1.50 metres tall. Calculate his height at the end of the year. Write 45 as a percentage of 150 Write 48 as a percentage of 60. A T shirt marked at £18 was sold for £14.40. What percentage discount was given on the marked price? During 2010, Callumʼs mass increased from 55 kilograms to 59.4 kilograms. Calculate the percentage increase in his mass. Mrs Collins sold a book for £60 to Mr Hudson. He later sold the book for cash for £87 Calculate his profit as a percentage of the price he paid for the book. 5|Page Maths Common Entrance revision Year 8 Topic/Objective To find percentage increase and decrease (Continued) To know fraction/decimal/percentage equivalents (Pupils should memorise these: ½=0.5=50% ¼ = 0.25 = 25% 1/10 = 0.1 = 10% 1/5 = 0.2 = 20% 1/100 = 0.01 = 1% Examples of questions Miss Venn sells calculators in the school Maths shop. Each calculator costs her £3.72 During 2008, she sold these calculators for £5.50 each. (i) How much profit did she make on each calculator? (ii) Calculate this profit as a percentage of the price she pays. Give your answer correct to 1 decimal place. Write these numbers in order, smallest first: Write these numbers in order, largest first: ⅓ = 0.3333... = 33 1/3 % ⅔ = 0.6666... = 66 2/3 %) To understand the order of operations (BIDMAS) Write 3/20 as a percentage Write 8% as a decimal Write 6/25 as a percentage Write 0.16 as a fraction in its simplest form. Calculate: i) 4+5x6 ii) To know what a prime number, square number and cube number is iii) 14 + 6 x 4 – 2 iv) 27 – 18 ÷ 32 From the list of numbers below, identify a square number, a cube number and a prime number: 13, 8, 12, 49, 120, 27 Why is 126 not a prime number? What is the first prime number? 6|Page Maths Common Entrance revision Year 8 Topic/Objective To know what a prime number, square number and cube number is (Continued) To write a number as the product of its primes Examples of questions 16 and 9 are both square numbers. If you add them together, the result is also a square number. Can you find two other square numbers that do the same? Write 60 as the product of its primes What is the largest odd number that divides into 60? Write 132 as the product of its prime factors, using index notation if necessary. Hence write 132 x 6 as a product of its prime factors, using indices Find the HCF of 48 and 36 Find the LCM of 48 and 36 Two numbers e and f are written as a product of their prime factors below: e = 22 x 34 x 5 x 73 f = 24 x 5 4 x 7 To collect like terms in an expression Find the HCF of e and f Simplify the following expressions: a) b) c) d) e) To simplify algebraic expressions, including using laws of index notation 2x + 3y + 5x + 4y 8p – 3q + 10p – 6q 5x + 7y – 3y + 8x – 4y 8x2 + 9x + 3x2 + 2x 3m – 9 + 2y + 16 – 15m – 3y Simplify: 2a x -3b 2a2 x 3a3 7|Page Maths Common Entrance revision Year 8 Topic/Objective To multiply out brackets and simplify Examples of questions Multiply out the bracket 3(4e - 5) Simplify 10 - 3(4e- 5) NB: REMEMBER NEGATIVE RULES!!! To factorise an expression To solve linear equations Multiply out the bracket and simplify 7p - 2(3p - 4) Multiply out the bracket and simplify 8x- 5(3 - x) Factorise fully: 12p + 18r 8g + 20 24 - 8y Solve the following equations: 2b - 7 =15 ¼ a = 12 3(2c + 1) =0 2 - d = 10 + 3d 3f - 2 = f +11 To form an algebraic expression and solve a related equation. 3(2y + 5) = 3 Mr and Mrs Ode have three daughters, Ann, Cath and Di. Let a represent Annʼs age in years. Cath is twice as old as Ann. (i) Write down Cathʼs age, in terms of a. Di is 2 years younger than Ann. (ii) Write down his age, in terms of a. The total of the ages of the three girls is 22 years. (iii) Form an equation, in terms of a, and solve it to find the value of a. (iv) What will the total of the girlsʼ ages be exactly 3 years from now? 8|Page Maths Common Entrance revision Year 8 Topic/Objective To substitute into an expression To use ratio and proportion to answer problems Examples of questions Given that x=3, y=-1 and z=4, find the value of: i) 3x + 2z ii) x2 - y2 iii) xyz iv) y+z y If c=5, d= -3 and e= -1, find the value of: i) 3c + d ii) d–c iii) (2d)2 iv) d+e d-e A formula used in physics is v = u + at Find v when u = 10, a = -2 and t = 8. To make custard for 4 people, Claire uses 300 millilitres of milk 2 eggs tablespoon of vanilla essence 80 grams of caster sugar (i) How much milk does Claire need to make custard for 10 people? (ii) Claire has plenty of milk, eggs and vanilla essence, but only 280 grams of caster sugar. If she uses all the caster sugar, for how many people can she make custard? Ianʼs packet of sweets contains only toffees and fruit centres in the ratio 3 : 5 There are 24 toffees in the packet. (i) How many sweets are there altogether in the packet? Ian eats 4 fruit centres from his packet of sweets. (ii) What is the ratio of toffees to fruit centres remaining in the packet? Give your answer in its simplest form. 9|Page Maths Common Entrance revision Year 8 Topic/Objective To use ratio and proportion to answer problems (continued) Examples of questions A large packet of sweets contains 50% more sweets than a standard packet. What is the ratio of the number of sweets in a large packet to the number of sweets in a standard packet? Give your answer in its simplest form. To know the sum of angles in a triangle is 180⁰ What is the size of each angle in an equilateral triangle? To know the properties of a right angled, isosceles and equilateral triangle. To know what vertically opposite angle are and that they are equal. To know what corresponding angles are and that they are equal. To know what alternate angles are and that they are equal To know that interior angles (supplementary angles) add up to 180° To know the sum of angles on a straight line is 180⁰ and around a point is 360⁰ 10 | P a g e Maths Common Entrance revision Year 8 Topic/Objective Examples of questions To use all of the above angle facts to calculate missing angles 11 | P a g e Maths Common Entrance revision Year 8 Topic/Objective To find the probability of an event occurring (and not occurring) To complete a possibility space and find related probabilities Examples of questions When a fair six-sided die is rolled, what is the probability of (a) scoring 4? (b) scoring a square number? (c) not scoring 6? Tim rolls his die 60 times. How many times would you expect him to score 4? Complete the table below to show all the possible outcomes. What is the probability that (a) the number on Hayleyʼs card is higher than the number on Jackʼs card? (b) both cards show a multiple of 3? Given that Hayleyʼs card shows a square number, what is the probability that the two numbers have a sum of 10? To find the mean, mode, median and range of a set of data. Find the mode of the following sets of data: A) 3, 2, 7, 6, 3 B) 4, 3, 0, 5, 0 C) 5, 6, 10, 6, 9, 3, 5, 4, 2 D) 2, 9, 7, 6, 5, 4 Find the range of the following data: A) 9, 3, 8, 7, 24 B) 7, 6, -3, 4, 2 C) 6, 3, 0, 10, 21 D) -5, -9, -20, -3 12 | P a g e Maths Common Entrance revision Year 8 Topic/Objective To find the mean, mode, median and range of a set of data (Continued) Examples of questions Find the median of the following sets of data: a) 3, 2, 7, 6, 9 b) 5, 10, 3, 7, 6, 12 c) 16, -2, 9, 8, -4, 7, 20 Find the mean of the following sets of data: a) 7, 3, 2, 6, 7 b) 6, 0, 0, 4, 5 c) -6, 2, 7, -3, 5 The eight swimmers in a 25 metres freestyle race recorded the following times in seconds: 16.5 21.2 17.5 15.9 17.7 18.3 17.5 17.8 To find the mean, mode, median and range from a frequency table (i) What was the range of the times? (ii) What was the median time? (iii) Work out the mean time to complete the race. (iv) By how much time did the winner beat the second-placed swimmer? Here is some information about the number of pets owned by each of the 18 children in David’s class. One child owns 7 pets. i) What is the modal number of pets owned? (ii) What is the median number of pets owned? (iii) (a) Calculate the total number of pets owned. (b) Calculate the mean number of pets owned. 13 | P a g e Maths Common Entrance revision Year 8 Topic/Objective Examples of questions To display data in graphical form e.g. pie chart Ma Bakerʼs Coffee Shop sells four different types of cake. One week, Ma Baker sold 90 cakes as follows: 35 were lemon ⅓ were chocolate 10% were ginger the rest were coffee How many degrees will show each cake on a pie chart? Draw a fully-labelled pie chart to show Ma Bakerʼs cake sales. Here is a pie chart showing the results of a survey carried out by Tim. He kept a record of the number of birds in his garden at 9 am each morning for a month. What was the modal number of birds during this month? To continue a given sequence Given that this month had 30 days, on how many days did he record 3 birds? Find the next 4 terms of the following sequences: a) 3, 5, 7, 9, .... b) 2, 7, 12, 17, ... c) 12, -5, -22, -39, ... 14 | P a g e Maths Common Entrance revision Year 8 Topic/Objective To continue a given sequence Examples of questions In this number sequence, the next term is obtained by adding the two previous terms. For example, 4 + 5 = 9 1, 4, 5, 9, 14, 23, 37 . . . (i) Write down the next two terms after 37 (ii) How many prime numbers less than 100 are there in the sequence? Another sequence follows the rule: halve the number, then add 1 (i) Write down the second and third terms if the first term is 62 Using the same rule with a different first term, Suzie noticed that the third term in her sequence was larger than the second term. (ii) Write down a number she could have used as the first term. To find any term of a sequence given its nth term To find the nth term of a linear sequence In each questions find the stated term given the nth term: a) 3n – 2 (12th term) b) 6 – 5n (8th term) c) 4n + 3 (29th term) Find the nth term of the following sequences: a) 2, 9, 16, 23, .... b) -4, -1, 2, 5, ... c) 20, 12, 4, -4, ... d) -13, -18, -23, -28, .... The 11th term of a sequence is 92. Given that the sequence increases in nines, can you write down the nth term? Using this nth term, what is the 19th term of the sequence? 15 | P a g e