KS3 Mathematics - 10 4 10 level 6 - Answers

Sheep and LambsOn a farm 80 sheep gave birth.30% of the sheep gave birth to two lambs.The rest of the sheep gave birth to just one lamb.In total, how many lambs were born?Show your working.30%2 Lambs10% of 80 sheep = 8 sheep30% of 80 sheep = 8 x 3 = 24so24 sheep have 2 Lambs whichmakes 48 Lambs altogether.ANSWERALTOGETHER48 + 56 = 104 LAMBS70%1 Lamb10% of 80 sheep = 8 sheep70% of 80 sheep = 8 x 7 = 56so56 sheep have 1 LambEquationsy121 08642BCA0– 4 –2 0 2 4 6 8 1 0 1 2 x– 2– 4Da.) Fill in the gaps to show which line has which equationLine ………E…….. has equation x = 10Some points/coordinateson line C are: (2,2); (10,10);(-3,-3). Notice that thefirst coordinate is thesame as the lastcoordinate. This meansthe x coordinate is thesame as the y coordinate x= y; y = xESome points/coordinates on line A are (0,10);(4,10); (8,10) y coordinate stays the same.These straight line graphs all pass through the point (10,10)This question is about graphsY = m x + cGRADIENTLine ………A…….. has equation y = 10Line ………C…….. has equation y = xSome points/coordinates on line E are(10,1); (10,6)(10,10) x coordinate stays the sameLine ………D……. has equation y = 3 x – 52Line ………B…….. has equation y = 1 x + 52This is the yintercept, thepoint at whichthe straight linemeets the y axisb.) Does the line that has the equation y = 2x – 5 passthrough the point (10,10)?Explain how you knowUsing the coordinate pair (10,10); x = 10 and y = 10.In the equation y = 2x – 5 we can replace y with 10 and x with 10 to give10 = (2 x 10) – 5 so the equation becomes 10 = 20 – 520 – 5 = 15 this is not equal to 10 so the equation does not passthrough the point (10, 10)

Mental Arithmetic QuestionsKS3 MATHEMATICS10 4 10Level 6 AnswersDay 51. What is three-fifths of forty pounds?One fifth (£40 ÷ 5) = £8Three fifths (£8 x 3) = £242. The longest bone in the human body is in the leg.The average length of this bone in a man is fiftycentimetres. In a woman it is ten per cent less.What is the average length of this bone in awoman?10% of 50cm (50 ÷ 10) = 5cmWoman’s bone = 50cm – 5cm= 45cm3. Using three as an approximation for pi, what isthe area of a circle with radius five centimetres?Area = 3 x 5²= 3 x 25= 75 cm²Learn thisArea = ∏r²4. I am thinking of a two-digit number that is amultiple of eight. The digits add up to six.What number am I thinking of?Multiples of 8: 8, 16, 24, 32, 40Which digits in each multiple add up to 6?Answer = 245. I am thinking of a number. I call it n. I add five tomy number.Write an expression to show the result.Answer: n + 5

Fractions(a) Add 106 and 56Thisbecomes3 + 65 5Both fractions must have thesame DENOMINATOR6 = 310 5Now use an arrow (↓) to show the result on the number line.0 1 2(b) How many sixths are there in 3 31 ?(c) Work out 3 31 ÷ 65Show your working.561525Notice the line is scaled in fifths3556455655657556859 = 1 45 595How many sixths are there in 1whole?6or 6 sixths in 1 whole6so 18 sixths in 3 whole.PuzzleYou can often use algebra to show why a number puzzle works.Fill in the missing expressions.This is an examplewith numbers.Example:591475Think of a numberAdd 4Now add the number youwere first thinking ofDivide by 2Subtract 2The answer is the number you were first thinking ofAlgebra:nn + 4n + 4 + nsimplified to2n + 4(2n+4) ÷ 2simplify ton + 2n + 2 - 2simpli fy tonTakethe lastanswern+4 andadd nNOT 5Take thelast answer2n+4 anddivideeverythingby 2Previous answer isn+2; now subtract2 to give n+2-2.Simplify to givethe answer nNOT 5-Remember 5 isjust an example0 1 2 3 3⅓Answer = 4

Mental Arithmetic QuestionsKS3 MATHEMATICS5% = 810% =1610 4 10 100% = 1601. Five percent of a number is 8. What is thenumber?Level 6 AnswersDay 62. A fair spinner has eight equal sections with anumber on each section. Five of them are evennumbers. Three are odd numbers. What is theprobability that I spin an even number?Probability = ⅝5 even numbers on the spinner8 equal sections of numbers on spinner3. I can make a three-digit number from the digitstwo, three and four in six different ways. Howmany of these three-digit numbers are even?234 324 423243 342 432 Even in bold4. What is the volume of a cuboid measuring fivecentimetres by six centimetres by sevencentimetres?Volume = length x breadth x height= 5cm x 6cm x 7cm= 210 cm³5. What is the remainder when you divide threehundred by twenty-nine?10 x 29 = 290 Therefore remainder is 10(300 – 290)

WheelchairWyn and Jay are using their wheelchairs to measure distances.The circumference (C) isthe distance the wheelmoves in one revolutionWedgesThis door wedge is the shape of a prism.Formula: aArea = (a+b) x h2 c hYou need to learn this formulabFormula:C = π x d(π = 3.142)You need to learn thisformula6.0cm2.5cm2.5cm3.5cmNOT TOSCALEThe shaded faceis called thecross-section.(a) The large wheel on Wyn’s wheelchair has a diameter of60cm. Wyn pushes the wheel round exactly once.(b)Calculate how far Wyn has moved. Show your working.C = π x d= 3.142 x 60= 188.52...................188.52 ............ cmThe large wheel on Jay’s wheelchair has a diameter of52cm. Jay moves her wheelchair forward 950cm.C = π x d = 3.142 x 52 = 163.28Calculate how many times the large wheel goes round.Show your working.Any would beacceptable950 ÷ 163.28 = 5.8182264You need to work outthe circumference first.or. 5.8 or 5.82 or 6............. timesFind how many times thecircumference fits intothe distance forward.(a)(b)The shaded face of the door wedge is a trapezium.Calculate the area of the shaded face.Show your working.(2.5 + 6.0) x 2.52= 8.5 x 2.52= 4.25 x 2.5 = 10.62510.625.................. cm²Calculate the volume of the door wedge.Show your working.10.625 x 3.5 = 37.187537.1875 or 37 or 37.2 or 37.19cm³Any would beacceptableFormula:Volume = area of Cross-section x depthYou need to learn this formula

Mental Arithmetic Questions25% = 750% = 14KS3 MATHEMATICS Therefore 100% = 281. Twenty-five per cent of a number is seven.What is the number?10 4 10 2. There are fourteen girls and thirteen boys in a class.What is the probability that a pupil chosen at randomLevel 6 AnswersDay 7will be a girl?Probability = 1427Total number of girlsTotal number of boys and girls3. The first even number is two.What is the hundredth even number?Answer 2004. The mean of two numbers is 8. One of the numbers istwo. What is the other number?Mean = 8, Total of 2 numbers is 16 because 16 ÷ 2 = 8If one of the numbers is 2 then the other number must be14 (16 – 2)5. How many edges are there on a square basedpyramid?Base has 4 edges (square) it also has 4 vertices which eachin turn join to form the peak of the pyramid (another 4edges. Total number of edges is 8.

Number CardsA newspaper wrote an article about public libraries in Englandand Wales. It published this diagram.James has these four number cards:The mean is 41 8 5 2 (1 + 8 + 5 + 2) ÷ 416 ÷ 4 = 41 8 5 2 ?Number of librariesopen for more than45 hours a weekReduction in opening hours7507006506005505004501988 1990 1992 1994 1996 1998Year(1 + 8 + 5 + 2 + ?) ÷ 5 = 4Bracket must = 2016 + ? = 20 Data on libraries from LISU (Library and Information Statistics Unit)? = 4James takes another card. The mean of the five cards is still 4. Whatnumber is on his new card? 4(b)Tara has these four number cards:You have to workout the mean of thefour cards first.Use the diagram to decide whether each statement below is true orfalse, or whether you cannot be certain.(a)The number of libraries open for more than 45 hours perweek fell by more than half from 1988 to 1998. Explainyour answer.You need 2numberswith adifferenceof 4 and atotal of 2010 3 2 5She takes = 5 another card. The mean goesup by 2. What number is on her new card?(c)10 3 2 5 ?Mean = (10 + 3 + 2 + 5) ÷ 4= 20 ÷ 410 10 10 10 ? ?Show your working.? = 15................................15Ali has six cards. The mean of the six cards is 10. The range ofthe six cards is 4.What are the numbers on the other two cards?(10 + 3 + 2 + 5 + ?) ÷ 5 = 7total must be 3520 + ? =35RANGE is the highestsubtract the lowest.(10 + 10 + 10 + 10 + ?+ ?) ÷ 6 = 10total must be 35(40 + ? + ?) = 60Libraries? + ? = 20As the range is 4, these numberscannot be the sameTrue False Cannot be certain√1988 about 725. ½ of this is about 362.1998 is 500 which is more than 362.Or it only dropped from 725 to 500, it should have dropped toabout 360.(b)In 2004 there will be about 450 libraries open in Englandand Wales for more than 45 hours a week.True False Cannot be certainExplain your answer.You cannot predict because:Data for 2004 is not givenThe trend might changeThere is not enough information given√You only need 1 reason forthe mark. Remember youcannot assume anythingunless it is given on agraph.You only get themark for1themarkexplanation. Makesure the scale is readcorrectly.

Mental Arithmetic Questions1. Multiply 8.7 by 2Think about doublingKS3 MATHEMATICS10 4 10Level 6 Answers8.7 x 2 = 17.42. A bat flies at an average speed of 32 kilometres anhour. At this speed, how far will it fly in 15 minutes?¼ of 32 = 32 ÷ 432 ÷ 4 = 8 km3. Multiply the brackets (2x + 1) (x – 1)Remember 15 mins is ¼of an hourDay 8 (2x + 1) (x – 1)= 2x 2 –x–1OR use amultiplicationgrid(2x + 1)(x - 1)2x 2 - 2x + x - 1= 2x 2 – x – 14. I’m thinking of a number. I call it t. I half it andsubtract five. Write an expression to show the result.t ÷ 2 – 5 or t -525. The first odd number is 1. What is the hundredth oddnumber?1992 x 100 – 1 = 199

EquationsSolve these equations.Show your working.8k - 1 = 15OR use matchednumber line2m + 5 = 10• 16 – 1 = 15, so 8k must equal 16• 8 x 2 = 16, so k must equal 2• 5 + 5 = 10, so 2m must equal 5.• 2 x 2 ½ is 5, so m must equal 2 ½k = ......2(a)Sibling agesPaul is 14 years old.His sister is exactly 6 years younger, so this year she is 8 yearsold.This year, the ratio of Paul’s age to his sister’s age is 14 : 814 : 8 written as simply as possible is 7 : 4When Paul is 21, what will be the ratio of Paul’s age to hissister’s age?RatioWrite the ratio as simply as possible.21-6 = 15 soPaul’s sister is 15years oldPaul : Sister21 (÷ 3): 15 (÷ 3)simplifies to7 : 53t + 4 = t + 13• Partition 3t + 4 to make 2t + t + 4• This gives 2t+t+4 = t + 13• Remove t from both sides thuskeeping the equation balanced.• 2t + 4 = 13• 9 + 4 = 13, so 2t = 9• t = 4 ½m = .......2 ½ .(b)When his sister is 36, what will be the ratio of Paul’s age to hissister’s age?Write the ratio as simply as possible.Paul : Sister? : 36Paul’s age is 6 more than hissister, so his age must be 42.42 (÷6) : 36 (÷6)7 : 6(c) Could the ratio of their ages ever be 7 : 7?Tick ( ) Yes or No.t = .......4 ½YesNo 2(3n + 7) = 8METHOD 1• 2x4=8. This means that3n+7 is equal to 4 and canbe written like this3n+7=4• -3+n=4 which means that3n=-3 i.e. 3xn=-3• n= -1METHOD 2• Multiply everything in thebracket by 2. This gives2x3n+2x7 = 8• → 6n+14 = 8• Subtract 14 from bothsides to give 6n+14-14 = 8-14• → 6n+-6, 6xn=-6 so n = -1Explain how you know.7:7 implies that the ages will be thesame at some point in their life. This isNOT true.They will never be the same age as Paulis always six years older.n = ........- 1

Mental Arithmetic Questions1. Add four to minus five.- 5 + 4 = -1Think of a numberlineKS3 MATHEMATICS2. What number should you add to minusthree to get the answer five?10 4 10-3 + 8 = 5Think of a number line again+3 +5-3 05Level 6 AnswersDay 93. How many nought point fives are there inten?10 ÷ 0.5 = 204. On average, the driest place on earthgets only nought point five millimetres ofrain every year.In total, how much rain would it expect toget in twenty years?0.5 x 20 = 10 mmRemember 0.5 = ½20 x ½ = 105. What is the sum of the angles in arhombus?360°A rhombus is a quadrilateral.All quadrilaterals have anangle sum of 360°

GlassesThere are 60 pupils in a school.6 of these pupils wear glasses.Find the fractionfor ‘wear glasses’.Total angles =360°Take care with decimals.Remember:Light Bulbs• Probabilities must add up to 1.The state of the company’s machines can be:• 0.09 + 0.03 is 0.12 NOT 0.012.(a)The pie chart is not drawn accurately.available for use and being usedWearglasses6 out of 60 = 6 / 601oravailable for use but not neededWhat should the angles be?Show your working.= / 100.93orbroken down.Do notwear glasses(a)The table shows the probabilities of the state of themachines in July 1994.Write in the missing probability.State of machines: July 1994Probability1 – (0.09 + 0.03)= 1 – 0.12Angle for glasses is 1 of 360 = 36°10Angle for no glasses is 360 – 36 = 324(b)……36°… and ……324°…Exactly half of the 60 pupils in the school are boys.From this information, what percentage of boys in this schoolwear glasses?Tick () the correct box below.Available for use, being usedAvailable for use, not neededBroken down(b)0.880.090.03During another month the probability of a machine beingavailable for use was 0.92.What was the probability of a machine being broken down?.... 0.08..........................Can be available and beingused OR available and notneeded.5% 6% 10%20% 50% not possible to tellThe 6 pupils who wear glasses couldall be girls. We don’t have enoughinformation about who wears√Brightlite calculated the probabilities of a bulb failing within 1000hours and within 2000 hours. Complete the table below to show theprobabilities of a bulb still working at 1000 hours and at 2000 hours.Time Failed Still workingAt 1000 hoursAt 2000 hours(c)0.070.570.431 – 0.07 = 0.931 – 0.57 = 0.43

Mental Arithmetic QuestionsKS3 MATHEMATICS10 4 10Level 6 AnswersDay 101. It takes someone one and a half minutes to swimthe length of the pool. How many lengths can Iswim in 15 minutes?10 x 1.5 = 15So 15 ÷ 1.5 = 102. Multiply minus eight by minus three.-8 x -3 = 243. If 4x + 3 = 23, what is the value of x?23 – 3 = 4x20 = 4x20 ÷ 4 = xx = 615 ÷ 1 ½= 15 ÷ 1.5=10 lengths8x3 = 24- 8x3 = -248x-3 = -24-8x-3 = 24OR use matchedline4. I have a fair eight sided dice numbered 12 to 19.What is the probability that I will throw a primenumber?P(prime number) = 3 / 8Remember a prime number hasonly 2 factors, itself and 1.Possible primes are 13, 17 & 19.5. What must I multiply n squared by to get n cubed?n 2 = n x nn 3 = n x n x nso n 2 x n = n 3

HedgingA garden centre sells plants for hedges.The table shows what they sold in one week.PlantsNumber ofplants soldTakingsBeech 125 £212.50Leylandii 650 £2437.50Algebra Pairs(a) Join pairs of algebraic expressions that have the same value when a = 3,b = 2 and c = 6This is very importanta = 3. b = 2. c = 6One pair is joined for you.ab means a x b. SubstituteA = 3 and b = 2 becomes 3 x 2 = 63c means 3 x c.Substitute c = 63 x 6 = 183cab3 c – 2bThis means(3xc) – (2xb)SubstituteC = 6 and b = 2so (3x6) –(2x2)18 – 4 = 14Privet 35 £45.50Hawthorn 18 £23.40Laurel 5 £32.252c + b means(2xc) + b.Substitute c=6and b=2 so (2x6)+ 2 = 142 c + b2a2a means 2 x a.Substitute a = 32 x 3 = 6Total 833 £2751.15a 2 means axa.Substitute a=3, 3x3 = 9a 2a + cSubstitute a=3 andc=6 so 3+6 = 9(a) What percentage of the total number of plants sold was Leylandii?(b)(c)Show your working.Number of Leylandii x 100Total number of plants= 650 x 100833= 78.03%rounds to 78%What percentage of the total takings was for Leylandii? Show yourworking.Total takings for Leylandii x 100 = 2436.50 x 100Total number of plants 2751.15= 88.6%Which is the cheaper plant, Beech or Privet?One beech = £212.50 ÷ 125 plants = £1.70One privet plant = £45.50 ÷ 35 plants = £1.30(so privet is the cheapest)Work out thecost of onebeech plantand one privetplant(b) Draw lines to join any pairs that will always have the same value when a =b = cReplace b with a so ab becomesaa to make a x a = a 2Replace c with aso 3c becomes3aReplace c and bwith a. So 2c +b becomes 2a +a whichsimplifies to 3a2 c +b3ca 2aba + cIMPORTANT a = b = c. ReplaceAll letters with a because theyall have the same value.3 c – 2bReplace c and b with a, so 3c-2bbecomes 3a – 2a = a.This is THE ODD ONE OUT2aReplace cwith a soa+cbecomesa+a whichsimplifiesto 2a