Fractions and indices Objectives © Consolidate writing a fraction in its © Use the order of operations, including simplest form by cancelling common brackets and powers. factors. © Use positive, negative and zero indices © Add, subtract, multiply and divide and the index laws for multiplication fractions, interpreting division as a and division of positive integer powers. multiplicative inverse, and cancelling common factors before multiplying or dividing. What's the point? Fractions are useful in mary different ateas of life. We can use fractions when decorating. We could mix a tin of white paint with a quarter of a tin of red paint to make pink, or we may use two thirds of a tin of paint to decorate a room. We can use fractions in farming. farmer could use three quarters of his land for crops and one quarter for animals. PCH elt) ad You should know ... Check in 1. Fractions ean be simplified by dividing. 1 Simplity For example: i 16 20 B 40 s ai . 3 5 96How i conven between mixed number and improper fractions, For example, W pseank8 + dremaneeraeen 1 _ 3X4) + L Gt o3p= B The meaning of indices, For example: 5 X 5 = Sorfive squared = 25 2% 2% 2=2' ortwo cubed = 8 PX PX pX p& p= posing index notation 1.1 Working with fractions Before you start working with fractions you need to be comfortable cancelling fractions by dividing numerators and denominators by common factors. You also nied 10 be able to convert between mixed numbers and improper fractions, Equivalent fraction puzzles 4am equivalent to 2 ancl my numerator s.a prime number What am I? 2: Tram equivalent to ©; and my denominators a prime numer. Wht am 1? 3 [am enuvalent to and my numerator and denominator are both arime numbers. What ami? 4. am equivalent to and the product of my rmierator and derominetoris 218, What ara '? 5) (af equvalentte 24 ane! as er mproper fraction the sum of my numerator and denominator is 40, What am 1? 6 am exuvelent to £2: and my denominators less than 20. What am |? 7am equivalent t 24 and as an improper fraction ry mumetater ts 9 more then ty denominator What arn? 8 | am equivalent to ® and my numerator s 72 jess than my denominator. What am I? 8 Jam equivalent to 2 and my numerator is smaler than 20 anda multiple of 3. What am 12 40 a When wing a fraction in its simplest ini ey ar rete rete denominator by a common factor, either the numerator or denominator or both wil be prime. Tue or false? What are the first eight prime numbers? hese puzzles will help you practise these skills. ‘Change these improper fractions to mixed numbers: 3 ' 20 Change these mixed numbers to improper fractions i a i 32 iti 5} Calculate: 12 wey th Writey X yX yX yX pK yX p using index notation, Adding and subtracting fractions Fractions with common denominators are easily adkled or subtracted For example: 3,1 5 L 5 Fractions without common denominators need to be changed to equivalent fractions with the same denominators. Addition and subtraction of mixed numbers is done in a similar manner.Exercise 1A 2 Work out: + e ele ein RI ein wie le 2 Caleulate: e Heit 2 a e 8-33 f beat ae Las mess bh 54-15 y dee jl aot 20 5 Write down three fractions, with different denominators, that add up to make 435. ‘Aittesh boughs ka or oranges He ga ha sister 12kg. What was the mass of the oranges he had left? 7A plank of wood is 3m in length, How long 201 willit be i'l curgmof wood from i? A F 8 Karen spent of her money on Monday. 474 She spent § of her money on Tuesday. What 5 fraction of money did she 9 spend on Monday and Tuesday? b have lett?” 9 A container holds litres of water. 1 3 How much water is left in the container if 2 Jerry drinks 2 litres? 7 4 Fill in the missing numbers: 9 a o+s-2 1 1 oon b 4-04 a= 84 jae (Ea Find the value of the letters in the following: pe a +2h=45 b b+ =at M2 ‘ 23 € 6g+ = 8 ech gromehes eteelet 2 42 a8 las 2 ls 5 it(Ona ‘unt fraction isa froction with a numerator of 2, for exemple $3, 3 ‘The ancient Egyptians didn’t write fractions with a riumetator greeter than 1. instead they would wite thenv aaa rior Bo or tions afer fractions. For example they wouldn't wrt 2 instead they would wnte # a8 2 + / (They wouldnt use 5+ Las ns involves repeating a uni ration ) investigate aitfeent non-unt fractions. Can ai nonunit fractions be writen as a sum of diferent unt fractions? lying and di \g fractions : To multiply fractions you just need to multiply the ‘Simplify again (these. simplifying steps can all be done in ore ge) riumeratons and the denominators For example: 2,.2.2%2_4 3°5 3xKS 15 13 3,3 3%3_9 XQ" 2%4" 2x48 You know that 3? means 3 x 3; so-we square fractions in the same way Often you can simplify the fractions, by cancelling ‘common factors, before doing the multiplication For example: (Change to improper fractions, Nh 4 2x7_ 14 sxa 15 ‘You can also uise the idea of cancelling when you are Write a5 a mixed number ‘multiplying more than two fractions together. Division of fractions is a little more tricky. Change: to improper Remember: fractions. 9=329x B+ 2=8x 6} _ Notice thatthe operation 1 © Fisthe same as X 5 the same as % 3 1 £ = Lis the same as xIn general, division by & number is the same as multiplying hy the number's inverse or reciprocal. Change unproper fractions, wt (enti) L (1 Lina (bx ; ‘ 2-1) 4 (3,4. Ba (xder Using this idea, all divisions of fractions can be turned into multiplications. The order of operations (BIDMAS) aso applies 10 fractions. Remember in calculations we do: Brackets first (note long dividing lines act like brackets) Lt ‘Then Indices (or powers) p ixt Then Division and Multiplication a Then Addition and Subtraction a ose 4°03 a4 1 7K Calculate: Tis long dividing line acts like a] ee : eased ialng ub 3 te ab 7s () [meron and denniaton fest axa bp eux 4 eax 4 g3ixa on3 Calculate: 7) Calculate: sph oon} “ae 8 Or 9G on. ens hs A pocket of Bacal ima a mast of Ex, Jason eats $ of the packet. What is the mass of biscuits left?” 9 Arectangle measures 4} em by 25 cm, What is the urea of the rectangle? abem 2 3, 2),1 : foe (@ o 8) +70 [£0 The area of a rectangular field is 1305 m*, ‘What is the field’s length if its width is 105 m?=o - PS) reciworoor Need help with fractions? Visit the website ‘www.cooimat.com ‘There is a complete course on fractions in the precigebra section. Choose the lessons you need further help on. Don't forget to do the questions! tre bottles of washing-up liquid are taken froma container holding 1644 litres, | (3>) INVESTIGATION i pie be (AES SERENE ‘Scientific calculators usually have a fractions key a which looks like: b When 75 bottles have been removed from “8 the container, what fraction of the original Examples of how to use it are as follows: amount of liquid remains in the container? FQ Work out a : wsimnis $.tevin( 3 (5) EIS) 2 5 You should see an answer of c 8 To do 43 x 2, you need to work out how to type mixed numbers into your calculator, or change 44 ee 1 @) =e) Houshould soe an answer of 2 or 22 or 3:7, {Some calulators willet you set up whether the answer wil be improper, @ mixed number ora decimal ~ see what yours ean do,) Or you could do 4! x 8 ae: ° ® (=) b xO Your calculator will add and subtract fractions too, a Lyeo ‘and you ean use calculator brackets. 3 ‘There are many different makes of calculators. @ There are many possible answers to Find out how to do calculations with fractions on part €. Find some more answer pairs and (youre: emo iced 1 ook ae vs stisetiore ‘write down the connection between them book, or you can ask your teacher. Repeat some of the questions from Exercises 1A and 18 using your calculator. (OMIM ‘a Find as many pairs of fractions as you can witn a product of 3 Ib) Fie ae inary pera ot Rector oa you aot when aided ged1.2 Indices A short way of writing 3K 3X3xX3i83 is 3 raised to the power 4, In the same way, 4° is 4 raised to the power 3, 4x4 xd = 6b ‘The power is also called the index. Notice that PX Pa dKAKAKEXS That is, 43 x 4? = 474? = 4 © So long as the two stumbers are powers of the same number you can multiply them by adding their indices. U nbols, Work out a x2 e 1xt x x x Notice also that axexe # aKa # ‘That is 4? + 4? = 43-2 = 4! © When two numbers are powers of the same number, you can divide them by subtracting their indices. Using symbols, Work out aves Exercise 1¢ 2 Copy and complete: a axax b 5K5x ... © a e f 8 3 h 10000 = 10 x 10 x 1 1000.00 = 10 10x 2 Copy and complete: Ge ae a B= 2x ai s a = PRI=TETX a= 5 Moe 3 Simplify: a @xGx6 b 7x7x7? © 3x3! x35 d 10 x 10x 10° a b © a 4 Simplify these, if possible, leaving the answer in index form. IF itis not possible, explain why. @ se2? b 2x3 © Pxaxs d axatxd 5 Copy and complete: 3 _3x3x a 33x gS} _5x5Sx.. SSX Goa qe Tage 6 Simpiity 2 a "3 “> 70 a = z f = eg ro S impli leavina yourmnewerin inte Tan a e+e b x45 12° + 12? mop ce d e 20° = 20% f g eee h i i k g+q 8p) = 2p! 10m! = 2aZero and negative indices What does 5° mean’? 5's 9° = 25+ 25 =1,bu Using symbols, d= Indices can also be negative, For example, = 8X 6X6 1 OT EK OX6XOK6 OXO but 6? = 6° = 6° = 67 Using the order of operations, cluding brackets and powers The order of operations (BIDMAS) also applies to indices. Remember, in calculations we do: Brackets first (note long dividing lines act like brackets) ‘Then Indices (or powers) ‘Then Division and Multiplication ‘Then Addition and Subtraction Simplify: 16 x 2° 5 1 16x29 = 16x =16x4=2 EXAMPLE 9 (x 3"? KS Simplity This long dividing line acts like brackets. Work out the pumerator and. devominator First. (brackets on the vumerater are worked. out first Exercise 1D 1 Copy and complete: a Ps? b 3+ 3° oc fer do! 2 G1 a 9} 2 Write down the value of: a 3 b 7? © (2) a st 3 Copy and complete: 4x a #aa°4) Write as a fraction: a7! b 3° ce 4! ad 3! e 5? £35 5S Write in index form: 1 1 1 ae > 2 © 1 1 1 @ 00 Oe ' 3 Find) a S85) b gis By considering 4” + 4°, show that 4 By considering 9 + 9°, show that 9° enor Look carefully at the first two examples below. ‘Then copy and complete the list. oO. =i 10 -t om oi = 7h = 10 lls 0.001 = 7000 Oo 90001 = 0 =O 0.00001 = O =O | (>) INVESTIGATION A3 x3 % 3 cube made up of 27 smaller cubes is painted green on its outside. How many of the smaller cubes ‘are unpainted? How many of the smaller cubes have 1 face painted? 2 faces painted? 3 faces painted? What about a4 4 x 4 cube made up of 64 smaller cubes? Copy and complete the table, Gan you spot any rules? What about a 13 x 13 x 13 cube? Exercise 1E 1A piece of ribbon is 50cm long. How many 11 cat lange of txbomcexb cit fom? How much ribbon is left over? 2 Twalk [km to school, then 2 km to the store, then 1) km back home, How far do I walk altogether? 3 On each bounce a ball rises to 3 ofits height at the start of the bounce. To what height will it rise afier the fourth bounce, if it was originally dropped from a height of 81 em? 4 A bottle of juice holds 2} litres. How many: glasses of juice ean be poured from the bottle ifeach glass holds 2 litre? 5 A }itre jug is filled with milk, Its used to fill, ipreusine tcltine! Ueetueniher’ tire ‘How much mitk remains in the jug? 33-3 6 Calculate the exact value of 7 a How many quarters in 16? How many thirds i © What isa tenth of 10? What is three tenths of 3? 8 a Find atenth of three quarters of $800. b Find three quarters of a tenth of $800. 9 Leon has made a mistake in his homework. The question ie: 22 x 35 Leon has written’ ‘What mistake has he made?Example 1 x2 _ QP _ 2 Work out: oF ee eat 2 9 =5+ Exercise 1 =s+ A. Work out: 1,2 44 oa Bg a y+z ob g+t 6 ate =5'+ 29 =5 62 _ 2 eo 6) - fea g 8 6— 2 c 6,2 es alae o8 Example 2 ay Work out = “3 a" s 4 Simplify: a 2x b st=s cea dg Pxaxe ” eae . oxo (@ x 6°) (x 8°)? Example 3 a easy winplite soa 5 A piece of cloth is4 m long. Amandeep used a YK? 5 LSS FoF it, What length did he use? ¢ 3 @x2y 6 It takes 33 hours to plastera wall, One third of a this time is spent mixing the plaster. How long does the actual plastering take? a UxPa7= 3" 7 Janice bought 6; metres of cloth. She used 1 metres. What fraction did she use? 4? x 4! 7 b 7 7ofcloense prliens Ibe bees ‘How many bags will be needed? 9 Atinholds 5} How many 3-litre cans can be filled from it? tres of oll 40 A piece of wood is 45 metres long. How many 1)-metre strips can be cut from it? 2A man’s work day is 8; hours. He spends 3{ hours dealing with customers and the remaining time working in his office. a What fraction of his work day does he spend with customers? b What fraction of the whole day is he in his office? (Hint: think about how many hours there are ina whofe day.) You should know ... A. How to add and subtract fractions. For example. 3,1_9,4 473 > 12" 1 How to multiply and divide fractions. For example: How to multiply and divide with indices, and about the zero index. For example: aoa i) 96 6 =6 42 A man spends ; of his weekly wage on food and ; on rent. If he spends $200 on food, how much does he spend on rent? (23) What are the numbers p, g, rand s if a Y= 41= 16 = 256 b 3°=9'= 81? = 6561 14 Yafeu says that if he starts with a number and squares it he ends up with the same number he started with. What could his number be? Is there more than one answer? Check out 2 Work ow Work these out, Write your answers in index form. a Pxe b sx co Px d xe eo se f 345 g2+e 1?