3 FRACTIONS. Before you start. Objectives

Transcription

1 FRATIONS Only one eighth of n iceberg shows bove the surfce of the wter, which leves most of it hidden. The lrgest northern hemisphere iceberg ws encountered ner Bffin Islnd in nd in 1. It ws 1 km long, 6 km wide nd hd height bove wter of bout 0 m. It hd mss of over 9 billion tonnes enough wter for everyone in the world to drink litre dy for over four yers. Objectives Before you strt In this chpter you will: dd, subtrct, multiply nd divide frctions nd mixed numbers find frction of quntity solve problems involving frctions. You need to be ble to: find the highest common fctor (HF) of two numbers find the lowest common multiple (LM) of two or three numbers simplify nd order frctions convert between improper frctions nd mixed numbers.

2 .1 Adding nd subtrcting frctions nd mixed numbers.1 Adding nd subtrcting frctions nd mixed numbers Objectives You cn dd nd subtrct frctions. You cn dd nd subtrct mixed numbers. Why do this? Mesurements re not lwys given in whole numbers. You my need to find the totl length of two distnces given s frctions, for exmple, _ km nd 1 km. Get Redy 1. Write 6 in its simplest form.. hnge _ to n improper frction.. hnge to mixed number. Key Points To dd (or subtrct) frctions, chnge them to equivlent frctions tht hve the sme denomintor. This new demonimtor will be the LM of the two denomintors (see Section 1.1 for LM). Then dd (or subtrct) the numertors but do not chnge the denomintor. To dd (or subtrct) mixed numbers, dd (or subtrct) the whole numbers, then dd (or subtrct) the frctions seprtely. Exmple Exmple Work out _ The LM of nd is. onvert _ to the equivlent frction with denomintor of. Subtrct the numertors only. Work out _ 6 Give your nswer s mixed number. Exminer s Tip If you could use severl numbers s the new denomintor, using the LM of the two denomintors mens you won t need to simplify lter The LM of 6 nd is 0. hnge ech frction to its equivlent frction with denomintor of 0. Simplify the frction. Divide ech number in the frction by. onvert the mixed number to n improper frction. 1 1 reminder. equivlent frctions mixed numbers 9

3 hpter Frctions D Exercise A Give ech nswer s frction in its simplest form. 1 Work out b 9 _ 9 c 1 1 d 1 Work out b c _ _ d _ 9 e _ f _ 9 g _ 9 6 h 0 _ Work out b _ c _ _ d _ 9 _ e _ f _ g 1 0 _ h _ 6 _ 9 Give ech nswer s frction or mixed number in its simplest form. Work out _ _ b _ _ c _ 6 _ e f _ _ g _ 6 9 Questions in this chpter re trgeted t the grdes indicted. d 9 h _ 9 _ 6 Exmple Work out _ 9 _ Add the whole numbers. onvert the frctions into equivlent frctions with denomintor of. 1 is n improper frction. hnge this into mixed number. Simplify. _ Exercise B 1 Work out 6 _ 1 b _ c _ 6 d 1 _ Becky cycled _ miles to one villge then further miles to her home. Wht is the totl distnce tht Becky cycled? A bg weighs _ lb. The contents weigh 1 lb. Wht is the totl weight of the bg nd its contents? 0

4 . Multiplying frctions nd mixed numbers Exmple Work out Method 1 _ will give negtive result. 0 Write 0 s Method _ onvert the mixed numbers to improper frctions. onvert the improper frction to mixed number. 1 0, reminder. Simplify the frction. Exercise 1 Work out 1 b _ 1 c 6 d _ Work out 1 b 1 _ c _ 1 _ d 9 _ A box contining vegetbles hs totl weight of _ kg. The empty box hs weight of 1 kg. Wht is the weight of the vegetbles? A tin contins _ pints of oil. Julie pours out pints from the tin. How much oil remins?. Multiplying frctions nd mixed numbers Objectives You cn multiply frctions. You cn multiply mixed numbers. You cn find frction of quntity. Why do this? Shops often dvertise discounts s _ off the norml price. To work out the discount you will need to multiply by _. Get Redy 1. Work out.. Work out 9.. hnge _ to n improper frction. 1. onvert to mixed number. 1

5 hpter Frctions Key Points To multiply frctions: onvert ny mixed numbers to improper frctions. Simplify if possible. Multiply the numertors nd multiply the denomintors. Exmple Work out _. Write s n improper frction. Write 1 s mixed number. Exmple Work out _ of 9 metres. 6 To find the frction of quntity, multiply the frction by the quntity _ metres Simplify by dividing the numertor nd denomintor by. Exmple Work out Simplify by dividing the numertor nd denomintor by. Simplify by dividing the numertor nd denomintor by. Exmple Work out _ 1 _ onvert ech mixed number into n improper frction. Divide the numertor nd denomintor by. onvert the improper frction into mixed number. improper frctions

6 . Dividing frctions nd mixed numbers Exercise D 1 Work out _ b _ e _ _ f _ _ c 11 _ d _ 6 1 g h 6 0 Work out b c 9 0 d _ Work out _ of kg b _ 9 of 1 m c _ of 1 litres d of pints Jomo delivers 6 newsppers on his round. On Fridys _ of the newsppers hve mgzine supplement. How mny supplements does he deliver? Brry erns.60 in one week. He pys of this in tx. How much money does he py in tx ech week? 6 Work out 1 b 1 _ c _ 1 1 d 1 _ e 1 f 1 g 6 _ 1 _ 9 h Kiern tkes minutes to complete one lp t the Go Krt entre. How long will it tke him to complete 6 lps? A melon weighs lb. Work out the weight of melons.. Dividing frctions nd mixed numbers Objectives You cn divide frctions. You cn divide mixed numbers. Why do this? A crpenter my wnt to work out how mny pieces of wood mesuring _ m he cn cut from m piece of wood. Get Redy 1. Work out _ _.. Work out _.. Write _ s n improper frction. Key Points To divide frctions: onvert ny mixed numbers to improper frctions. onvert divide to multiply nd invert the second frction (inverted mens turned upside down). Multiply the top numbers nd multiply the bottom numbers. inverted

7 hpter Frctions Exmple 9 Work out _ Multiplying by _ is the sme s dividing by. _ is clled the reciprocl of. 1 Write the whole number s n improper frction. _ 1 Exmple Work out _ 6 _. Give your nswer in its simplest form hnge to nd turn the second frction upside down. Multiply the frctions. Turn upside down to get. Divide the numertor nd denomintor by. 1 _ Write the improper frction s mixed number. 9 Exmple 11 Work out _ _ Write the mixed numbers s improper frctions. Turn 1 upside down to get 1. Divide top nd bottom by nd by. Write the improper frction s mixed number. Exercise E 1 Work out _ 6 b _ c _ e f _ g 0 d 9 16 _ 1 1 h 1 16 Work out b _ 1 c _ 1 _ d 6 _ _ 9 e 1 _ f _ 9 _ g 1 h A tin holds litres of methylted spirit for lmp. How mny times will it fill lmp holding litre? A metl rod is _ metres long. How mny short rods metre long cn be cut from the longer rod? Tr nd Stone cn resurfce km of rod in dy. How mny dys will it tke them to resurfce rod of length _ km?

8 . Frction problems. Frction problems Objective You cn solve problems involving frctions. Why do this? A vet my need to work out frctions of dosge depending on the size of the niml in comprison to the stndrd. Get Redy 1. Work out _ _. Work out 9 _. Work out Key Point You cn use your knowledge of frctions to solve problems from rel life. Exmple 1 In cinem _ of the udience re women, of the udience re men. All the rest of the udience re children. Wht frction of the udience re children? A0 A0 _ Add nd _ to find the frction of the udience who re women or men Subtrct 1 from 1 to find the frction 0 of the udience who re children of the udience re children. Exmple 1 A school hs 100 pupils. 60 of these pupils re girls. of the girls like swimming. of the boys like swimming. Work out the totl number of pupils in the school who like swimming. A0 A Work out the number of girls who like swimming. Work out the number of boys in the school. Work out the number of boys who like swimming. Work out the totl number of pupils who like swimming. 1 pupils like swimming.

9 hpter Frctions Exercise F D 1 Simon spends of his money on rent nd of his money on trnsport. Wht frction of his money does he spend on rent nd trnsport ltogether? b Wht frction of his money is left? _ 9 of n iceberg lies below the surfce of the wter. The totl volume of n iceberg is 990 m. Wht volume of this iceberg is below the surfce? DVDs re sold for 1 ech. _ of the 1 goes to the DVD compny. How much of the 1 goes to the DVD compny? AO AO AO AO An MP plyer usully costs. In sle ll prices re reduced by _. Work out the sle price of the MP plyer. A fctory hs 1 workers. 60 of the workers re femle. _ of the femle workers re under the ge of 0, of the mle workers re under the ge of 0. How mny workers in totl re ged under 0? 6 There re 6 students in clss. Jved sys tht _ of these students re boys. Explin why Jved cnnot be right. Tmmy wtches two films. The first film is 1 _ hours long nd the second one is hours long. Work out the totl length of the two films. of squre is shded. of the shded prt is shded blue. Wht frction of the whole squre is shded blue? 9 Alison, Becky nd rol tke prt in chrity rely rce. The rce is over totl distnce of _ km. Ech girl runs n equl distnce. Work out how fr ech girl runs. In book, _ of the pges hve pictures on them. Given tht pges hve picture on, work out the number of pges in the book. 11 Alex spent _ of his pocket money on computer gme. He spent of his pocket money on sweets. He sved the rest. Given tht Alex sved.0, work out how much pocket money he got. hpter review To dd (or subtrct) frctions, chnge them to equivlent frctions tht hve the sme denomintor. This new denomintor will be the LM of the two denomintors. Then dd (or subtrct) the numertors but do not chnge the denomintor. To dd or subtrct mixed numbers, dd or subtrct the whole numbers, then dd or subtrct the frctions seprtely. To multiply frctions, convert ny mixed numbers to improper frctions, simplify if possible, then multiply the numertors nd multiply the denomintors. To divide frctions, convert ny mixed numbers to improper frctions, convert divide to multiply nd invert the second frction, then multiply the numertors nd multiply the denomintors. You cn use your knowledge of frctions to solve problems from rel life. 6

10 hpter review Review exercise 1 Simplify these frctions 1 1 b c d 96 1 e 1 hnge _ 1 to n improper frction. b hnge to mixed number. Mny wge erners work fixed number of hours, typiclly hours week. If they re required to work more thn this, they re pid overtime. For exmple, overtime pid t time nd hlf would men someone normlly erning per hour would receive 1 per hour for the extr hours. opy nd complete the tble for the following workers. Nme Aron.0 hi 1.00 Mhmood 1.0 Hourly rte 6 Overtime t time nd hlf Overtime t double time A0 Work out _ b _ 6 _ c d 1 Exm Question Report % of students nswered this sort of question poorly becuse they simply dded the numertors nd dded the denomintors. D Using the tble bove find Aron s weekly wge if he works hours t.0 per hour nd 6 hours overtime t time nd hlf. hi normlly works 6 hours week nd is pid for ny overtime t time nd qurter. b One week she erned. How mny extr hours did she work? AO 6 Work out _ _ b _ 6 c _ _ 6 1 m Digrm NOT ccurtely drwn Exm Question Report 1% of students nswered this sort of question well becuse they delt with the integers nd frctions seprtely. 1 m Work out the re of this rectngle. Hint: Are of rectngle length width. b Work out the perimeter of this rectngle. c Work out the difference in lengths between the shortest nd the longest side.

11 hpter Frctions Digrm NOT ccurtely drwn AO 1 cm cm The digrm represents prt of mchine. In order to fit the mchine, the prt must be between cm nd 6 cm long. 16 Will the prt fit the mchine? You must explin your nswer. June On frm, _ of the lnd re is used to keep sheep. Hlf of the rest of the lnd re on the frm is used to grow crops. The lnd re of the frm used to grow grss is m. Work out the lnd re of the frm used to keep sheep. June 009 The distnce from Grnby to Hightown is _ miles. The distnce from Hightown to Islely is miles. Jim wlks from Grnby to Islely vi Hightown. He stops for rest when he hs wlked hlf the totl distnce. How fr hs he wlked when he stops for his rest? 11 A 1 x 1 B 1 x D 1 Here is design for book cse with two shelves. All the mesurements re in inches. The gp AB is the sme s the gp D. Work out the vlue of x. b In nother design the bookcse is the sme except the middle shelf hs been moved so tht the gp AB is twice the gp D. Find the size of the gp AB. AO 1 1 _ 1 c b 1 d 1 This is mgic squre. The sum of the three numbers in ech row, ech column nd ech digonl is the sme. Work out the vlue of, b, c nd d. AO 1 A scientist wnts to estimte how mny fish there re in lrge pond. He ctches 0 one dy, tgs them nd puts them bck into the pond. Next week he gin ctches 0 fish, of which re tgged. Estimte how mny fish there re in the pond. 1 There re 960 pupils in school. _ of the pupils re in lower school. of the pupils in the lower school re girls. 1 Work out the number of girls in the lower school.

12 hpter review 1 The digrm shows squre ABD. The points E nd F re the midpoints of sides AD nd D respectively. Wht frction of the squre re the tringles: ABE b DEF c BF d BEF? A B AO B E D F 9