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GCSE: Fractions Skipton Girlsβ High School Objectives: Be able to add, subtract, multiply and divide fractions, whether improper fractions or mixed numbers. Find fractions of an amount and solve problems involving successive fractions of an amount. Teacher Guidance Possible lesson structure: Lesson 1: Equivalent fractions, adding/subtracting fractions. Go > Lesson 2: Mixed numbers, adding/subtracting mixed numbers. Simple multiplication/cross-cancelling. Go > Lesson 3: Multiplying mixed numbers. Dividing fractions. Go > Lesson 4: Fractions of amounts, reverse fractions of amounts. Problem solving involving fractions of amounts. Go > STARTER :: Equivalent Fractions Give an equivalent fraction which is as simple as possible. 2 1 = ? 4 2 6 2 = ? 9 3 4 2 = ? 10 5 45 5 = ? 54 6 Adding Fractions Adding fractions is very simple if the denominators are the same. 2 3 5 + = ? 7 7 7 βI ate 2 sevenths of the pizza followed by 3 sevenths. How much have I eaten?β Therefore what should be our strategy if the denominators are different? ? 2 1 8? 3? 11 + = + = 3 4 12 12 12 ? Use equivalent fractions! Find a number both of the denominators go into (preferably the smallest). More Examples 1 1 2 1 π + = + =? 3 6 6 6 π 3 4 21 20 ππ + = + ?= 5 7 35 35 ππ 5 8 15 16 ππ + = + ?= 6 9 18 18 ππ Check Your Understanding 1 3 1 3 2 π + = + =? 8 4 8 8 π 2 7 6 49 36 ππ ? β = β = 6 7 42 42 ππ 3 1 1 1 1 30 20 15 12 ππ + + + = + + +? = 2 3 4 5 60 60 60 60 ππ N 1 1 π π π+π + = + =? 2 π ππ ππ ππ A Quicker (Mental) Method We can multiply the numerators diagonally. 5 1 40 + 6 + = 48 6 8 Step 1: Multiply the denominators (note: this guarantees you get a number both 6 and 8 go into, but it may not be the smallest!) Step 2: Since the 6 got multiplied by 8, so does the 3. i.e. We are multiplying diagonally. 46 23 = = 48 24 Disadvantages of this method: β’ Because weβre not finding smallest denominator, further simplification may be required. β’ Doesnβt extend to more than two fractions. Step 3: And repeat with the other numerator. ? A Quicker (Mental) Method Another example: 2 3 10 + 27 37 ? = + = 9 5 45 45 Quickfire Questions (in your head!) 3 2 ππ + = ? 5 3 ππ 3 3 ππ + = ? 4 5 ππ 5 1 ππ + = ? 8 3 ππ 1 1 π β = ? 2 3 π 4 2 ππ + = ? 7 5 ππ 7 1 ππ ππ β = =? 9 6 ππ ππ Exercise 1 No calculators! 1 Calculate the following, simplifying your fractions 6 Add the following fractions, giving your result where possible: a c e 2 1 3 3 8 6 5 1 3 1 4 5 4 3 2 + = + = + = 7 12 13 8 27 10 ? ? ? 1 5 5 6 7 6 b d f 1 5 π ππ β β β in terms of any variables given. 1 1 = 10 10 3 7 = 30 5 2 17 = 9 18 ? ? ? a c ? ? 16 1 2 3 16 8 1 4 1 8 ? ? ?167 16 0 a b c ππ πππ ππ ππ ? ? 2 1 1 ππ β + = 3 4 5 ππ 8 3 πππ β 49 = ππππ 37 4 2 8 2 β β β 9 7 7 9 Complete the magic square (where the total of each row, column and long diagonal is the same). ? ? d 1 3 1 8 + + 1 27 + = π π π+ππ ππ ? ? π 2 b π 3 + = ππ π ? 1 ? 3 Identify the missing fraction. 1 1 π 7 11 + β‘ = b a 4+β‘=3 ππ ? 11 15 1 5 π 3 7 1 d + ββ‘= c 12 + β‘ = 36 ππ ? 5 6 2 1 3 5 5 Calculate: 4 2 π 2 π + = N Unit fractions are fractions where the of my bananas are yellow and green. The rest are pink. What fraction are pink? 1 π 1 π ππ = β? ππ = πππ πππ ? numerator is 1, e.g. . Egyptian fractions are a 5 sum of unit fractions where all denominators are different. Can you express each of these unit fractions as Egyptian fractions? There may be multiple ways. 1 π π 1 possibility = + 2 π π 1 π π = + 1 possibility 3 π ππ 1 π π π π = + 2 =possibilities + 4 ππ π π ππ 1 π π = + 1 possibility 5 π ππ 1 π π π π π π π π = + = 4 possibilities + = + = + 6 ππ ππ ππ π π ππ π ππ Do the same for larger denominators. Can you identify when youβll have only 1 possibility? (Note to teachers: I proved it in footnotes of this slide) Mixed Numbers β Improper Fractions Improper fractions are where the numerator is greater than the denominator. Mixed numbers have an integer and fractional part. 1? ?3 4 13 4 We could make up 3 wholes using quarters (because 4 goes into 13 three times). The remainder was 1, so we have 1 quarter left over. Mixed Numbers β Improper Fractions 14 4 β 2? 5 5 17 2 β 5? 3 3 13 1 β 6? 2 2 24 β 7 3 3? 7 41 1 β 4? 10 10 How many thirds does the 4 wholes give you? ? 1 13 4 β 3 3 3 13 2 β ? 5 5 1 15 7 β ? 2 2 2 26 8 β ? 3 3 6 83 11 β ? 7 7 Adding Mixed Numbers To add mixed numbers: 1 1 3 ?5 2 3 +5 =3 + 2 3 6 6 π =π ? π 5 1 π π 2 + 5 = π +?π 6 3 π π π ? =π π π =π ? π Make fractional parts the same. Add whole parts and fractional parts separately. If weβve βoverflownβ into the next whole, we need to carry. Subtracting Mixed Numbers To subtract mixed numbersβ¦ 1 1 2 3 9 β 4 = 9 β?4 3 2 6 6 8 3 = 8 β?4 6 6 5 =4 ? 6 Weβd end up with a negative number for the fractions. Can we borrow a whole? (In the same way we borrow in column subtraction) (Alternatively you can just convert both to improper fractions and subtract as normal) 1 2 π π 7 β3 =π β?π 4 3 ππ ππ ππ π =π β?π ππ ππ π =π ? ππ 2 3 56 β 40 5 7 ππ ππ = ππ β?ππ ππ ππ ππ ππ = ππ β?ππ ππ ππ ππ ? = ππ ππ Check Your Understanding 1 2 π 2 + 142 = πππ? 2 3 π 1 1 ππ 76 β 30 = ππ ? 4 3 ππ Multiplying Fractions Skills to do with multiplying fractions: β’ Multiply simple fractions. β’ Multiply a mixture of whole numbers, proper fractions and mixed numbers. β’ Understand βcross cancellingβ with fractions, including with several fractions where there may be a pattern of cancelling. β’ Find a fraction of an amount (including a fraction of a fraction). β’ Understand how to deal with fractions nested inside fractions. β’ Solve puzzles involving fractions, particularly involving successive fractions of an amount and whatβs left. Multiplying fractions is fairly easy. Simply multiply numerators and denominators. 3 5 15 × = ? 4 7 28 Cross-Cancelling Recall that we can simplify fractions by dividing top and bottom by a common factor. Since numerators are being combined and denominators likewise, thereβs nothing preventing us cancelling any numerator with any denominator. 1 1 5 6 55 π × = ? 11 18 π 3 10 14 π × = ? 21 15 π 1 2 3 πβ1 π ? × × × β―× = 2 3 4 π π We can see the numbers between 2 and π β 1 are common to top and bottom. This leaves just 1 in the numerator and π in the denominator. Test Your Understanding Ensure you simplify your fraction. A 3 5 ππ ? × = 7 11 ππ B 13 20 π × = ? 80 39 ππ C 5 21 π × = ? 14 15 π N 1 2 3 4 5 998 π×π π × × × × × β―× = ?= 3 4 5 6 7 1000 πππ × ππππ ππππππ All numbers between 3 and 998 will cancel top and bottom. This leaves just 1 and 2 in the numerator and 999 and 1000 in the denominator. Exercise 2 1 Convert the following to mixed numbers: 8 3 a 8 β 1 1? β 1 b 7 7 5 5? c 11 3 59 6 2 β 3 3? 5 Bob and Dave go to PizzaScoff. Starting with 35 4 5 pizzas, Bob eats 6 pizzas and Dave eats 8 19 3 β 4 4 4 111 1 β 22 5 5 ? ? d 5 pizzas. How many are left? ππ f β9 ? e 6 2 Convert the following to improper fractions: 1 5 5 32 3 β a 22 β 2 ? b 9 9 ? 2 c 73 β 23 3 4 ? d 35 β 19 5 ?ππππ 1 3 6 The perimeter of a rectangle is 20. If its width is 3 7 3 , what is its height? π ? π π ? 7 Calculate and simplify: 3 16 15 a 4× 5 × 9 =π 3 Calculate the following (leave answer as mixed number) 1 2 3 98 π 1 3 1 3 2 3 × × × β― × = b a 2 + =3 b 5 4 + 2 5 = 8 20 3 4 5 100 ππππ 2 4 4 5 7 9 11 101 1 2 8 1 5 2 N × × × ×β―× c 4 β =3 d 8 β4 =3 = ππππ 5 3 15 2 6 3 1 3 5 7 97 e 7 3 + 10 1 = 17?16 7 3 21 1 7 f ππ ππ g 68 β 17 = ππ ? 4 Calculate (and simplify): 4 3 12 π a 5 × 2 = 10 = π? b c e 10 11 54 55 × × ? ? 33 π = 100 ππ 77 ππ = 27 π ? ? ? ? ? 3 5 ? d f ? 2 3 13 10 5 β 7 4 = 2 20 ? 1 5 1 4 ππ ππ h 39 β 29 = π? ? ? ? 7 24 π × = 6 21 π 18 39 π × = 13 12 π 81 17 ππ × = 51 56 ππ N Simplify: 3 5 4 6 5 7 πβ1 π+1 × × × × × β¦× × 4 4 5 5 6 6 π π After the first fraction, each pair of fractions cancel. This leaves just the first and last fraction, giving π π+π ππ ? Multiplying Mixed Numbers If the numbers are not improper fractions, convert them into improper fractions. Examples: 1 1 π π ππ 3 × 2 = × ?= 2 4 π π π 2 π ππ ππ 4 × 3 = × ?= 5 π π π 2 ππ π ππ 7 ×2= × ?= 3 π π π Quickfire Questions: Schoolboy Error: thinking the 8 answer is . 4 π 2× = ? 3 π 5 ππ ×2= 7 π 6 ? 1 4 π π ππ 2 × = × ? = 3 5 π π ππ 1 3 π ππ ππ 1 × 2 = × ?= 3 5 π π ππ 1 3 π ππ πππ 3 × 4 = × ?= 2 4 π π π Dividing Fractions Mental division 1 1 2 ÷ = ππ ? 2 4 1 5 ÷ = ππ ? 4 1 100 ÷ = πππ ? 3 What if this was phrased as βhow many quarter pizzas go into two and a half pizzasβ? What appears to be the effect of 1 dividing by 4? 1 Or by 3? Dividing Fractions More generallyβ¦ Reciprocate (i.e. βflipβ) the second fraction, and use multiplication instead. 4 2 ÷ 5 3 4 3 π = × = ? 5 2 π 1 1 2 ÷1 2 3 5 4 = ÷ 2 3? 5 3 ππ = × = 2 4 π 3 4 ÷ 7 9 3 9 ππ = × = ? 7 4 ππ 1 2÷3 2 2 7 = ÷ 1 2 2 2? π = × = 1 7 π 4 8÷1 5 8 9 = ÷ 1 5 8 5 ? ππ = × = 1 9 π Test Your Understanding A D 2 3 ×4 5 17 4 = × 5 1 ππ? = π B 1 8÷ 5 = ππ ? C [JMO 1996 A4] Evaluate 1 2 π ? ππ + 1 3 + 1 4 3 6÷1 4 6 7 = ÷ 1?4 π π = × π π 1 5 × β 1 6 Exercise 3 1 Calculate the following, simplifying/crosscancelling where possible. a 4×1=π b 1 1 × 4 = ππ c e g ? ? ? ? 5 π 25 21 ππ × = 7 10 π 1 1 1 ×1 =π 2 3 1 8 4 × = ππ 2 3 d f h ? ? ? ? 3 π 18 11 π × = 55 24 ππ 12 39 π × = 13 8 π 2 3 ππ 8 ×1 = 3 4 π π 1 π e 6 ÷ 42 = π? g 10 ÷ 9 9 10 = πππ ππ ? 1 1 ππ 1 2 π 2 5 3 1 4 4 π π h ÷ = 4 [JMC 1997 Q9] What is the value of 4 3 1β 4 2 β ππ 3 ÷ 1 4 ? 4 × + = ππ ? ? 16 ? ? ? 5 [IMC 2012 Q15] Which of the following has a value that is closest to 0? 1 1 1 1 1 1 A + × B + ÷ C 2 1 × 2 1 β 2 3 4 1 1 ÷ 3 4 1 1 × 3 4 D 2 1 2 3 1 β 3 4 1 ÷ 4 Solution: E ? [Kangaroo Pink 2003 Q15] What is the value of: 1 1 1 1 + × 1 + × β―× 1 + ? 2 3 2003 π π π ππππ ππππ × × ×β―× = = ππππ π π π ππππ π ? ? πππ πππ 7 N1 f 39 ÷ 23 = π? 1 6 π2 + π = π β ππ = ππ ππ [JMO 2010 A1] What is the value of 1 2 3 4 5 + 1 + 1 + 1 + 1 ? Solution: 55 1 3 E d 4 3 ÷ 2 2 = ππ? 1 7 If π = , π = β2, π = 1 determine: ? 1 1 8 6 8 2 [JMC 2010 Q6] Which of the following has the largest value? 1 1 1 1 1 A6÷ B 5÷ C4÷ D 3÷ E 2÷ 2 3 4 5 6 Solution: C c 1 2 ÷ 5 = ππ? 1 10 Calculate 1 3 Calculate the following: 3 π 4 6 ππ a 1÷4=π ? b 7 ÷ 11 = ππ? 1 9 5 N2 Calculate 1 β 1 1β = 1 1β 1 1 1β3 π π ? Embedded Fractions [JMO 2004 A1] Write 1 1 1+3 as a decimal. Thereβs two ways you could deal with this: 1 3 fraction 3 by denominator Multiply top and bottom of outer =inner one.= of 1 3+1 4 1+ 1 3 ? 1 4 = =1÷ 4 1 3 Simplify denominator then treat outer fraction as a division. 1+ 3 3 1 3 ?3 = × = = 0.75 1 4 4 Check Your Understanding A What is 1 3 ? 1 + 4 3 π π π π ππ ππ π÷ + = π ÷? = × = π π ππ π π π N What is 1 1β 1 ? 1 1β3 π π π = = = = βπ π π π ? πβ π πβ β π π π Fractions of Amounts 3 4 Find of 80 Find 2 4 of 3 5 Find 1 2 of 2 3 3 4 of of 4 5 β π π β π π π π ? × = π ππ β π π π π π π ?× π π You can think of the word βofβ as × × ππ? = ππ × × = π π [JMC 2004 Q15] Granny spends one third of her weekly pension on Thursday night, and one quarter of what remains on Friday. What fraction of the original amount is left for her big night out on Saturday? 3 2 She retains 4 of the 3 remaining. π π π π β × = π π ? [JMO 2014 A10] My four pet monkeys and I harvested a large pile of peanuts. Monkey A woke in the night and ate half of them; then Monkey B woke and ate one third of what remained; then Monkey C woke and ate one quarter of the rest; finally Monkey D ate one fifth of the much diminished remaining pile. What fraction of the original harvest was left in the morning? β π π π π π π π π × × × = ? π π Reverse Fractions of Amounts 3 of a number is 18. What 7 π is 6. Thus number is 42. π was the original number? ? [Kangaroo Grey 2014 Q4] A bucket was half full. A cleaner added two litres of water to the bucket. The bucket was then three-quarters full. How many litres can the bucket hold? π is 2 litres. Thus whole ? bucket is 8 litres. π 2 Bob has just been on a diet and managed to lose of his body weight. He 5 is now 15 stone. How heavy was he pre-diet? π π of his original weight is 15kg. is 5kg. Thus his original weight was π π ? 25kg. Exercise 4 1 a b 2 4 If of a number is 20, what is the number? 35 7 5 2 6 3 If of a number is 30, what is of it? 24 ? 7 [IMC 2015 Q3] What is a half of a third, plus a third of π a quarter, plus a quarter of a fifth? Solution: ? 3 ? ππ [JMC 2013 Q13] When painting the lounge, I used half of a 3 litre can to complete the first coat of paint. I then used two thirds of what was left to complete the second coat. How much paint was left after both coats were complete? Solution: 500ml ? 8 ? 4 [IMC 2007 Q4] Between them, Ginger and Victoria eat two thirds of a cake. If Ginger eats one quarter of the cake, what fraction of the cake does Victoria eat? ? Solution: 5 π ππ [IMC 1997 Q15] On the first day after the flood, half of Noahβs animals escaped. On the second day one third of the remainder wandered off. On the third day one quarter of the rest hopped it. What fraction of π Noahβs original menagerie was then left? π 6 ? [JMO 1997 A6] One half of the class got As. One third of the rest got Bs. One quarter of the remainder got Cs. One fifth of the others got Ds. What fraction of the π class got Es or worse? Solution: ? ππ [JMO 2007 A4] The hobbits Frodo, Sam, Pippin and Merry have breakfast at different times. Each one takes a quarter of the porridge in the pan, thinking that the other three have not yet eaten. What fraction of the porridge is left after all four hobbits had their breakfast? ππ Solution: πππ [TMC Regional 2013 Q10] Dean spent one fifth of the amount of money in his wallet and then one fifth of what remained. He spent a total of £72. Find the amount of money in his wallet to start with. £200 ? 9 [TMC Regional 2009 Q10] Three squirrels, Steve, Keith and Benjamin, have spent all day collecting nuts. At the end of the day they are very tired and go to bed. During the night, Steve wakes up and eats a nut. He then decides to take half of the remaining pile and hides it before going back to sleep. Keith then wakes up. He too is hungry, so eats a nut. He also takes half of the remaining pile and hides it before going back to sleep. Finally, Benjamin wakes up, eats a nut and hides half of the remaining pile. In the morning, all of the squirrels share the remaining nuts equally between them and each goes off with four more nuts. Each squirrel then eats all the nuts he now has. How many nuts has Steve eaten in total? Solution: 56 ?