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Strategy 1: Guess and Check

Strategy 1: Guess and Check. Q1. Peter keeps some chickens and goats on his farm. There are altogether 16 heads and 44 legs of the animals. How many chickens and goats are there? 1st Guess: Use half of them as chicken and half of them goats                         Heads     Legs

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Strategy 1: Guess and Check

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  1. Strategy 1: Guess and Check Q1. Peter keeps some chickens and goats on his farm. There are altogether 16 heads and 44 legs of the animals. How many chickens and goats are there? 1st Guess: Use half of them as chicken and half of them goats                         Heads     Legs 8 chickens           8          8x2 = 16 8 goats                8          8x4 = 32                          -----                -------                           16                    48 (X) 2nd Guess: Use more chickens and less goats as there are too many legs                         Heads     Legs 9 chickens          9          9x2 = 18 7 goats               7          7x4 = 28                        ------                -----------                           16                   46 (X) 3rd Guess: Use more chickens and less goats again as there are still too many legs                        Heads        Legs 10 chickens       10             10x2 = 20 6 goats               6               6x4 = 24                        ------                     ---------                        16                        44 (Correct) Answer: There are 10 chickens and 6 goats.

  2. Strategy 2 :Work From Behind( Work Backwards) Q2. Stacy and Huiyu had 400 stickers between them. Stacy gave 3/7 of her stickers to Huiyu. Huiyu then gave 3/8 of her stickers to Stacy. After that, they had an equal number of stickers. How many stickers did Stacy had at first? Working out the question from backwards, 400 divided by 2 = 200 stickers5 units -> 200 stickers (Huiyu)1 unit -> 40 stickers3 units -> 40 x 3 = 120 stickersTherefore Huiyu gave 120 stickers to Stacy200 - 120 = 80 stickers 4 units -> 80 stickers (Stacy)1 unit -> 20 stickers7 units -> 20 x 7 = 140 stickers Answer: Stacy had 140 stickers at first. (Huiyu had 260 stickers at first) 

  3. Strategy 3 : Simplify the problem  Q3. A shopkeeper had a total of 120 blue and black bags in the ratio 3:5. After he sold an equal number of each colour of bag, the ratio of blue to black bags was 3:8. How many bags did he sell altogether. Total: 3 + 5 = 8 units120 bags divided by 8 units = 15 bags15 x 3 = 45 blue bags, 15 x 5 = 75 black bagsDifference in number of bags = 75 - 45 = 30 bagsAfter selling an equal number bags of each colour, the difference is the same at 30 bags.8 - 3 = 5 unitsThere are 5 more units of black bags than blue bags 5 units -> 30 bags1 unit -> 6 bags11 units -> 66 bags (both blue and black)120 - 66 = 54 bags Answer: The shopkeeper sold 54 bags. (27 blue and 27 black bags)

  4. Strategy 4: Trial & Error, Act it Out Q4. John is given a large container filled with 2.4 litres of orange juice. He has 3 empty containers whose capacities are 1.3 litres, 1.1 litres and 0.5 litres. What can John do to get 3 equal portions of 0.8 litres of orange juice each Answer: • Step 1. Pour 2.4 litres container of juice into 1.1 litres and 0.5 litres container 2.4 - 1.1 - 0.5 = 0.8 litres (in 2.4 litres container) • Step 2. Pour 0.5 litres container of juice completely into 1.3 litres container. Fill up 1.3 litres container completely with juice from 1.1 litres container. There will be 0.3 litres of juice left in 1.1 litres container. • Step 3. Next, pour 1.3 litres container of juice into the empty 0.5 litres container. 1.3 - 0.5 = 0.8 litres (in 1.3 litres container) 2.4 - 0.8 - 0.8 = 0.8 litres. (in the other 2 containers)

  5. Strategy 5: Draw A Model Q5. Mrs Ang has some children. Each of her daughters has twice as many brothers as sisters. Each of her sons has the same number of brothers as sisters. How many daughters does Mrs Ang have ? Answer: Mrs Ang has 3 daughters (and 4 sons)

  6. Strategy 6: Make an organised list  Q6. When a number is divided by 3, the remainder is 1. When the same number is divided by 5, the remainder is 3. What is the number if it is between 20 and 30? Make a list which is a multiple of 3 and then add 1 3+1= 4, 6+1 = 7, 9+1 = 10.... => 4, 7, 10, 13, 16, 19, 22, 25, 28, (stop at 30) Make a list which is a multiple of 5 and then add 3 5+3= 8, 10+3 = 13, 15+3= 18.... => 8, 13, 18, 23, 28 (stop at 30) There are two answers, 13 and 28. But the number is between 20 and 30, therefore the correct answer is 28. Answer: The number is 28.

  7. Strategy 7: Ratio Method  Q7. 3 children can carry 21 books. How many books can 8 children carry if each child can carry the same number of books? Using ratio method, put the unknown to be solved on the right hand side (RHS). Put the known item on the left hand side (LHS). LHS              RHS 3 children -> 21 books 8 children -> (21/3) x8  (RHS divided LHS x below LHS)                     = 7 x 8                    = 56 The number of books that 8 children can carry is 56 books. (Notice that this ratio method allows you to skip one step to find how many books one child carries) Answer: The number of books that 8 children can carry is 56 books.

  8. Strategy 8: Total Amount (Quantity) of Work Rule.  Q8. It takes 2 men 6 days to paint 3 houses. How long does it take 3 men to paint 6 houses? Calculate the total amount of work the two men does = 2 x 6 = 12 mandays Use the ratio method, 3 houses -> 12 mandays 6 houses -> 12/3 x 6 = 24 mandays 24 mandays divided by 3 men = 24/3 = 8 days. It takes 8 days for 3 men to paint 6 houses. (this is the concept of "more hands make the work lighter") Answer: It takes 8 days for 3 men to paint 6 houses.

  9. Strategy 9: Compare and Eliminate method  Q9. 3 diamond rings and 2 gold rings cost $1900. A diamond ring and a gold ring cost $700. How much does a gold ring cost? (A) DR DR DR  + GR GR = $1900 (B) DR + GR = $ 700 (C) DR + GR +  DR + GR     = $700 x 2 sets = $1400 compare (A) and (C), (A) has one more DR than (C) 1 DR = $1900 - $1400 = $500 Therefore 1 GR = $700- $500 = $200 Answer: A gold ring cost $200.

  10. Strategy 10: Substitution method  Q10. A box with 50 rubber balls in it weighs 1150 g. Another identical box with 30 rubber balls weighs 750 g. Find the weight of each rubber ball. 1 Box + 50 rubber balls -> 1150 g 1 Box + 30 rubber balls -> 750 g  1 box + 30 balls + 20 balls -> 750 g  + 20 rubber balls = 1150 g 20 rubber balls weighs 1150 -750 = 400 g 1 rubber ball weighs 400/ 20 = 20 g. Each rubber ball weighs 20 g. Answer: The weight of each rubber ball is 20g.

  11. Strategy 11. Simplify the Problem by finding a pattern.  Q11. Add up the even numbers from 2 to 100. Write out the number sequence given in the question; 2 + 4 + 6 + ... + 96 + 98 + 100 Notice the pattern; 2 + 100 = 102 4+ 98 = 102 6 + 96 = 102 and so on... There are 100/2 = 50 even number from 2 to 100. There are 50/2 = 25 pairs of even numbers which adds up to 102. 102 x 25 = 2550. Answer: The answer is 2550.

  12. Strategy 12. Simplify the Problem (Challenging).  Q12. There are 1000 lockers in a school with 1000 students. The first student opens all the lockers. The second one closes lockers 2,4,6,8,10 till 1000. The third one changes the state (close opened one and open closed one) of lockers 3,6,9,12,till 1000. The fourth one changes the state of lockers 4,8,12,16 till 1000. How many lockers are opened at the end? 1st student open 1000 lockers   2nd student closed the even number lockers, so only odd number left open. 1, 3, 5, 7, ...999 => there are 500 odd number lockers left open 3rd student change state for: 3, 6, 9, 12, 15... 999 => multiples of 3 (total 333 numbers, 167 odd and 166 even numbers) 500 - 167 (odd) + 166 (even) = 499 lockers left open 4th student change the state for:   4, 8, 12, 16, 20... 1000 => multiple of 4 (total 250 numbers) since 499 locker included some even number lockers which are multiple of 3 and 4 => these even number lockers are multiple of 12. (4x3=12)   12, 24, 36, ...996 => 83 multiples of 12.   250-83 = 167 (multiple of 4 but not multiple of 12)   499 + 167 - 83 = 583 lockers left open Answer: There are 583 lockers left open in the end

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