Mathematical Puzzles These puzzles have been adapted from various sources to be used with pupils who finish classwork early. Most of the questions were chosen with enthusiastic, bright 11 year olds in mind. Some of the puzzles are also appropriate for class work - an initial worked example on the board will help a lot. There are a few trick questions. Some questions can be quickly answered if you chance upon the right approach, but the 'long' solution isn't too arduous. Remove Answers Show Answers
Scales and Vessels 1. How can you measure out exactly 4 litres of water from a tap using a 3 litre and a 5 litre bucket? Ans 2. 3. 4. 5. 6. 7. 8. 9.
3litre ----0 3 0 2 2 3
5litre -----5 2 2 0 5 4
10. A 24 litre bucket is full of lemonade. 3 men want to have equal amounts of it to take home, but they only have a 13 litre, a 5 litre and an 11 litre bucket. How do they do it? Ans 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.
24 13 11 5 ---------24 0 0 0 11 13 0 0 6 13 0 5 6 2 11 5 8 0 11 5 8 5 11 0 8 13 3 0 8 8 3 5 8 8 8 0
22. A Queen (78kg), the Prince (36kg) and the King (42kg) are stuck at the top of a tower. A pulley is fixed to the top of the tower. Over the pulley is a rope with a basket on each end. One basket has a 30kg stone in it. The baskets are enough for 2 people or 1 person and the stone. For safety's sake there can't be more than a 6kg difference between the weights of the baskets if someone's inside. How do the people all escape? Ans
23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33.
Basket 1 Basket 2 --------------------Stone up Prince down King down Prince up nothing up Stone down Queen down Stone and King up nothing up Stone down Prince down Stone up nothing up Stone down King down Prince up Stone up Prince down
34. One out of 9 otherwise identical balls is overweight. How can it be identified after 2 weighings? Ans: Weigh 3 against 3, then you'll know which group of 3 contains the heavy ball. Pick 2 balls from that group and weigh one against the other. 35. One out of 27 otherwise identical balls is overweight. How can it be identified after 3 weighings? Ans: Weigh 9 against 9, then 3 against 3. 36. How many ways can you put 10 sweets into 3 bags so that each bag contains an odd number of sweets? Ans 15 solutions. The first trick is to realise that if you put one bag inside another, then sweets in the inner bag are also in the outer bag. The only workable configuration is to put one bag inside another and leave the third alone. The answers can be obtained using the following octave script, where bag b is inside bag a 37. for a=0:10 38. for b=0:(10-a) 39. c=10-a-b; 40. if (rem((a+b),2)==1 && rem(b,2)==1 && rem(c,2)==1) 41. fprintf('a=%d b=%d c=%d\n',a,b,c) 42. end 43. end 44. end
Ferries 1. A man has to take a hen, a fox, and some corn across a river. He can only take one thing across at a time. Unless the man is present the fox will eat the hen and the hen eat the corn. How is it done? Ans 2. 3. 4. 5. 6. 7. 8.
MAN AND HEN -> <- MAN MAN AND FOX -> <- MAN AND HEN MAN AND CORN -> <- MAN MAN AND HEN ->
9. 3 missionaries and 3 obediant but hungry cannibals have to cross a river using a 2-man rowing boat. If on either bank cannibals outnumber missionaries the
missionaries will be eaten. How can everyone cross safely? Ans 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
CANNIBAL and MISSIONARY -> <- MISSIONARY CANNIBAL and CANNIBAL -> <- CANNIBAL MISSIONARY and MISSIONARY -> <- CANNIBAL and MISSIONARY MISSIONARY and MISSIONARY -> <- CANNIBAL CANNIBAL and CANNIBAL -> <- CANNIBAL CANNIBAL and CANNIBAL ->
21. 2 men and 2 boys need to cross a river in a boat big enough for 1 man or 2 boys. How do they do it? Ans 22. 23. 24. 25. 26. 27. 28. 29. 30.
BOY and BOY -> <- BOY MAN -> <- BOY BOY and BOY -> <- BOY MAN -> <- BOY BOY and BOY ->
SMP and CSE 1974 extend this to cover the case of n men.
Picking Captains 1. 6 boys pick a captain by forming a circle then eliminating every n'th boy. The 2nd boy in the counting order can choose n. If he wants to be captain what's the smallest n he should pick? Ans: 10 2. 12 black mice and 1 white mouse are in a ring. Where should a cat start so that if he eats every 13th mouse the white mouse will be last? Ans: If the white mouse is 1st in the counting order, the cat should start at the 7th mouse : hint - start anywhere, see how far out you are, then make the necessary correction 3. 20 passengers are in a sinking ship. 10 are mathematicians. They all stand in a ring. Every 7th climbs into the lifeboat which can only hold 10 people. Where should the mathematicians stand in the ring? Ans: 1 4 5 7 8 9 14 15 16 17 4. 30 passengers are in a sinking ship. They all stand in a circle. Every 9th passenger goes overboard. The lifeboat holds 15. Where are the 15 lucky positions in the circle? Ans: 1 2 3 4 10 11 13 14 15 17 20 21 25 28 29 The last of these questions can be answered using the following octave script howmanyatstart=30; howmanyatend=15;
first=1; leap=9; x=first-1; a=1:howmanyatstart; while (length(a)>howmanyatend) x=rem(x-1+leap,length(a)); a(x+1)=[]; end a
By changing the initial values in this script you can solve questions 2 and 3.
Incomplete Sums (Worked Examples in J.A.H. Hunter's "Mathematical Brain Teasers"). 1. Each letter represents a different digit 2. 3. 4. 5.
SEND +MORE ----MONEY
Ans: 9567+1085 = 10652 6. This sum uses all the digits 7. 8. 9. 10.
28* +**4 ---****
Ans: 289+764 = 1053 11. This subtraction sum uses all the digits from 1 to 9. 12. 9 * * 13. - * 4 * 14. ----15. * * 1
Ans:927 - 346 = 581 16. O represents odd digits E represents even digits 17. EEO 18. xOO 19. ----20. EOEO 21. EOO 22. ----23. OOOOO
Ans: 285x39
24. P represents prime digits 25. 26. 27. 28. 29. 30. 31.
PPP xPP ----PPPP PPPP ----PPPPP
Ans: 775x33 32. Some more additions o o o o o o o o o o o o o o o o o o o o o o o o o o
THE TEN MEN ---MEET
Ans: 490+407+107=1004 SLOW SLOW OLD ---OWLS
Ans: 4712 SAL SEE THE SUEZ ----CANAL
Ans: 920+977+547+9876=12320 FIVE FIVE NINE ELEVEN -----THIRTY
Ans: 4027+4027+5057+797275
Letters Agree on a font of capital letters. • • • • • •
Put the letters into sets according to line symmetry Put the letters into sets according to rotational symmetry Put the letters into sets according to topology How many letters only use straight lines? There is only one number whose English name uses as many straight lines to write as the number itself. Think of a number. Write it out in words. Write in words the number of letters you've used (E.g. SIXTEEN-SEVEN-FIVE-FOUR). Continue do so and see what
happens. Try 3 other numbers. > You always end at FOUR. In german you end up at VIER.
Numbers 1. Alan, Bill and Chris dug up 9 nuggets. Their weights were 154, 16, 19, 101, 10, 17, 13, 46 and 22 kgs. They took 3 each. Alan's weighed twice as much as Bill's. How heavy were Chris's nuggets? Ans: 272 2. The product of 3 brothers' ages is 175. Two are twins. How old is the other one? Ans: 7 3. A man has 2 bankcards, each with a 4 digit number. The 1st number is 4 times the 2nd. The 1st number is the reverse of the 2nd. What is the first number? Ans: 8712 4. Tom has 7 sandwiches, Jan has 5, Simon has none. They share them out equally. Simon leaves, paying for his sandwiches by leaving 12 biscuits. What's the fairest way for Tom and Jan to share out the biscuits? Ans: 3 to Jan 5. A cyclist buys a cycle for 15 pounds paying with a 25 pound cheque. The seller changes the cheque next door and gives the cyclist 10 pounds change. The cheque bounces so the seller paid his neighbour back. The cycle cost the seller 11 pounds. How much did the seller lose? Ans: 21 pounds? 6. Using four "4"s and common symbols (including the square root, factorial and recurring decimal symbols), make sums whose answers are 0, 1, 2....100 (See Mathematical Bafflers) 7. Make fractions (each using all the digits from 1 to 9) with these values 1/2, 1/3....1/9 Ans: 8. 6729/13458, 5823/17469, 3942/15768, 2697/13485, 2943/17658, 9. 2394/16758, 3187/25496, 6381/57429
10. A greengrocer was selling apples at a penny each, bananas at 2 for a penny and pears at 3 for a penny. A father spent 7p and got the same amount of each type of fruit for each of his 3 children. What did each child get? Ans: 1 apple, 2 bananas and 1 pear. 11. A woman bought something costing 34c. She only had 3 coins: $1, 2c and 3c. The shopkeeper had only 2 coins: 25c and 50c. Fortunately another customer had 2 10c coins, a 5c coin, 2 2c coin and a 1c coin. How did they sort things out? Ans: They pool the money. The woman takes 74, the shopkeeper takes 109 and the customer 28. 12. Mr and Mrs A are 120 km apart. A bee is on Mr A's nose. The couple cycle towards each other, Mr A at 25km/h and Mrs A and 15km/h. The bee dashes from Mr A's nose to Mrs A's nose and back again and so on at 60km/h. How far does the bee travel before the cyclists crash? Ans: The cyclists crash after 3 hours so the bee flies 180km.
13. Pick a number. If it's even, divide by 2. If it's odd multiply by 3 and add 1. Continue this until you reach "1". Eg 3-10-5-16-8-4-2-1. Which integer less than 100 produces the longest chain? Ans: 97 leads to a 119 link chain. The lowest number that causes a long chain is 27 (112 links). 14. Pick a number. Multiply the digits together. Continue until you get a single digit. What is the only 2 digit number which would require more than 3 multiplication? Ans: 77 15. Starting with 1, place each integer in one of 2 groups so that neither contains a 3 term Arithmetic Progression. How far can you go? Ans: Up to 8. 16. The following 2 questions use the following result - "Given integers a and b the biggest number that can't be expressed in the form ia + jb is ab - a - b." o Apples are packed in boxes of 8 and 15. What is the biggest number of apples that would require loose apples? Ans: 97 o A country only has 5p and 7p coins. Make a list of prices that you could give exact money for. What is the highest prices that you couldn't give exact money for? Ans: 23 17. If D = the day (1-366) in year Y, then the day of the week can be calculated using 18.
d = D+Y+(Y-1)/4 - (Y-1)/100 + (Y-1)/400 mod7
where d=1 would mean Sunday, etc. Can the first day of each century (e.g. 1st Jan 2001, 1st Jan 1901) be any day? Ans: No. Just Monday, Tuesday, Thursday or Saturday. See "Mathematical Bafflers" 19. Pick 3 digits (not zero) and make 6 2-digit numbers from them. Add up all these numbers, add up all the original digits and divide the first total by the second. Ans: 22 20. How many presents did the "true love" send during the 12 days of Christmas? Ans: 364
21.
At a fairground stall there are 3 piles of cans. You get 3 throws. You can only knock off the top can of a pile. The 2nd throw counts double, the 3rd triple. How do you get exactly 50? Ans: 7, 8, 9
Enumerations 1. Holding it's hands out, palms upward a child starts counting on all its fingers and thumbs, going to and fro. If it counts up to 1982 which finger does counting end on? Ans: (middle finger is 3 or 7 (mod8)). 2. You have 3 bricks, each measuring 18 x 9 x 6 cm. How many different heights can you build up with them? Ans: 14 3. How many right-angled triangles with integral sides have one side of 15? Ans 4. 5. 6. 7. 8.
9, 12, 15 15, 20, 25 15 112 113 8 15 17 15 36 39
9. Minibuses seating 10, 12 and 15 passengers can be used to convey 120 passengers. There are 5 of each size of bus. How many different ways can the buses be used so that all the ones used are full? Which way uses the least buses? Ans 10. 11. 12. 13. 14.
10 12 15 -------3 5 2 4 3 3 5 4
15. Roosters cost 5 pounds, hens 3 pounds and 3 chicks cost 1 pound. Buying at least one of each type of bird how can you buy 100 birds for exactly 100 pounds? Ans 16. 17. 18. 19.
Rooster Hen Chick 4 18 78 8 11 81 12 4 84
20. A woman puts 120p on the counter. "some 4p stamps, 6 times as many 2p stamps and the rest in 5p stamps please". What does she need? Ans: 5 4p's. 21. A cube is painted white and cut into 27 small cubes. How many of these little cubes are painted on i) 1, ii) 2, iii) 3, iv) 0 sides? See SMP E for answers and developments
Geometry 1. 9 geraniums have to be planted so that there are 3 plants in each row. How can they be planted so that there are a) 8 b) 9 c) 10 lines.
Ans:
2. o o
How many straight lines of unique lengths can you draw from dot to dot? Ans: 5 How many non congruent triangles can you draw? Ans: 8
Miscellaneous 1. A man buys 5 cigarettes a day for 25 days. He keeps the stubs. From 5 stubs he made a new cigarette. How many cigarettes does he smoke during this period? 2. The bacteria in a test-tube double each minute. It is full in one hour. When was it half full? Ans: 59 mins. 3. In a knockout with 39 players, how many games are played? Ans: 39-1 4. In a knockout with 39 players, how many byes? Ans: 3 (find 2**n such that 2**n>39 then express 2n-39 in base 2 and count 1's in the binary number) 5. A necklace is made from 4 chains each of 3 links. It costs 1p to cut a link and 2p to resolder it. How can a necklace be made for 9p? Ans: Cut one chain into 3 links. Use these to connect up the rest. 6. If a crocus costs 8p, a tulip 7p and a daffodil 11p, how much does a hyacinth cost? Hint: count the vowels and consonants 7. Arrange a line of 5 coins as follows 8.
Head Head Head Tail Tail Tail
In 3 moves (each move consisting of turning over a pair of adjacent coins), arrange the coins so that Heads and Tails alternate. Ans: Swop 3 and 4, 4 and 5, then 2 and 3
9. Town A and town B are 99 km apart. There are 98 km posts between them. On each post is the distance to A and the distance to B. On how many posts are only 2 different digits used? (for example, the post that says 'A 33, B 66' only uses 2 digits). From SMP. 18 posts: 11, 18, 22, 27, 33, 36, 44, 45... 10. Clocks o How many times is a stopped clock correct each week? o A clock goes at the right speed, but backwards. How many times each week does it tell the right time?
Knights
1. If you follow the jumps of a chess knight from word to word you can make a 20 word sentence about the ages of Sue and Sal. Sue is in her teens, so how old is Sal? Ans: Start at Sal in the bottom row. Sal is 8 (thanks to Jenny Leppington-Clark for discussing this puzzle, and thanks to Sweta Singh for pointing out the right answer). The sentence is "SAL IS NOW HALF AS OLD AS SUE WAS WHEN SAL WAS A THIRD AS OLD AS SUE IS NOW". So let's create some variables SueNow and SalNow are the ages now, and SueThen and SalThen are the ages X years ago. Then SalNow = SueThen/2 and SalThen = SueNow/3 Also SueNow = SueThen + X SalNow = SalThen + X Substituting, we get SalThen + X = SueThen/2 and SalThen = (SueThen + X)/3 Subtracting, we get X = SueThen/2 - SueThen/3 - X/3
so
4X/3 = SueThen/6 and hence 8X = SueThen So, assuming X is an integer and Sue is now a teenager, X=2 and SueThen will be 16. That makes SueNow 18 and SalNow 8.
2. On a 5 by 5 chessboard, a knight on the centre square would have 8 possible moves. Draw a 5 by 5 grid and put "8" in the middle square. Fill in the other squares with the number of moves that a knight would have from that square. 3. Draw a 5 by 5 chessboard. Put a "0" in the top left corner square. Imagine a knight there. Put a "1" in all the squares that it can reach in 1 go, a "2" in all the squares that it can reach in a minimum of 2 goes, etc, until all the squares are filled in.
Digital Displays _ _ _ _ _ _ _ _ | | | _| _| |_| |_ |_ | |_| |_| |_| | |_ _| | _| |_| | |_| _|
• • • • • • • •
Make a table showing how many lines are used for each digit. While changing from 0 to 1, four bars are turned off. No new ones are turned on. Make a table of what happens as the digits change 0->1 1->2 ... 9->0
ON 0
OFF 4
Add up the "OFF" and the "ON" columns. What do you notice?
Palindromes 1. A car milometer shows 15951. After 2 hours it shows another palindromic number. How fast was the car going on average? 2. Pick a 2 digit number. Reverse it then add to the original number. Have you got a palindrome? If not then repeat the process. Try all the 2 digit numbers (as a class activity) (from Mathematical Bafflers). Ans 3. Sum of digits 4. Num of additions
1-9 10 11 12 13 14 15 16 17 18 1 2 1 2 2 3 3 6 24 6
References 1. Mathematical puzzles and pastimes, A. Bakst, Princeton,N.J, 1965. 2. More mathematical activities: a resource book for teachers, Brian Bolt, Cambridge University Press, 1985. 3. The Corgi Book of Problems, Jameson Erroll, Transworld Publishers Ltd., 1964. 4. Mathematical Bafflers, Angela Dunn, Dover Publications, 1980. 5. Mathematical puzzles & diversions, M. Gardner, London, 1961. 6. More mathematical puzzles & diversions, M. Gardner, London, 1963. 7. Mathematical puzzles for the connoisseur, P.M.H. Kendall and G.M. Thomas, London, 1962. 8. Mathematical games and puzzles, T. Rice, London, 1973. 9. The Canterbury Tales, Henry Dudeney, Thomas Nelson and sons, 1949 See also Mathematical games Updated May, 2009
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Grade Five Problems 1) How many addition signs should be put between digits of the number 987654321 and where should we put them to get a total of 99? 2) Divide the face of the clock into three parts with two lines so that the sum of the numbers in the three parts are equal. 3) According to experts the first 4 moves in a chess game can be played in 197299 totally different ways. If it takes 30 seconds to make one move, how long would it take one player to try every possible set of 4 moves? 4) Batman has been imprisoned by the Riddler. To escape he must find the quickest way to move the tower of plutonium disks from one post to another so that the disks have the same arrangement as on the original post. He may move only one disk at a time. What is the minimum number of moves he must make in order to move the ten disk tower and have it appear the same?
5) A man has to be at work by 9:00 a.m. and it takes him 15 minutes to get dressed, 20 minutes to eat and 35 minutes to walk to work. What time should he get up?
6) Replace each blank with the correct digit. 4 3 _ 2 4 5 _ _ 1 2 7 --------8 8 9 3
7) In the first year of production a play sells 1572 tickets, in its second year it sells 1753 tickets, in its third year it sells 152 less than in its second year. How many tickets are sold in 3 years? 8) The Riddler has left a clue for Batman to follow at the scene of each crime. These are the clues that Batman has found: • • • • •
(1) There is a 1 in the thousands place. (2) The digit in the tens place is 9 times the digit in the thousands place. (3) Multiply the digit in the thousands place by 2. (4) The digit in the ones place is a hand without a thumb. (5) The digit in the hundreds is 2 less than the number in the tens. Solve the riddle to find the number and help Batman stop the Riddler.
9) Scientists have just discovered people on Neptune. There are 3 villages, 2 cities and 1 super city on Neptune. These are the populations in 1994 and 1996. 1994 Eilosa Vertu Pridi Dedrun Antran Maran
129 308 90 500 700 1200
1996 204 292 50 600 693 1500
List the places in order of increasing size in 1994 and 1996. In which year was the population of the planet greater?
10) Rachel, Kim and Shawn went to Cuba for a vacation. On the way to Cuba, the plane made the trip in 315 minutes. On the return trip, the flight took 216 minutes. They stayed in Cuba for 3629 minutes. Estimate how long the trip took to the nearest 10. How long did the trip really take?
11) The mass of the Great Pyramid is 557t greater than that of the Leaning Tower of Pisa. Stone Henge has a mass of 2695t which is 95t less than the Leaning Tower of Pisa. There once was a Greater Pyramid which had a mass twice that of the Great Pyramid, what was the mass of the Greater Pyramid.
12) Find the number of faces, edges and vertices on the figure shown.
• •
What would this figure be called? What would the two figures contained in it be called?
13) Given a cube, draw a second figure with a square base, having 8 edges, 5 vertices and 5 faces. Need A Hint?
Answer
14)
Given a cone having one edge and one vertex and two faces, make two intersecting cuts through the flatside and find the number of vertices, edges and faces each piece will have. Need A Hint? Answer
15) How many different cubes can you make if each face of a given cube has a line connecting the center points of two opposite edges?
Need A Hint? Answer
16) If every vertex of a regular pentagon is connected to every other vertex, how many triangles are formed? Need A Hint? Answer
17) The weather during Pith Possum's vacation was strange. It rained on 15 different days, but it never rained for a whole day. • • • •
Rainy mornings were followed by clear afternoons. Rainy afternoons were preceeded by clear mornings. There were 12 clear mornings and 13 clear afternoons in all. How long was the vacation?
Need A Hint? Answer
18) Festival Antigonish keeps a set of five counter weights to help stagehands move heavy props. The weights can balance exactly any load that is a multiple of 10kg, from 10kg up to a total of the five weights. Give the masses of weights which meet these conditions and allow the stagehands to counterbalance the max possible load. Need A Hint? Answer
19) If you begin with a one digit integer, multiply by 3, add 8, divide by 2 and subtract 6, you will get the integer back. Find the number. Need A Hint?
Answer
20) This stairway is made of cubes. How many cubes would be needed to make the steps 9 steps high?
Need A Hint? Answer
21) Look at the chart and fill in the numbers. The key is to find a pattern. 4 2 2 4
Need A Hint? Answer
5 2 3 6
9 4 5 20
16 6 42
5 25
7 8
7 1
63
2
4
22) This is a turtle graphics problem. A bug is walking across the doorstep of a house. When he starts out he is facing the house at the opposite side of the doorstep. If he goes 3 units left, 1 unit right, 2 units left, 8 units left, what will the bug have to do to get back where he started as quickly as possible without retracing his steps? Need A Hint? Answer
23) If Jane is older than Kim, Kim is older than Shawn. Shawn is younger than Jane and Rachel is older than Jane List the people from oldest to youngest. Need A Hint? Answer
24) There are 12 people in a room. 6 people are wearing socks and 4 people are wearing shoes, 3 people are wearing both. How many people are in bare feet? Need A Hint? Answer
25) One morning grasshopper fell down a hole 2 metres deep. He would climb 1/4 of a metre every day but at night he slid down 1/8 pf a metre. At this rate, how many days until the grasshopper gets out? Need A Hint? Answer
26) Kim, Raelene, Jim and Rick finished first, second, third and fourth in a motor bike race. If their numbers were 5, 17, 1 and 7, use the following clues to find out who had what number and what order they finished in. • • • • •
17 said she would have placed higher if her bike had not stalled at the start. Rick finished before 7 but after Raelene. 1's father, said he was very proud of his daughter's finish. Kim finished after 5. Jim wasn't third.
Need A Hint? Answer
27) An ice cream stand has nine different flavours. A group of children come to the stand and each buys a double scoop cone with two flavours of ice cream. If none of the children choose the same combination of flavours, and every different combination of flavors is chosen, how many children are there? FLAVOURS
1 2 3 4 5 6 7 8 9
Vanilla Maple Chocolate Tiger Raspberry Strawberry Coffee Moon Mist Cherry Vanilla
Need A Hint? Answer
28) Place the digits 9,4, 7, 6, 5, 1, in the boxes in order to get the largest result. [ ][ ] x [ ][ ] + [ ] x [ ] =
Need A Hint? Answer
?
29) Four boys work together painting houses for the summer. For each house they paint they get $256.00. If the boys work for 4 months of summer and their expenses are $152.00 per month, how many houses must they paint for each of them to have one thousand dollars at the end of the summer? Need A Hint? Answer
30) In her first year a dog breeder's dogs produce 2 puppies. In her second year her dogs produce three-times as many puppies. In her third year her dogs produce 5 times as many puppies as the first year. How many puppies will the breeder have produced in her first 3 years? If she sells the puppies for $200.00 each, how much will she have made? Need A Hint? Answer
31) Two frogs are hopping along a path one lily pad wide, going in the same direction. There are 18 lily pads and one frog jumps 3 lily pads at a time, the other jumps two lily pads at a time. If the frogs leave the shore one after the other, how many lily pads will be jumped on by both frogs? Need A Hint? Answer
32) A grocery store has a sale on bananas. If you buy six bananas you get the sale price. If the grocer has 489 bananas how many bunches of six can he sell at his sale price? In this case how many can be sold at the regular price? Need A Hint?
Answer
33) Some students hold a bake sale to raise money for their school library. They sold fudge and squares and cookies. One of each type of treat was put into 95 paper bags and the students were allowed to keep the left overs. If they started out with 110 cookies, 130 pieces of fudge and 116 squares, the students put the same number of each kind of treat in each bag(eg. 2cookies, 1 fudge, 1 square. How many of each treat did they keep? Need A Hint? Answer
34) Describe the shaded area of the following figure in terms of a fraction.
Need A Hint? Answer
35) Sixteen players enter a tennis tournament. If there can be no draws how many games must be played if each player can be eliminated by one loss. Need A Hint?
Answer
36) If a man can walk 4km in 1 hour and his wife can walk 5km in 1 hour, how many minutes will it take them to meet if they begin walking at the same time at opposite ends of a 12km road, and have agreed to meet at the half way point?
Need A Hint? Answer
37) In how many different ways can the cars of a three car train be arranged? Need A Hint? Answer
38) In 1989 the price of a soccer ball was $8.00. A financial analyst predicted that the price would rise $0.25/year for the next 10 years. In what year will the price be $9.75? Need A Hint? Answer
39) A math student interviewed 50 fifth graders. 41 said they like peanut butter sandwiches, 35 liked jam sandwiches and 30 liked both on their sandwiches. How many students liked neither? Need A Hint?
Answer
40) A unit fraction is a fraction where the numerator is 1 and so it looks like 1/n. Can you express the fraction 19/72 as the sum of two unit fractions? Need A Hint? Answer
41) How many unit cubes are needed to build this solid figure?
Need A Hint? Answer
42) See how many triangles you can draw using any three of the five points on the circle as vertices.
Need A Hint? Answer
43) Nobody knows how old Aunt Helen is but she gave a few hints. She had passed 1/20 of her life before she started school. She spent 3/20 of her life in school; She worked for 1/10 of her life before she got married. She was married for 2/5 of her life. Her husband died after 7/10 of her life. From reading Uncle Harry's gravestone you find out that she has been a widow for 24 years. How old is Aunt Helen? Need A Hint? Answer
44) If a ball is dropped from a height of 100m, each time it hits the ground it bounces 3/5 of the height it fell. How far will the ball have travelled in the 5th bounce? Need A Hint? Answer
45) Two girls went on a vacation together and they agreed to split all expenses. They trip was prepaid so Mary wrote a cheque for food and one for lodging. Clare wrote the cheques for gas and entertainment. How much does Clare owe Mary? Mary:
Clare:
FOOD $56.97
LODGING $83.29
GAS
ENTERTAINMENT
$41.96
$37.47
Need A Hint? Answer
46) The record for the greatest number of consecutive jumping jacks is 14500 in 5 hours and 35 minutes. If Joe does 55 jumping jacks per minute, how long will it take Joe to tie the record? Need A Hint? Answer
47) A line of length 4 is divided into nine equal segments. Find a fraction to describe the length AB.
Need A Hint? Answer
48)
Rachel opened her math book and found that the sum of the facing pages was 243. What pages did she open to? Need A Hint? Answer Answers
Grade Five Answers 1) Answer: 1)
9+8+7+65+4+3+2+1 = 99
-> 7 addition signs.
2)
9+8+7+6+5+43+21 = 99
-> 6 addition signs.
2) Answer:
26+26+26 = 78
3) Answer: 30 seconds * 4 * 197299 = 23675880 seconds.
4)
Answer: The quickest way is to move each disk onto the first post except for the last disk -> which is 9 moves. Then move the last disk to the second post which is one move. Then move each of the disks on the first post to the second post which is an additional 9 moves. 9 + 9 + 1 = 19 moves
5) Answer: 9 hours - (15 + 20 + 35) 9 hours - 70 minutes 7 hours + 120 minutes - 70 minutes = 7 hours and 50 minutes = 7:50 AM
6) Answer: 4 3 1 2 4 5 4 4 1 2 7 --------8 8 9 3
7) Answer: +1572 +1753 +1601 -----4926
8) Answer: 2794
9)
1753 -152 -----1601
Answer: 1994 Eilosa Vertu Pridi Dedrun Antran Maran
1996
129 204 308 292 90 50 500 600 700 693 1200 1500 ----------------2927 3339
3339 - 2927 = 412 therefore the population in '96 is greater
10) Answer: ESTIMATE 320 220 3630 -----4170
ACTUAL 315 216 3629 -----4160
11) Answer: Mass of Stone Henge + 95t = Mass of Leaning Tower Mass of Leaning Tower + 557t = Mass of Great Pyramid Therefore the mass of the Great Pyramid
= 2(2695t + 95t + 557t) = 6694t
12) Answer: 17 edges
9 faces
Pentagonal Prism contains a trianglular prism.
13)
10 vertices rectangular prism and a right
Answer: 4 edges, 4 vertices and one face come from square base. 12E 8V 6F
8E 5V 5F
14) Answer: There are many solutions. Here we have one solution.
4V 6E 4F
15) Answer: 23 = 8
16) Answer: 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10
= 10 triangles
1,2 / 2,3 / 6,4 / 4,1 / 10,7 / 7,6 / 9,8 / 8,10 / 9,5 / 5,3 = 10 triangles 7,0,5 / 4,0,8 / 2,0,7 / 5,0,4 / 8,0,2 = 5 triangles 1,2,3 / 1,4,6 / 6,7,10 / 10,8,9 / 9,5,3 = 5 triangles 2,0,7,8,10 / 4,0,8,5,9 / 7,0,5,2,3, / 8,0,2,4,1 / 5,0,4,7,6 = 5 triangles Answer:
35 triangles
17) Answer: 5 totally clear days and 15 half and half. In the 15 half and half days: 7 <= 12 of the clear mornings were followed by rain, 8 <= 13 of the clear afternoons were preceeded by rain. 20 days of the vacation.
18) Answer: counter weights 10kg 20kg 40kg 80kg 160kg
19) Answer:
Possible loads 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 : : 310
Trial and error [(2 [(6 [(8 [(4
* * * *
3 3 3 3
+ + + +
8)/2] 8)/2] 8)/2] 8)/2]
-
6 6 6 6
= = = =
1 7 10 4
<<<<-
NO NO NO YES!!
The number is 4
Algebraically: [(3*x + 8)/2] -6 = 4
20) Answer: Find a pattern. 2 3 4 5 + 1)
steps steps steps steps
3 cubes 6 cubes 10 cubes
2 + 1 3 + (2 + 1) 4 + (3 + 2 + 1)
9 steps
45 cubes
9 + (8 + 7 + 6 + 5 + 4 + 3 + 2
You will need 45 cubes.
21) Answer: 4 = 2 2 5 = 2 2
7 3 4
+ * + *
2 2 = 4 3 3 = 6
4
5
9
13
10
15
16
3
2
2
4
7
5
7
7
1
2
3
5
6
5
8
9
2
4
6
20
42
25
56
63
2
12
22) Answer: 1 unit left and 3 units left.
23)
Answer:
24)
Answer: 12 - 7 = 5 people in bare feet!
25) Answer: Each day the grasshopper goes 1/8 m until the day when the grasshopper is at 1.75m in morning, he gets out of the hole that day. 1.75/(1/8)=14 and 14 + 1 = 15 days.
26) Answer: Race Number
Position
K RA J
5 5 5
(17) (17) 17
(1) 2 3 4 1 (2) 3(4) 1 2 3(4)
Ri
5
17
(1) 7 (1) 7 1 7 1
7
1
2 (3)4
1 and 17 are female -- from clue 1 and 3 5 and 7 are male -- from above J did not come 3rd -- from clue 5 17 is not first -- from clue 1 Kim is not first. Rick finished before 7, therefore he must be 5 after Raelene who was first, therefore, Raelene is #1 and first Rick is #5 and second Kim is #17 and third Jim is #7.
27) Answer: The number of children = number of different combinations of two different flavours. 1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9
2,3 2,4 2,5 2,6 2,7 2,8 2,9
3,4 3,5 3,6 3,7 3,8 3,9
4,5 4,6 4,7 4,8 4,9
5,6 5,7 5,8 5,9
6,7 6,8 6,9
36 = 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 There are 36 children.
28) Answer: [9][5] * [7][6] + [4] * [1] = 7224
29) Answer: Must earn 4 x $1,000.00 + 4 x $152.00= 4608 Number of houses = 4608/$256/house = 18 houses.
30)
7,8 7,9
8,9
Answer: 2 + 3(2) + 5(2) = 18 18($200.00) = $3600.00
31) Answer: How many numbers between 1 and 18 are multiples of 2 and 3? 6, 12, 18 [ ][o][x][o][ ][x/o][ ][o][x][o][ ][x/o][ ][o][x][o][ ][x/o] 1
2
3
There will only be 3 lily pads that both frogs jump on - Number 6, 12 and 18.
32) Answer: 489 divided by 6 equals 81 with a remainder of 3 He casn sell 81 bunches, which wuld leave him 3 to sell at regular price.
33) Answer: 110 130 116 110 130 116
divide divide divide - 95 = - 95 = - 95 =
by by by 15 35 21
95 equals 1 with a remainder of 15 95 equals 1 with a remainder of 35 95 equals 1 with a remainder of 21 cookies fudge squares
cookies fudge squares
The students kept 15 cookies, 35 pieces of fudge and 21 squares.
34) Answer: A rectangle = (5-1)(4-1) = (4)(3) = 12 - each shaded area.
(1) (2) (3) (4)
(3-1)(2-1) (4-2)(3-2) (5-4)(4-3) (3-2)(4-3)
= = = =
(2)(1) (2)(1) (1)(1) (1)(1)
= = = =
2 2 1 1 6
Area shaded / Area of rectangle = 6/12 = 1/2
35) Answer: ROUND 1 2 3 4
PLAYERS REMAINING 16 16 - 8 8 - 4 4 - 2
GAMES 8 4 2 1 ---15
36) Answer: Time for Man
Time for Wife
6km / 4km/hr = 3/2 * 60min = 90 minutes
6km / 5km/hr = 6/5 * 60min = 72 minutes
As the woman will get there first, she will have to wait for the man. The time taken to meet will be 90 minutes.
37) Answer: (1) [1] - [2] - [3] (2) [2] - [3] - [1] (3) [3] - [2] - [1] (4) [2] - [1] - [3] (5) [3] - [1] - [2] (6) [1] - [3] - [2] There are three different ways.
38) Answer: $9.75 - $8.00 = $1.75 $1.75/$0.25/yr = 7 years Therefore, 1989 + 7 = 1996
39) Answer: Only Jam 35 - 30 5
Only Peanut Butter 41 - 30 11
Jam/Peanut Butter 30
Number that liked neither = 50 - (5+11+30) = 4
40) Answer: 72/18 = 4 18/72 = 1/4 18/72 + 1/72 = 1/4 + 1/72 = 19/72
41) Answer: In a full cube 3 x 3 x 3 = 27 There are 5 missing cubes, therefore there are 22 cubes needed to build this figure.
42) Answer:
123 124 125 134 145
234 235 245
345
There are 10 possible triangles. Each will be used 6 times.
43) Answer: (3/10)*? = 24
and 24/(3/10) = 80.
Aunt Helen is 80 years old.
44) Answer: 3/5(3/5)(100)
3/5(3/5)(3/5)(100)
(1) (2) (3) 100 + 2(3/5)(100) + 2(3/5)(3/5)(100) + 2(3/5)(3/5)(3/5)(100) + 2(3/5)(3/5)(3/5)(100) = 100 + 600/5 + 1800/25 + 5400/125 + 16200/625 = 361.12m
45) Answer: $56.97 83.29 41.96 + 37.47 --------$219.69 1/2(219.69) = 109.85 $109.85 - (Clare's payment) = $30.42 = amount Clare owes Mary.
46)
Answer: 1min/55 * 14500 = 263.64 min = 4h 23.64 min.
47) Answer:
4 * 3/9 = 12/9 = 4/3 = AB 48) Answer: 243/2 = 121.5 121.5 pages are whole numbers therefore the pages are 121 and 122
