This is the Math Kangaroo Solutions Video Library, presenting solution suggestions for Levels 3 and 4 from the year 2013. These solutions are presented by Lucas Nielskowski. The purpose of the Math Kangaroo Solutions Video Library is to help you learn how to solve math problems, such as those presented in the Math Kangaroo competition. It is important that you make sure to read the problem as well as listen as I read the problem. After reading and listening to the question, pause the video and try to solve the question. Question 1. In which figure is the number of black kangaroos larger than the number of white kangaroos? So if we take a look at the first option, A, we can count out the number of black and white kangaroos. We see that there are 1, 2, 3, 4 white kangaroos and 1, 2, 3 black kangaroos. So this means in letter A, there are more white kangaroos than there are black kangaroos. So this will not be the answer. Answer B, then we can start counting. There is 1, 2, 3, 4 white kangaroos as well. And for black kangaroos, there are 1, 2, 3, 4 black kangaroos. Again, we are not given the correct solution since the number of white and black kangaroos is the same. Moving on to answer C, we start counting 1, 2, 3, 4 white kangaroos and 1, 2, 3, 4 black kangaroos, the same as letter B. So this will not be our answer. Moving on to D, we can count out 1, 2, 3, 4 white kangaroos and then 1, 2, 3, 4, 5 black kangaroos. So in answer D, there are more black kangaroos than there are white kangaroos. So the answer will be letter D. Question 2. Alleyne writes a correct calculation. Then she covers two digits, which are the same with stickers. See the picture. Which digit is under the stickers? So we have 40 something plus 50 something equals 104. Now, once we take this question, we can say 40 plus 50 plus a number plus another number equals 104, since this is the same. And we know that both the new numbers will have to be the same number. This is the same as 2 times a number plus 90, which is 40 plus 50 equals 104. We subtract 90 from both sides, we get 2 times the number equals 14. And then to get our mystery number, we just have to divide both sides by 2. Our mystery number is 7. So the answer is letter D. Question 3. In what way should the last four circles be shaded so that the pattern is continued? So looking at the pattern, we start off with one black circle, then one gray circle, followed by three black circles, then three gray circles, and then four black circles. So we want to know which ones will be next. If we look at the pattern, we notice that it goes black, white, black, white, and so on. Now, we also know that the previous number of black circles will be telling us how many gray circles there are. So if there are three black circles, then there will be three gray circles afterwards. Knowing this, we have four black circles, so we must end up with four gray circles. And this is shown in answer E. So the correct answer is letter E. Question 4. How many triangles can be seen in the picture to the right? If we take a look at the picture to the right, we can outline each triangle. We have 1, 2, 3, 4, 5, 6, 7, 8 small triangles. However, we can find some more triangles if we look larger. There's 1 and 2. So 8 and 2 give us a total of 10 triangles. So the correct answer is letter B. Question 5. In the London 2012 Olympics, USA won the most medals, 46 gold, 29 silver, and 29 bronze. China won the second most medals with 38 gold, 27 silver, and 23 bronze medals. How many more medals did USA win than China? Now if we make a table for China and the USA, we can look how many medals each won. We know that China won 38 gold, 27 silver, and 23 bronze. USA won 46 gold, 29 silver, and 29 bronze. Now we want to figure out the difference. So 46 minus 38 will give us 8. That means the US won 8 more gold medals than China. We won 2 more silver medals than China. And finally, the USA won 6 more bronze medals than China. Now we want to total up the number of medals that the US has more than China. So 8 plus 2 plus 6. When we do so, we get our answer. 16. Letter C. Question 6. Daniel had a package of 36 pieces of candy. Without breaking any pieces of candy, he divided all the candy equally among each of his friends. Which of the following was definitely not the number of his friends? We know that Daniel has 36 pieces of candy and that none of them are broken and all candy is split evenly between his friends. We assume that Daniel has two friends and we can split the candy 36 divided by 2, meaning each friend would receive 18 pieces of candy and they would be equally split. So letter A cannot be an answer. If we assume that Daniel has three friends, then we do 36 divided by 3, giving us 12. Once again, there is an equal split. If we divide by 4, assuming there are 4 friends, we get 9 pieces of candy each. This means that C cannot be the answer. If we divide by 6, assuming there are 6 friends, each friend would get 6 pieces of candy. The only answer left is 5. 36 cannot be divided by 5, so the answer is letter D. Question 7. Vera's mom prepares sandwiches with two slices of bread each. A package of bread has 24 slices. How many sandwiches can she prepare from two and a half packages of bread? Now, we know that one package has 24 slices. We want to figure out how many slices there are in two and a half packages. We can do so by multiplying 24 slices by 2.5, and doing so means that two and a half packages will contain 60 slices of bread. Now, we want to find out how many sandwiches 60 slices makes. We know that two slices of bread make one sandwich. Now, if we multiply both sides by 30 so that we get 60 slices, we get 30 sandwiches. So, with two and a half packages of bread, Vera's mom can make 30 sandwiches. So, the answer is letter B. Question 8. About the number 325, five boys said, Andy, this is a three-digit number. Barry, all the digits are different. Charlie, the sum of the digits is 10. Danny, the one's digit is five. Eddie, all the digits are odd. Which of the boys was wrong? So, if we look at the number 325, we know that Andy is telling the truth, and he is correct, since it is a three-digit number, 3, 2, and 5. So, we can cross off Andy. Barry says that all the digits are different. We can see that all of them are different, in fact, 3, 2, and 5. So, we can cross off Barry. Charlie says that the sum of the digits is 10. We do 3 plus 2 plus 5, we get the sum of 10. So, Charlie is also correct. We can cross him off. Danny says the one's digit is five. If you look at the one's digit, we see that it is five. So, Danny is correct. Finally, all we have left is Eddie. Now, Eddie says that all the digits are odd. This is incorrect, since while three and five are odd numbers, two is an even number. So, the boy that is wrong is Eddie. Answer E. Question 9. A rectangular mirror was broken. Which of the following pieces is missing from the picture of the broken mirror? If we draw out a rectangle to signify the original mirror before it was broken, and then we take the broken mirror and draw this design, we can see what the piece missing looks like, outlined in yellow. Now, if we take away all of the rest of the mirror, we're left with what the piece looks like. If we simply just rotate it 180 degrees, we see that it looks exactly like the missing piece. Our answer will be letter B. Question 10. Each time Pinocchio lies, his nose gets six centimeters longer. Each time he tells the truth, his nose gets two centimeters shorter. After his nose was nine centimeters long, he told three lies and made two true statements. How long was Pinocchio's nose afterwards? Now, we know that Pinocchio's nose gets six centimeters longer every time he lies, and two centimeters shorter each time he tells the truth. We start off with it being nine centimeters long, and he tells three lies, meaning we add six centimeters one, two, three times. Now, nine plus six plus six plus six gives us 27. So, after Pinocchio tells three lies, his nose will be 27 centimeters long. However, we also know that he made two true statements, so we can take away two centimeters and another two centimeters. So, we do 27 minus 2 minus 2, which gives the total length of Pinocchio's nose, which is the answer 23 centimeters, letter D. Question 11. In a shop, you can buy oranges in boxes of three different sizes. Boxes of five oranges, boxes of nine oranges, or boxes of ten oranges. Pedro wants to buy exactly 48 oranges. What is the smallest number of boxes he can buy? We want to buy 48 oranges. We need to do this by buying boxes of the largest amount of oranges. So, if we start off, and let's say we buy five boxes of ten oranges, we would end up with 50 oranges, which is too many oranges. However, if we bought four boxes of ten oranges, we would end up with 40 oranges, which isn't enough. So, if we did buy three boxes of ten oranges, we would get 30 oranges. However, if we bought nine more oranges in a nine orange box, plus nine more, we would end up with exactly 48 oranges. So, the smallest number of boxes Pedro can buy to have 48 oranges is five, letter D. Question 12. Anne starts walking in the direction of the arrow. At every intersection of streets, she turns either to the right or to the left. First, she goes to the right, then to the left, then again to the left, then to the right, then to the left, and finally again to the left. Then, Anne is finally walking towards. So, to solve this problem, we just have to look at what the diagram looks like. We know that Anne is starting off in the direction of the arrow. First, she will turn to the right, like so. Then, she turns to the left, she gets to the basket, and she turns to the left once more, once more, during which she turns to the right, gets to the traffic light. Then, she turns left at the envelope, and finally, she turns left one more time, reaching the train. So, this means that Anne will be walking towards the train, letter A. Question 13. Classmates Andy, Betty, Kathy, and Danny were born in the same year. Their birthdays were on February 20th, April 12th, May 12th, and May 25th. Not necessarily in this order. Betty and Andy were born in the same month. Andy and Kathy were born on the same day of different months. Who of these schoolmates is the oldest? We make ourselves a little table with all the different dates. February 20th, April 12th, May 12th, and May 25th, we can start putting in different people in different boxes. Start off by putting Andy and Betty in both the May boxes since we know that Betty and Andy were born in the same month. Now, Betty could have been born on May 12th and Andy on May 25th. However, the reverse could have been possible as well. Andy being born on May 12th and Betty on May 25th. Next, we know that Andy and Kathy were born on the same day of different months. So that means we can put Andy in April 12th and Kathy in May 12th. And we can also do the reverse and put Kathy in April 12th and Andy is already in May 12th. Finally, we have one person, Danny, who wasn't mentioned at all. And we have February 20th, which doesn't fit any of the descriptions. This means that Danny will be born on February 20th. And also, we notice that Andy and Kathy are born on the same day of different months. However, Betty and Andy are born in the same month. So, Andy cannot be born on April 12th. So, we can put Danny in the box for February 20th and cross Andy out of April 12th. Now, this means that Kathy was born on April 12th. We know that Andy and Kathy are born on the same day and that means Andy had to have been born on May 12th. So, that means Andy will be born on May 12th and that leaves Betty for May 25th. So, we want to find out which of the schoolmates is the oldest. The correct answer will be D, Danny, since she was born on February 20th, the earliest date out of all four schoolmates. Question 14. 30 children going to Adventure Park took part in at least one of two events. 15 of them took part in the moving bridge contest and 20 of them went down the zipline. How many children from Adventure Park took part in both events? Now, we know that there were 15 on the moving bridge and 20 went down the zipline. If we add this number together, we get 35. However, we know that we don't have this many students since there are only 30 children. This means that if we take 35 and subtract 30 from it, we will get our answer. 35 minus 30 gives us our answer of 5. Answer E. Question 15. Which of the following pieces fits with the piece in the picture to the right so that together they form a rectangle? So, if we start off by drawing a rectangle, we can place the piece that we are given on the right of the picture like so. All we have to do is fill in the empty spot so that we have a rectangle. Doing so, we would get a shape that looks as such. Now, this is the same shape that we see in one of our answers and that is our answer. Letter B. Question 16. The number 35 has the property that it is divisible by the digit in the ones position because 35 divided by 5 is 7. The number 38 does not have this property. How many numbers greater than 21 and smaller than 30 have this property? If we draw a number line with the ends of 21 and 30, we can list out all the numbers and test if they have this property. So, after 21, we'll have 22, then 23, 24, 25, 26, 27, 28, 29. So, to test the property, we just have to divide the number by the number in the ones digit. So, starting off with 22, we will divide by 2. Doing so, it gives us 11. So, this number does have this property. If we divide 23 by 3, we do not get a whole number. It does not have this property. 24 divided by 4 gives us 6. It will have the property. 25 divided by 5 also has this property since we received the number 5. 26 cannot be divided by 6. 27 is not divisible by 7. 28 cannot be divided by 8, and 29 cannot be divided by 9. So, that means there are only three numbers with this property that are greater than 21 and smaller than 30. So, the answer is 3. Letter B. Question 17. Joining the midpoints of the sides of the triangle in the drawing, we obtain a smaller triangle. We repeat this one more time with the smaller triangle. How many triangles of the same size as the smallest resulting triangle fit in the original drawing? So, if we start off with the triangle, and then we join the midpoints of the sides of the triangle, we get something like this. And then we have to do it one more time, resulting like so. Now, we want to know how many of these smaller triangles will fit in the large triangle. So, we just need to redraw these triangles like so, connecting the midpoints. Now, all we have to do is count how many triangles we ended up with. So, we have 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, and 16 small triangles. So, the answer will be letter D. Question 18. After the 1st of January, 2013, how many years will pass before the following event happens for the first time? The product of the digits in the notation of the year is greater than the sum of these digits. So, what the question is asking is, for example, in the year 2013, if we multiply the three digits, which number would be greater? The product of multiplying, or the sum of adding all the different digits? So, we know that the year is 2013, and we want to know when this will happen for the first time after this. So, if we look at 87 years in the future, we get the year 2100. Now, if we multiply all these digits, we get 2 times 1 times 0 times 0, which means the product will be 0, and the sum will be 2 plus 1 plus 0 plus 0. So, the sum is 3. So, the product is not greater than the sum. Moving on, 98 years in the future, we get the year 2111. The product in this will be 2, or the sum will be 5. So, once again, this is not the solution we're looking for. We go 101 years in the future, and we get 2114. The product will be 2 times 1 times 1 times 4, or 8. The sum will be 2 plus 1 plus 1 plus 4, also 8. Here, the product and sum are equal. This is still not the answer we're looking for, since the product is not greater than the sum. 102 years in the future, we get the year 2115. We multiply these out, we get 2 times 1 times 1 times 5, or 10. And if we take the sum, we get 2 plus 1 plus 1 plus 5. That gives us 9. So, 102 years in the future, in the year 2115, the product of the digits will be greater than the sum of the digits of that year. So, that means the correct answer is 102, letter D. Question 19. In December, Tasha the cat slept for exactly 3 weeks. How many minutes did she stay awake during this month? Now, we know that December has 31 days in it. We know that Tasha the cat slept for 3 weeks. So, we can do minus 3 weeks. However, we cannot subtract weeks from days. So, to make weeks into days, we can multiply by 7. So, we get 31 minus 3 times 7 equals the days awake. We want to know how many minutes Tasha stayed awake for. So, to do this, we have to multiply the days awake by 24 hours, because each day has 24 hours. However, we still want to have the answer in minutes. Knowing each hour contains 60 minutes, we can just multiply this once more by 60 minutes. So, we get our solution. We get 31 minus 3 times 7, then multiplied by 24, and once more multiplied by 60. When we see this answer, this is answer letter B. Question 20. Basil has several domino tiles, as shown below. He wants to arrange them in a line according to the following domino rule. In any two neighboring tiles, the neighboring squares must have the same number of dots. What is the largest number of tiles he can arrange in this way? Now, we have domino tiles that have 1 and 2, 1 and 3, 1 and 4, 2 and 3, 3 and 4, 2 and 6, and 4 and 5. We notice that the last two tiles, 2 and 6, as well as 4 and 5, only have a 5 and a 6, which doesn't repeat anywhere else. That means that these domino tiles would only be able to be placed as n tiles. So, these will be useless, since we already have an odd number of each other domino. Now, if we set up our dominoes, like so, starting off with the 1 and 2, we can attach a domino with the 2 tile and the 3 tile on the other side. Next, we will place down our domino with the 3 and 4, and then we can take our domino with the 4 and 1 and flip it upside down, so that way it can connect like so. And finally, we can connect our domino with the 1 and 3. This gives us the largest number of dominoes that we can arrange, using the domino rule, 5. So, the answer will be letter C. Question 21. Christy has to sell 10 glass bells, which vary in price. $1, $2, $3, $4, $5, $6, $7, $8, $9, and $10. In how many ways can Christy divide all the glass bells into 3 packages, so that each of the packages has the same price? If we take the total of the 10 bells, and we add 1 plus 2 plus 3 plus 4 plus 5 plus 6 plus 7 plus 8 plus 9 plus 10, we get a value of 55. Now, this means that the total value of all 10 glass bells is $55. However, 55 cannot be divided by 3 into equal numbers. So, that means it is impossible for Christy to divide all the glass bells into 3 packages, with each package having the same price. So, our answer will be letter E. Question 22. Peter bought a rug 36 inches wide and 60 inches long. The rug has a pattern of small squares containing either a sun or a moon, as can be seen in the figure. You can see that along the width there are 9 squares. When the rug is fully unrolled, how many moons can be seen? Now, if we take 36 inches divided by 9 squares, we get 4. That means each of the squares has a side length of 4. So, if we take a closer look at the diagram, we know that the rug will be 36 inches wide and 60 inches long. If each of the squares is 4 inches on each side, then we can start dividing up each of these. We know that within 8 inches within the length, we will have 9 moons. Now, we just have to repeat this 7 times, which will mean 56 inches. Now, if we multiply, we get 56 inches is equal to 9 times 7 moons, which is how many moons will be in the rug if it was 56 inches by 36 inches. Then, we just have to add 4 more moons for the remaining 4 inches. So, we get 63 plus 4 moons, giving us our total of 67, or letter B. Question 23. Baby Roo wrote down as few numbers as possible, using only the digits 0 and 1 to get 2013 as the sum. How many numbers did Baby Roo write? Now, since Baby Roo is only using the digits 0 and 1 to get 2013, that means Baby Roo has to use at least 3 different numbers to get the number totaling 3. So, it'll be something ending with 1 plus something ending with 1 plus something ending with 1 to get a value that is something ending with 3, in our case 2013. To get to 2013, we can do 1011 plus 1001 plus 1. This gives us the sum of 2013 using as few numbers as possible. This means that Baby Roo wrote down 3 numbers, letter B. Question 24. Beatrice has many pieces like the gray one in the picture. At least how many of these gray pieces are needed to make a solid gray square? If we start taking these gray pieces and arrange them in such a fashion, then we find out that we can make a 6 by 6 solid square by using 4 pieces. So, that means our answer will be letter B.

Math Kangaroo

Lucas Nielskowski

problem-solving

mathematics

video library

Levels 3 and 4

2013 competition