This is the Math Kangaroo Solutions Video Library, presenting solution suggestions for Levels 3 and 4 from the year 2019. These solutions are presented by Lucas Naleskowski. 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 on your own. Question 1. The higher the step on the podium, the higher the rank of the runner. Who finished 3rd? If we take a closer look at the podium, we see there are 5 runners. We know that the person that is highest on the podium is in first place, since that is the highest rank. So we just see who is on the highest podium. It's D, so this is the first person who finished the race. Next, B is the second highest, so B finished the race second. Then we have E, who finished the race third, C, who finished the race fourth, and A, who finished the race fifth. Since we wanted to know who finished third, we just go back and see, and we notice that our answer is E. Question 2. In the pictures, each dot stands for 1, and each bar stands for 5. For example, 3 dots and 1 bar stands for 8. Which picture stands for 12? Now, we can break 12 down to 2 plus 10. This means we will have 2 dots, and since 10 is made up of 2 fives, 2 bars. So with this, we know our answer is C. Question 3. Yesterday was Sunday. What day is tomorrow? We make ourselves a calendar, starting with Sunday being the first day, then Monday, Tuesday, Wednesday, Thursday, Friday, and Saturday, and we can go ahead and proceed with the problem. If we know that yesterday was Sunday, then we know that the day today is Monday. We want to know what day is tomorrow, and the day after Monday is Tuesday. So our answer is A, Tuesday. Question 4. There are two holes in the cover of a book. When the book is open, it looks like this. Which picture does Olaf see through the holes when he closes the book? Take a closer look at the book. We can start labeling which square will correspond with which square on the cover. So when we close the book, squares 1 on each side will cover each other. And so we can keep doing this like so, seeing which part of the book will overlap. Now, if we look at squares 2, 3, 7, and 6, we will see when the book is closed, they will cover the book. So squares 2 and 3 will cover the red car, and squares 6 and 7 will cover the other car. We are left with the blue, orange, and pink cars when the book is closed, which means that our answer is D. Question 5. Karina cuts one piece like this, from the sheet shown to the right. Which piece can she get? If we look at the sheet that Karina has, and we start to look at the different possibilities she can get by cutting out two squares next to each other, then we quickly realize that if she cuts this piece out, as outlined by the red, then we can get our answer A. Question 6. Three people walked across a field of snow wearing muddy shoes. In which order did they do this? If we take a closer look at the tracks of the three people, we notice that the largest footprints, the gray shoes with the circles, tread across both types of shoes. So that means they are the freshest footprints, and they will have to be last. Then if we take a look at the smallest footprints, the gray ones, while they have been treaded on by the larger shoes, they also are on top, or have treaded on the gray shoes with the checkerboard pattern. So this person was second. And if we notice the last person we haven't analyzed, their footprints have been stepped on by both people, so they must have been the first to walk across the snow. So this is what our order looks like, so the answer is A. Question 7. Pia is making shapes using the connected sticks shown in the picture. Which of the following shapes uses more sticks than Pia has? If we look at the sticks that Pia has, we must count all the segments they have. There are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Then if we count the number of segments each of the shapes in the answers, we can find out which one has too many. A will have 10. B also has 10. C is also 10. D is made up of 12. And E is made up of 10. This means that shape D uses more sticks than Pia has, so the answer is D. Question 8. What number should replace the question mark when all the calculations are completed correctly? If we take a look at our given calculation sheet, we can start off with 2 plus 1, which will give us 3. Then we can go ahead and do 0 plus 3, which gives us 3. Then 1 plus 1 will give us 9, so 1 plus 8 will give us 9. And finally we are left with 8 minus 3 equals question mark. And 8 minus 3 gives us our answer, B, 5. Question 9. Linda pinned up 3 photos in a row on a cork board using 8 pins. Peter wants to pin up 7 photos in the same way. How many pins does he need? If we look at the 3 photos that are pinned up, we notice that only 8 pins are used for 3 photos. Using this method, we know the first photo in the front will use 4 pins, and every photo after that will take up 2 pins. Now, if Peter wants to set up 7 photos using the same method, we must divide this up. We'll have the one first photo, and then we will have 6 additional photos. We know that the first photo will take 2 pins, and every additional photo takes 2 pins. So, we'll have to add 6 times 2, which will give us 4 pins plus 12 pins to set up 7 photos. So our answer is B, 16. Question 10. Dennis wants to remove one cell from the shape. How many of the shapes below can he get? If we take a closer look at the shape, we know we can only remove one cell at a time to get any of the given top shapes. If we remove the cell that is most on the right, we will get the first shape. If we remove the cell that is most on the left, we can get the next shape. And then finally, if we remove the top cell, we get the third shape. So out of this one shape, we have gotten 3 different shapes. So our answer is 3, C. Question 11. 6 strips are woven into a pattern as shown. What does the pattern look like from the back? We look at the given pattern, and we focus on the middle strip like so. We notice that the first yellow strip is behind the blue one, the second yellow strip is also behind the blue strip, and the third yellow strip is in front of the blue strip. And when we look at this shape from the back, these will all be reversed like so, with the yellow being in the front, the second yellow being in the front, and the last yellow being behind. And when we look at all our options, A, B, C, D, and E, we notice that only one of the options has the middle blue strip with two yellow strips in front of it, and that one is C, which is our answer. Question 12. The weight of a dog toy is a whole number. How much does one dog toy weigh? If we say that a dog toy is equal to D kilograms, then we can set up some equations. D kilograms is less than 12 kilograms, since the dog toy goes up on a scale when compared to a 12 kilogram weight. We know that two dog toys weigh more than 20 kilograms, since they are on the bottom of the scale. Now, with this piece of information, we can divide by two and get one dog toy is more than 10 kilograms. Looking at all our answers, only one is more than 10 kilograms, and that answer is E, 11 kilograms. Question 13. Sarah has 16 blue marbles. She can trade marbles in two ways. Three blue marbles for one red marble, and two red marbles for five green marbles, which is the maximum number of green marbles she can get. We draw 16 blue marbles. We can start switching them out for red marbles. Each three blue marbles will be transferred into one red marble, like so. We can do this five times. Now we have one blue marble left remaining, but one blue marble cannot be exchanged for anything. We also have five red marbles left, and we know that two red marbles can be switched for five green marbles. So if we take away two red marbles, we can get five green ones, and then we can do this one more time. With this, we are left with one red marble, one blue marble, and the rest are green marbles. There are no further exchanges we can make, so once we count up our green marbles, which is 10, we get the maximum number of green marbles, B, 10. Question 14. Steven wants to write each of the digits 2, 0, 1, and 9 in one of the boxes of the sum. He wants to get the largest possible answer. Which digit can he write instead of the question mark? So if we take a look at the equation, we have a three digit number plus a single digit number. Using the digits 2, 0, 1, and 9, we have to get the largest possible answer. So first off, we must put our largest number in the first digit slot, being a 9, since this number will be 900 something plus something. Next, in the 10 spot, we want to put our next highest number, a 2. Now when we get to the third slot, the single digit number, we will be adding 0 or 1. If we do 920 plus 1, or 921 plus 0, it will give us the same result regardless, 921. So to get the largest possible answer, 921, Steven will have to write either a 0 or a 1, or answer A. Question 15. A glass full of water weighs 400 grams. An empty glass weighs 100 grams. How many grams does a glass half full with water weigh? So first, we must find out how much the water itself weighs. To do this, we take 400 grams, which is a glass filled with water, minus 100 grams, which is the weight of the glass, which gives us 300 grams. This means that a glass full of water is 300 grams. Now we can divide this by half to find out half of a glass full of water. This gives us 150 grams. Since we know that an empty glass weighs 100 grams, we have to add 150 grams plus 100 grams to get the weight of half a glass of water plus the weight of a glass. This gives us our answer, which is D, 250. Question 16. Together, the apple and pear cost 5 cents. The banana and apple cost 7 cents, and the pear and banana cost 10 cents. How much do all three cost together? We'll call the banana B, the pear P, and the apple A. From the question, we know that A plus B equals 5, since together the apple and pear are 5 cents. B and A, banana and apple, together cost seven cents. P and B, pear and banana, cost ten cents. Now we can rewrite these formulas. A, apple, equals five minus P, pear. And A, apple, equals seven minus B, banana. And then if we set these equal to each other, we can get five cents minus the cost of a pear is equal to seven cents minus the cost of a banana. Or a pear is equal to a banana minus two cents. If we go back to our first formula and put that in there, then we can get banana minus two cents plus a banana is equal to ten cents. When we add the two and add the bananas together, we get two bananas cost 12 cents. We divide by two, we get a banana cost six cents. If we put that back into the formula of a pear equals a banana minus two, we get pear equals six minus two, or the cost of a pear is four cents. If we put that back into one of our first formulas, apple plus pear equals five, then we get apple plus four cents is equal to five cents, or apple equals one cent. Now we want to find out how much a banana, pear, and apple cost. And we can just substitute our numbers. We get one plus six plus four. And when we do this, we got our answer, which is the 11 cents. Question 17. Each shape stands for a different number. The sum of the three numbers in each row is shown to the right of the row. Which number does the star stand for? If we look at our grid with all our shapes and numbers, we can conclude the following. A circle plus a star plus a heart is equal to 15. A circle plus a circle plus a circle is equal to 12. If we divide 12 by 3, we do three circles is equal to 12, or a circle is equal to four. If we put that back into our first formula, we can get four plus a star plus a heart is equal to 15. Now we know that a star plus the heart will give us 11. If we look at our last formula, a star plus a heart plus a heart is equal to 16. We can substitute the star plus a heart equals 11 in and get 11 plus a heart is equal to 16, or a heart is equal to 5. Next, if we put the 5 back into that formula, we get star plus 5 equals 11. If we subtract 5, we get star and we get our answer is equal to 6. So the answer is E6. Question 18. Anna used 32 small white squares to frame a 7 by 7 picture. How many of these small white squares does she need to frame a 10 by 10 picture? To frame a 7 by 7 picture, Anna needs 4 times 7 white squares, which is 28, plus 4 more for each corner, so 32 in total. Following the same principle for a 10 by 10 frame, she'll need 10 times 4 white squares, which is 40, plus 4 more corner squares. So 40 plus 4 gives us our answer, which is C44. Question 19. The pages of a book are numbered 1, 2, 3, 4, 5, and so on. The digit 5 appears exactly 16 times. What is the maximum number of pages this book could have? So to solve this problem, we just have to go down numerically and make note of every 5 we see. So if we start counting 1, 2, 3, 4, 5, we take note that we just saw a 5 digit, and we keep going 6, 7, 8, 9, 10, 11, 12, 13, 14, 15. This has another 5 digit. If we keep doing this, we'll get to 25, 35, 45. Now once we're at 45, if we keep going 46, 47, 48, 49, 50. This has a 5 digit. 51, also 52, 53, 54, 55 has two 5 digits. 56, 57, 58, 59, and then 60 doesn't. So we can keep going. Now since we've already gotten our 16 5s, we can't get the 65, since that would be our 17th 5 digit. But since we want the maximum number of pages the book can have, we want to go one page before we reach 65, which would be 64. So that means the maximum number of pages the book with 16 5 digits would be 64. Question 20. A hallway has the dimensions shown in the picture. A cat walks on the dashed line along the middle of the hallway. How many meters does the cat walk? If we take a closer look at the hallway, it is important to start looking at how far the dashed line is from the walls. Since we know it is in the middle, we can start putting values. The first part of the hallway is 8 meters wide, so that means the dashed line is 4 meters away from the wall. The last hallway is 6 meters wide, so halfway in is 3 meters. And now the second, the middle part of the hallway, will be equal to 4 meters. This is because 40 meters minus 36 meters is 4, and then halfway in is 2 meters. With this we can get on to the problem. The first section will be 40 meters minus 2 meters, since that is the length of the first section the cat walks, or 38 meters. The second length will be 20 minus 4 plus 3, which is the length that we see 20 meters minus 4, the distance from the wall, plus 3, the distance from crossing on to the third part of the hallway. The last distance is simply 28 minus 2, or 26. To find out how many meters the cat walked, we just have to add these three values together, and this gives us our answer, which is 83, or E. Question 21. In a park there are 15 animals, cows, cats, and kangaroos. We know that precisely 10 are not cows, and precisely 8 are not cats. How many kangaroos are there in the park? Since we know there are 15 animals, cows, cats, and kangaroos, when we are told 10 are not cows, we know that 15 minus 10 gives us 5, so there has to be 5 cows. Since 8 are not cats, the rest are cats, so 15 minus 8 is 7, so there are 7 cats. Now that we know that there are 5 cows and 7 cats, we can do 15 total animals minus 7 minus 5 to get the number of kangaroos, and with that we get our answer, which is B, 3. Question 22. Mary has nine small triangles. Three of them are red, three are yellow, and three are blue. She wants to form a big triangle by putting together these nine small triangles so that any two triangles with an edge in common are different colors. Mary places some small triangles as shown in the picture. Which of the following statements is true after she has finished? So if we look at the triangle she's made, we're missing five more triangles. We've used two blue, one yellow, and one red, so that means we need two more red, one more blue, and two more yellow. Now we know that 5 must be red since it is next to a blue and a yellow triangle, so that is the only piece it can be. Next, we know that 4 has to be blue since it is adjacent to yellow and red. Next, we have spots 1, 2, and 3. However, we notice that since we've used three blues, none of these can be blue. Next, we can see that we've used two reds so far, so one of the slots will have to be red and we've only used one yellow, so that means two of those will have to be yellow. The only way to make this work and to have them not touching edges with a triangle of the same color is the yellows will have to be in spots 1 and 3. If we look at our answers, that is exactly answer E. 1 and 3 are yellow, so the answer is E. 1 and 3 are yellow. Question 23. There are five children, Alec, Bartek, Czarek, Darek, and Edek. One of them ate a cookie. Alec says, I did not eat the cookie. Bartek says, I ate the cookie. Czarek says, Edek did not eat the cookie. Darek says, I did not eat the cookie. And Edek says, Alec ate the cookie. One child is lying. Who ate the cookie? Now, since we know that only one of them is lying, let's look at Alec first. He says, I did not eat the cookie. Now, if he was lying, that means he ate the cookie. However, only one person ate the cookie. And since Bartek also says he ate the cookie, if Alec ate the cookie, he'd be lying. But that means Bartek would also have to be lying since they both could not have eaten the cookie. So that means Alec is telling the truth. He did not eat the cookie. But if we go down to Edek, who says Alec did eat the cookie, he is lying because we know that Alec did not eat the cookie. We go next to Bartek, who says, I ate the cookie. That means he's telling the truth since we've established that Edek is lying. So with this, we know that the one child lying is Edek. And that means Bartek, who said he ate the cookie, is telling the truth. So who ate the cookie? Bartek. Question 24. Emil started to hang up towels using two pegs for each towel as shown in figure one. He realized that he would not have enough pegs and began to hang up the towels as shown in figure two. Altogether, he hung up 35 towels and used 58 pegs. How many towels did Emil hang up in the way shown in figure one? So if we let T represent towels, and P represent pegs, then we can write out formulas for figure one and two. Figure one will be two pegs for every one towel. Figure two will be pegs are equal to towels plus one. Now, to solve this problem, we just start looking at the possible answers and crossing them out. So if we assume A is the correct answer, that means Emil used 12 towels in the figure one. So if you put in 12 for T, we have 12 towels and 24 pegs. Since we know that he used 35 towels in total, then he'd have to have 13 more towels in figure two, or 24 more pegs. If Emil used 12 towels in figure one, he used 23 towels in figure two. Altogether, that would mean he used 48 pegs. However, we know that he used 58 pegs, so A cannot be the answer. Next, we can assume that B is the correct answer. So let's say he used 13 towels in figure one, meaning 26 pegs, and also 22 towels in figure two, and 23 pegs, which again in total gives us 49 pegs, which cannot be the answer since we know he used 58 pegs. Next, let's assume Emil used 21 towels in figure one. That would mean 42 pegs. Then in figure two, that would mean he used 14 towels and 15 pegs, which gives us 57 pegs. While close, it is not the correct answer since we know he used exactly 58 pegs. So we can cross C off. Once we get to D, 22 towels, we say he started off with 22 towels in figure one. That's 44 pegs, and then 13 more towels in figure two, which is an additional 14 pegs. When we add 44 plus 14 pegs, we get 58 pegs, and 22 towels plus 13 towels is 35 towels. And since we know Emil hung up 35 towels and 58 pegs, this is our correct answer, D, which is 22.

Math Kangaroo

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Lucas Naleskowski

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math competition

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