This is the Math Kangaroo Solutions Video Library, presenting solution suggestions for Levels 3 and 4 from the year 2015. These solutions are presented by Lucas Nowskowski. 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. Given these clouds, what will we get at the end if we start off with 2? So we start off with 2, and we subtract 0. So 2 minus 0 gives us 2. Next, we add 1, which gives us 3. Finally, we multiply by 5. So 3 times 5 will give us our answer, which is E, 15. Question 2. Eric has 10 identical metal strips. He used screws to connect pairs of them together into 5 long strips. Which strip is the longest? Since we know that each of the metal strips is identical in length, we just have to count the number of holes in between where the strips will overlap. So strip A only has 1, strip B has 5, strip C has 3, strip D has 2, and strip E has 4. Now the strip that is longest will have the least overlap. So the one with the least is the one with only 1, which is answer A. Question 3. Which number is hidden behind the square in the equation to the right? To do this, we look at our equation. Triangle plus 4 equals 7. If we subtract 4 from both sides, we get triangle equals 3. Now we take a look at our second equation, which involves the square, which is square plus triangle equals 9. And we know since the triangle equals 3, we can say square plus 3 equals 9. If we subtract 3 from both sides, we get our answer, which is square is equal to 6. So our answer is E, 6. Question 4. Which of the expressions below has the greatest value? So if we take a look at each of the expressions and solve them, we will find which one gives the greatest value. The first one, which is the difference between 1,000 and 100 divided by 10. We first do what is inside the parentheses and get 900 divided by 10. Solution is 90. The next is the difference between 1,000 and 100 divided by 9. We get 900 divided by 9, which gives us 100. The next will be 1,000 minus 1 all over 9, which gives us 999 divided by 9, which gives us the solution 111. The next is the difference between 1,000 and 10 all divided by 9, which gives us 990 divided by 9, or 110. The final is the difference between 1,000 and 10 all divided by 10, which gives us 990 divided by 10, or 99. The greatest value of these is 111. So the expression 1,000 minus 1 all divided by 9 will be the correct answer, also known as answer C. Question 5. We start drawing segments connecting every other dot on the circle until we are back at the number 1. The first two segments are already drawn. Which figure do we get? Take a look at the circle with the lines already drawn. We keep the pattern going, skipping every other dot and connecting them. We'll go from 1 to 3, 3 to 5, 5 to 7, 7 to 9, 9 to 2, 2 to 4, 4 to 6, 6 to 8, and 8 to 1, stopping at 1. When we look at all of the possible solutions, we see this only matches to one of them, which is answer E. Question 6. A certain whole number has two digits. The product of the digits of this number is 15. The sum of the digits of this number is blank. So since we know the whole number is two digits, and when you multiply these two digits, we get 15. Now, the only numbers that fill this requirement are 3 and 5. When we add 3 and 5 to get the sum of the digits, we get 3 plus 5, which gives us our answer, E, 8. Question 7. In the picture, we see an island with a highly indented coastline and several frogs. How many of these frogs are sitting on the island? We look at the picture. We can fill in the parts where the water is, like so. We know that the island has the tree, so that is the actual island. The frogs that are in the water will not be counted. So when we count the frogs that are on the island, or in the white background, we start off 1, 2, 3, 4, 5, 6. There is a total of 6 frogs on our island, so the answer is B, 6. Question 8. This year, March 19th, falls on a Thursday. What day of the week will it be in 30 days? So if we draw ourselves a little calendar, like so, going from Monday to Sunday, and we start off on March 19th, Thursday, we can write in 19 and Thursday. Now, we can keep adding days until we get to the 31st, which is the last day of March, in which case we reset and start going from 1. Now, when we do so, we just have to count 30 days away from Thursday the 19th. When we do so, we get to the 18th, which is a Saturday, so our answer is D. Question 9. My umbrella has kangaroo written on top. It is shown in the picture on the right. Which of the pictures below also shows my umbrella? If we look at the umbrella on the right, which has the kangaroo written down, K-A-N-G-A-R-O-O, from the top perspective, we can compare it to the different solutions. For example, solution B has G-A-N. However, when we look at the kangaroo umbrella from the top, there is no way that G-A-N go in that order. There is only K-A-N or G-A-R. The only umbrella that has this is N-G-A, which we see on the top right umbrella in the top half, N-G-A. So, the correct answer will be A. Question 10. Basil wants to cut the shape shown in figure 1 into identical triangles as shown in figure 2. How many triangles will he get? We take a look at the figures. We notice we can start putting in the small triangles in figure 2 into figure 1 as such. And if we continue to do this, and we start counting how many triangles we put in, we can see that we get a total of 15 triangles. And no more can be fit into this shape. So, our answer will be D, 15. Question 11. Louis had 7 apples and 2 bananas. He gave 2 apples to Yuri, who in return gave some bananas to Louis. Now, Louis has as many apples as bananas. How many bananas did Yuri give to Louis? So, if we just look at Louis' fruits, we notice that he has 7 apples and 2 bananas. Now, he gives 2 apples to Yuri, so he has 2 apples less, which means 5 apples left. And since Yuri gives him a number of bananas, so he has an equal number, Yuri has to give him 3 bananas. So, our answer is B, 3. Question 12. Grandma bought some candy. She gave each of her grandchildren 4 pieces of candy and had 2 pieces left. If she wanted to give each of them 5 pieces of candy, she would be 2 pieces short. How many grandchildren does she have? So, if we say G is for grandchildren and C is for candy, we can set up some equations. Say 4G plus 2 is equal to C, and 5G minus 2 is equal to C. Now, both of these are equal to C, candy, which means we can set them equal to each other. 4G plus 2 is equal to 5G minus 2. Now, when we simplify this, we get 4G plus 4 equals 5G. When we simplify this further by subtracting 4G from both sides, we get 4 equals G, which means there are 4 grandchildren, or answer B, 4. Question 13. In a speed skating competition, 10 skaters reached the finish line. The number of skaters that came in before Tom was 3 less than the number of skaters who came in after him. In which place did Tom end up in? If we take a look at the 10 spots and assume that they are lined up as such, our difference solutions are 1, 3, 4, 6, and 7. Now, if Tom is the red spot, and let's say we're testing for 1st place, this cannot be the case. Since we know that Tom had skaters in front and behind him, 3 less than the number of skaters came in before him than they did after him. If we take a look at the 3rd spot, this can also not be the case, since only 2 people came in before him. Now, if we look at the 4th spot, we see that 3 people came in before him, and 6 people came in after him, which is 3 less. So, our answer is going to be C, 4. Question 14. Joseph has 4 toys. A car, an airplane, a ball, and a ship. He wants to put them all in a row on a shelf. Both the ship and the airplane have to be next to the car. In how many ways can he arrange the toys so that this condition is fulfilled? Now, we can label these C for car, A for airplane, B for ball, and S for ship. Now, if we look at our shelf and make 4 spots, we can start off with the airplane in the first spot. We know that the airplane has to be next to the car, so the car will have to be in the next spot over. We also know that the ship has to be next to the car as well, so that will have to be in the 3rd spot. The last spot will be left for the ball. We can do the same thing, however, with just the ball on the other end of the shelf, like so. Or, we can position these, switching the position of the airplane and the ship, since they will still be next to the car. So, we start off with the ship, continue with the car, and then airplane, and then put the ball last. Or, do the same thing, however, put the ball first. These are the only ways to fulfill the conditions, so the answer will be B, 4. Question 15. Pete rides a bicycle in a park that has paths as shown in the picture. He starts from point S and goes in the direction of the arrow. At the first crossroad, he turns right. Then, at the next crossroad, he turns left. Then, right again. Then, left again. And so on, in this order. Which letter will he not pass? So, if we take a closer look at the diagram, and we know we start off at point S, so we will go this way, and get to point B. We know at the first crossroad, he will turn right, like so, passing E and C. Next, Pete will turn left, like so, after which Pete will start following his pattern, which is right, left, right, left, and so on. So next he turns right, and finally left, passing through A. Now we know that we end up where we started. And since he follows the same pattern every time, Pete will continue on this same stretch of road, not going in any other direction. The only place he doesn't come remotely close to is D, and that is our answer. Question 16. There are five ladybugs in the picture on the right. Two ladybugs are friends with each other, only if the number of spots that they have differ exactly by one. On kangaroo day, each of the ladybugs sent one text greeting to each of her friends. How many text greetings were sent? Now if we look at the five ladybugs, we can look at how many spots each one has. So the first ladybug has two spots. The next ladybug has three. The ladybug after that will also have three. The next ladybug has five spots, and the final ladybug has six. Now we know that these ladybugs will send text greetings only to their friends, and are only friends with ladybugs whose number of dots differ by one. So on kangaroo day, we know that the first ladybug with two dots will send a text to the first ladybug with three dots, and the other ladybug with three dots, and we know that they will do the same. This ladybug will send a text to the ladybug with two dots, and this ladybug will send a text to the ladybug with two dots. And there are also the ladybugs with five and six dots, which are also friends, since five and six are within one of each other. So the ladybug with five dots will send a text to the ladybug with six dots, and the ladybug with six dots will send a text to the ladybug with five dots, which will give us a total of six text greetings. So our answer is C. Question 17. The figure shown to the left is divided into three identical pieces. What does each of the pieces look like? If we take our shape, and we start trying to fill it out with the correct pieces, and we start off with trying A, then we can put them in like this, this, and this. And that is using three identical pieces A. So our solution is A. A. Question 18. Jack built a cube using 27 small cubes, which are colored either gray or white. No two of the small cubes, which are the same color, share a common face. How many white cubes did Jack use? Now if we look at the front of the cube, we see that the pattern looks like this. And since we know that no two cubes, which have the same color, share a face, then the next plane will look like this, having it alternated. And then the last plane in the back will look like this, like the first plane. Now to find out how many white cubes Jack used, we just have to count the white spots. There's 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13. So our answer is C, 13. Question 19. We can fill a certain barrel with water if we use water from six small pitchers, three medium pitchers, and one large pitcher. Or from two small pitchers, one medium pitcher, and three large pitchers. If we only use large pitchers of water, how many of them do we need to fill the barrel? So let's go ahead and start off by saying small pitchers will be S, medium pitchers M, and large pitchers L. Now we know that one barrel is six small pitchers, three medium pitchers, and one large pitcher. Or 6S plus 3M plus 1L. We also know that a barrel can be made by two small pitchers, one medium pitcher, and three large pitchers. Or 2S plus 1M plus 3L. Now since we know that both of these are equal to a barrel, these two values will be equal to each other. Next we want to simplify the equation. So we get 6S plus 3M plus L equals 2S plus M plus 3L. Next we can subtract two small pitchers from each side to get 4S plus 3M plus L equals M plus 3L. Afterwards we can now subtract one medium pitcher from each side. We get 4S plus 2M plus L equals 3L. Then we subtract L from both sides. 4S plus 2M equals 2L. And then we can divide both sides by two. So we know that one large pitcher is equal to two small plus one medium pitchers. This information we can go back to our equation 2S plus 1M plus 3L. And since we know 2S plus M is equal to L, we can put that in as L. So one large pitcher plus three large pitchers equals a barrel. So to fill a barrel we need answer A, four pitchers. Question 20. The numbers 2, 3, 5, 6, and 7 are written in the squares of the cross in such a way that the sum of the numbers in the row is equal to the sum of the numbers in the column. Which of the numbers can be written in the center of the cross? If we take a look at our cross, we can start putting in all the different numbers. If we start off with 2, then we have to set an equation. Disregarding 2, since it will be both in the row and column, we can just look at the other four numbers, which in this case are 3, 5, 6, and 7. And whenever we do this, we want to add the one smallest and largest numbers together, in this case 3 and 7, and then add the other two middle numbers, in this case 5 and 6 together, and see if they will be equal. In this case, if 2 is in the middle, then we will have 3 and 7 in the row and 5 and 6 in the column. However, 3 plus 7 does not equal 5 plus 6, so 2 cannot be in the middle. Another possible solution is putting 3 in the center. However, 2 plus 7 will not equal 5 plus 6. However, if we put 5 in the middle, we can have 2 plus 7 equal to 3 plus 6, so 5 can be a solution. Next, if we put 6 in the middle, we get 2 plus 7 is not equal to 3 plus 5, so this cannot be a solution. Finally, we have 7 in the center, and we have 2 plus 6 is equal to 3 plus 5, which means 7 can also be a solution. That means there are two possible solutions, and our answer is D, 5 or 7. Question 21. Peter has 10 balls, numbered from 0 to 9. He gave 4 of the balls to George and 3 to Anne. Then each of the three friends multiplied the numbers on their balls. As a result, Peter got 0, George got 72, and Anne got 90. What is the sum of the number on the balls that Peter kept for himself? So, we know that Peter will end up with 3 balls, George will have 4, and Anne will have 3. Now, since the multiplication of Peter's balls results in 0, Peter has to have the ball with the 0 on it. We also know that Peter has to have the ball with the 7 on it, since neither George nor Anne's value of 72 and 90 are divisible by 7. Next, Anne will have a 5, since her number is the only one divisible by 5, and George will have a 4, since his number is the only one divisible by 4. Next, we look at 90 divided by 5, which will give us 18. Now, we want to find out what this could possibly give us. This can either be achieved by 3 times 6 or 2 times 9, so that means George will have the 1, and George and Anne will either have 3 and 6, and the other will have 2 and 9, or it'll be reversed, and one will have 2 and 9, and the other 3 and 6. Regardless, this leaves Peter with only one more ball, the 8. When we add all of Peter's balls, 0 plus 7 plus 8, we get our answer, E, 15. Question 22. Three ropes are laid down on the floor as shown on the right. You can make one big, complete rope by adding one of the sets of rope ends shown in the pictures below. Which of the sets will make one complete rope? So, this is the first part we are given, and these are all our options. One of these will connect in such a way that it'll make one long, continuous rope. If we use option A, this will not create a continuous rope. It'll create two separate ropes, so this is off the table. Option B also does not complete one singular rope. It actually creates three distinct ropes. However, when we look at option C, we can follow the pattern with our fingers and notice that it creates one long strand of rope that loops all around, and each point is connected. So, the answer will be C. Question 23. We have three transparent sheets with the pattern shown to the left. We can rotate the three sheets but not turn them over. Then, we put them all one exactly on top of another. What is the maximum possible number of black squares seen in the square obtained in this way if we look at it from above? So, if we draw our three sheets, we can start trying to stack them on top of each other. If we stack the first sheet on the second sheet like so, it will cover an additional four spots, as many as it would possibly be able to. Now, the only way for the next sheet to cover more spots is like so, and this will leave one white spot and eight black squares. So, the maximum number of squares that can be covered is D, eight. Question 24. Anna, Berta, Charlie, David, and Alyssa were baking cookies on Friday and and Saturday. Over the two days, Anna made 24 cookies. Berta made 25, Charlie 26, David 27, and Alyssa 28. Over the two days, one of them made twice as many cookies on Friday. One made three times as many, one made four times as many, one made five times as many, and one made six times as many. Who baked the most cookies on Friday? So, we know Anna has 24, Berta 25, Charlie 26, David 27, and Alyssa 28. Now, Berta's is the only number that is divisible by five. So, when we do that, we get five cookies baked on Friday. Next, we know that David is the only one divisible by three, which gives us nine cookies on Friday. Anna is divisible by six, so it'll be four cookies, and Alyssa is the only one divisible by four, which is seven cookies. And finally, Charlie is the only one left, which is divided by two, and 26 divided by two gives us 13 cookies on Friday. So, who baked the most cookies on Friday? The answer is C, Charlie.

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