Hello and welcome to the Math Kangaroo Media Library. You are about to view interactive solutions to levels 3 and 4 of the 2011 competition. You will likely notice that some of the solutions presented here are slightly different from the suggested solutions you may have already reviewed. So as you follow along, compare your own solutions and the suggested solutions with the current presentation. Please make sure that you understand the differences. If you have any questions or comments, feel free to contact me at the address provided here at the bottom of the screen. My name is Luke and I'm a past Math Kangaroo participant and I hope that this presentation you will find useful in preparing for your next Math Kangaroo competition. Solution number 1. Which of the numbers below is the greatest? We can simply compute the sum or difference in each of the given expressions and find our largest quantity. So looking at A, we will add 20 to 11. So we have 0 plus 1 in the first column, that's 1. And then moving on to the left, 2 plus 1 gives us 3. So 31 is the answer and we see that the answer in B has to be less than the answer in A because now we're subtracting the 11 from 20. So we have 20, subtract 11, we make this 2 a 1, make this 0 a 10. So we obtain a 9, 1 minus 1 is a 0. When we move on, we see that in C, that's the number 20 together with two 1's. And when we make the addition, 0 plus 1 plus 1 is 2, we obtain a 22. In D, this is subtraction now. So 20 take away 2 is 18 and here in E, we have a 2 plus 0 plus 1 plus 1. That sum over here is equal to 4. So the answer in A gives us the greatest number of all the possible choices. We choose the answer 20 plus 11, A. Question number 2. Michael is painting the word kangaroo on a poster. Each day he paints one letter. He painted the first letter on a Wednesday. What day of the week will it be when he paints the last letter? Here I have written out the days of the week and we can simply do what Michael does and start writing one letter per day successively until we are done writing the word kangaroo. So beginning with the K, the first letter on a Wednesday, we have a K, then the next letter gets painted on a Thursday, A, Friday gets the letter N, Saturday is a G, Sunday again an A, and following that we have on Monday the letter R, and Tuesday gets an O, so does a Wednesday. And that's the whole word kangaroo. So Wednesday is the right day, the day on which Michael will be finished painting the whole poster. Alternatively, we can see that the word kangaroo consists of 1, 2, 3, 4, 5, 6, 7, 8 letters. So since the day that he begins painting on is a Wednesday, that's his first letter, and then he needs 7 more days to finish painting, and that's exactly one week. Seven days is one week, so starting on a Wednesday, he will also have to finish on a Wednesday. So there are two ways at least to solve this problem, but either way we choose the correct answer as C, Wednesday. Question number 3. Which of the stones below needs to be added to the box on the right side of the scale right here in order for the two boxes on the scale to weigh the same? And we see that the numbers on the stones indicate their weight in kilograms, and so what we have is an imbalance on the left-hand side of the scale. We have more weight than on the right-hand side of the scale, and our job would be to add just the right-sized weight stone to balance the scales so they're even here. So first, let's find out how much weight is in the box on the left, and we add up individually all the weights, we have 26 kilograms, 8 kilograms, and 12. We add that together, 2 plus 8 is 10 plus 6, we have 16, carry over the 1, and we have 46 kilograms on the left-hand side. On the right-hand side, we do the same, we have a 20-kilogram stone, a 17-kilogram stone, and adding those together, we obtain 7 plus 30, 37 kilograms, and what we need to make sure is that we add just the right amount here to the right-hand side, so out of 37 kilograms, we obtain 46 kilograms. So let's subtract the difference, we need the difference between 46 and 37, so we can subtract that, here we have to make this a 3 and this a 16, 16 minus 7 is 9, and then the 3s cancel, so that's our missing weight. So here we need to throw in a 9-kilogram stone, which is the stone pictured in C, and that is our answer, C. Question number 4. The train to Atlanta leaves 3 and a half hours from now. Paul got up 2 and a half hours ago. How many hours before the train leaves did Paul get up? Let's sketch a little diagram to keep track of this information. We can have the time now, and in the future, we know that 3 and a half hours away, so plus 3.5, we can represent our time like this, as a decimal, 3.5 hours from now the train leaves. And so now we keep adding on to this diagram, from now, 2 and a half hours in the other direction, minus 2.5 hours, that's when Paul got up. So we can say Paul wakes. And our job here is to compute the difference in these times between when the train left or is going to leave, and between the time that Paul got up. So we have to subtract. We need the difference again. And so what we have is 3.5 hours. We subtract negative 2.5 hours, and that gives us 3.5 hours plus 2.5 hours. And when we add that up, we have 3 plus 2, 5 hours, a half plus a half plus one more hour, for a total of 6 hours. And we can now answer the question, we know exactly that from the time the train left, it has been 6 hours since Paul woke up. So that's answer E. Question number 5. A toy is placed in one of the squares of a grid, as shown in the picture. So we have a 4 by 4 grid, and here in one of the cells is a kangaroo toy. Now a child moves the toy from one square to the next. He first moved it one square to the right, then one square up, then one square to the left, then one square down, and then again one square to the right. Which of the following pictures shows where the toy will be at the end? So to solve the problem, we have to follow the same pattern as the child and move the same toy in exactly the same way as he did, and I can do that here for you on the computer. I can make a copy of the board with the kangaroo, and I can pick up the kangaroo and move him exactly in the same way the child does, and we follow the directions now. We see that he first moved it one square to the right, so we do that, one square to the right. He then moved it one square up, then one square to the left, then one square down. We end up at our initial position, but there is one move remaining, and we know that the last move is again one square to the right. So now we leave the kangaroo alone, and we see that this position here is exactly the configuration we see in picture B. So that has to be our answer. The kangaroo ends up exactly in the spot here in diagram B after he is being done, played with. Question number six. Ala, Lenka, and Miso went out for dessert. Lenka paid $4.50 for three scoops of ice cream. Miso paid $3.60 for two cookies. How much did Ala pay for one scoop of ice cream and one cookie? If we divide the total price paid for ice cream by the amount that was purchased, we can find out the price of one scoop. So we have that three times one scoop is equal to exactly $4.50. So dividing, we see that one scoop has to be equal to exactly $1.50. We do the same thing for cookies. We have a total of two cookies being purchased. So two times one cookie comes out to $3.60. And so dividing by two, we see the price of one cookie is exactly $1.80. So now what we have to do is add these prices together for one ice cream and one cookie. And that is exactly how much Ala spent on dessert. So we have Ala's dessert here, or Ala's bill, comes out to $1.50. That's for one scoop of ice cream. Add that to $1.80. That's for one cookie. And we have then addition, zero plus zero is zero, five plus eight gives us 13. So copy the three, carry over the one, and that gives us a total of $3.30. So that is the amount in A. A is the answer. To question number six. Question number seven. Susan described one of the figures below. She said, this figure is gray and is not a rectangle. Which figure did Susan describe? So we have figures that are either shaded gray or they are not shaded at all. So let's first make sure that the figure we are looking for is gray. That means we get to discount all the figures that are not gray. So in A we have a hexagon that is not shaded. In C we have a square that is not shaded. And in D we have a circle that is not shaded. But our figure has to be shaded so these three cannot be what Susan described. Now moving on, we are down to two shaded figures. And we also know that what we are looking for cannot be a rectangle. And that's exactly the figure in E. This is a rectangle and our figure is not. Which leaves us with exactly one choice. And now we know that Susan was describing, in fact, this gray triangle. So that is the answer we choose, B. Question number 8. A tower clock strikes on the hour as many times as the hour. So for example, it strikes 8 times at 8 o'clock, 9 times at 9 o'clock, and so on. It also strikes once when it is half past the hour. That is at 8.30, 9.30, 10.30, and so on. How many times will the clock strike between 7.45 and 10.45? So let's keep track of time somehow. What I will do is I will start at 7.45 and move 15 minutes at a time until it is 10.45. And in between, whenever there is an opportunity for the clock to strike, we will count how many times it actually strikes. So beginning with 7.45, we have no clock strikes all the way up until 8 o'clock. And at 8 o'clock exactly, we will have 8 clock strikes. So let's count clock strikes. So here we have 8 of them. And then moving on at 8.15, nothing happens. But at 8.30, past the hour, we have exactly 1 clock strike. And we keep going. Next is 8.45, at 9 o'clock. At the hour, we have as many clock strikes as the hour, so we add 9 to our count. And we keep going at 9.15, nothing happens. At 9.30, half past the hour, we have plus 1 clock strike. And then moving on, we have 9.45, 10. And we have to add 10 clock strikes. 10.15, nothing happens. 10.30, we have another clock strike at half past the hour. And then we end at 10.45. So those are all the clock strikes that we have. And we can add them up. We have 10 plus 9 plus 1, so that's 20. 8 plus 2 is exactly 30 clock strikes. And that is answer D.

Math Kangaroo

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competition 2011

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