Question 17. Natalie folded a piece of paper in half and cut out a shape, as shown in the picture to the right. Which of the pictures below shows the piece of paper after it was unfolded? So let's imagine that we have a sheet of paper that we fold, and then from the folded edge we cut away the following shape. We see that there are no breaks here in the cutaway shape. So if we were to unfold this sheet of paper, we expect that the hole that we made in it will not have any breaks in it. So unlike the situation in picture D or in C where there is a gap between the parts we cut away, and also in B, we have to obtain a nice figure that we cut away without any gaps in the missing region here. So that's why the shape in A is the correct figure. And because I'm working on a computer, I can show you exactly how this situation would look like. I can take this sheet of paper here, make a copy of it with the computer, scale it so it is easier to decide how it looks, move it over a little bit, and then make another copy of it to simulate the unfolding process. I will have to move it over, and imagine that after it is rotated like this, I am exactly unfolding it like so. And we see that after doing this operation, the picture I obtained looks very much like the picture in A. So that is another reason why we have chosen the correct answer. Question number 18. Mr. and Mrs. Taylor have three daughters. The youngest is five years old. The middle daughter is four years younger than the oldest daughter, and also six years older than the youngest daughter. How old is the Taylor's oldest daughter? Let's keep track of the situation here by looking at three daughters. So here they are. We have maybe the youngest one here on the left, the middle daughter is here, and the oldest one is on the right. So we know that the youngest is exactly five years old. And now let's read the problem again. We know that the middle daughter here is four years younger than the oldest daughter, but we have no idea how old the oldest daughter is. And at the same time, the middle daughter is six years older than the youngest. So now we can use that information to say that the middle is six years older, six plus five, six years older than the youngest daughter, and that gives us 11 years old. And finally, here we have the oldest daughter. She is, as we know from the problem, she is going to be four years older than her younger sister. So we have, on top of the 11 years, another four years, and that is 15 years old. And now let's read the problem one last time to make sure that we have all the information here correctly. The youngest daughter is five years old, we know that. The middle daughter is four years younger than the oldest daughter, so 11 versus 15 is four years younger. And at the same time, six years older than the youngest daughter, so 11 versus five is a difference of six years. So we have found the age of the oldest daughter. She is 15 years old, and the answer here is D. Question number 19. The flowers in the flower shop were kept in three vases. There were 16 flowers in the first vase, 11 flowers in the second vase, and 17 flowers in the third vase. The owner decided to sell only bouquets of five flowers each. After selling some bouquets, she noticed that she did not have enough flowers to make another bouquet. How many flowers did she have left? So let's imagine that we have some flowers in a vase, maybe these are 16 red flowers in one vase, and let's arrange them in groups of five so they're ready for sale in bouquets. And we have one, two, three, four, five, that's one bouquet, five more makes 10, 15, and 16. Let's go on to the next flowers, maybe these are blue, and in a separate vase there would be 11 of them. Now we have one, two, three, four, five, that's one bouquet, five more makes 10, and we have 11 with an extra flower remaining. And maybe purple flowers, there are 17 of them in the last vase. Let's arrange them into bouquets of five, that's one bouquet, two bouquets, three makes 15 flowers, and two remain. Now all the bouquets can be sold, but we have to pay attention to how many flowers are remaining, and we see that there are a total of one from the red flowers, one from the blue flowers, and two from the purple flowers, for a total of four leftover flowers. And these cannot be arranged and sold in a bouquet of five, so they are just simply left over, and the answer to the problem is, therefore, D, four flowers are left over. Question number 20. Simon has two identical aquariums. There are 26 quarts of water in one and 42 quarts of water in the other. I have drawn two aquariums to represent Simon's situation, and we have to answer the question, how many quarts of water does Simon need to pour from the second aquarium into the first in order to have the same amount of water in both? So we imagine that Simon is transferring water out of the aquarium on the right to the aquarium on the left. And we see that the difference between them can be computed as the difference between 42 and 26 quarts. So we do that, 42 quarts minus 26 comes out to exactly 16 quarts. And if we were to transfer 16 quarts out of where there is more water into where there is less water, we would exactly reverse the situation. One aquarium would still contain 42 quarts, and the other one would contain 26. So we want to transfer exactly half the difference so that we are pouring 8 quarts out of the 42-quart aquarium, and then here we would obtain after subtracting 42 minus 8, that gives us 34 quarts, and 26 plus 8 also is equal to 34 quarts. And then we would obtain equal volumes in both aquariums. So the key is to transfer half the difference between the volumes, and that is exactly 8 quarts, so the answer is D. 21. Fido the dog, Filomen the cat, and four monkeys together weigh 24 pounds. We also know that Fido and one monkey together weigh 11 pounds. In addition to that, Filomen the cat and two monkeys together weigh one pound less than Fido the dog and one monkey weigh together. Each of the monkeys weighs the same. And our question here is, how much does Filomen the cat weigh? So let's keep track of the animals here with some pictures. We're going to have a dog, that's Fido, a cat, that's Filomen, and then a monkey, which is not named. And now let's draw enough of them to represent our situation. If I have the dog, the cat, and one monkey, and then add to this three more monkeys, so like this, that's the cat, the dog, and four monkeys, I can group them together, and according to the first sentence in the problem, together these animals weigh exactly 24 pounds. Let's group them in another way. Just Fido, the dog, and a monkey together weigh 11 pounds. So let's have a copy of the dog, and let's have a copy of the monkey. Let's group them together, and now these together weigh exactly 11 pounds. We also know that together, Fido and one monkey are exactly one pound heavier than Filomen and two monkeys. So the cat and two monkeys, here we can find them, make copies of these pictures, and these together, according to the second sentence in the problem, Filomen and two monkeys together weigh one pound less than Fido and one monkey. So let's group in blue the cat together with two monkeys, like that, and then we can have a little equation. This is going to be not quite equal, because we know that these weigh less, so add one to the right-hand side of the equation and then we have exactly what we need okay so this tells us that the cat together with two monkeys must weigh 10 pounds so this is this is 10 10 plus 1 is 11 and that's how much the dog and the monkey weigh and then we can take away these animals from our initial collection in just the right way the dog and one monkey here in in red let's underline them the dog and one monkey that together is 11 pounds and then the cat let's do that in blue the cat and two monkeys together that's exactly 10 and the remaining one monkey we can solve for by subtraction so let's make a copy of a monkey and we see that in green this monkey is going to have the following weight we have to subtract from 24 the weight of all the animals grouped together in red so that's minus 11 and then also subtract from that the weight of all the animals in blue so that's minus 10 and that gives us after performing this calculation 24 minus 11 is 13 and then minus 10 from 13 we get 3 so each monkey weighs 3 pounds and now we can calculate how much filament weighs filament together with 10 monkeys weighs uh excuse me filament together with two monkeys weighs 10 pounds so if we subtract the weight of two monkeys here we would see that finally the cat by himself if we were to copy him over here finally the cat by himself weighs uh what we know is true when he's together with two monkeys minus the weight of two monkeys so minus 3 and minus 3 that gives us finally something we can compute 10 minus 6 is equal to 4 and we have our answer after all this calculating the cat weighs exactly four pounds so the answer here is b question number 22 anita clara michael and daniel had an apple eating contest the person who ate the most apples won daniel ate more apples than clara and michael ate fewer apples than anita we also know that daniel did not win so who ate the most apples let's keep track of our winners and our losers in the following way i'll have a table here where on the top of the table i will write in the names of the people who ate the most and on the bottom the people who ate the least apples and i have the following names there is a daniel then we have a clara we have a michael and we also have anita now let's arrange them in the order that makes sense according to the information that's given we see that daniel ate more apples than clara so in our chart here he should appear above clara and let's move him next to clara but higher than clara then we also know that michael ate fewer apples than anita so in our ordering from least to greatest michael should have eaten fewer apples than anita so he goes below in the ordering so at least we have so far the names arranged like this and then finally we know that daniel did not win so daniel has to be positioned in the final bookkeeping towards the bottom towards where the people that ate less apples should be written in and the remaining here names have to be written in closer to the top and so then we have our answer it turns out that anita appears towards the very top of the chart where the people who ate more apples should be and since she's the first we choose her as the winner she must have eaten the most apples and clara ate the least it turns out but the winner was anita the answer is a question number 23 what number do we need to put in the first square in order to get 100 as the result after doing all the operations shown below so what happens is we insert a number into this machine that we imagine first multiplies the number by 4 the answer is deposited here in the circle then we add 2 to this number and in the rectangle we have the sum and finally we multiply by 2 and obtain 100 so there are two ways to solve the problem we can either guess and check start with 11 then multiply by 4 add 2 multiply by 2 and see if the answer is 100 which is certainly a good way to solve the problem except possibly time-consuming we might have to check several answers so let's see how that would work with 11 if we let 11 be in the square then 11 times 4 is 44 plus 2 we would have 46 and then times 2 we would have 80 92 which is not the right answer so 11 cannot be the correct answer so then let's think what would happen if we worked backwards maybe that is a little quicker and that is another way to solve the problem so let's see what happens if we were to undo all the operations if i were to begin with 100 and instead of multiplying by 2 i would divide by 2 then 100 divided by 2 is 50 then instead of adding 2 i would subtract 2 and from 50 i would obtain 48 and finally instead of multiplying by 4 i would divide by 4 and 48 divided by 4 comes out to be 12 so 12 should be the answer but let's check our work let's erase here our previous calculation and let's try this again with 12 so if 12 goes in the box then 12 times 4 is 12 plus 12 plus 12 plus 12 24 plus 24 gives us 48 48 plus 2 is 50 and 50 times 2 is 100 so we see that either method would have worked except we would have to check 11 9 14 and finally arrive at 12 so it's the most time-consuming method working backwards gave us the answer quicker and then we checked our work anyway just to be extra sure but the answer comes out to be 12. Question number 24. Paul and John were building using identical cubed blocks. Paul made the building shown in picture 1 and picture 2 shows Paul's building as seen from above where we note that the numbers in the squares indicate how many blocks were stacked one on top of another as viewed from above then we know that picture 3 shows John's building as seen from above and our task here in this question is to figure out which of these answers is John's building so the way this works is that when Paul looked at picture 2 that was essentially his blueprint for building the construction in picture 1 and if we were to go and write in on top of each block the number of blocks that are stacked in a tower below that block we would get exactly the diagram in picture 2 so let's see how that works if I go to the very left here there is just one block and I have to put a one on top of it next to it is just one block stacked so I put a one on top of that but just on top of or to the north of that block I have two blocks stacked together so on top of here the tower of two I have to put a two next to it then is a tower of one a tower of one and a tower of one but this tower of three blocks has to be labeled with the three and so that's exactly the picture that we would see from above one one one one four ones next to each other at the very end is a three below that is a one and over here that's where the two comes from so let's study picture number three the highest tower that we see would be over here corresponding to three blocks stacked together so we notice that in picture a we have a tower of four blocks so that's not possible we also see that in picture d the highest towers are of height two so that is also not possible the question then reduces to figuring out which of the pictures in b and c represent the diagram in picture c both of them have a tower of three blocks at the very end and let's continue with our analysis here let's label all the towers with two blocks in picture b there are three of them and also in picture c there are three of them and we are supposed to have exactly three of them so so far we are undecided finally let's label all the single blocks we have one two three of them in picture b and one two three four of them in picture c but in our blueprint diagram we have exactly one two three single blocks so we have to also discount picture c and the only one that remains by the process of elimination is the diagram here in picture b so that is john's building

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