Question number 9. At the end of the skiing season, there were 12 pairs of ski boots left at the store. How many ski boots, counted one by one, were left at the store? We have to count pairs of objects, so let's recall that a pair is two items. And we have 12 pairs. Therefore, to count all individual items, we should multiply 12 pairs by 2 items per pair, and our equation for the total number of ski boots would look like this. We multiply 12 by 2, and we obtain 24 as the product, and we conclude that there are 24 boots altogether. And that is answer C. Question number 10. The picture below shows a puzzle with one piece missing. Which of the pieces below needs to be added to the puzzle in order for it to make a picture of a cat? So here in our puzzle, we have an almost complete cat. Its legs are visible, its tail is visible, its whiskers and nose are visible, its left eye and ear are here, but its right ear and its right eye are partly missing. So we should be looking for a piece that contains a little green and a little black to fill in this missing eye. That looks to us like the piece D and like the piece A. So now we have to decide which one is the correct choice. We see also that a piece of the cat's ear is missing from the puzzle, and the picture in D does not contain any portion of the cat's ear, but the picture in A does. So we must choose A as the answer to the problem. And now I can even check. Because I'm working on the computer here, I can move this piece. So imagine that you are cutting this out of your paper and moving it over to fill in the missing gap, and clearly this is not the right choice. Let's try the same idea with piece A, and if I move this piece into position, we see that this is exactly the missing piece of the puzzle. And so we choose A as the answer. Question number 11. Today is 3-12-2011. No item can be sold after the date shown below it. Which of these items cannot be sold? Let's look at each item and decide on the month and the year past which it cannot be sold. In our notation here, the month is indicated by the number to the very left before the first dash. So we have the cheese with an expiration date of the 9th month, which is September, and the year is still 2011. So past September of 2011, this item may no longer be sold. Moving on, in B we have a carton of juice, and the expiration date is set for the 3rd month of 2012, so that is March. March is the 3rd month, and the year is 2012. Now in item C, we have 7 coming first, standing for the 7th month, which is July, and the year is 2011. And finally, the last item may not be sold past the 2nd month of year 2011. So the 2nd month would be February, and the year is 2011. Now today is 3-12-2011, and that corresponds to March, the 3rd month of the year 2011. So any item whose expiration date falls before March of 2011 may not be sold. We see that the only item here in the list is item D that may not be sold past February of 2011. February comes before March, and that is the only month here in 2011 that does indeed come before March and is associated to some item for sale. So we choose D as our answer. Question number 12. In 36 years, Mark's grandmother will celebrate her 100th birthday. How old is Mark's grandmother now? We have to imagine that from the age of 100, we will subtract 36 years and thus arrive at Mark's grandmother's age. So let's write down the equation. We have 100 years old, and then we subtract 36 years, and we obtain grandmother's current age. So solving this equation, we have 100, we have to subtract 36, and to do that, we can write 36 as 30 plus 6 to make the subtraction a little bit easier. So the problem boils down to subtracting from 100, first 30, and then 6 so that altogether we subtract 36, and then we can group this into two operations. First subtract 30, then subtract 6. So 100 minus 30 is 70, and when we subtract 6 from 70, we obtain 64. So that is the grandmother's age, and we can check our work. We can say that 64 years plus 36 years is equal to, and then we have our calculation, 60 plus 4 plus 30 plus 6, that gives us 60 plus 30 plus 6 plus 4, that is 60 plus 30 gives us 90, 6 plus 4 is 10, and that really does come out to 100. So we have checked our work, and we're sure that grandmother's current age has to be 64 years. Question number 13. Anne has several dogs and four cats. The number of her cat's ears is equal to the number of her dog's paws. How many dogs does Anne have? To better keep track of the ears and the paws, let's draw a picture of a dog and of a cat. So here we go, and we know in addition to this that there are a total of four cats. So I will make four copies of the picture of the cat that I have here to represent all of the cats that belong to Anne. Now let's count the number of ears. So here we have the number of ears, and each cat has two ears. Two over here, two more, and two more. So to tally all of these up, we would say that's 2 plus 2 plus 2 and plus 2 for a total of 8. And we can even count them on our pictures, 1, 2, 3, 4, 5, 6, 7, 8. Now in addition to this, this number of ears has to match the number of paws on the dogs, and here we have a picture of a dog which has exactly four paws. But that is not enough to match eight ears, so let's have one more dog. So let's make a copy of this picture here, and now we have two dogs. Together we see that the dogs have 4 plus 4 paws, and that is 8. And it happens to match the number of ears on the cats. So we have our desired number of animals, and it turns out that with four cats, Anne must also have four dogs if the number of paws on the dogs and the number of ears on the cats match. So the answer is B, two dogs. Question number 14. To find her toy, Marie needs to follow the path which is marked by the following signs in this order. Which toy belongs to Marie? So what we have to do is imagine that we are here standing where Marie is standing and walking along these possible paths all the way from left to right until we reach the correct toy. And as we do that, we see that at some signs there are several different ways to proceed. So we will always try to choose the path that takes us closer to the toys. So let's begin. We start with the triangle here, of which there are two to begin with. And we have to proceed to the heart in several possibly different ways. So in first instance, we have one heart over here connected to the triangle. And in the second instance, we have two hearts connected to the triangle. So that's what we will do. We will imagine that we take these three paths all at once. Each of them takes us in the direction of the toys. Next is the square. So from our current position here, we have two different choices. From our position over here, we have one choice. And also below, we have one choice. But of these three squares, only one is closer to the toys. So I will choose to follow this path instead of the other two. And then we keep going. The triangle is next. And from my current position over here, I have just one triangle available. So I walk down. Next after the triangle is the heart. And the heart, again, I have two choices to make. I can go to the right or I can go to the left. I choose not to go to the left because that is farther away from the toys. So I go to the right. Next, the correct sign would be the square again. And again, I have two choices. I can go to the right or I can go up. So I will choose to go to the right as that is the closer path to the toys. So let's do that. Let's go to the right. Afterwards, I have a triangle to follow. Here is the triangle. And from my position here, I have a triangle. So that's where I go to the triangle. And the next item here is a heart. Now we kind of run into trouble. We see that there is no heart over here. So we have to backtrack. We have to backtrack to where we had a choice. And that was one, two steps back. We were at the heart and the next item was the square. We had two choices to go to the right or to go up. We went to the right but that was a bad choice. So let's go this time up. Up to the square. After that, again, the triangle is next. Here I have a choice of one triangle. That is where I will go. After that is the heart. And now I do have a choice of a heart. So let's go in that direction. And finally, after the heart is again the square. And that is over here. So we have found the correct toy. It turns out to be this bear. And Marie has found her toy, which is answer A. Question number 15. The picture below shows part of a train schedule. Right now it is 8.45. Mr. Smith will go from Chicago to Indianapolis on the next train. The trip will take two hours and 45 minutes. What time will Mr. Smith arrive in Indianapolis? First, looking at the departure schedule here from Chicago to Indianapolis, we have to find the next train Mr. Smith will be able to take from Chicago to Indianapolis. The time right now is 8.45, which falls somewhere in between the departure time here of 8.30 and 9.15. This shows us that Mr. Smith has, in fact, missed the 8.30 train. And he must wait for the next train at 9.15. So assuming that he does take the next train at 9.15, we have to add to this the travel time of two hours and 45 minutes. And if we do that, we see that 45 minutes and 15 minutes makes one hour. So he will arrive at 10 plus two hours. And that is exactly 1,200 hours, or noon. Now, looking at our answers, we have such a possibility. And we choose answer B for the arrival time in Indianapolis. Question number 16. Katie bought three identical pencils, two identical pens, and two identical erasers, and paid a total of $11.60. Hannah bought one pencil, two pens, and two erasers, and she paid $8.40. How much does one pencil cost? One pencil cost. Let's separately keep track of Katie's purchase. We have here on the left Katie. And she purchased three pencils. She purchased two pens, and also two erasers. And the total she paid was $11.60. Now, separately from that, Hannah, she bought two pens. She bought one pencil, two pens, and finally two erasers. She paid a total of $8.40. Now, we see that between the two of them, they do have exactly the same number of erasers, exactly the same number of pens, but Katie has two more pencils than Hannah. So that must account for the difference in the price. So let's subtract. We have $11.60. Let's subtract from that $8.40. And what we see is $11 take away 8 is $3, and $0.60 take away $0.40 is $0.20. So the difference in price comes out to $3.20. And that is the difference between purchasing three pencils and one pencils. So we conclude that, therefore, two pencils cost exactly $3.20. But we are interested in the price of a single pencil. We divide by 2. So 1 half of $3.20 is $1.60, and it is. And so that is answer C $1 and 60 cents

math reasoning

arithmetic problems

logical reasoning

puzzle solving

age calculation

symbolic paths