#$&* course MTH 151 8:55pm, 2/18/14 If your solution to stated problem does not match the given solution, you should self-critique per instructions at
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Given Solution: These numbers can be paired as follows: 1 with 100, 2 with 99, 3 with 98, etc.. There are 100 number so there are clearly 50 pairs. Each pair adds up to the same thing, 101. So there are 50 pairs each adding up to 101. The resulting sum is therefore total = 50 * 101 = 5050. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I’m not understanding how you got that each of the numbers would add up to 101, unless I missed that information. I just added each number, like 1 + 2 + 3…100. Were we supposed to add them to see if they all get the same sum, like 1 + 100 = 101, 2 + 99 = 101? This was a little confusing to me. ------------------------------------------------ Self-critique Rating: 3
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Given Solution: Pairing 1 with 2000, 2 with 1999, 3 with 1998, etc., and noting that there are 2000 numbers we see that there are 1000 pairs each adding up to 2001. So the sum is 1000 * 2001 = 2,001,000. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I’m starting to understand more of how you are getting the pairs of numbers, but I did not go up to 2001. I stopped at 2000, so I guess that really is the only thing that I didn’t do according to how you solved the problem. I also didn’t figure out the sum, but understand that the number of pairs * the number of numbers = the sum, correct? ------------------------------------------------ Self-critique Rating: 3
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Given Solution: We can pair 1 with 501, 2 with 500, 3 with 499, etc., and each pair will have up to 502. However there are 501 numbers, so not all of the numbers can be paired. The number in the 'middle' will be left out. However it is easy enough to figure out what that number is, since it has to be halfway between 1 and 501. The number must be the average of 1 and 501, or (1 + 501) / 2 = 502 / 2 = 251. Since the other 500 numbers are all paired, we have 250 pairs each adding up to 502, plus 251 left over in the middle. The sum is 250 * 502 + 251 = 125,500 + 251 = 125,751. Note that the 251 is half of 502, so it's half of a pair, and that we could therefore say that we effectively have 250 pairs and 1/2 pair, or 250.5 pairs. 250.5 is half of 501, so we can still calculate the number of pairs by dividing the total number of number, 501, by 2. The total sum is then found by multiplying this number of pairs by the sum 502 of each pair: 250.5 * 502 = 125,766. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I think I am forgetting to go one step further, and go to 502. For example, I stopped at 501. Would this always be the requirements to follow when the number of elements is odd? My first instinct is to just to pair up each number (1 + 501 + 502, 2 + 500 = 502) and so on. ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q004. Use this strategy to add the numbers 1 + 2 + ... + 1533. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: I would start by adding 1 + 1533 = 1534, 2 + 1532 = 1534, and so on. There would be 766.5 pairs, because 766.5 * 2 = 1533. The sum is 766.5 * 1534 = 1,175, 811. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: Pairing the numbers, 1 with 1533, 2 with 1532, etc., we get pairs which each adult to 1534. There are 1533 numbers so there are 1533 / 2 = 766.5 pairs. We thus have a total of 1534 * 766.5, whatever that multiplies out to (you've got a calculator, and I've only got my unreliable head). &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I think I’m getting the hang of it now! I got the same answers you did as far as the number of pairs. I do see how these questions can be confusing, and do require a little extra thinking. ------------------------------------------------ Self-critique Rating: 3 ********************************************* Question: `q005. Use a similar strategy to add the numbers 55 + 56 + ... + 945. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: I would just add up the numbers according to how we have for the previous problems. Instead of beginning at 1, however, we’re beginning with 55, and in order to see the difference, have to subtract 55 from 945. 945 - 55 = 890, so 890 + 55 = 945, and 889 + 56 = 945, and so on. I am, however, a little confused on how to figure out the number of pairs. My guess would be because we subtracted 55 from 945, and got 890, that we’d divide 890 / 2 and get 445, meaning that there are 445 pairs. And this would mean that our sum would be 396,050, but I am not feeling too confident in my answers for the number of pairs or the sum. confidence rating #$&*: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: We can pair up 55 and 945, 56 and 944, etc., obtaining 1000 for each pair. There are 945 - 55 + 1 = 891 numbers in the sum (we have to add 1 because 945 - 55 = 890 tells us how many 1-unit 'jumps' there are between 55 and 945--from 55 to 56, from 56 to 57, etc.. The first 'jump' ends up at 56 and the last 'jump' ends up at 945, so every number except 55 is the end of one of the 890 'jumps'. But 55 is included in the numbers to be summed, so we have 890 + 1 = 891 numbers in the sum). If we have 891 numbers in the sum, we have 891/2 = 445.5 pairs, each adding up to 1000. So we have a total of 445.5 * 1000 = 445,500. STUDENT COMMENT I got very confused on this one. I don’t quite understand why you add a 1. INSTRUCTOR RESPONSE For example, how many numbers are there in the sum 5 + 6 + 7 + ... + 13 + 14 + 15? 15 - 5 = 10. However there are 11 numbers in the sum (5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15). &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I also was a bit confused on this one, but am hoping to get a better understanding of it after looking over this chapter again in the textbook. ------------------------------------------------ Self-critique Rating: 3
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Given Solution: Pairing 4 with 900, 8 with 896, etc., we get pairs adding up to 904. The difference between 4 and 900 is 896. The numbers 'jump' by 4, so there are 896 / 4 = 224 'jumps'. None of these 'jumps' ends at the first number so there are 224 + 1 = 225 numbers. Thus we have 225 / 2 = 112.5 pairs each adding up to 904, and our total is 112.5 * 904. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I’m having trouble understanding the ‘jumps’, and how they’re affecting the answers. I’m confused as to what you meant by “None of these ‘jumps’ ends at the first number so there are 225 numbers”? ------------------------------------------------ Self-critique Rating: 3
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Given Solution: We can pair 1 and n, 2 and n-1, 3 and n-2, etc., in each case obtaining a sum of n + 1. There are n numbers so there are n/2 pairs, each totaling n + 1. Thus the total is n/2 * (n+1). &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): This question was very confusing to me. Why wouldn’t you just take n, and say n - 1, n - 2, etc. like we have in the previous problems? I also don’t understand how you got n/2 * (n+1). ------------------------------------------------ Self-critique Rating: 3