course MTH 151
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10:36:11 1.3.6 9 and 11 yr old hosses; sum of ages 122. How many 9- and 11-year-old horses are there?
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RESPONSE --> There's no way of knowing without more information which combination of 11 and 9 would be the right one.
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10:36:53 ** If there was one 11-year-old horse the sum of the remaining ages would have to be 122 - 11 = 111, which isn't divisible by 9. If there were two 11-year-old horses the sum of the remaining ages would have to be 122 - 2 * 11 = 100, which isn't divisible by 9. If there were three 11-year-old horses the sum of the remaining ages would have to be 122 - 3 * 11 = 89, which isn't divisible by 9. If there were four 11-year-old horses the sum of the remaining ages would have to be 122 - 4 * 11 = 78, which isn't divisible by 9. If there were five 11-year-old horses the sum of the remaining ages would have to be 122 - 5 * 11 = 67, which isn't divisible by 9. The pattern is 122 - 11 = 111, not divisible by 9 122 - 2 * 11 = 100, not divisible by 9 122 - 3 * 11 = 89, not divisible by 9 122 - 4 * 11 = 78, not divisible by 9 122 - 5 * 11 = 67, not divisible by 9 122 - 6 * 11 = 56, not divisible by 9 122 - 7 * 11 = 45, which is finally divisible by 9. Since 45 / 9 = 5, we have 5 horses age 9 and 7 horses age 11. **
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RESPONSE --> Okay..now I understand what the question wanted me to do.
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10:38:15 Query 1.3.10 divide clock into segments each with same total
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RESPONSE --> These aren't even the same questions that we did for homework. They aren't even the right number in the book. I don't understand what the question wants me to do.
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10:38:47 ** The total of all numbers on the clock is 78. So the numbers in the three sections have to each add up to 1/3 * 78 = 26. This works if we can divide the clock into sections including 11, 12, 1, 2; 3, 4, 9, 10; 5, 6, 7, 8. The numbers in each section add up to 26. To divide the clock into such sections the lines would be horizontal, the first from just beneath 11 to just beneath 2 and the second from just above 5 to just above 8. Horizontal lines are the trick. You might have to draw this to see how it works. **
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10:45:42 Query 1.3.18 M-F 32 acorns each day, half of all acorns eaten, 35 acorns left after Friday
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RESPONSE --> They would have started the week with 128 acorns.
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10:45:59 ** You have to work this one backwards. If they were left with 35 on Friday they had 70 at the beginning (after bringing in the 32) on Friday, so they had 70 - 32 = 38 at the end on Thursday. So after bringing in the 32 they had 2 * 38 = 76 at the beginning of Thursday, which means they had 76 - 32 = 44 before the 32 were added. So they had 44 Wednesday night ... etc. **
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10:46:56 Query 1.3.30 Frog in well, 4 ft jump, 3 ft back.
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RESPONSE --> Twenty days. The four foot progress has the three foot lapse subtracted, making the frog actually move 1 foot forward each day.
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10:47:39 ** COMMON ERROR: 20 days CORRECTION: The frog reaches the 20-foot mark before 20 days. On the first day the frog jumps to 4 ft then slides back to 1 ft. On the second day the frog therefore jumps to 5 ft before sliding back to 2 ft. On the third day the frog jumps to 6 ft, on the fourth to 7 ft., etc. Continuing the pattern, on the 17th day jumps to 20 feet and hops away. The maximum height is always 3 feet more than the number of the day, and when max height is the top of the well the frog will go on its way. **
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10:48:44 Query 1.3.48 How many ways to pay 15 cents?
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RESPONSE --> 15 pennies 1 nickel and 10 pennies 2 nickels and 5 pennies 1 dime and 5 pennies 1 dime and 1 nickel 3 nickels
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10:48:58 ** To illustrate one possible reasoning process, you can reason this out in such a way as to be completely sure as follows: The number of pennies must be 0, 5, 10 or 15. If you don't use any pennies you have to use a dime and a nickle. If you use exactly 5 pennies then the other 10 cents comes from either a dime or two nickles. If you use exactly 10 pennies you have to use a nickle. Or you can use 15 pennies. Listing these ways: 1 dime, 1 nickel 1 dime, 5 pennies 2 nickels, 5 pennies 3 nickels 15 pennies 1 nickel 10 pennies **
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10:50:52 Query 1.3.52 Given 8 coins, how do you find the unbalanced one in 3 weighings
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RESPONSE --> Divide the 8 coins into groups of four. Weigh one group of four against the other . The ligher four has the fake in it. Then divide that group into two and weigh them against one another. One of the lighter two is the fake. Then weigh those two coins against one another. The lighter coin is the fake. In two weighs? I don't have a clue how to do that.
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10:50:59 ** Divide the coins into two piles of 4. One pile will tip the balance. Divide that pile into piles of 2. One pile will tip the balance. Weigh the 2 remaining coins. You'll be able to see which coin is heavier. **
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}]{{wׁc} assignment #011 ѯ}ifǣNd Liberal Arts Mathematics I 10-11-2005
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10:54:39 Query 1.4.24 1 to any power is what?
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RESPONSE --> 1.
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10:54:41 ** The meaning is 1^2, 1^3, 1^4. We take a power of the base. The result is always 1. **
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10:55:49 Query 1.4.30 sqr of neg gives what?
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RESPONSE --> A square can't be negative. It would give an error message.
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10:55:58 ** The square root of a negative will often give you an error (what can you square to get a negative, since any number squared is positive?), but on certain calculators it gives a complex number (actually two complex numbers). These are not real numbers; for the purposes of this course there is no real square root of a negative number. There is no real number that can be squared to give a negative. If you square a negative number you get a negative times a negative, which is positive. If you square a positive number you get a positive number. So a negative number has no real square root. **
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10:56:58 Query 1.4.42 drawer has 18 compartments; how many drawers to hold 204 tapes?
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RESPONSE --> Eleven drawers and one extra to hold the last 6.
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10:57:05 ** 204 / 18 = 11 with remainder 6. If we had 11 drawers they would hold all but 6 of the tapes. The leftover tapes also have to go into a drawer, so we need a 12th drawer. **
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fz|DϞȠ assignment #012 ѯ}ifǣNd Liberal Arts Mathematics I 10-11-2005
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10:57:28 ANSWERS/COMMENTS FOR QUERY 12
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10:58:21 Query 3.1.10 Mary is top grossing film. Is this a statement?
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RESPONSE --> Yes, it is a statement.
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10:58:36 ** A sentence is a statement if it is true or false. Otherwise it is not a statement. 'There goes a big one' is not a statement because the word 'big' is open to interpretation so is not a statement. 'There are 3.87 * 10^89 particles in the universe at this instant' is a statement: it is either true or it isn't, though we don't know enough to tell which. The gross receipts for a film can be regarded as hard facts--unlike opinions on whether a film is good, or artistic. If 'a top grossing film' is defined as, say, a top-10 film in gross receipts, then we could ascertain whether it is true or false and we would have a the statement. However, 'top grossing' isn't defined here--does it mean one of the top three for the week, top 10 for the year, or what?--and for that reason we can't decide for sure whether it is true or false. So this sentence couldn't be regarded as a statement. **
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RESPONSE --> Oh...okay.
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10:58:56 Query 3.1.12 sit up and behave Is this a statement?
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RESPONSE --> No, it's not something that can be verified.
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10:59:01 ** This is not a statement. It is a command. You could evaluate the truth of the statement 'you sat up', but not the truth of the command to sit up. **
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10:59:13 Query 3.1.18 Is this a compound statement or not: 'calif or bust'.
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RESPONSE --> Yes, it is.
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10:59:19 ** The meaning of the phrase is actually 'we're gonna get to California or we're gonna break down trying', in which context it is a compound statement involving 'or'. **
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11:00:57 Query 3.1.30 negate 'some people have all the luck
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RESPONSE --> No one has all the luck.
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11:01:09 ** The negation would be 'all people don't have all the luck', which means 'nobody has all the luck'. The negation of 'some do' is indeed 'all do not', which is the same as 'none do'. The negation of 'all do' is 'some do not'. The negation of 'none do' is 'some do'. COMMON ERROR: Not everyone has all the luck, or equivalently some people do not have all the luck. This is not incompatible with the original statement, and the negation must be incompatible. Both would be true if some do have all the luck and some don't. **
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11:02:17 Query 3.1.42 p: she has green eyes q: he is 48. What is the statement (p disjunction q)?
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RESPONSE --> She has green eyes and he is 48.
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11:02:41 ** The statement is 'She has green eyes or he is 48 yrs. old' **
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RESPONSE --> Oh, I looked at it wrong.
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11:03:17 Query 3.1.48 What is the statement -(p disjunction q)
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RESPONSE --> It is not the case that p or q.
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11:03:29 ** The correct translation is 'It is not the case that she has green eyes or he is 48 yrs. old'. An equivalent statement, using deMorgan's Laws, would be 'she doesn't have green eyes and he is 48 years old' COMMON ERROR: She doesn't have green eyes or he is not 48 years old. This statement negates p V q as ~p V ~q, which is not correct. The negation of p V q is ~p ^ ~q. **
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RESPONSE --> Okay.
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11:04:38 Query 3.1.54 Jack plays tuba or Chris collects videos, and it is not the case that both are so
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RESPONSE --> Jack plays the tuba V Chris collects videos, ~ both.
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11:04:58 ** The statement 'jack plays or Chris collects' is symbolized by (p U q). The statement that it is not the case that both are so is symbolized ~(p ^ q). The entire statement is therefore (p U q) ^ ~(p ^ q).**
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11:05:17 3.1.60 true or false: there exists an integer that is not a rational number.
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RESPONSE --> No, that's false. Integers by definition are real numbers.
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11:05:21 ** A rational number is a number that can be written as p / q, with p and q both integers. Examples are 2/3, -5489/732, 6/2, etc.. Other examples could be 5/1, 12/1, -26/1; these of course reduce to just 5, 12, and -26. The point is that any integer can be written in this form, with 1 in the denominator, so any integer is in fact also a rational number. Thus there is no integer that is not a rational number, and the statement is false. **
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11:06:01 Query 3.1.66 true or false: each rat number is a positive number
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RESPONSE --> True. Negative numbers cannot be rational.
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11:06:17 ** This is false, and to prove it you need only give an example of a rational number that is negative. For example, -39/12 is a rational number (integer / integer) and is negative. **
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RESPONSE --> Oh...I thought negative numbers couldn't be rational.
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11:07:06 Query 3.1.75 difference between 'all students did not pass the test' is the statement ' not all students passed the test'
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RESPONSE --> ""All students did not pass the test"" means that there was not a single student who passed. ""Not all students passed the test"" means that there were at least one student who did not pass the test.
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11:07:12 ** The first statement says that 'all students did not pass', meaning every student didn't pass, i.e., nobody passed the test. The second statement says that not everyone passed--at least one student didn't pass. The second statement doesn't address the question of whether anyone passed or not, so it doesn't necessarily say that some students did pass, but it leaves open the possibility that some did. Since the second statement contains possibilities the first does not the statements are not equivalent. **
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