Intro Quant Reasoning
Intro Quant Reasoning MATH 1030
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This 3 page Class Notes was uploaded by Miss Noel Mertz on Monday October 26, 2015. The Class Notes belongs to MATH 1030 at University of Utah taught by Staff in Fall. Since its upload, it has received 8 views. For similar materials see /class/229949/math-1030-university-of-utah in Mathematics (M) at University of Utah.
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Date Created: 10/26/15
MATH 103002SPRING 2005 7 NOTES ON PERCENTAGE INCREASE Consider the following FALSE statements 1 lfl earn 20 less than you do then you earn 20 more than 1 do 2 1f the actual price of a car is 66 more than the labeled price due to tax then the labeled price is 66 less than the actual price We have discussed in class why the the rst statement is false The second statement is false for the same reason One way to see this is to substitute actual numbers into the statement Suppose that 1 make 20 less than you do and you make 10hri Then my wage is given by the formula 10h r 7 10h r 10h r 7 2h7 8h7quot because my wage is found by taking your wage 10hr and subtracting 20 of that which is 2hri On the other hand 20 more than 8hr is 8h7quot 8h7quot 8h7quot 7 1160h7quot 9160h39ri This difference is accounted for by the fact that a percentage increase or decrease depends on the starting value 20 of the large number is more than 20 of the smaller number This can be explained by the following fact Percentage change is a relative change ie it depends on the starting value Quiz 3 problem 3 asked you to calculate the pretax cost of a car given the price after taxes Funda mentally this problem just statement 2 above Several students calculated the pretax price of the car as if statement 2 were correct but for the same reason as before statement 2 is not true So the calculation 19187i997010661918719917921158 is incorrect We can check this assuming the pretax cost was 17921158 the actual cost would be that amount plus tax which is 17921158 0106617 921158 19 104140 19 187199 A good mental exercise would be to try and explain why the number we arrived at 19104i40 using the wrong method is smaller than the given number 19187i99i Hint consider what the starting values are and try to put the problem into the context of the boxed fact above By the way the correct solution to problem 3 can be found with the rest of the quiz solutions on our webpage MATH 103002SPRING 2005 7 INDUCTIVE VS DEDUCTIVE ARGUMENTS As the title suggests these notes are meant to be a supplement to the lecture on inductive and deductive reasoning An argument is composed of a series of premises or supposed facts and a conclusion which should be derived from the premises The textbook gives formal de nitions of the two types of arguments An inductive argument makes a case for a general conclusion from more speci c examples A deductive argument makes a case for a speci c conclusion from more general premises These are nice de nitions but in practice it is dif cult to determine if a speci c argument is inductive or deductive Consider the following situation Linda is 31 years old single outspoken and very bright She majored in philosophy As a student she was deeply concerned with issues of discrimination and social justice and also participated in antinuclear rallies Rank the following possibilities from 1 most likely to 5 least likely 7 Linda is a teacher 7 Linda works in a bookstore and takes yoga classes 7 Linda is a bank teller 7 Linda sells insurance 7 Linda is a bank teller and is active in the feminist movement A response to this question would be a conclusion The reasons for such a conclusion would constitute the premises For example I could argue as follows Premise Teachers are bright Premise Teachers are college graduates Premise A college degree isn7t needed to work in a bookstore Premise Linda graduated from college Conclusion Linda is more likely to be a teacher than to work at a bookstore In other words Linda is a teacher77 ranks higher than Linda works in a bookstore and takes yoga classes77 The following is another possible argument This is patterned what a group in our class did Premise There are ve possible scenarios Premise I wrote each on a sheet and randomly picked that Linda works in a bookstore Conclusion Linda is more likely to work in a bookstore than anything else These are both examples of inductive arguments This can be ascertained using the de nitions but perhaps an easier way to know that these are inductive arguments is to ask How can I analyze the argument An inductive argument can only be analyzed in terms of its strength It requires personal judgment as to whether or not the conclusion follows from the premises ln examples above do you think the arguments are sound 2 MATH 1030702SPRING 2005 7 INDUCTIVE VS DEDUCTIVE ARGUMENTS On the other hand a deductive argument can be analyzed in terms of its validity whether or not the conclusion is direct result of the premises and its soundness in addition to being valid the premises are true Contrast the previous two inductive arguments with the following deductive arguments Premise All people who are bank tellers and active feminists are bank tellersi Premise Linda is a person Conclusion It is more likely that Linda is a bank teller than that she is a bank teller AND active in the feminist movement Premise All women who sell insurance are deeply concerned with issues of social justice Premise Linda is deeply concerned with issues of social justice Conclusion Linda sells insurance With each of these examples we can analyze their validity and soundnessi Given the premise all bank tellers who are active in the feminist movement are bank tellers77 the conclusion is correct Therefore the argument is valid Moreover the premise is true so we conclude that the argument is sound The second argument is invalid as can be veri ed by drawing a Venn Diagrami Thus even though the conclusion may still be correct the argument is not sound Exercise 1 Analyze the argument Premise All women who are deeply concerned with issues of social justice sell insurance Premise Linda is deeply concerned with issues of social justice Conclusion Linda sells insurance Exercise 2 Come up with three inductive arguments and three deductive arguments related to what Linda does For each of the inductive arguments analyze the strength of the argument For each of the deductive arguments draw the accompanying Venn diagram and determine the validity and soundness possible of the argument Mathematicians tend to be skeptical of inductive argumentsi Even if the argument is very strong they are never satis ed until a deductive proof is given Consider the following example Conclusion If n is a natural number then the sum of the rst it numbers is nltn1gt 2 ile123n71ni A possible inductive argument could be 22 l Premise For n 2 we have 1 2 3 7 Premise Forn3wehavel236 4 4 l Premise For n 4we havel234 10 Premise I plugged it into my calculator for lots of other numbers and it always worked If you7re interested in seeing a deductive proof I would be more than happy to show you one