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Copyright © Cengage Learning. All rights reserved. CHAPTER 6 Proportional Reasoning

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Copyright © Cengage Learning. All rights reserved. SECTION 6.2 Percents

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3 What Do You Think? What does percent mean? To what other mathematical concepts is percent related? In what real-life contexts have you encountered percents?

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4 The Origin of Percent

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5 The word percent literally means “per hundred” and comes directly from the Latin per centum. There are direct and immediate connections among percent, equivalent fractions, and proportions. For example, let’s say that a student correctly answered 12 out of 16 questions on a quiz. To determine the student’s score as a percentage, we are asking what fraction with a denominator of 100 is equivalent to.

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6 The Origin of Percent This question can be stated as a proportion: That is, what value of x makes and equivalent fractions? Solving for x, we find the student’s grade is 75%.

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7 Uses of Percent

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8 We use percents for a variety of purposes: To communicate—for example, we hear on the news that a fire is 30% contained. To make sense of situations—a manufacturer may say that the germination rate of a grass seed is 80%. To make decisions—should I refinance my house if the interest rate is down to 7.5%? To compare—in one high school of 345 there were 12 dropouts, and in another high school of 567 there were 17 dropouts. The first school has a dropout rate of 3.5% and the second school has a dropout rate of 3.0%.

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9 Investigation A – Who’s the Better Free-Throw Shooter? You are the coach of the girls’ basketball team at a local middle school. It is the fifth game of the season, the game is tied, and there are only 5 seconds left on the clock. The referee has called a technical foul on the other team. You get to choose any one of your girls to take the shot. Basing your decision only on their free-throw shooting thus far this season, whom would you pick? Becky has made 8 of 12 free throws. Rachel has made 15 of 20 free throws.

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10 Investigation A – Discussion Strategy 1: Use Fractions Represent the players’ free-throw shooting as fractions. At this rate, Becky would make of 20 shots, which is Thus Rachel is doing better.

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11 Investigation A – Discussion Strategy 2: Make a Proportion Some students feel more comfortable with proportions. How would you describe why this proportion will tell us the answer? In this case, x represents how many baskets Rachel must have made in order to have the same ratio as Becky. When we solve for x, we get x =. Because Rachel has made 15 out of 20, which is better than, she is doing better than Becky. cont’d

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12 Investigation A – Discussion Strategy 3: Convert Fractions to Decimals or Percents ≈ 0.67 = 67% whereas = 0.75 = 75% In real life, what factors other than the players’ free-throw percentages might a coach consider before making this decision? cont’d

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13 Investigation B – Understanding a Newspaper Article A newspaper story reports that 8% of the 7968 students at Midvale College work fulltime. How many students work full-time? First try to estimate the number of students and then determine the exact answer. Discussion: Strategy 1: Use 10% as a Benchmark A very rough estimate: 8% is close to 10%, which is, a fraction that we can do mental multiplication with rather simply. If we round 7968 to 8000, then of 8000 is 800.

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14 Investigation B – Discussion Strategy 2: Use 1% as a Benchmark We could mentally find 1% and use simpler numbers to build up to 8%: 1% of 7968 is about 80. 8% of 8000 is 80 8 = 640. cont’d

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15 Investigation B – Discussion Strategy 3: Find a Close Unit Fraction We could use for 8%. In this case, we can use the compatible-numbers strategy When we compare this to, we conclude that the actual number of students working full-time is between 600 and 700. cont’d

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16 Investigation B – Discussion Converting among percents, decimals, and fractions When we estimate the answers to percent problems, it is helpful to know the basic conversions among percents, fractions, and decimals. Table 6.3 shows some of the more commonly used conversions. Table 6.3 cont’d

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17 Investigation B – Discussion With this in mind, let us examine different ways to determine the “exact” number of students working full-time at Midvale College. Strategy 1: Connect to the Meaning of Percent We can represent the ratio of students working full-time to the total number of students as a fraction—that is, cont’d

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18 Investigation B – Discussion If we let x represent the number of students working full-time, we have Now we can solve for x and see that x = 637.44. Strategy 2: Use an Appropriate Procedure 8% of 7968 = 0.08 7968 = 637.44 About 637 students work full-time. cont’d

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19 Investigation B – Discussion An answer of 637.44 or even 637 is too precise; it is an example of what I call pseudo-precision. When the person in the college determined that 8% of the students work full-time, he or she took the number of students who work full-time and divided by the total number of students, and the number that appeared on the calculator was closer to 0.08 than it was to 0.07 or to 0.09. In other words, the actual number of students who work full-time is between 7.5% of 7968 and 8.5% of 7968. That is, the actual number is between 598 and 677. cont’d

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20 Investigation B – Discussion We have just found that any number of full-time students between 598 and 677 will round to 8% of 7968. Because we know only that 8% was reported, it would be more accurate to say not that 637 students work full-time but, rather, that the number of students who work full-time is between 598 and 677, or 637 40. cont’d

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21 Investigation C – Buying a House The Benanders are going to buy a house. Before giving a family a mortgage to buy a home, banks generally require that the total monthly payment (including property taxes) be no more than 28% of the family’s gross monthly income. What must the Benanders’ monthly income be in order for them to buy a home on which the monthly payment will be $800?

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22 Investigation C – Discussion Strategy 1: Use Guess–Check–Revise One way of paraphrasing the problem is to say that if the bank computes 28% of the Benanders’ monthly pay, this number must be at least $800 or the bank will turn them down. In other words, the bank takes their monthly income and multiplies it by 0.28. If the product is greater than $800, they qualify.

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23 Investigation C – Discussion For many students, this line of reasoning leads to guess–check–revise, as shown in Table 6.4. Table 6.4 cont’d

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24 Investigation C – Discussion Strategy 2: Rewrite the Problem as an Equation Another set of students paraphrases the problem something like this: 28% of their salary must be at least 800. Changing the wording to be more “mathematical,” we have 28% of what is $800? The jump to an equation now is not as great a leap to make: 0.28 times x = 800, where x = their monthly salary cont’d

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25 Investigation C – Discussion That is, if 0.28x = 800, solve for x, and see that x = 2857.14. That is, their monthly income must be at least $2857. Strategy 3: Use a Diagram In Figure 6.3, the whole box represents their monthly income, and the shaded area represents 28% of their income. cont’d Figure 6.3

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26 Investigation C – Discussion If we look at the problem from a part–whole perspective, We can also interpret the figure in the following way: If 28 boxes has a value of 800, then 100 boxes has a value of what? If 28 boxes represents a value of $800, then 1 box represents a value of 100 boxes represents a value of $2857.14 cont’d

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27 Connections Between Percent and Other Mathematical Topics

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28 Connections Between Percent and Other Mathematical Topics How would you represent the three previous investigations in terms of part–whole relationships? How would you interpret them in terms of proportions? One of the keys to seeing the interconnectedness of all three problems is to realize that in percent problems, there are two parts and two wholes. For example, in Investigation A, the 12 shots Becky attempted are the whole and 8 are the part she made.

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29 Connections Between Percent and Other Mathematical Topics When we determine that this part–whole relationship is equivalent to 67%, we are saying that this is equivalent to 67 parts in a whole of 100 [Figure 6.4(a)]. Figure 6.4(a)

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30 Connections Between Percent and Other Mathematical Topics To determine 8% of 7968 in Investigation B, we need to find the part out of the whole of 7968 that is equivalent to 8 parts out of 100 [Figure 6.4(b)]. Figure 6.4(b)

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31 Connections Between Percent and Other Mathematical Topics In Investigation C, to determine the Benanders’ monthly payment, we ask, 800 is 28% of what whole [Figure 6.4(c)]? Figure 6.4(c)

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32 Connections Between Percent and Other Mathematical Topics If we now examine these problems in terms of proportions, we can see that the differences among the three problems have to do with what part or whole is missing. Now that we have investigated some of the basic aspects of percents, the next investigations can serve both as stretching problems and as a self-assessment of how well you “own” the concept (that is, how well you can apply your knowledge to nonroutine problems).

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33 Investigation D – Sale? Jorge is excited. He just saw in the newspaper that Showroom Appliances is celebrating 20 years in business by offering 20% off on all merchandise. He just went there yesterday to buy a television set that was regularly priced at $260. What is the sale price of the television?

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34 Investigation D – Discussion Probably the most common solution I see to this problem looks like this: 0.20 $260 = $52 $260 – $52 = $208 That is, you determine the discount and then subtract the discount from the regular price. Jorge will pay $208. However, Randi says that she solved the problem in one step: 0.80 $260 = $208

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35 Investigation D – Discussion One way to get beyond the “how” of this shortcut and into the “why” is to represent this problem with a diagram. In Figure 6.5, if the whole box represents $260 (the original price), what do the two shaded areas represent (that is, 20 boxes and 80 boxes)? Figure 6.5 cont’d

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36 Investigation D – Discussion The 20-box shaded region represents the discount—that is, the amount by which the store will reduce the price, or how much Jorge will save. The 80 boxes therefore represent the sale price of the television. Because it is the sale price that we are looking for, we do not need to find 20% of the price and subtract it from the original price; we can determine the sale price directly. cont’d

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37 Investigation D – Discussion We can show that the two procedures are mathematically equivalent: 260 – 0.20(260) = 260(1 – 0.20) = 260(0.80) Thus, 260 – 0.20(260) = 260(0.80) cont’d We are using the distributive property. Because 1 – 0.20 = 0.80

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38 Percent Change

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39 Percent Change Many problems involving percents in real life are not simple problems like the ones we have investigated thus far but, rather, involve change. Such comparisons are called percent increase, percent decrease, percent change, percent error, percent faster or slower, or percent more or less.

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40 Investigation E – What Is a Fair Raise? Let’s say I am the president of a small company, Bassarear’s Bagels, that has done very well in the last year. I have decided that I want to share my good fortune with my hardworking and devoted employees. I announce that I will be giving a $2-an-hour raise to everyone. Two days later I become aware of some grumbling. A number of employees are complaining that this is not fair.

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41 Investigation E – What Is a Fair Raise? I am stunned. I thought that giving everyone the same raise was the epitome of fairness. Why are some employees grumbling? How would you explain this to me if you were one of my dissatisfied employees? cont’d

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42 Investigation E – Discussion When I do this investigation with my own students, I receive many different reasons for the complaints: Workers with more seniority should receive “bigger” raises. Full-time workers should receive “bigger” raises. Hard-working workers should receive “bigger” raises. This raise is not fair to the people making more money.

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43 Investigation E – Discussion For the concept of percent increase, I want to focus on the last reason. Why would a higher-paid employee feel that my raise was not fair to him or her? Imagine you are such an employee. How might you help me to see your point? Let’s say the janitor is making $8 an hour and the manager of one of the stores is making $20 an hour. From a proportional perspective, the janitor is pretty happy because $2 an hour represents of her salary. cont’d

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44 Investigation E – Discussion The manager, however, is not very happy, because $2 represents only of her salary. From the proportional perspective, a “fair” raise would be one in which the ratio of raise to present salary would be equal for everyone; that is, all the raises would be proportional. cont’d

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45 Investigation E – Discussion Additive versus multiplicative increases The chart below shows the difference between adding the same amount to each person’s wage (an additive increase) and multiplying each person’s wage by the same amount (a proportional increase). cont’d

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46 Investigation E – Discussion If we multiply the janitor’s salary by 1.2, we are in effect increasing her salary by. That is, 8(1.2) = 8 + 0.2(8) = 8 + 1.60 = 9.60 If we multiply the manager’s salary by 1.2, we are in effect increasing her salary by. That is, 20(1.2) = 20 + 0.2(20) = 20 + 4 = 24 cont’d

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47 Investigation E – Discussion Because is equivalent to, we say that both employees have received a 20% raise; that is, their salaries are 20% greater than they were before. In order to understand better the difference between additive and multiplicative changes, look at Figure 6.6, which shows the “before” and “after” salaries using both methods. What do you notice? Figure 6.6 cont’d

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48 Investigation E – Discussion In the additive case (at the left in Figure 6.6), what is equal is the amount of the raise (represented by the white boxes). That is, both people received a raise of $2 per hour. In the multiplicative case (at the right in Figure 6.6), what is equal is the ratio of the increase to the wage. In both cases, the amount of the raise (represented by the white boxes) is equal to of the wage. As I hope you are seeing, the idea of “equality” is as complex in mathematics as it is in history and political science. cont’d

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49 Investigation E – Discussion It is also important to know that not everyone would agree that the 20% raise is fair. Many people interpret this scenario as “the rich getting richer” because the difference between the janitor’s and the manager’s hourly pay was $12 and is now $14.40. cont’d

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50 Investigation F – How Much Did the Bookstore Pay for the Textbook? When college bookstores purchase textbooks, they generally sell the books for 20% to 25% more than they paid for them. In other words, their markup is between 20% and 25%. Let’s say that you paid $95 for a textbook and your bookstore marked up the price by 25%. How much did the bookstore pay for the textbook?

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51 Investigation F – Discussion Strategy 1: Act it Out Let’s say you are working in the bookstore. You would take the price of the book (which we don’t know at this point) and find 25% of that number, and then you would add that to the price of the book. This sum would be $95. We can solve the problem by using guess–check–revise (Table 6.5) or by forming an equation.

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52 Investigation F – Discussion Strategy 2: Use Guess–Check–Revise Table 6.5

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53 Investigation F – Discussion Strategy 3: Form an Equation If we let x = the price the bookstore paid for the book, many students can “see” the equation emerging by acting out the process. The bookstore employee finds 25% of x and then adds this amount to the amount the bookstore paid for the book; this sum must be $95. That is, 0.25x + x = 95 1.25x = 95 x = 76 cont’d

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54 Percents Less than 1 or Greater than 100

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55 Percents Less than 1 or Greater than 100 In real life, we frequently encounter percents greater than 100% or less than 1%. Researchers and teachers know that when students do not thoroughly understand the concept of percents, their success rate with such problems goes down dramatically. Clark Elementary School has decided to buy a new copying machine.

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56 Investigation G – The Copying Machine One of the selling points of the new machine is that the manufacturer advertised that the school can expect about 2 paper jams per 1000 copies. After six months, the faculty wanted to see how the copier was actually doing. They had run 42,164 copies and had encountered 96 paper jams.

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57 Investigation G – The Copying Machine A. Do you think the ratio of paper jams at Clark is consistent with the advertised paper jam rate? Is the school averaging about 2 paper jams per 1000 copies? B. Represent the advertised paper jam rate as a ratio, a fraction, a decimal, and a percent. C. Which one would you use if you were writing the ad for the copier? D. How do you think the manufacturer determined the figure of 2 paper jams per 1000 copies? cont’d

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58 Investigation G – Discussion A. One way (of several) to answer this question is to set up the following proportion: By setting the ratio equal to, we are saying the rate of 96 jams per 42,164 copies is equivalent to how many jams per 1000 copies? Solving for x, we find x 2.3,which means that Clark is averaging about 2.3 paper jams per 1000 copies. Because the paper jam rate for Clark is closer to 2 jams per 1000 than to 3, we would conclude that the copier is acting as advertised.

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59 Investigation G – Discussion B. All of the following are mathematically equivalent to a paper jam rate of 2 per 1000: 1 in every 500 copies will jam. of the copies will jam. 0.2% of the copies will jam. One-fifth of 1% of the copies will jam. cont’d

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60 Investigation G – Discussion C. As an advertiser, I think I would say that the paper jam rate is one-fifth of 1%. This sounds smaller than 2 copies per 1000. However, I also know that most people will understand 2 copies per 1000 much better than one-fifth of 1%. D. The manufacturer probably divided the total number of jams by the total number of copies. I would hope that this was done with many different copiers rather than just one copier, and I would hope that these copiers were not all brand new. The ratio that resulted from this calculation was closer to 2 per 1000 than to 1 per 1000 or 3 per 1000. cont’d

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61 Investigation H – 132% Increase? In 2005, I read that “the percent increase since 1975 in births by Cesarean section in the United States is 170.” The article went on to say that 640,000 Cesarean sections had been done in 1975. I immediately wondered how many had been done in 2005. A. Before you solve the problem, try to make a rough or refined estimate. B. Then solve the problem.

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62 Investigation H – Discussion A. A rough estimate: Because a 100% increase is double the original number, the number of Cesarean sections more than doubled, so the answer will be more than 1,280,000 Cesarean births in 2005. A refined estimate: An increase of 170% = original amount + 100% + 70% 600,000 + 600,000 + 400,000 = 1,600,000 Cesarean births in 2005.

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63 Investigation H – Discussion B. Strategy 1: Break the problem into parts A 170% increase can be broken into a 100% increase and a 70% increase. A 100% increase is equivalent to doubling, which gives us 1,280,000. A 70% increase is 0.7(640,000) = 448,000. Therefore, a 170% increase brings us to 1,728,000 Cesarean births in 2005. cont’d

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64 Investigation H – Discussion Strategy 2: Connect to a Similar, Simpler Problem For example, if the problem had said that there had been a 25% increase, many students would have found the answer by multiplying 640,000 by 1.25. If a 25% increase is determined by multiplying by 1.25, what would we do to determine a 170% increase? We would multiply 640,000 by 2.7. cont’d

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65 Investigation H – Discussion Without an estimate, many students arrive at the wrong answer of 1,088,000 because they multiply 640,000 by 1.7 or they add 70% of 640,000 to 640,000. In this case, even a very rough estimate serves many students nicely: It tells them that the number of Cesarean births in 2005 must be at least 1.2 million. cont’d

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66 Interest

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67 Interest One of the ways in which almost everyone encounters percents is with interest—when you buy a car, when you buy a house, and when you don’t pay off your entire credit card balance at the end of the month, you pay interest on the amount that you owe. Most people have a very limited understanding of the consequences of interest. Let’s say you buy a house for $225,000 and make a down payment of $25,000.

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68 Interest You go to the bank and take out a 30-year mortgage for $200,000, and you are told that your monthly payment will be $1151. Multiply this monthly payment by 360 (there are 360 months in 30 years) to determine how much you actually pay for the house (the $200,000 you borrowed plus all the interest). You actually paid $459,360 for the house! Thus, over 30 years, you pay $239,360 of interest! Let us examine how interest is determined, first in a simple case and then in some instances that are more realistic.

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69 Interest Several variables affect how much interest you receive (on an investment) or pay (on a loan): the original amount, called the principal (P), the annual interest rate (r), specified in percent, and the time of the investment/loan (t), specified in years. Although simple interest—when the principal does not change over the course of the loan—is not common, it is a good starting point for understanding how interest is determined. For example, let’s say that Jenna borrows $2000 from a relative and agrees to repay the loan after 1 year at the rate of 6%.

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70 Interest At the end of the year, she pays $2000 + 6% of $2000; that is, she pays $2000 + 120 = $2120. More commonly, interest is compounded; that is, the interest is determined at specified intervals and added to the principal at those times. For example, if Jenna’s relative had specified that the loan be compounded semiannually, then to determine how much Jenna would owe at the end of the year, the interest would be determined every 6 months and added to the principal.

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71 Interest Thus, after 6 months, she would owe $2000 + 0.03(2000) = $2060. The annual interest rate is 6%, so the semiannual rate is of 6%, or 3%, because 6 months is year. We determine how much she owes after the next 6 months as follows: $2060 + 0.03(2060) = 2060 + $61.80 = $2121.80.

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72 Interest In this particular case, a 1-year loan compounded only semiannually, the difference between simple and compound interest is not huge. However, on most deposits and loans, the interest is compounded daily; that is, the annual interest rate is divided by 365.

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73 Investigation I – Saving for College When Emily was born, her grandparents decided to contribute toward her college education by opening a savings account for $1000. If they add no other money, and if the account earns 6% compounded annually, how much money will there be in the account after 18 years?

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74 Investigation I – Discussion This is a nonroutine, multistep problem. Therefore, let us go one step at a time. After 1 year How much money would there be in the account after 1 year? There are several different methods that students will use. We can find 6% of $1000 and then add that to 1000; that is, $1000 + 0.06($1000) = $1060.

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75 Investigation I – Discussion Or, we can obtain $1060 in one step: $1000(1.06) = $1060. This connection between procedures is important because in order to understand this problem, you need to be able to understand the second procedure; the first one is too cumbersome. After 3 years Now determine how much money would be in the account after 3 years, cont’d

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76 Investigation I – Discussion $1000(1.06) = $1060 after 1 year $1060(1.06) = $1123.60 after 2 years $1123.60(1.06) = $1191.02 after 3 years Let us stop and analyze this strategy. In the first step, we multiplied 1000 by 1.06 to get 1060. In the second step, many students clear the calculator and then multiply 1060 by 1.06. Then, in the third step, they clear the calculator and then multiply 1123.60 by 1.06. cont’d

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77 Investigation I – Discussion We could do this on the calculator: 1000 1.06 1.06 1.06 We can represent what we have learned as $1191.02 = 1000(1.06)(1.06)(1.06) We can now use our knowledge of exponents to say $1191.02 = 1000(1.06) 3 cont’d

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78 Investigation I – Discussion The third, and most efficient, method for finding the amount after 3 years uses this knowledge. How would we enter the last expression in a scientific calculator? Enter 1000 1.06 y x 3 to get the same answer. Before we move on to solve the original question, this is a good time to understand the basic formula for use with interest. cont’d

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79 Investigation I – Discussion Look at how we determined the value (amount) in the bank after 3 years. The money after 3 years = (1000)(1.06) 3 = 1000(1 + 0.06) 3 That is, A = P(1 + r) t cont’d

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80 Investigation I – Discussion After 18 years This discovery makes the original problem much less tedious to solve. Strategy 1: Use the y x Button The most straightforward solution is to use the y x button on your calculator: 1000 1.06 18 = $2854.34 cont’d

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81 Investigation I – Discussion Strategy 2: Use the Memory Key If your calculator lacks an exponentiation key, you could use the memory key M+ by doing the following: First, activate the memory by pressing 1.06 and then M+. Now press 1000 MR = (you should see 1060, the amount after 1 year). To get the amount for each succeeding year, you simply press MR that many times. cont’d

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82 Investigation I – Discussion That is, you press Strategy 3: Use a Spreadsheet A spreadsheet (on a computer) can also be used. When you open the spreadsheet, the columns are marked with letters—A, B, C, and so on—and the rows are marked with numbers. cont’d

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83 Investigation I – Discussion In column A, we enter the numbers 0 through 18 (Figure 6.7). cont’d Figure 6.7

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84 Investigation I – Discussion In column B, we enter 1000 into the first row, signifying the amount we have at the beginning. In the B2 cell, we now enter “= B1*1.06”; that is, we tell the computer to multiply the amount in the B1 cell by 1.06. Now we will see 1060 in the B2 cell. Now, we highlight the B column from row 2 through row 19 and select “Fill down.” This command essentially tells the computer to repeat the computation—that is, to multiply the previous amount by 1.06. cont’d

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85 Investigation I – Discussion After we do this, the computer will display the amount at the end of each year! One advantage of the spreadsheet is that we can play the “what if” game: what if they had started with $3000, what if the interest rate had been 7%, and so on. cont’d

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86 Investigation J – How Much Does That Credit Card Cost You? Most credit cards require a minimum monthly payment. If the customer is not able to pay the entire balance (for example, after a vacation or after Christmas), the customer pays at least this minimum amount; then a fee called a finance charge is determined on the basis of the interest rate the company is charging. Let’s say George found himself in just that situation. His VISA bill has come in, and the balance is $761.34. He decides he can pay $61.34. If the bank issuing the VISA card determines the finance charge at the annual rate of 18%, called the APR (annual percentage rate), what will his finance charge be?

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87 Investigation J – Discussion Because the balance is compounded monthly, the bank will determine George’s finance charge by multiplying his unpaid balance ($700) by 0.015. Do you see where the 1.5% comes from? To find the monthly rate, we divide the yearly rate by 12 (months): 18% 12 = 1.5%, and 1.5% = 0.015. Thus George’s finance charge will be $700(0.015) = $10.50.

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