1. Key Stone Problems… Key Stone Problems… next Set 1 © 2007 Herbert I. Gross

2. You will soon be assigned five problems to test whether you have internalized the material in Lesson 1 of our algebra course. The Keystone Illustrations below are prototypes of the problems you'll be doing. Work out the problems on your own. Afterwards, study the detailed solutions we've provided. In particular, notice that several different ways are presented that could be used to solve each problem. Instructions for the Keystone Problems next © 2007 Herbert I. Gross

3. As a teacher/trainer, it is important for you to understand and be able to respond in different ways to the different ways individual students learn. The more ways you are ready to explain a problem, the better the chances are that the students will come to understand. next © 2007 Herbert I. Gross

4. The first solution we present is the one we feel is the "simplest". However when it comes to teaching, there is no "one size fits all" that will help every student. These other methods are there to be used as supplementary approaches, that is, to be used to help students who do not grasp the problem at first sight. next Our first problem illustrates what we mean by a direct computation, and our second illustration illustrates what we mean by an indirect computation. © 2007 Herbert I. Gross

5. next Problem #1 The price of an object is marked “$67 plus tax”. How much must you pay for the object if the tax is 7% of the marked price? Keystone Illustrations for Lesson 1 next Answer: $71.69 © 2007 Herbert I. Gross

6. Solution for Problem 1: A 7% tax means that for every $1 of the marked price, you have to pay $1.07. Since the marked price is $67, you would have to pay $1.07 sixty-seven times. That is, you would have to pay… 67 × $1.07 or $71.69. next © 2007 Herbert I. Gross

7. This is what is meant by a direct computation. In our PowerPoint presentation, it is what we called the “plain English” model. That is, we start with the marked price, then multiply by $1.07, and the product is our answer. next © 2007 Herbert I. Gross Note 1

8. Other Solutions for Problem 1 Using a Formula The process we used in solving this problem did not depend on the marked price being $67. More specifically, in terms of a formula if we let T denote the price (in dollars) including the 7% tax and M the marked price also in dollars), the formula would be… T = 1.07 × M (1) In this example we would replace M by 67 to obtain… T = 1.07 × 67 (2) next © 2007 Herbert I. Gross

9. So if we wanted to use a calculator we would simply enter the following sequence of key strokes. next 1.07 × 67 = © 2007 Herbert I. Gross 71.69 after which the display would show 71.69. next

10. Because the multiplication sign (×), and the letter (x) are easy to confuse, we usually eliminate the times sign whenever possible by writing, say, 1.07(M) or simply 1.07M rather than 1.07 × M. This is consistent with writing 3 × 1 apple as 3 apples. next © 2007 Herbert I. Gross Note 1

11. Other Solutions for Problem 1 Computing the Tax First We could also have approached the problem by first computing 7% of $67 (that is 0.07 × $67) to conclude that the tax was $4.69, which we would then add to the marked price ($67) to obtain $71.69. next © 2007 Herbert I. Gross

12. Computing the Tax First That is, another way to express equation (2) (T = 1.07 × 67) is: T = 67 + 0.07(67) (8) Notice that equations (2) and (8) are equivalent. More generally… M + 0.07M = 1M + 0.07M = (1 + 0.07)M = 1.07M next © 2007 Herbert I. Gross

13. Other Solutions for Problem 1 The “Corn Bread” Model Whenever we're talking about percent we may think of the total quantity as being a corn bread that is pre sliced into 100 equally sized pieces. In this way each piece represents 1% of the whole quantity. So we divide the corn bread that represents the marked price into 100 equally sized pieces and then to represent the 7% tax we annex an additional 7 of the equally sized pieces to obtain… next © 2007 Herbert I. Gross

14. next © 2007 Herbert I. Gross next The “Corn Bread” Model (drawn to scale) Since the marked price is $67, each of the 100 pieces that represents the marked price represents $67 ÷ 100 or $0.67. Therefore the tax, which is represented by 7 of these pieces is 7 × $0.67 or $4.69. The Marked (pretax) Price $67 100 pieces The Tax ? 7 pieces $4.69

15. The “Corn Bread” Model Hence the total cost is… next © 2007 Herbert I. Gross next Based on the above diagram since there are now 107 pieces, and each piece represents $0.67, the total price is also represented by 107 × $0.67 which again yields $71.69 as the answer. $67 + $4.69 Total Price $71.69 100 pieces + 7 pieces107 pieces × $0.67 / 107pieces Total Price $71.69

16. To add realism to our discussion the above diagram was drawn to scale. However, if one can visualize what is contained in the diagram, there is no need to draw it to scale. For example, suppose that we know that the marked price is $67, we could have written… next © 2007 Herbert I. Gross Note 1 The Marked (Excluding Tax) Price 100 pieces $67.00 The Tax 7 pieces ?

17. Since the marked price is $67, each of the 100 pieces that represents the marked price represents $67 ÷ 100 or $0.67. Therefore the tax, which is represented by 7 of these pieces is 7 × $0.67 or $4.69. Hence the total cost is $67 + $4.69 or $71.69. next © 2007 Herbert I. Gross Note 1 The Marked (Excluding Tax) Price 100 pieces $67.00 The Tax 7 pieces ? $4.69 The Marked Price + Tax Price $67.00 + $4.69 = $71.69 next = Total Price

18. In summary, once we recognized that 100 pieces represented $67, elementary arithmetic would have told us that since the total cost was represented by 107 pieces, it would be 107 × $0.67 regardless of what scale we had used. next © 2007 Herbert I. Gross Note 1 The Marked (Excluding Tax) Price 100 pieces The Tax 7 pieces Total Price 107 pieces 107 × $.67 = $71.69

19. A rate is usually expressed as a phrase consisting of two noun phrases separated by the word “per”. For example... miles per hour (speed) miles per hour per hour (acceleration) dollars per person apples per lawyer In doing computations the word “per” is replaced by “ ÷ ”. For example, miles per hour means miles ÷ hours or miles hours Brief Review of Constant Rates next © 2007 Herbert I. Gross

20. If an automobile traveling at a constant speed goes 60 miles in 2 hours, its speed is 60 miles ÷ 2 hours (60 miles/2 hours ) or 30 miles per hour. If the speed remains constant the answer will be 30 miles per hour no matter what time interval we use. To generalize this idea, if the automobile travels m miles in h hours, we divide m by h to find the speed. In this case we know that the speed is 30 miles per hour. Hence, m/h = 30 or m = 30 × h Example next © 2007 Herbert I. Gross

21. To most students a phrase such as “miles per hour” seems less threatening than the equivalent fraction form “miles/hours”. However, being able to switch from one form to another can be very advantageous. For example, given a rate such as 3/7 of a mile per minute, we can paraphrase it by recognizing that in 7 minutes it travels 3/7 of a mile 7 times, or 3 miles. More visually… Psychological Note next © 2007 Herbert I. Gross 3 7 miles minutes (which we read as 3 miles per 7 minutes). next

22. In other words, 3 miles per 7 minutes is the same rate as 3/7 of a mile per minute. When we say 3 miles per 7 minutes, we do not see any fractions. Psychological Note next © 2007 Herbert I. Gross

23. We are quite comfortable with our understanding that 6 ÷ 2 = 3. What might not be as obvious is that if 6 and 2 modify the same noun, the answer remains the same. For example, 6 apples ÷ 2 apples = 3 (not 3 apples!). Namely, 6 apples ÷ 2 apples means the number we have to multiply 2 apples by to obtain 6 apples as the product. A Note On Canceling “Common” Rates next © 2007 Herbert I. Gross

24. Clearly 3 × 2 apples = 6 apples. Do not confuse 6 apples ÷ 2 apples with 6 apples ÷ 2. Namely, 6 apples ÷ 2 = 3 apples because 2 × 3 apples = 6 apples. So in the language of common fractions 6 apples/2 apples = 3 and 6 apples/2 = 3 apples. A Note On Canceling “Common” Rates next © 2007 Herbert I. Gross

25. In other words if the same noun occurs in numerator and denominator, we may “cancel” it in the same way that we can cancel the same numerical factor if it occurs in both the numerator and denominator. This idea can play an important role when we deal with problems that involve constant rates. A Note On Canceling “Common” Rates next © 2007 Herbert I. Gross

26. Other Solutions for Problem 1 Using Ratio and Proportion From the given information, we know that for every $1 of the marked price, the price including tax will be $1.07. This is a constant rate problem and the rate is… next $1.07 after tax (3) $1.00 pre tax and because “dollars” is in both the numerator and denominator we may write this as… 1.07 after tax 1.00 pre tax © 2007 Herbert I. Gross 1.00 pre tax 1.07 after tax or equivalently next

27. Observe the adjective/noun theme here. next © 2007 Herbert I. Gross Namely… $1.07 $1.00 $1.00 $1.07 ≠ However… $1.07after tax $1.00 pre tax $1.00 pre tax $1.07 after tax = Note 1 next

28. © 2007 Herbert I. Gross So if the pre-tax cost is $67.00, and if we let A stand for the after-tax cost, the rate is also given by… A after tax 67.00 pre tax (4) However since the rate (ratio) is constant, the rates expressed in expressions (3) and (4) are the same. In other words… 1.07 after tax 1.00 pre tax A after tax 67.00 pre tax = (5) next or… A after tax per pre tax 67.00 1.07 after tax per pre tax 1.00 =

29. And since the nouns on both sides of equation (5) are the same, we may rewrite the equation as… © 2007 Herbert I. Gross 1.07 1.00 A 67 = We can then multiply both sides of equation (6) by 67 to obtain the same result we obtained using other methods. Namely: A = 67 × 1.07 = 71.69 Note 1 next = 1.07 (6)

30. An Alternative Way to View Ratio and Proportion next © 2007 Herbert I. Gross While an expression such as $1.07after tax $1.00 pre tax looks like a fraction, we may read it as $1.07 $1.00 after tax dollars per pre tax dollars. next Note 1

31. In this context, using our adjective/noun theme we may view equation (5) as our noun phrase being “after-tax dollars per pre-tax dollars” And since the nouns on both sides of the equation are the same, the adjectives must also be the same. Hence: next © 2007 Herbert I. Gross 1.07 1.00 A 67 = And this is equivalent to the result we obtained using ratio and proportion. next Note 1

32. For students who are comfortable with “filling-in-the-blank” questions, we may reword this question in the form… next © 2007 Herbert I. Gross next $67 before tax = $___ after tax (7) Since the blank” is modifying “after tax”, in order to use our adjective/noun theme the noun on the left hand side of the equation must also be “after tax”. Note 1

33. To obtain this form we use a “cute” way of multiplying by 1. Namely, since $1.07 after tax is equivalent to $1.00 before tax, we may view next © 2007 Herbert I. Gross $1.07 after tax $1.00 before tax to obtain the equivalent expression… next as being another name for 1. Hence we may multiply the left hand side of equation (7) by $1.07 after tax $1.00 before tax $67 before tax 1 $1.07 after tax $1.00 before tax × Note 1

34. and after canceling the common denominations, we see that the left side of equation (7) is equivalent to $67 × 1.07 after tax; whereupon we obtain… next © 2007 Herbert I. Gross next $67 before tax 1 $1.07 after tax $1.00 before tax × = $_____after tax 67 next Note 1

35. Other Solutions for Problem 1 The “Function Machine” (Computer Program Model): Using this method the input is the marked price, the output is the total cost and the program is “Multiply by 1.07”. That is… next Marked Price inputprogramoutput Total Cost × 1.07 next © 2007 Herbert I. Gross

36. The “Function Machine” (Computer Program Model): So in this case we obtain: next Marked Price inputprogramoutput next $67 Total Cost $71.69 × 1.07 $67 © 2007 Herbert I. Gross next

37. Problem #2 The sales tax on an object is 7% of the marked price. How much was the marked price if the price including the tax was $74.90? Answer: $70.00 next Keystone Illustrations for Lesson 1 © 2007 Herbert I. Gross

38. Solution for Problem 2 In terms of the “plain English” model, the answer was $74.90 after multiplying the marked price by 1.07. That is, we pick a number and then multiply it by 1.07. So in terms of the calculator it would seem that the sequence of key strokes should be: next ? × 1.07 = 74.90 © 2007 Herbert I. Gross next

39. Other Solutions for Problem 2 Using a Formula The formulas we use in both Problem 1 and Problem 2 are the same. That is, using the same notation as before, the formula here is also T = 1.07 × M (9) However in Problem 2, it is T that is replaced by $74.90 and we thus obtain… 74.90 = 1.07 × M (10) next © 2007 Herbert I. Gross

40. Using a Formula In this case 74.90 was obtained after we multiplied by 1.07. Therefore to paraphrase the indirect equation... 74.90 = 1.07 × M (10) into an equivalent form that can be solved by a direct computation, we would have to rewrite it as… 74.90 ÷ 1.07 = M (11) next © 2007 Herbert I. Gross

41. Reading comprehension is very important. For example in using the formula T = 1.07 × M, it is important to know whether a given number (such as 74.90 or 67) represents T or whether it represents M. next © 2007 Herbert I. Gross next A main point here is that equation (10) involves an indirect computation (algebra) while the equivalent equation (11) involves a direct computation (arithmetic). Note 2

42. Other Solutions for Problem 2 The “Corn Bread” Model The corn bread model in this instance is the same as it was in our solution to problem 1, except now the corn bread consists of the total cost. next © 2007 Herbert I. Gross next The Tax 7 pieces The Marked (pretax) Price 100 pieces That is, 100 pieces represent the marked price, and 7 pieces represent the tax. So our picture, becomes… Total Price ($74.90) 107 pieces next

43. The “Corn Bread” Model Hence the total cost consists of 107 pieces, collectively worth $74.90. next © 2007 Herbert I. Gross next The Tax 7 pieces The Marked (pretax) Price 100 pieces Since the pieces are of equal size, the size of each piece is given by $74.90 ÷ 107 = $0.70. 100 × $0.70 = $70. Because the marked price is represented by 100 of these pieces, the marked price is 100 × $0.70 or $70.

44. Other Solutions for Problem 2 The “Function Machine” (Computer Program Model): Using this method the input is the marked price, the output is the total cost and the program is “Multiply by 1.07”. That is… next Marked Price inputprogramoutput Total Cost × 1.07 next © 2007 Herbert I. Gross

45. The “Function Machine” (Computer Program Model): So undoing the program would look like this… next Marked Price inputprogramoutput next Total Cost x 1.07 © 2007 Herbert I. Gross outputprograminput ÷

46. The “Function Machine” (Computer Program Model): In this problem the total cost (that is, the output) is $74.90; and so we see that… next Marked Price inputprogramoutput next $70 Total Cost $74.90 ÷ 1.07 $74.90 next © 2007 Herbert I. Gross

47. Other Solutions for Problem 2 Using Ratio and Proportion In this exercise we know that for every $1 of the marked price, the price including tax will be $1. This is a constant rate problem and the rate is: next $1.07 pre tax (12) $1.00 after tax © 2007 Herbert I. Gross

48. Observe that when we used this method for solving Problem 1, rather than using expression (12) we used the expression next © 2007 Herbert I. Gross The reason is that it is computationally simpler to put the “unknown” in the numerator (thus avoiding the “cross multiplication” algorithm which is usually done by rote). $1.07after tax $1.00 pre tax. Note 2

49. next © 2007 Herbert I. Gross In problem 1 the “unknown” was the “after tax” while in problem 2 the unknown” is the “pre tax”. $P pre tax $79.40 after tax. (13) Note 2 So if the after-tax cost is $74.90 and if we let P stand for the pre-tax cost, the rate is also given by… next

50. © 2007 Herbert I. Gross However since the rate (ratio) is constant, the rates expressed in expressions (12) and (13) are the same. In other words… $P pre tax $79.40 after tax Note 2 next $1.00 pre tax $1.07 after tax (14) =

51. © 2007 Herbert I. Gross and since the nouns on both sides of equation (14) are the same we may rewrite the equation as. P 79.40 Note 2 next 1.00 1.07 (15) =

52. © 2007 Herbert I. Gross We can then multiply both sides of equation (7) by 74.90 to obtain the same result we obtained using other methods, namely… P = 74.90 × Note 2 next = 1.00 1.07 74.90 1.07 = 74.90 ÷ 107

53. © 2007 Herbert I. Gross In trying to solve this problem it might have been tempting to take 7% of $74.90 and call this the tax. However the $74.90 was obtained after we added the sales tax. For this reason it was better to view formula (1) as T = 1.07 × M rather than as T = M + 0.07M, since the $74.90 represents the entire cost (that is, including the 7% tax). Final Caution

54. next © 2007 Herbert I. Gross There are times when algebraic means are either too cumbersome or else nonexistent for solving certain types of problems. In such cases, we often use trial and error (sometimes referred to as numerical analysis) to solve the problem. Trial and Error or Estimation

55. next © 2007 Herbert I. Gross We saw in part (a) that if the pre-tax cost of the object was $67, the after-tax cost was $71.69. Since in part (b) the after-tax cost was $74.90, we know that the pre-tax cost has to be greater than $67. Trial and Error or Estimation Example

56. next © 2007 Herbert I. Gross As another example, suppose there was a collection of 40 coins consisting solely of nickels and dimes, the value of which was $3.40, and you wanted to know how many of the coins were dimes. Trial and Error or Estimation

57. next © 2007 Herbert I. Gross Trial and Error or Estimation A rather simple first step is to notice that if all 40 coins were dimes the value would have been $4.00; and if all 40 coins had been nickels, the value would have been $2.00. In other words, before you are even told what the total value is you would know that it had to be more than $2.00 but less then $4.00

58. next © 2007 Herbert I. Gross Trial and Error or Estimation The fact that the total value (i.e., $3.40) is closer in value to $4 than to $2 tells us that there are more dimes than nickels. As a guess we might assume that there are 30 dimes and 10 nickels. If this had been the case the total value would have been $3.00 + $0.50 or $3.50. Since this exceeds the total value, we would need more nickels and fewer dimes.

59. next © 2007 Herbert I. Gross Trial and Error or Estimation If we had assumed that there had been 25 dimes and 15 nickels, the total value would have been $2.50 + $0.75 or $3.25. And since this is less than the given total value we know that we need more dimes and fewer nickels. Combining our previous two steps we see that we need more than 25 dimes but less than 30. Continuing this way we would eventually hit upon the correct answer. next