KeyTrain Level 7. For. Level 7. Published by SAI Interactive, Inc., 340 Frazier Avenue, Chattanooga, TN

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1 Introduction For Level 7 Published by SAI Interactive, Inc., 340 Frazier Avenue, Chattanooga, TN Copyright 2000 by SAI Interactive, Inc. KeyTrain is a registered trademark of SAI Interactive, Inc. Work Keys is a registered trademark of ACT, Inc., used by permission. This document may contain material from or derived from ACT s Targets for Instruction, copyright ACT, Inc., used by permission. Portions copyright Advancing Employee Systems, Inc., used by permission. Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 1

2 Introduction Level 7 Introduction Welcome to Level 7 of. Problems in Level 7 are the most difficult in the Work Keys system. The mathematics used is still fairly straightforward. However, more emphasis is placed on being able to understand the problem. You will have to read them carefully. There many be many details and steps of reasoning involved. There also may be additional information that is not actually needed to solve the problem. Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 2

3 Introduction Skills Required in Level 7 Include: - Performing several steps of reasoning and multiple calculations - Solving problems involving more than one unknown and/or non-linear functions - Calculating the percentage of change - Calculating multiple areas and volumes of sphere, cylinders and cones - Setting up and manipulating complex ratios and proportions - Determining the best economic value of several alternatives - Finding mistakes in multiple-step calculations. Complex Formulas or Ratios In Level 7, some problems will require you to manipulate more complex formulas or ratios. For example, instead of finding the area of a circle from its diameter, you may need to find the diameter given its area. This may also be combined with other details such as converting units of measurement to solve the problem. The concept of non-linear functions will be introduced. An example is determining the gas mileage of a car at different speeds. You will not need to create such equations, but you will need to understand information such as this. Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 3

4 Introduction Some other problems may require setting up simple equations. The equations may involve two or more unknown quantities that you must solve for. Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 4

5 Introduction This lesson is divided into seven topics: Multiple Step Problems Areas and Volumes Ratios and Proportions Best Deals Multiple Unknowns Troubleshooting Non-Linear Functions Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 5

6 Multiple Steps ½ % x Level 7 Multiple Steps In these multiple step problems, you may be required to solve several intermediate problems. For instance you may need to calculate the gas mileage of your car during a trip. But first you would need to calculate the length of the trip from several odometer readings or from a map. Then you would need the amount of gas used by looking at gas receipts and the gas gauge. Finally, you could determine the mileage by dividing the number of miles by the number of gallons of gas used. Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 6

7 Multiple Steps Types of Multiple Step Problems Types of multiple step problems you may see include: - Whole Number Math - Fractions - Decimals - Percentages In these multiple step problems, you need to break up the problem into smaller parts. Careful emphasis must be placed on methodically working through the problem. Recall the standard procedure for word problems: 1) First, read the problem carefully. What is the problem asking? For each step in the problem, determine the number of facts you are trying to find. 2) What are the facts? Lay out the known facts. Do you have the facts you need to solve the problem? You may not have them immediately -- you may need to solve another hidden problem first to get the facts you need. There may be additional unneeded information. 3) Set up and solve intermediate (hidden) problems. Solve the hidden problems to get the facts you need. 4) Solve for the answer. Now that you have the information you need, you can solve the original problem for the information that you were actually asked. 5) Check that the answer is reasonable. Be sure to check your answer! Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 7

8 Multiple Steps Percentages Recall that a percentage means the number of parts out of 100. Percentages can also be written as a fraction or a decimal. Some common examples are: % % % Problems in Level 7 may deal with more complicated percentages. Some percentages may be less than 1% or more than 100%. To work with these percentages, use the same rules as with percentages between 1% and 100%. Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 8

9 Multiple Steps Multiple Step Word Problem Here is an example of a word problem with a percentage of less than 1%: The table below shows how a leading automobile dealer plans to spend its advertising dollars for the coming year. If this company plans to increase its budget by 5% to $85,000 next year, how much will be spent on farm journal ads? Advertising Budget 48.5% Television 23% Magazines 20% Newspapers 7% Radio 1% Billboards 0.5% Farm Journals 1) First, read the problem carefully. What is the problem asking? How much will be spent on farm journal ads? 2) What are the facts? $85,000 total budget 0.5% on farm journals (The fact that the budget is 5% over last year s is not needed information.) 3) Set up and solve the problem. Farm ads 0.5% of $85, % $425 $85,000 $85,000 4) Check that the answer is reasonable. 1 1% of $85,000 would be $850, so % is $ Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 9

10 Multiple Steps Multiple Step Word Problem Here is another example of a multiple step problem involving percentages: You select several items from a supply catalog. The list price of the items is $4.29, $2.13, $1.25, $17.95 and $0.85. Your company gets a 10% discount on all items in the catalog. How much should you make out a purchase order for, including the $10 shipping charge and 5.25% sales tax? 1) First, read the problem carefully. What is the problem asking? What is the total cost of the order? 2) What are the facts? Item costs are as shown. 10% discount on all items Add $10 for shipping Add 5.25% sales tax 3) Setup and solve the problem. First you need the total cost of items : $ $ $ $ Second, apply 10% discount by subtracting 10% from total of items : $ (0.10 $26.47) $ $2.65 $23.82 (round to nearest cent.) Figure 5.25% tax : 5.25% of $ $23.82 $1.25 Figure total : Total Items + Tax + Shipping $0.85 $ $26.47 $ $10 $35.07 Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 10

11 Multiple Steps Decimals Numbers can be expressed in different forms. One of these is called decimals. Look at the following charts: thousands hundreds tens ones tenths hundreths thousandths ten thousandths Decimal Fraction Percent 1, , ,000% 10,000% 1,000% 100% 10% 1% 0.1% 0.01% The value of a single digit depends on its place in the number. Each decimal place in the number is worth ten times the value of the decimal place to its right. For instance, 100 is ten times as much as 10. Likewise, 0.01 is ten times as much as Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 11

12 Multiple Steps Rounding Decimals As was first discussed in Level 3, rounding is the process of estimating a number to a particular decimal place. This is especially common in dealing with money. In the previous example, a 10% discount of $26.47 (2.647) was rounded to the nearest penny ($2.65). To round a number to a given place: Step 1) Find the rounding place (the decimal place you want to round to) Step 2) Look at the digit to the right of the rounding place. If it is less than 5 - leave the digit in the rounding place unchanged. If it is 5 or more - increase the digit in the rounding place by one. Step 3) Remove all digits to the right of the rounding place. For instance, round to the nearest thousandth: thousandths place is digit to the right is 5, so round the 7 up to 1. So the number rounded to the nearest thousandth is Round to the nearest thousandth: thousandth place is digit to the right is a 4, leave the rounding digit alone. So the number rounded to the nearest thousandth is Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 12

13 Multiple Steps Multiple Steps Problem 1 In the next 10 problems you will practice rounding numbers to the place shown. Remember that a number ending in 5 is normally rounded up, not down. Round this number to the nearest 0.1 (tenth): 3, Answer: Multiple Steps Problem 2 Round the number to the place shown. Remember that a number ending in 5 is normally rounded up, not down. Round this number to the nearest 0.01 (hundreth): 8, Answer: Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 13

14 Multiple Steps Multiple Steps Problem 3 Round the number to the place shown. Remember that a number ending in 5 is normally rounded up, not down. Round this number to the nearest 1,000: 6, Answer: Multiple Steps Problem 4 Round the number to the place shown. Remember that a number ending in 5 is normally rounded up, not down. Round this number to the nearest 0.01 (hundreth): 7, Answer: Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 14

15 Multiple Steps Multiple Steps Problem 5 Round the number to the place shown. Remember that a number ending in 5 is normally rounded up, not down. Round this number to the nearest 1: 2, Answer: Multiple Steps Problem 6 Round the number to the place shown. Remember that a number ending in 5 is normally rounded up, not down. Round this number to the nearest 0.1 (tenth): 7, Answer: Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 15

16 Multiple Steps Multiple Steps Problem 7 Round the number to the place shown. Remember that a number ending in 5 is normally rounded up, not down. Round this number to the nearest 100: 9, Answer: Multiple Steps Problem 8 Round the number to the place shown. Remember that a number ending in 5 is normally rounded up, not down. Round this number to the nearest 100: 9, Answer: Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 16

17 Multiple Steps Multiple Steps Problem 9 Round the number to the place shown. Remember that a number ending in 5 is normally rounded up, not down. Round this number to the nearest 0.01 (hundreth): 6, Answer: Multiple Steps Problem 10 Round the number to the place shown. Remember that a number ending in 5 is normally rounded up, not down. Round this number to the nearest 1: 5, Answer: Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 17

18 Multiple Steps Multiple Steps Problem 11 A fish and shrimp company has the yearly catches as listed in the table. Fred s Fish Yearly Catch (Thousands of Kilograms) Fish Shrimp , In which year did they have the highest total catch? Check the correct answer. A B C D. Not enough information Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 18

19 Multiple Steps Math Operations Adding and Subtracting Fractions: If the fractions have the same denominator, just add or subtract the numerators with the same denominator: If the denominators are different, you must convert one or both fractions to same denominator. Then add or subtract the numerator: Multiplying Fractions: To multiply fractions, simply multiply the numerators together, and multiply the denominators together: Dividing Fractions: To divide, invert the dividing fraction and then multiply: Mixed Numbers: Convert the mixed numbers to fractions and then proceed as above: Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 19

20 Multiple Steps Here is Another Word Problem Example: A jeweler is making a copy of a necklace that is 13 ½ inches long. If each separate link is ¾ inch long, how many links are needed for the new necklace? 1) First, read the problem carefully. What is the problem asking? How many links are needed for the new necklace? 2) What are the facts? Original necklace is 13 ½ inches long. Each link is ¾ inch long. 3) Set up and solve the problem: To determine the number of links, you must divide chain length by length of a link : Convert 13 to an improper fraction : 2 1 (26 + 1) Divide : Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 20

21 Multiple Steps Multiple Steps Problem 12 An operator-assisted telephone call from New York to Paris costs $6.75 for the first 3 minutes and $1.20 for each additional minute. If a New York to Paris call costs $15.15, how long was the call? Check the correct answer. A. B. C. D. 5 minutes 9 minutes 10 minutes 13 minutes Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 21

22 Multiple Steps Multiple Steps Problem 13 The ABC Overnight Express company will deliver a package on the next day to anywhere in the country for $14.00 up to 1 lb., $25.00 up to 2 lbs., and $3.00 for each additional lb. up to 10 lbs. If a 7 ½ lb. package is to be sent from Denver to Boston, how much would it cost? A. $22.50 B. $43.00 C. $47.00 D. $55.00 Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 22

23 Multiple Steps Multiple Steps Problem 14 To date a toy manufacturer has sold over 2,000,000 stuffed dolls. Three out of four of the dolls had blue eyes. About how many of the dolls had eyes colored other than blue? A. 50,000 B. 500,000 C. 1,000,000 D. 1,500,000 Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 23

24 Multiple Steps Multiple Steps Problem 15 A vendor at a fair sold 5 knit hats for $9.50 each. She then sold the sixth and seventh for $11.25 each, and the eighth and ninth for $8.75. She was charged 15% of her gross for booth space. How much money did she make after other expenses? A. $41.31 B. $66.95 C. $74.37 D. $87.50 Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 24

25 Multiple Steps Summary Multiple Steps % ½ x In many real-life problems, you will not be given the exact numbers you need to solve the problem. You may have to make several different calculations to get the data you need. The key to solving these problems is to break the larger problem down into smaller ones. Determine the pieces of information you need to solve the problem asked. Do you have these pieces? If not, can you calculate them from the information you do have? You have practiced these skills here. You will use these techniques again in the other sections of this lesson. Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 25

26 Volumes and Areas Level 7 Volumes and Areas Volume is the space enclosed or capacity within a three-dimensional figure, such as a box or room. Volume is measured in cubic units. It tells you how much the figure will hold. Level 6 introduced the method for calculating the volume of rectangular boxes. This section will also show how to calculate the volume of some more complex shapes. These shapes are cylinders, cones and spheres. This section will also include exercises in calculating the volumes and areas of more complex shapes by dividing them into simple shapes. Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 26

27 Volumes and Areas Volume of Rectangular Solids Recall how to determine the volume of a rectangular box: The volume length x width x height L x W x H Note that this is the same as the area of the base times the height: V L x W x H Area x H height 3 depth 3 width 4 Volume 4 x 3 x 3 36 Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 27

28 Volumes and Areas Volume Word Problem Here is an example of a word problem with volume: A room is 20 feet long and 10 feet wide. The ceiling is 8 feet high. How much air does the room hold? 1) First, read the problem carefully. What is the problem asking? How much air does the room hold? In other words, what is its volume? 2) What are the facts? Length: 20 feet Width: 10 feet Height: 8 feet 3) Set up and solve the problem. Volume length x width x height 20 ft. x 10 ft. x 8 ft. 1,600 ft. 3 or 1,600 cubic feet 4) Check that the answer is reasonable. 2 x 8 16, then add two zeros for 1,600. Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 28

29 Volumes and Areas Volume of a Cylinder Recall that the volume of a rectangular box is the area times the height. This is the same for all straight-walled solids. A cylinder is an example of this. The volume of a cylinder area x height In this case, the base is a circle. Therefore: Volume Area of a circle x height pi x r x r H π r 2 H height diameter 2 x radius Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 29

30 Volumes and Areas Cylinder Volume Word Problem Here is another example of a volume word problem. A grain silo is formed in the shape of a cylinder. It is 10 feet in diameter and 30 feet high. How much grain can the silo hold if filled completely full? 1) First, read the problem carefully. What is the problem asking? How much grain does the silo hold? In other words, what is its volume? 2) What are the facts? Diameter: 10 feet (so the radius 5 feet) Height: 30 feet 3) Set up and solve the problem. 2 Volume pi r r height π r H ,355 ft or 30 2,355 cubic feet Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 30

31 Volumes and Areas Volume of a Sphere The volume of a sphere is: Volume 4 3 pi r r r 4 π r 3 3 radius Example: If you have a ball 1 ft. in diameter, how much air does it contain? V V 4 π r 3 3 (The radius is half the diameter, so r 1 ) cubic feet Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 31

32 Volumes and Areas Volume of a Cone The volume of a cone is: 1 Volume Area of circle Height (pi r r) H π r H radius Example: You have a cone shaped object with a diameter of 4 inches and a height of 10 inches. What is its volume? V 1 2 π r H 3 (The radius is half the diameter, so r 2") V cubic inches 10 Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 32

33 Volumes and Areas Tips to Remembers 1) π (or pi) is the ratio of the circumference of a circle to its diameter. For the Work Keys assessment, use ) The radius of a circle (r) is the distance from the center of the circle to a point on the circle. It is one-half of the diameter. 3) Squaring a number means to multiply the number by itself. So the radius squared means multiply the radius by the radius (5 2 5 x 5 25) 4) Cubing a number means to use the number as a factor 3 times. (5 3 5 x 5 x 5 125) Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 33

34 Volumes and Areas Volumes and Areas Problem 1 For a new home, a circular hole has been dug for the septic tank. The hole measures 6 feet across and is 9 feet deep. How many cubic feet of dirt was removed? Check the correct answer. A. B. C. D cubic feet cubic feet cubic feet cubic feet Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 34

35 Volumes and Areas Volumes and Areas Problem 2 A large conical mound of sand has a diameter of 45 feet and a height of 19 feet. Find the volume of sand. Check the correct answer. A. B. C. D. 10,068 cubic feet 13,077 cubic feet 20,135 cubic feet 30,203 cubic feet Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 35

36 Volumes and Areas Volumes and Areas Problem 3 A cylindrical column is to be built out of concrete. The column has a diameter of 3 feet and is 10 feet tall. How many yards of concrete (cubic yards) will be needed? Check the correct answer. A. B. C. D cubic yards 7.85 cubic yards cubic yards cubic yards Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 36

37 Volumes and Areas Volumes and Areas Problem 4 A cylindrical bin with a diameter of 15 feet 3 inches and a height of 24 feet 4 inches is used to hold wheat. One cubic foot holds bushels. How many bushels of wheat can be stored? Check the correct answer. A. B. C. D. 469 bushels 3,573 bushels 2,605 bushels 5,528 bushels Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 37

38 Volumes and Areas Volumes and Areas Problem 5 A roll of copper tubing has an outside diameter of 1 inch and an inside diameter of ¾ inch. How much refrigerant can 12 feet of the tubing hold? Check the correct answer. A. B. C. D cubic feet cubic feet cubic feet 5.30 cubic feet Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 38

39 Volumes and Areas Multiple Areas Many problems in the workplace involve finding the area of an irregular (odd shaped) figure. This can be quite difficult. However you might be able to break the irregular figure into several parts which are regular figures. This would be like making a puzzle where all the pieces are square, rectangles, triangles or circles. Then you can find the area easily. To find the area of an irregular figure: 1) Break the irregular figure down into several standard figures, 2) Find the area of each standard figure, and 3) Add the area of the standard figures together. Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 39

40 Volumes and Areas Example of a Problem with Multiple Areas Consider the wall shown below. How many square feet of wallpaper would be required to cover the wall? Ignore any wasted paper. 4 ft. 4 ft. 3 ft. 12 ft. Window 5 ft. 10 ft. There are two ways to figure the area of the wall: 1) Figure the area of the entire wall including the window, and then subtract the area of the window. 2) Divide the area around the window up into four rectangles, and add the area of the rectangles. Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 40

41 Volumes and Areas Method 1: Figure the area of the entire wall including the window, and then subtract the area of the window. 4 ft. 4 ft. 3 ft. 12 ft. Window 5 ft. 10 ft. The entire wall is 12 feet high by 10 feet wide. By subtracting the area around the window, the window must be 3 feet high by 3 feet wide: 12 ft. high - 4 ft. - 5 ft. 3 ft. high window 10 ft. wide - 4 ft. - 3 ft. 3 ft. wide window Area of wall total area - window area (12 x 10 ) - (3 x 3 ) 120 sq. ft. - 9 sq. ft 111 sq. ft. Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 41

42 Volumes and Areas Method 2: Divide the area around the window up into four rectangles, and ad the area of the rectangles. 3 ft. 4 ft. 4 ft. 3 ft. 12 ft. 5 ft. 3 ft. Area of the four rectangles: x 4 48 sq. ft x 4 12 sq. ft x 3 36 sq. ft x 3 15 sq. ft. Area of wall total area of rectangles 48 sq. ft sq. ft sq. ft sq. ft. 111 sq. ft. Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 42

43 Volumes and Areas Volumes and Areas Problem 6 Compute the area of the shape shown. 5 in. 3 in. 3 in. 1 in. 1 in. 3 in. What is the area of the shape shown? Check the correct answer. A. B. C. D. 15 sq. in. 30 sq. in. 36 sq. in. 39 sq. in. Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 43

44 Volumes and Areas Volumes and Areas Problem 7 A customer wants to recarpet a living room and hallway. You charge $5.99 per square yard to remove old carpet and $4.50 per sq. yd. to install the new carpet. The carpet and pad they selected are $16.00 and $3.00 per sq. yd., respectively. The living room is 17 ½ by 20 ft. and the hall is 4 by 12 ft. How much should you charge for this job? Check the correct answer. A. $1, B. $1, C. $1, D. $1, Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 44

45 Volumes and Areas Volumes and Areas Problem 8 A ceiling measures 24 feet by 26 feet. You need to install ceiling tiles measuring 2 feet by 4 feet in size. How many ceiling tiles are required? Check the correct answer. A. 8 B. 78 C. 104 D. 624 Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 45

46 Volumes and Areas Volumes and Areas Problem 9 You must panel 10 rooms in an apartment building. Each room is 12 ft. by 16 ft. with 8 ft. ceilings. Paneling sheets are 4 by 8. Assume that any paneling cut out for doors and windows is waste. How many panels are required for all of the rooms? Check the correct answer. A. 14 B. 70 C. 140 D. 448 Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 46

47 Volumes and Areas Volumes and Areas Problem 10 You are painting 5 rooms. Each room measures 11 ft. by 12 ft. and is 8 ft. high. Each room also has two windows measuring 4 ft. by 6 ft. and a door of 3 ft. by 6 ft. 8 in. If a gallon of paint covers 440 sq. ft., how many gallons do you need? Check the correct answer. A. 2 B. 5 C. 6 D. 10 Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 47

48 Volumes and Areas Summary Volumes and Areas This section discussed methods for calculating the volume of common shapes. These include boxes, spheres, cylinders and cones. The volume is used to determine how much space the shape contains. The section also discussed how more complicated shapes may be broken down into a collection of simple shapes. In this way you can calculate the area or volume of many different kinds of figures. As you have seen, these skills can be useful for planning many different kinds of projects. By determining the area or volume, you can predict the amount of materials available or needed. Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 48

49 Ratios and Proportions Level 7 Ratios and Proportions A ratio is a comparison of numbers. For example, the number of minutes in an hour to the number of minutes in a day can be said as: and can be written as: 60 to :1440 or 60/1440 which simplifies to 1/24. Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 49

50 Ratios and Proportions Proportions A proportion is a statement that two ratios are equal. For example, suppose you can assemble 2 machine parts in 30 minutes. Then you know that 4 machine parts would take 60 minutes. This is because the ratio of parts to minutes is the same: 2 parts 30 minutes 4 parts 60 minutes This is a proportion. In your mind, you used this proportion to determine that 4 parts would take 60 minutes. You can use proportions to resize, or ratio, many tasks you see in the workplace. Cross Multiplication Ratios were used in earlier levels of this course. In this level, the problems may - contain a mixture of fractions and decimals - contain different units of measurement - contain more difficult computations. However the basic calculations remain the same. These usually involve crossmultiplication. In a proportion the cross products of the ratios are equal: 2 parts 4 parts 30 minutes 60 minutes Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 50

51 Ratios and Proportions Solving Ratio/Proportion Problems This problem leads you through the process of solving ratio/proportion problems. Fine Floors can install 15 square yards of carpeting in 4 hours and 30 minutes. At this rate, how long would it take to install carpeting in a room that measures 11 ft. 9 in. by 13 ft. 4 in.? 1) First, read the problem carefully. What is the problem asking? How long will it take to install the carpet? 2) What are the facts? 15 square yards in 4 hours 30 minutes New room 11 9 by ) Set up and solve the problem. First you need to find the area of the new room : Area 11'9" 13'4" (9" 9 ft ' 13.34' 0.75', 4" 4 ft sq. ft. 0.67") Now set up a proportion to solve the problem. 15 sq. yd. 4.5 hr sq. yd. n Ratio is sq. yds. to hours spent : Solve by cross multiplying : 15 n n ( ) hr.or 5 hr.13 min. Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 51

52 Ratios and Proportions Ratios and Proportions Problem 1 Some new business phones ring 2 short rings every 5 seconds to let you know that a call is coming from outside of the building. How many rings would you count in 35 second? Check the correct answer. A. B. C. D. 13 rings 14 rings 70 rings 88 rings Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 52

53 Ratios and Proportions Ratios and Proportions Problem 2 A secretary types 4,160 words in one hour and 20 minutes. At the same rate, how many words can be typed in an 8-hour day, assuming no breaks? A. 594 B. 2,912 C. 24,960 D. 27,733 Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 53

54 Ratios and Proportions Ratios and Proportions Problem 3 It takes 830 bricks to construct a wall that measures 14 feet 9 inches long and 6 feet high. How many bricks will be needed to build a wall 36 6 long and 6 high? Check the correct answer. A. 336 B. 2,039 C. 2,053 D. 2,054 Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 54

55 Ratios and Proportions Ratios and Proportions Problem 4 You need 2 ¾ wheelbarrows of sand to make 8 wheelbarrows of concrete. How much sand will you need to make 248 cubic feet of concrete? A. B. C. D. 84 cubic feet 85 ¼ cubic feet 682 cubic feet Not enough information Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 55

56 Ratios and Proportions Ratios and Proportions Problem 5 The pitch of a roof (the ratio of the rise to the run of the rafters) is 1/3. Find the rise in a roof with a horizontal run of 15 ¾ ft. Check the correct answer. A. B. C. D. 5 ¼ ft. 5 ¾ ft. 47 ¼ ft. 47 ¾ ft. Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 56

57 Ratios and Proportions Summary Ratios and Proportions A proportion is a comparison of two equal ratios or rates. Using proportions allows you to scale up or down information. This is useful to predict how things can change as a business grows. The units of measurement used in a ratio do not need to all be the same, as long as they are consistent. When the cross product is multiplied, the units for both cross products must be the same. Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 57

58 Best Deals Level 7 Best Deals Best Deal problems involve making comparisons between different options. The best deal is the option that fulfills the goal of the situation better. It may be the option that costs less, makes more money, or uses less energy. In the workplace, employees may often need to do several calculations to compare costs and then choose the best deal. In this section, the problems will involve several calculations to be able to determine the best option. Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 58

59 Best Deals Solving Best Deal Problems Solving best deal problems involves several basic steps: Read the problem Break the problem into smaller problems Compute the different options Compare each option and determine the best one. Example: Power company A sells electricity for $0.04/kwhr. Company B sells for $0.03/kwhr plus a $100/month charge. If your business uses 4000 kwhr per month, which company should you use? Compute one company at a time: Company A: $0.04/kwhr x 4000 kwhr $160 Company B: $0.03/kwhr x 4000 kwhr + $100 $120 + $100 $220. Company A will supply the required electricity for less. Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 59

60 Best Deals Best Deals Problem 1 The next 10 problems show two different prices for the same goods. Determine which is cheaper, or if they are the same. Which is cheaper? Check the correct answer. A. 63 quarts of oil for $56.70 B. 34 quarts of oil for $37.74 C. They are the same Best Deals Problem 2 Below are two different prices for the same goods. Determine which is cheaper, or if they are the same. Which is cheaper? Check the correct answer. A. 54 boxes of pens for $ B. 42 boxes of pens for $ C. They are the same Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 60

61 Best Deals Best Deals Problem 3 Below are two different prices for the same goods. Determine which is cheaper, or if they are the same. Which is cheaper? Check the correct answer. A. 86 lbs. of hamburger for $ B. 2 lbs. of hamburger for $5.26 C. They are the same Best Deals Problem 4 Below are two different prices for the same goods. Determine which is cheaper, or if they are the same. Which is cheaper? Check the correct answer. A. 84 gallons of gas for $86.52 B. 70 gallons of gas for $72.10 C. They are the same Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 61

62 Best Deals Best Deals Problem 5 Below are two different prices for the same goods. Determine which is cheaper, or if they are the same. Which is cheaper? Check the correct answer. A. 33 copies for $3.30 B. 32 copies for $3.84 C. They are the same Best Deals Problem 6 Below are two different prices for the same goods. Determine which is cheaper, or if they are the same. Which is cheaper? Check the correct answer. A. 40 cases of soda for $ B. 54 cases of soda for $ C. They are the same Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 62

63 Best Deals Best Deals Problem 7 Below are two different prices for the same goods. Determine which is cheaper, or if they are the same. Which is cheaper? Check the correct answer. A. 62 cases of paper for $ B. 39 cases of paper for $ C. They are the same Best Deals Problem 8 Below are two different prices for the same goods. Determine which is cheaper, or if they are the same. Which is cheaper? Check the correct answer. A. 69 boxes of labels for $1, B. 58 boxes of labels for $1, C. They are the same Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 63

64 Best Deals Best Deals Problem 9 Below are two different prices for the same goods. Determine which is cheaper, or if they are the same. Which is cheaper? Check the correct answer. A. 97 liters of acetone for $ B. 77 liters of acetone for $ C. They are the same Best Deals Problem 10 Below are two different prices for the same goods. Determine which is cheaper, or if they are the same. Which is cheaper? Check the correct answer. A. 45 cans of tuna for $33.30 B. 39 cans of tuna for $26.13 C. They are the same Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 64

65 Best Deals Best Deal Problems Best Deal problems in Level 7 include these additional details: Calculating the possible economic value of the deal Determining the unit cost Finding the difference and the percent difference between options. Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 65

66 Best Deals Example of Level 7 Best Deal Word Problem Here is an example of a Level 7 best deal problem: Cereal at Store A costs $4.95 for 20 ounces. The same cereal costs $4.25 for 18 ounces at Store B. How much more do you save per ounce at the cheaper store? 1) First, read the problem carefully. What is the problem asking? Where do you get the best deal and how much better? 2) What are the facts? Store A: $4.95 for 20 ounces Store B: $4.25 for 18 ounces 3) Set up and solve the problem. First, find the cost per ounce at each store: Store A : $4.95 Store B: $ ounces $0.248 per ounce 18 ounces $0.236 per ounce Store B is cheaper : It is cheaper by $ $0.236 $0.012 per ounce. Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 66

67 Best Deals Example of Level 7 Best Deal Word Problem Here is another example of a Level 7 best deal problem: After taking inventory at Fancy Fabrics you determine that there is a need to order more thread. You note in the records that last year you purchased thread from Ted's Threads, who sold you cartons of 12 spools for $27. You recently received a notice from Wade's Warehouse that says you can buy thread from them for $2.30 each, and $2.20 a spool for spools over 2 dozen. You need to order 3 dozen spools. What percent can you save by going with the lower price? 4) First, read the problem carefully. What is the problem asking? Where do you get the best deal and by what percent? 5) What are the facts? Ted s: 12 spool for $27 Wade s: $2.30 each and $2.20 each for over 2 dozen 6) Set up and solve the problem. Find the cost at each, then compare: Ted's - - Use ratio : ; n $81 27 n Wade's - - Buy 24 at $2.30 and12 at $2.20 ($ ) + ($ ) $81.60 Percent saving change higher price ($ $81) $ % Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 67

68 Best Deals Best Deals Problem 11 Two different ski resorts are checked to find the best deal for ski lessons. Super Ski charges $525 for 2 hours lessons on 12 days during December through February. Monster Ski offers lessons for $24 an hour. Super Ski is 10 miles from your home, and Monster Ski is 15 miles. It will cost $0.27 per mile to drive to the lessons. Which would be the least expensive for the same number of hours? Check the correct answer. A. B. C. D. Super Ski Monster Ski They both give the same deal Not enough information to tell Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 68

69 Best Deals Best Deals Problem 12 You start a new job where you are paid $450 per week. Your previous job paid $10.75 per hour for a 40-hour work week. What is the percent raise when you begin your new job? Check the correct answer. A. 4.4% B. 4.7% C. 47.0% D. Not enough information Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 69

70 Best Deals Best Deals Problem 13 You are in charge of purchasing stationary for your company. Paper Factory sells a ream (500 sheets) of paper for $37.50; Papers Inc. sells 5 reams for $38 each and $37 each thereafter, and Sales Warehouse sells 750 sheets for $ If you need 20,000 sheets, where will you get the best deal? Check the correct answer. A. B. C. D. Papers, Inc. Paper Factory Sale Warehouse Cannot tell some prices by ream Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 70

71 Best Deals Best Deals Problem 14 A calculator originally marked $65 is on sale at Store A for 25% off. Employees of the store receive an extra 15% discount off of the marked price. The same $65 calculator can be purchased at Store B for 40% off. If you are an employee of Store A, which store should you purchase the calculator from? A. B. C. D. Store A Store B Stores A and B are the same Cannot tell Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 71

72 Best Deals Summary Best Deals These best deal problems have combined best deal calculations with unit conversions, discounts, and other complicating factors. These factors are typical of what you might see in real life. In fact, some stores may use complicated language or formulas on purpose. This may make it difficult to determine which store is actually cheaper. But with the skills you have learned here, you can find who is really cheaper. Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 72

73 Multiple Unknowns Level 7 Multiple Unknowns Most problems require that you solve for one answer. Some problems, however, have more than one number that you must find. These are known as multiple unknowns. Earlier sections have had problems with multiple unknowns. In these earlier problems, you could solve for one unknown, and then solve for the other. For instance, you could find the sum of several costs, then find the sales tax due on the total order. In other types of problems, the two or more unknowns are linked more closely. It can be difficult to solve for one without the other. Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 73

74 Multiple Unknowns Problems with Multiple Unknowns This section deals with types of problems where the multiple unknowns are closely linked. For instance: Width? If the perimeter of a rectangle is known to be 26 ft., and the length is 5 ft. longer than the width, what is the width and length? Length? Perimeter 26 ft. The length is 5 ft. more than the width. You cannot directly solve for the length without knowing the width. Neither can you directly solve for the width without knowing the length. However there are methods to solve this kind of problem fairly easily. Solving Multiple Unknown Problems These types of problems can be solved using two different methods: Substitution Method Both unknowns are represented in terms of one variable (or letter). You substitute one variable in terms of the other variable to solve the equation. Creating a New Equation Use two different variables and two equations. You can then create a new equation by adding or subtracting the two equations. The goal in both cases is to eliminate one variable in order to solve for the other variable. Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 74

75 Multiple Unknowns Solving with Substitution Width? Length? Perimeter 26 ft. The length is 5 ft. more than the width. You know that the perimeter of a rectangle is: Perimeter (2 x Width) + (2 x Length) 26 or 2W + 2L 26 For the substitution method: Use the letter W to represent the width. Using the letter W like this is called a variable. Then you can write the length as W + 5, so Perimeter 2W + 2(W + 5) 26 Multiplying out: 2W + 2W Collecting like terms: 4W 16 Dividing both sides by 4: W 4 Then you know that the length Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 75

76 Multiple Unknowns Solving by Creating a New Equation Width? Length? Perimeter 26 ft. The length is 5 ft. more than the width. You know that the perimeter of a rectangle is: Perimeter (2 x Width) + (2 x Length) 26 or 2W + 2L 26 Also you know that the length is 5 more than the width: L W + 5 Now the trick is to add or subtract the tow equations to eliminate one variable. Multiply the second equation by 2: 2L 2W + 10 Subtract 2W from each side: 2L - 2W 10 Now subtract: 2L + 2W 26 - (2L - 2W 10) 4W 16 So W 4 and L Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 76

77 Multiple Unknowns Sample Problem with Multiple Unknowns The sum of two numbers is 18. Twice the first number plus three times the second number equals 40. Find the two numbers. 1) First, read the problem carefully. What is the problem asking? Find the two numbers. 2) What are the facts? If the numbers are X and Y, then: X + Y 18 2X + 3Y 40 3) Set up and solve the problem: Method 1: Substitute for X: Method 2: Subtract Equations: From the first equation, X 18 - Y Multiply the first equation by 3 Substitute into the second equation: (this means multiply each item by 3), then subtract the second equation: 2(18 - Y) + 3Y 40 3X + 3Y Y + 3Y 40 (Multiply out) - 2X - 3Y -40 3Y - 2Y (Combine like terms) X 14 Y 4 Then Y 18 - X 14 X 18 Y Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 77

78 Multiple Unknowns Multiple Unknowns Problem 1 The manager of a theater knows that 900 tickets were sold for $2300. There were two ticket prices, one for the first floor and one for the balcony. The first floor tickets sold for $3 each and balcony tickets were $2. How many of each type of tickets were sold? Check the correct answer. A. B. C. D. 300 floor, 600 balcony 400 floor, 500 balcony 500 floor, 400 balcony 600 floor, 300 balcony Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 78

79 Multiple Unknowns Multiple Unknowns Problem 2 The perimeter of a rectangle is 54 feet. Twice the length is 3 feet more than the width. What is the size of the rectangle? Check the correct answer. A. B. C. D. length 10 ft., width 17 ft. length 17 ft., width 10 ft. length 19 ft., width 35 ft. length 35 ft., width 19 ft. Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 79

80 Multiple Unknowns Multiple Unknowns Problem 3 An engineer worked for 6 days and her assistant worked for 7 days on a project. Together they received a salary of $4500. The next week the engineer worked 5 days and the assistant 3 days, and earned a combined salary of $2900. What is the daily salary for each? Check the correct answer. A. B. C. D. Engineer $200/day, Assistant $500/day Engineer $300/day, Assistant $400/day Engineer $400/day, Assistant $300/day Engineer $500/day, Assistant $200/day Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 80

81 Multiple Unknowns Multiple Unknowns Problem 4 A total of 60 gallons of gas is to be allotted to two vehicles. One is to receive 12 gallons less than the other does. How many gallons will each receive? Check the correct answer. A. B. C. D. 15 gallons and 45 gallons 20 gallons and 40 gallons 22 gallons and 38 gallons 24 gallons and 36 gallons Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 81

82 Multiple Unknowns Multiple Unknowns Problem 5 You received two receipts for servicing the company cars. In one, four quarts of oil and 40 gallons of gas cost $92. On the other, six quarts of oil and 52 gallons of gas cost $126. What is the cost of a quart of oil and a gallon of gas? Check the correct answer. A. B. C. D. $1.50 for a quart of oil, $8.00 for a gallon of gas $4.00 for a quart of oil, $1.90 for a gallon of gas $8.00 for a quart of oil, $1.50 for a gallon of gas $12.33 for a quart of oil, $1.00 for a gallon of gas Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 82

83 Multiple Unknowns Multiple Unknowns Problem 6 Your company borrows money from two banks. It borrows $300 more from Fifth Bank, which charges 7% interest, than from Sixth Bank, which charges 8% interest. If your interest payments for one year are $1,260, how much does your company borrow at each bank? Check the correct answer. A. Fifth Bank $700, Sixth Bank $1,000 B. Fifth Bank $1,000, Sixth Bank $700 C. Fifth Bank $8,260, Sixth Bank $8,560 D. Fifth Bank $8,560, Sixth Bank $8,260 Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 83

84 Multiple Unknowns Summary Multiple Unknowns Solving multiple unknown problems can be confusing. However you may run into situations where this must be done. The key to solving these problems is to express the information you know in equation form. Then substitute one equation into another, or subtract the equations to eliminate one unknown. Then you can go back and figure the other value. Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 84

85 Troubleshooting Level 7 Troubleshooting Mistakes are often made in solving problems. Many mistakes can be avoided if you always check the answer. See if the answers are reasonable, or use estimate the answer and see if it is close to the answer you have. At times, you may need to check the work of other people for mistakes. It may also be important for you to figure out how the mistake was made. This section will focus on these types of problems. Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 85

86 Troubleshooting Level 7 Troubleshooting In Level 7 of : You will be asked to find mistakes in multiple-step calculations. You may be asked to find the correct answer only. You may also be asked to decide where and how the mistake was likely made. Finding Mistakes Finding a mistake can often be a process of trial and error. Here are some common things to look for when trying to find a mistake: To find the correct answer, resolve the problem to determine the correct solution. Check for possible errors in unit conversions, proportions, or operations. For instance, was a factor multiplied instead of divided? Check for possible errors in entering numbers into a calculator. Could the decimal place have been entered wrong? If the problem involves calculation area or volume, were the right dimensions used? For instance, was radius used instead of diameter? Was the right formula used? Were steps performed in the correct order? If discounts are involved, were they taken in the correct order? Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 86

87 Troubleshooting Sample Troubleshooting Problem It takes a carpet layer 3 hours to install 12 square yards of carpeting. He bids a job installing carpeting in a room that measures 11 6 by 12 9, figuring that it will take him 12.3 hours to install the carpet. Is his bid correct? If not, what error was made? 1) First, read the problem carefully. What is the problem asking? Did the installer figure the time required to install the carpet correctly? 2) What are the facts? 12 square yards took 3 hours to install. New room is 11 6 by ) Set up and solve the problem. First calculate the area of the new room : 11'6" 12'9" 11.5 ft ft sq. ft sq. ft. Now see if he was of sq. yds. to hours to install: 3 hrs. Are the ratios equal? 12 sq. yds. Check the cross product : (9 sq. ft. / sq. yd.) right by checking the proportion of? 16.3 sq. yd hrs sq. yds ? ratios 48.9? 148 No,12.3 hrs. is not correct Solving the proportion for the correct number of hours gives 4.1. This is about 3 times less than he had figured, so he might have converted sq. yd. to sq. ft. using 3 instead of 9. Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 87

88 Troubleshooting Troubleshooting Problem 1 A circular patio was being installed. In order to purchase the materials, the area of the patio must be computed. It was requested that the diameter of the patio be 12 yards. The area was figured to be sq. yards. Was the area correct? If not, why? Check the correct answer. A. No, forgot to multiply by pi (3.14) B. C. D. No, used diameter instead of the radius No, calculated the circumference instead of the area Yes, the answer is correct as is Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 88

89 Troubleshooting Troubleshooting Problem 2 It takes 7 yards of material to make 3 jackets. You bought 15 yards of material to make 7 jackets. Did you buy the right amount of material? If not, how much were you over or under? Check the correct answer. A. B. C. D. No, you bought 2 yards too much material. Yes, you bought just enough. No, you bought 1 yard too little. No, you bought 1 1/3 yards too little. Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 89

90 Troubleshooting Troubleshooting Problem 3 At a recent sale, a stereo system was marked down to $1,550. They claimed that this was a 75% decrease from the original price of $2,067. Did you get the 75% discount? If not, why? Check the correct answer. A. B. C. D. No, you got 75% of the price, not a 75% discount. No, they only gave 7.5% off No, they added the discount instead of subtracting. Yes, they charged you the correct price. Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 90

91 Troubleshooting Troubleshooting Problem 4 A contractor buys four items from a local lumber yard. As a contractor, she gets a 7% discount on all purchases. The items total $ The lumber yard charges $44.95 before tax. Is the charge correct? If not, why? Check the correct answer. A. B. C. D. No, the yard gave him a 7% discount 4 times. No, the yard charged 70% of the full price. No, the yard gave $7 off of each item. Yes, the charges were correct, a 7% discount. Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 91

92 Troubleshooting Troubleshooting Problem 5 A repairman charges $18 per hour to repair appliances plus $0.27 per mile to drive to the house and back. It took the repairman 2 hours and 15 minutes to fix the Andersen's washing machine, and he drove 21 miles to get to their home. He charged $51.84 for the visit. Was the charge correct? If not, why? Check the correct answer. A. B. C. D. No, he charged for 3 hours instead of 2 hours 15 minutes No, he charged for the mileage twice. No, he converted minutes to hours wrong. Yes, the charge is correct as is. Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 92

93 Troubleshooting Summary Troubleshooting As you have seen, troubleshooting is the process of finding and fixing errors in calculations. If you find that an answer is wrong, you may have to try several different guesses to figure out exactly what mistake was made. As you gain more responsibilities in your job, you may find that you will need to do more troubleshooting. You may be responsible for ensuring that work is done correctly. Therefore you must be able to spot errors that have been made, and to correct them. Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 93

94 Nonlinear Functions Level 7 Nonlinear Functions People in some occupations use nonlinear equations. An equation describes how one thing changes with another. For instance, an equation might describe the distance it takes to stop a car traveling at different speeds. The difference in a nonlinear equation is that the ratio between the two is not constant. Therefore it would not take exactly twice the distance to stop a car going 60 miles per hour as one going 30 miles per hour. It would actually take more than twice the distance. In this lesson you will not be expected to create nonlinear equations. However you should be able to use graphs, tables and formulas that represent this type of information. An example of a nonlinear equation might be the gas mileage of a car at different speeds. Given a table or graph showing the gas mileage at different Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 94

95 Nonlinear Functions speeds, you should be able to determine the amount of gas a car will consume at a given speed for a given number of miles. Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 95

96 Nonlinear Functions Examples of Nonlinear Functions Some other examples of nonlinear functions include: Braking distance for a car traveling at various speeds on dry concrete. Distance versus time for a car accelerating at a constant rate. Voltage versus time to discharge for a capacitor. Money earned from investments over time. Income tax and sales commissions. Nonlinear equations represent situations that, when graphed, appear as curved lines. These formulas have variables that are multiplied by themselves, or are raised to a power. Examples of terms like this are: x 2 1 x The figure at right is a graph of the nonlinear equation: y x For instance, when the variable x is 3, then: y is x 2 11 Y f X Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 96

97 Nonlinear Functions Using Nonlinear Formulas to Solve Problems In this example, a nonlinear formula can be used to solve a problem. If a car is traveling at 85 kilometers per hour, about how many meters will the car require to stop after the driver steps on the brake? The distance required to stop can be estimated from: 110 d v 2 (where d is the distance in meters and v is the velocity or speed in kph.) 1) First, read the problem carefully. What is the problem asking? What distance will the car travel before stopping? 2) What are the facts? The nonlinear equation to find the distance is given as shown. 3) Set up and solve the problem. Substitute 85 for v in the equation and solve for d : 110 d 110 d d v 7, , meters The stopping distance formula in this example is a nonlinear function because the speed, v, is squared. This means that it is multiplied by itself, v2 v x v. You can see that this is a nonlinear equation by generating a list of distances for various speeds as shown below. Then plot the points on a graph like that shown at right. The resulting graph is a curve, not a straight line. v d d, meters Stopping Distance for a Car V, kilometers per hour Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 97

98 Nonlinear Functions Nonlinear Functions Problem 1 The braking distance of a pickup on dry concrete can be estimated as: b 0.074v 2 (Where b is the braking distance in feet, v is the speed in miles per hour.) Find the braking distance for a pickup traveling at 60 mph. A. B. C. D. 4.4 ft. 266 ft. 1,622 ft. 2,664 ft. Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 98

99 Nonlinear Functions Nonlinear Functions Problem 2 A photographer must evaluate the lighting in a pose. At first, the subject was 5 meters from the light. The light intensity can be found from: l 72 / d 2 (where l is the light intensity in lumens/m, and d is the distance from the light in m.) How far away should the light be to double the light intensity? Check the correct answer. A. B. C. D. 2.5 meters 3.5 meters 10 meters 20 meters Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 99

100 Nonlinear Functions Nonlinear Functions Problem 3 You need to evaluate your staff for your company's insurance plan. The plan requires you to give the average age of your employees. You have the chart below showing the ages of the staff. Age Number of Staff According to the table, what is the average age of the employees? A. 25 B. 39 C. 40 D. 41 Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users.page 100

101 Nonlinear Functions Nonlinear Functions Problem 4 The balance of your savings account can be predicted each month using the formula: B P (1+i) n where B is the balance after n months, P is the starting balance, i is the monthly interest (as a decimal). If you start with $1,500 and get 5% annual interest, what is the balance after 5 years? Check the correct answer. A. $1, B. $1, C. $15, D. $28, Copyright 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users.page 101