Applied Mathematics Level 7 - Weeblyslccareerreadiness.weebly.com/uploads/5/4/8/8/54881327/...Problems in Level 7 are the most difficult in the Work Keys system. The mathematics used

KeyTrain Level 7 Applied Mathematics Introduction

Copyright © 2000, SAI Interactive, Inc. For use only by KeyTrain licensed users. Page 1

For

Applied Mathematics

Level 7

Published by SAI Interactive, Inc., 6403 Sail Pointe, Hixson, TN 37343. Copyright © 2000 by SAI Interactive, Inc. KeyTrain is a 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.



Level 7

Applied Mathematics

Introduction Welcome to Level 7 of Applied Mathematics. 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.



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.



Some other problems may require setting up simple equations. The equations may involve two or more unknown quantities that you must solve for.



This lesson is divided into seven topics:

¾ Multiple Step Problems

¾ Areas and Volumes

¾ Ratios and Proportions

¾ Best Deals

¾ Multiple Unknowns

¾ Troubleshooting

¾ Non-Linear Functions

KeyTrain Level 7 Applied Mathematics Multiple Steps


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% x

Level 7

Applied Mathematics

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.

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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!



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%.

0.25 41

10025

25% 0.5 21

10050

50% 1.0 100100

100% ========



Multiple Step Word Problem Here is an example of a word problem with a percentage of less than 1%:

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.


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

$425

$85,000 0.005

$85,000 1000.5%

$85,000 of 0.5% ads Farm

=

×=

×=

=

$425. is %21

so $850, be would$85,000 of 1%



Multiple Step Word Problem Here is another example of a multiple step problem involving percentages:

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.

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?

$35.07 $10 $1.25 $23.82 Shipping Tax Items Total : totalFigure

$1.25 $23.82 0.0525 $23.82 of 5.25% : tax5.25% Figure

cent.)nearest to(round $23.82 $2.65 - $26.47 $26.47) (0.10 - $26.47

:items of totalfrom 10% gsubtractinby discount 10%apply Second,

$26.47 $0.85 $17.95 $1.25 $2.13 $4.29 :itemsofcost total theneedyou First

=++=++=

=×=

==×

=++++



Decimal

Fraction

Percent

Decimals Numbers can be expressed in different forms. One of these is called decimals. Look at the following charts:

1,000 100 10 1 0.1 0.01 0.001 0.0001 1,000

100

10

1

100,000% 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 0.001.

10000

1

1000

1

100

1

10

1

thou

sand

s

hund

reds

tens

ones

tent

hs

hund

reth

s

thou

sand

ths

ten

thou

sand

ths



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 2.63751 to the nearest thousandth: thousandths place is 7 2.63751 digit to the right is 5, so round the 7 up to 1. So the number rounded to the nearest thousandth is 2.638. Round 2.63749 to the nearest thousandth:

thousandth place is 7

2.63749 digit to the right is a 4, leave the rounding digit alone.

So the number rounded to the nearest thousandth is 2.637.



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.

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 0.1 (tenth): 3,554.1114 Answer:

Round this number to the nearest 0.01 (hundreth): 8,087.0444 Answer:





Round this number to the nearest 1,000: 6,133.8095 Answer:






Round this number to the nearest 1: 2,278.6042 Answer:

Round this number to the nearest 0.1 (tenth): 7,964.6811 Answer:















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

1996 4.203 3,834 1997 5.024 1.972 1998 5.97 2.05

In which year did they have the highest total catch? Check the correct answer. _____ A. 1996 _____ B. 1997 _____ C. 1998 _____ D. Not enough information



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:

53

51 -

54 1

33

31

32

43

42 -

45

21

42

41

41

===+

===+

121

129 -

1210

43 -

65

65

62

63

31

21

==

=+=+

21

126

43

32

==×

53

1 58

12

54

21

54

==×=÷

151

8 15121

3

11

511

32

3 51

2 ==×=×



Here is Another Word Problem Example: 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:

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?

18 6

108

34

2

27

43

2

27

:Divide2

27

2

1) (26

21

13

:fractionimproper an to21

13Convert 43

21

13

:link a oflength by length chain dividemust you links, ofnumber thedetermine To

==×=÷

=+

=

÷



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. 5 minutes _____ B. 9 minutes _____ C. 10 minutes _____ D. 13 minutes



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



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



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 before other expenses? _____ A. $41.31 _____ B. $66.95 _____ C. $74.37 _____ D. $87.50



½

% x Summary – Multiple Steps 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.

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KeyTrain Level 7 Applied Mathematics Volumes and Areas


Level 7

Applied Mathematics

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.



height = 3

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

Volume = 4 x 3 x 3 = 36

width = 4

depth = 3



Volume Word Problem Here is an example of a word problem with volume: 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


Volume = length x width x height = 20 ft. x 10 ft. x 8 ft. = 1,600 ft.3 or 1,600 cubic feet


2 x 8 = 16, then add two zeros for 1,600.

A room is 20 feet long and 10 feet wide. The ceiling is 8 feet high. How much air does the room hold?



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 r2 H

height

diameter = 2 x radius



Cylinder Volume Word Problem Here is another example of a volume word problem. 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


feet cubic 2,355or ft 2,355 30 5 3.14

Hr height r r pi Volume

3

2

2

=

××=

=×××= π

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?



Volume of a Sphere The volume of a sphere is:

Example: If you have a ball 1 ft. in diameter, how much air does it contain?

3r 34 r r r pi

34 Volume π=××××=

feet cubic 0.52

0.5 0.5 0.5 3.14 34 V

)21 r so , thehalf is (The

3r 34 V

=

××××=

=

=

diameterradius

π

radius



Volume of a Cone The volume of a cone is:

Example: You have a cone shaped object with a diameter of 4 inches and a height of 10 inches. What is its volume?

Hr 31 H r) r (pi

31

circle of 31

2π=××××=

××= HeightAreaVolume

inchescubic 41.9

10 2 2 3.14 31 V

)2" r so , thehalf is (The

Hr 31 V 2

=

××××=

=

=

diameterradius

π

radius



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 3.14. 2) 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 (52 = 5 x 5 = 25) 4) Cubing a number means to use the number as a factor 3 times.

(53 = 5 x 5 x 5 = 125)



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. 84.78 cubic feet _____ B. 254.34 cubic feet _____ C. 284.34 cubic feet _____ D. 1017.36 cubic feet



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. 10,068 cubic feet _____ B. 13,077 cubic feet _____ C. 20,135 cubic feet _____ D. 30,203 cubic feet



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. 2.62 cubic yards _____ B. 7.85 cubic yards _____ C. 70.65 cubic yards _____ D. 282.6 cubic yards



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 0.804 bushels. How many bushels of wheat can be stored? Check the correct answer. _____ A. 469 bushels _____ B. 3,573 bushels _____ C. 2,605 bushels _____ D. 5,528 bushels



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. 0.037 cubic feet _____ B. 0.065 cubic feet _____ C. 0.147 cubic feet _____ D. 5.30 cubic feet



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.



Window

4 ft.

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. 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.

4 ft.3 ft.

5 ft.

10 ft.

12 ft.



Window

4 ft.

Method 1: Figure the area of the entire wall including the window, and then subtract the area of the window. 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.

4 ft.3 ft.

5 ft.

10 ft.

12 ft.



4 ft.

Method 2: Divide the area around the window up into four rectangles, and ad the area of the rectangles. Area of the four rectangles:

1. 12’ x 4’ = 48 sq. ft. 2. 3’ x 4’ = 12 sq. ft. 3. 12’ x 3’ = 36 sq. ft. 4. 5’ x 3’ = 15 sq. ft. Area of wall = total area of rectangles = 48 sq. ft. + 12 sq. ft. + 36 sq. ft. + 15 sq. ft. = 111 sq. ft.

4 ft.3 ft.

5 ft.

3 ft.

12 ft.

3 ft.



Volumes and Areas Problem 6 Compute the area of the shape shown.

5 in. What is the area of the shape shown? Check the correct answer. _____ A. 15 sq. in. _____ B. 30 sq. in. _____ C. 36 sq. in. _____ D. 39 sq. in.

3 in.

1 in.

3 in.

1 in.

3 in.



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. $1,068.67 _____ B. $1,082.46 _____ C. $1,214.62 _____ D. $1,262.14



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



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



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



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.

KeyTrain Level 7 Applied Mathematics Ratios and Proportions


Level 7

Applied Mathematics

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:

60 to 1440 and can be written as:

60:1440 or

60/1440 which simplifies to

1/24.



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:

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 cross- multiplication. In a proportion the cross products of the ratios are equal:

minutes60parts 4

minutes30parts 2

=

120 120 30 4 60 2

minutes 60parts 4

minutes 30parts 2

=×=×

=



Solving Ratio/Proportion Problems This problem leads you through the process of solving ratio/proportion problems.


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 13’4”


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.?

min. 13 hr. 5or hr. 5.22 15 17.4) (4.5 n 17.4 4.5 n 15

:gmultiplyin crossby Solve

nyd. sq. 17.4

hr. 4.5yd. sq. 15

:spent hours toyds. sq. is Ratio problem. thesolve toproportion a upset Now

)0.67" ft.124 4" ,0.75' ft.

129 (9"

ft. sq. 156.67 13.34' 11.75' 13'4" 11'9" Area :roomnewtheofarea thefind toneedyou First

=÷×=×=×

=

====

=×=×=



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. 13 rings _____ B. 14 rings _____ C. 70 rings _____ D. 88 rings



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



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



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. 84 cubic feet _____ B. 85 ¼ cubic feet _____ C. 682 cubic feet _____ D. Not enough information



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. 5 ¼ ft. _____ B. 5 ¾ ft. _____ C. 47 ¼ ft. _____ D. 47 ¾ ft.



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.

KeyTrain Level 7 Applied Mathematics Best Deals


Level 7

Applied Mathematics

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.



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.



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.

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. 63 quarts of oil for $56.70 _____ B. 34 quarts of oil for $37.74 _____ C. They are the same

Which is cheaper? Check the correct answer. _____ A. 54 boxes of pens for $145.80 _____ B. 42 boxes of pens for $139.86 _____ C. They are the same





Which is cheaper? Check the correct answer. _____ A. 86 lbs. of hamburger for $204.68 _____ B. 2 lbs. of hamburger for $5.26 _____ C. 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





Which is cheaper? Check the correct answer. _____ A. 33 copies for $3.45 _____ B. 32 copies for $3.84 _____ C. They are the same

Which is cheaper? Check the correct answer. _____ A. 40 cases of soda for $207.60 _____ B. 54 cases of soda for $165.78 _____ C. They are the same





Which is cheaper? Check the correct answer. _____ A. 62 cases of paper for $976.50 _____ B. 39 cases of paper for $557.31 _____ C. They are the same

Which is cheaper? Check the correct answer. _____ A. 69 boxes of labels for $1,745.70 _____ B. 58 boxes of labels for $1,109.54 _____ C. They are the same





Which is cheaper? Check the correct answer. _____ A. 97 liters of acetone for $909.86 _____ B. 77 liters of acetone for $462.00 _____ C. 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



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.



Example of Level 7 Best Deal Word Problem Here is an example of a Level 7 best deal problem:


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


First, find the cost per ounce at each store:

ounce.per $0.012 $0.236 - $0.248by cheaper isIt :cheaper is B Store

ounceper $0.236 ounces 18 $4.25 :B Storeounceper $0.248 ounces 20 $4.95 :A Store

=

=÷=÷

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?



Example of Level 7 Best Deal Word Problem Here is another example of a Level 7 best deal problem:


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


0.7% 0.0074 $81.60 $81) - ($81.60 pricehigher change savingPercent

$81.60 12) ($2.20 24) ($2.30 $2.20at 12 and $2.30at 24Buy -- s Wade'

$81 n ;n36 12 :ratio Use-- sTed'

:compare then each,at cost theFind

==÷=÷=

=×+×

==27

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?



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. Super Ski _____ B. Monster Ski _____ C. They both give the same deal _____ D. Not enough information to tell



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



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 $55.50. If you need 20,000 sheets, where will you get the best deal? Check the correct answer. _____ A. Papers, Inc. _____ B. Paper Factory _____ C. Sale Warehouse _____ D. Cannot tell – some prices by ream



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. Store A _____ B. Store B _____ C. Stores A and B are the same _____ D. Cannot tell



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.

KeyTrain Level 7 Applied Mathematics Multiple Unknowns


Level 7

Applied Mathematics

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.



Problems with Multiple Unknowns

This section deals with types of problems where the multiple unknowns are closely linked. For instance:

Width = ?

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.

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?

Perimeter =

26 ft. Length = ?



Solving with Substitution Width = ? Length = ? 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 + 10 = 26

Collecting like terms: 4W = 16 Dividing both sides by 4: W = 4 Then you know that the length = 4 + 5 = 9

Perimeter = 26 ft.



Solving by Creating a New Equation Width = ? Length = ? 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 = 4 + 5 = 9

Perimeter = 26 ft.



Sample Problem with Multiple Unknowns 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 = 54 36 - 2Y + 3Y = 40 (Multiply out) - 2X - 3Y = -40 3Y - 2Y = 40 - 36 (Combine like terms) X = 14 Y = 4 Then Y = 18 - X = 14 X = 18 – Y = 18 – 4 = 14

The sum of two numbers is 18. Twice the first number plus three times the second number equals 40. Find the two numbers.



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. 300 floor, 600 balcony _____ B. 400 floor, 500 balcony _____ C. 500 floor, 400 balcony _____ D. 600 floor, 300 balcony



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. length 10 ft., width 17 ft. _____ B. length 17 ft., width 10 ft. _____ C. length 19 ft., width 35 ft. _____ D. length 35 ft., width 19 ft.



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. Engineer $200/day, Assistant $500/day _____ B. Engineer $300/day, Assistant $400/day _____ C. Engineer $400/day, Assistant $300/day _____ D. Engineer $500/day, Assistant $200/day



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. 15 gallons and 45 gallons _____ B. 20 gallons and 40 gallons _____ C. 22 gallons and 38 gallons _____ D. 24 gallons and 36 gallons



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. $1.50 for a quart of oil, $8.00 for a gallon of gas _____ B. $4.00 for a quart of oil, $1.90 for a gallon of gas _____ C. $8.00 for a quart of oil, $1.50 for a gallon of gas _____ D. $12.33 for a quart of oil, $1.00 for a gallon of gas



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



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.

KeyTrain Level 7 Applied Mathematics Troubleshooting


Level 7

Applied Mathematics

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.



Level 7 Troubleshooting In Level 7 of Applied Mathematics:

• 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?



Sample Troubleshooting Problem


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 12’9”.


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?

9.of instead 3 using ft. sq. toyd. sq. converted havemight he so figured, had he than less times3

about is This 4.1. gives hours ofnumber correct for the proportion theSolving

correctnot is hrs. 12.3 No, 148 ? 48.9

12.3 12 ? 16.3 3 :product cross Check the

yds. sq. 16.3hrs. 12.3 ?

yds. sq. 12hrs. 3 equal? ratios theAre

:install tohours toyds. sq. of ratios of proportion thecheckingby right washe if see Now

yd. sq. 16.3 yd.) sq. / ft. sq. (9 ft. sq. 146.625

ft. sq. 146.625 ft. 12.75 ft. 11.5 12'9" 11'6" :roomnew theof area thecalculateFirst

=

×=×

=

=÷

=×=×



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 37.68 sq. yards. Was the area correct? If not, why? Check the correct answer. _____ A. No, forgot to multiply by pi (3.14) _____ B. No, used diameter instead of the radius _____ C. No, calculated the circumference instead of the area _____ D. Yes, the answer is correct as is



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. No, you bought 2 yards too much material. _____ B. Yes, you bought just enough. _____ C. No, you bought 1 yard too little. _____ D. No, you bought 1 1/3 yards too little.



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. No, you got 75% of the price, not a 75% discount. _____ B. No, they only gave 7.5% off _____ C. No, they added the discount instead of subtracting. _____ D. Yes, they charged you the correct price.



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 $72.95. The lumber yard charges $44.95 before tax. Is the charge correct? If not, why? Check the correct answer. _____ A. No, the yard gave him a 7% discount 4 times. _____ B. No, the yard charged 70% of the full price. _____ C. No, the yard gave $7 off of each item. _____ D. Yes, the charges were correct, a 7% discount.



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. No, he charged for 3 hours instead of 2 hours 15 minutes _____ B. No, he charged for the mileage twice. _____ C. No, he converted minutes to hours wrong. _____ D. Yes, the charge is correct as is.



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.

KeyTrain Level 7 Applied Mathematics Nonlinear Functions


Level 7

Applied Mathematics

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



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.



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:

The figure at right is a graph of the nonlinear equation:

For instance, when the variable x is 3, then:

x x1 x 2

2 xy 2 +=

11 2 9 2 3 isy 2 =+=+

02468

101214161820

0 1 2 3 4 5

X

Y

f



Using Nonlinear Formulas to Solve Problems In this example, a nonlinear formula can be used to solve a problem.

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.


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 0 0.00

20 3.64 40 14.55 60 32.73 80 58.18

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 = v2 (where d is the distance in meters and v is the velocity or speed in kph.)

meters 65.7 110 7,225 d 7,225 d 110

85 v d 110 :dfor solveandequation in the for v 85 Substitute

22

=÷==

==

Stopping Distance for a Car

0.00

10.00

20.00

30.00

40.00

50.00

60.00

70.00

80.00

0 20 40 60 80 100

V, kilometers per hour

d, m

eter

s



Nonlinear Functions Problem 1 The braking distance of a pickup on dry concrete can be estimated as: b = 0.074v2 (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. 4.4 ft. _____ B. 266 ft. _____ C. 1,622 ft. _____ D. 2,664 ft.



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 / d2 (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. 2.5 meters _____ B. 3.5 meters _____ C. 10 meters _____ D. 20 meters



Nonlinear Functions Problem 3

Age Number of Staff 23 32 36 39 41 45 50 52

2 3 2 3 4 6 2 3

According to the table, what is the average age of the employees? _____ A. 25 _____ B. 39 _____ C. 40 _____ D. 41

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.



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. $1,905.96 _____ B. $1,914.42 _____ C. $15,302.40 _____ D. $28,018.77



Nonlinear Functions Problem 5 You operate an electric gas compressor. The electricity to run the compressor costs $0.08 for every 1,000 cu. ft. of gas you make. The compressor currently operates at 50% of full load. The graph below shows how much electricity the compressor uses for every 1,000 cu. ft. of gas made.

If you double production by running the compressor at full speed, what will the electricity cost? _____ A. $0.04 per 1,000 cubic feet _____ B. $0.06 per 1,000 cubic feet _____ C. $0.08 per 1,000 cubic feet _____ D. $0.12 per 1,000 cubic feet

Electricity Consumption per 1000 cu.ft. Gas Produced

0

1

2

3

4

5

6

7

0 20 40 60 80 100% Full Load

Elec

tric

ity U

sed

(Kw

h)



Summary – Nonlinear Equations You can tell that equations are nonlinear when they have a variable that is multiplied by itself or another variable, or is raised to a power other than 1. These types of equations govern common things in our life such as interest calculations, or the motion of objects from gravity or other forces. You should be able to use simple nonlinear formulas to find needed information. You can also plot points from the equation on a graph to create a curved line.

KeyTrain Level 7 Applied Mathematics Answers


Level 7

Applied Mathematics

Answers

KeyTrain Level 7 Applied Mathematics Multiple Steps - Answers


Multiple Steps – Answers Multiple Steps Problem 1: Round 3,554.1114 to 3,554.1 (nearest 0.1) Multiple Steps Problem 2: Round 8,087.0444 to 8,087.04 (nearest 0.01) Multiple Steps Problem 3: Round 6,133.8095 to 6,000 (nearest 1,000) Multiple Steps Problem 4: 7,081.5907 to 7,081.59 (nearest 0.01) Multiple Steps Problem 5: Round 2,278.6042 to 2,279 (nearest 1) Multiple Steps Problem 6: Round 7,964.6811 to 7,964.7 (nearest 0.1) Multiple Steps Problem 7: Round 9,919.4417 to 9,900 (nearest 100) Multiple Steps Problem 8: Round 9,050.6931 to 9,100) (nearest 100) Multiple Steps Problem 9: Round 6,199.7209 to 6,199.72 (nearest 0.01) Multiple Steps Problem 10: Round 5,439.6989 to 5,440 (nearest 1)



Multiple Steps Problem 11: The correct answer is A. What is the problem asking? Which year had the highest catch? What are the facts? 1996: 4,203 and 3,834 kilograms 1997: 5,024 and 1,972 kilograms 1998: 5,970 and 2.050 kilograms Set up and solve the problem: Add up the amount for each year:

1996: 4,203 + 3,834 = 8,037 kilograms 1997: 5,024 + 1,972 = 6,996 kilograms 1998: 5,970 + 2,050 = 8,020 kilograms 1996 had the highest total catch with 8,037 kilograms Multiple Steps Problem 12: The correct answer is C. What is the problem asking? How long was the phone call? What are the facts? First 3 minutes are $6.75 Each additional minute is $1.20 The call cost a total of $15.15 Set up and solve the problem: Solve by guess and testing: First 3 minutes cost $6.75 Add another 5 minutes: $6.75 + (5 x $1.20) = $12.75 – Need more Add 7 minutes to initial 3: $6.75 + (7 x $1.20) = $15.15. – Correct Therefore the call was 3 + 7 = 10 minutes total



Multiple Steps Problem 13: The correct answer is B. What is the problem asking? How much will the shipment cost? What are the facts? A package under 1 lb. Costs $14.00 A package between 1 and 2 lbs. Costs $25.00 Over 2 lbs. and up to 10 lbs. Costs $3.00 for each lb. over 2 lbs. Package to be mailed is 7 ½ lbs. Set up and solve the problem: Cost = Cost for 2 lbs. + $3.00 for every lb. over 2 lbs. Amount over 2 lbs. = 7 ½ - 2 = 5 ½ lbs. (In this case, charges would be rounded up to next higher even lb. – 6 lbs. extra) Cost = 2 lb. charge + 6 extra lbs. charge Cost = $25.00 + (6 x $3.00) Cost = $25.00 + $18.00 = $43.00 Multiple Steps Problem 14: The correct answer is B. What is the problem asking? How many dolls did not have blue eyes? What are the facts? 2,000,000 total dolls. 3 out of 4 had blue eyes. Set up and solve the problem:

3 out of 4 had blue eyes. This means 1 out of 4 did not have blue eyes. This can be expressed as:

Fraction: ¼ Decimal: 0.25 Percentage: 25% The total number without blue is then: 2,000,000 x 0.25 = 500,000



Multiple Steps Problem 15: The correct answer is C. What is the problem asking? How much money did she make after paying the fair for the booth? What are the facts? 5 hats sold for $9.50 2 hats sold for $11.25 (6th & 7th) 2 hats sold for $8.75 (8th & 9th) Fair gets 15% for the booth. Set up and solve the problem: Total money collected = (5 x $9.50) + (2 x $11.25) + (2 x $8.75) = $87.50 Booth charge = $87.50 x 15% = $87.50 x 0.15 = $13.13 Net money = $87.50 - $13.13 = $74.37

KeyTrain Level 7 Applied Mathematics Volumes and Areas - Answers


Volumes and Areas – Answers Volumes and Areas Problem 1: The correct answer is B. What is the problem asking? What is the volume of the cylindrical hole? What are the facts? Diameter is 6 feet. Depth (height) is 9 feet Set up and solve the problem:

feetcubic254.34 3ft. 254.34 ft. 9 ft. 3 ft. 3 3.14 =

H r r π= Volume9 = (H)Height

3 = 2 6 =diameter of half is Radius

==×××

×××

÷

Volumes and Areas Problem 2: The correct answer is A. What is the problem asking? What is the volume of the conical mound? What are the facts? Diameter of base is 45 feet. Height is 19 feet. Set up and solve the problem:

feet cubic 10,068 =

19 22.5 22.5 3.14 31

=

H r r 31

= Volume

feet 19 = (H)Height ft. 22.5 = 2 45 =diameter of half is Radius

××××

××××

÷

π



Volumes and Areas Problem 3: The correct answer is A. What is the problem asking? How much concrete is needed, or what is the volume of the column? What are the facts? Diameter of base is 3 feet. Height is 10 feet Set up and solve the problem:

yards cubic 2.62 = yd.) cu. / ft. cu. 27 feet cubic 70.65 ft./yd.) 3 3 (3ft cubic 27 = yard cubic 1

:yards cubic convert to toNeedfeet cubic 70.65 =

10 1.5 1.5 3.14 = H r r π= Volume

ft. 1.5 = 2 3 =diameter of half is Radius

÷××

××××××

÷

Volumes and Areas Problem 4: The correct answer is B. What is the problem asking? What is the volume of the cylindrical bin? What are the facts? Diameter of base is 15 feet 3 inches. Height is 24 feet 4 inches Set up and solve the problem:

bushels 3,573 = ft. cu.bus. 0.804

x ft. cu. 4,443.5

:bushels Convert tofeet cubic 4,443.5 =

24.34 7.625 7.625 3.14 = H r r π= Volume

ft. 24.34 = (H)Height ft. 7.625 = 2 15.25 =diameter of half is Radius

ft. 24.34 = in. 4 ft. 24 ft. 15.25 = in. 3 ft. 15

:feet decimal Convert to

××××××

÷



Volumes and Areas Problem 5: The correct answer is A. What is the problem asking? What is the volume of the interior of 12 feet of tubing. What are the facts? Inside diameter is ¾ inch (do not need outside diameter). Length is 12 feet. Set up and solve the problem:

ft.cu 0.037 = ft.) cu.per in. cu. (1,728 ft. cu. Convert toin. cu. 63.59 =

144 0.375 0.375 3.14 = H r r π= Volume

in. 144 = in./ft.) (12 ft. 12 :inches length toConvert in. 0.375 = 2 0.75 =diameter of half is Radius

cylinder. a is tubeof inside of Shape

××××××

×÷

Volumes and Areas Problem 6: The correct answer is D. What is the problem asking? What is the area of the figure? What are the facts? Dimensions as shown on the figure Set up and solve the problem: Divide the figure into 3 parts as shown: Add the three areas together: 15 sq. in. + 9 sq. in. + 15 sq. in. = 39 sq. in.

5 inches by 3 inches, Area = 5 x 3 = 15 sq. in.





Volumes and Areas Problem 7: The correct answer is C. What is the problem asking? What is the total cost for removing the old carpet and installing the new carpet? What are the facts? Remove old carpet: $5.99 per sq. yd. New carpet: $16.00 per sq. yd.

Padding: $3.00 per sq. yd. Installation: $4.50 per sq. yd. Set up and solve the problem:

$1,214.62 = $27.48 yd. sq. 44.2 =Cost

yd. sq. 44.2 = ft. sq. 9yd. sq. 1 ft. sq. 398

:yards Convert toft. sq. 398 = 48 + 350 = Area Total

ft. sq. 48 = 12 4 = Hall of Areaft. sq. 350 = 20 17.5 = Room Living of Area

yard. sq.per $27.48 is charges above of Total

×

×

××

Volumes and Areas Problem 8: The correct answer is B. What is the problem asking? How many ceiling tiles are needed? What are the facts?

Ceiling is 24 ft. by 26 ft. Tiles are 2 ft. by 4 ft.

Set up and solve the problem:

26 ft.

13 tiles

24 ft. 6 tiles

tiles78 = 13 6 = square tiles13by tiles6 = tilesTotal tiles13 = per tile ft. 2 ft. 26

26ft wall. thealong tilesoflength side ft. 2 arun can then You tiles6 = per tile ft. 4 ft. 24

evenly).fit will tiles(6 walllong ft. 24 thealong tilesoflength side ft. 4 arun you ifevenly installcan You edges. at the somecut tohavemay -- ceiling in theevenly fit tiles theif see toNeed

×÷

÷



Volumes and Areas Problem 9: The correct answer is C. What is the problem asking? How many pieces of paneling are needed? What are the facts?

10 rooms Rooms are 12 feet by 16 feet by 8 feet high Paneling pieces are 4 feet by 8 feet


panels 140 = 14 10 need rooms, 10For panels 14 = 4) (2 + 3) (2 needs roomEach

4) = 4 (16 panels 4 needs wall16'Each 3) = 4 (12 panels 3 needs wall12'Each

height 8'an and wallof width 4' acover willpanelEach high. 8'by wide16' are that walls2

andhigh 8'by wide12' are that walls2 has roomEach

×××÷÷

Volumes and Areas Problem 10: The correct answer is B. What is the problem asking? How many gallons of paint are needed? What are the facts?

5 rooms (each 11 feet by 12 feet by 8 feet high) Each room has 2 windows (4 feet by 6 feet) Each room has 1 door (3 feet by 6 feet 8 inches)


gallons 5about or gallons 4.9 = gal.per ft. sq. 440 ft. sq. 2.160ft. sq. 2,160 = 432 5 = rooms 5

ft. sq. 432 = 20 - 48 - 132 + 368 = roomEach ft. sq. 132 = 12'by 11' = Ceiling

ft. sq. 20 = 6.67 3 = 8" 6'by 3' =Door ft. sq. 48 = 6 4 2 = 6'by 4' are 2 = Windows

ft. sq. 368 = 8) 12 (2 + 8) 11 (2 = roomeach in 8'by 12' are 2 and 8'by 11' are 2 = Walls

doors - windows- ceiling + walls== area Total

÷×

×××

××××

KeyTrain Level 7 Applied Mathematics Ratios and Proportions - Answers


Ratios and Proportions Ratios and Proportions Problem 1: The correct answer is B. What is the problem asking? How many rings in 35 seconds? What are the facts?

2 rings every 5 seconds Need to count for 35 seconds


rings 14 n 5 35) (2 n

35 2 n 5 :multiply Cross

seconds 35ringsn

seconds 5

rings 2

secondstorings of ratio of proportion a Use

=÷×=

×=×

=

Ratios and Proportions Problem 2: The correct answer is C. What is the problem asking? How many words can be typed in 8 hours? What are the facts?

4,160 words are typed in 1 hour 20 minutes Set up and solve the problem:

24,960 80 480) (4,160 n 480 4,160 n 80

minutes 480n words

minutes 80

words4,160

: time to wordsof proportion upSet minutes 480 60 8 hours 8

minutes 80 20 60 minutes 20hour 1 minutes -- units same the to timesallConvert

=÷×=×=×

=

=×==+=



Ratios and Proportions Problem 3: The correct answer is D. What is the problem asking? How many bricks for the new wall? What are the facts?

830 bricks for a 14 foot 9 inch long wall New wall is 36 feet 6 inches long


bricks) 2,054 toup (round bricks 2,053.9 n 14.75 36.5) (830 n 36.5 830 n 14.75

:multiply Cross

36.5ft.bricksn

ft. 14.75bricks 830

feet). to(convertedlengthsonly theratiocan you soheight, same theare Both walls

=÷×=×=×

=

Ratios and Proportions Problem 4: The correct answer is C. What is the problem asking? How much sand is needed for 248 cubic feet of concrete? What are the facts?

concrete. of ows wheelbarr8for sand of ows wheelbarr43

2


sand offeet cubic 85.25 n 8 248) (2.75 n

248 2.75 n 8 :multiply Cross

concretefeet cubic 248sand ft. cubicn

concrete ows wheelbarr8

sand ows wheelbarr2.75

ratios.different twoin the units same thehave toneednot doYou ratio.given ain units same theuseyou as long as concrete tosand of volume theratiocan You

=÷×=

×=×

=



Ratios and Proportions Problem 5: The correct answer is A. What is the problem asking?

roof?run ft. 43

15 ain rise theisWhat

What are the facts?

feet. 43

15 is roof theofRun

.31

is run) to(risePitch


ft. 41

5 ft. 5.25 n

3 15.75 n 1 15.75 n 3 :Multiply Cross

run ft. 15.75rise ft.n

31

.roof new to)31

(pitch of proportion a Use

==

÷=×=×

=

KeyTrain Level 7 Applied Mathematics Best Deals - Answers


Best Deals – Answers Best Deals Problem 1: The correct answer is A.

Best Deals Problem 2: The correct answer is A.


Best Deals Problem 4: The correct answer is C.


Best Deals Problem 6: The correct answer is B.


quartper $1.11 34 $37.74quartper $0.90 63 $56.70

=÷=÷

boxper $3.33 42 $139.86boxper $2.70 54 $145.80

=÷=÷

poundper $2.63 2 $5.26poundper $2.38 86 $204.68

=÷=÷

gallonper $1.03 70 $72.10gallonper $1.03 84 $86.52

=÷=÷

copyper $0.12 32 $3.84copyper $0.10 33 $2.97

=÷=÷

caseper $3.07 54 $165.78 caseper $5.19 40 $207.60

=÷=÷

caseper $14.29 39 $557.31 caseper $15.75 62 $976.50

=÷=÷






Best Deals Problem 11: The correct answer is A. What is the problem asking? Which option is the least expensive? What are the facts?

Monster Ski: $525 for 2 hours per day for 12 days, 10 miles away Monster Ski: $24 per hour; 15 miles away Driving cost: $0.27 per mile


$673.20

$589.80

=+==×××=

==××=

××==

=+==×××=

=+=

$97.20 $576 Total SkiMonster $97.20 $0.27 12 2 15

mileper $0.27at days 12for each way miles 15 Driving $576 12 2 24

days 12 day per hrs. 2 hour per $24 Lessons days 12for costs driving plus costslesson SkiMonster

$64.80 $525 Total SkiSuper $64.80 mileper $0.27 12 2 10

mileper $0.27at days 12for each way miles 10 Driving days 12for cost driving $525 SkiSuper

:days) and lessons ofamount same the(assumeoption each for cost total thefind toNeed

boxper $19.13 58 $1,109.54box per $25.30 69 $1,745.70

=÷=÷

literper $6.00 77 $462.00liter per $9.38 97 $909.86

=÷=÷

canper $0.67 39 $26.13can per $0.74 45 $33.30

=÷=÷



Best Deals Problem 12: The correct answer is B. What is the problem asking? What is the percent raise? In other words, what percent higher is the new than the old job? What are the facts?

New job -- $450 per week Old job -- $10.75 per hour for 40 hours per week


4.7% 100% 0.047 100%by gmultiplyinby percent Convert to

0.047 $430 $430) - ($450 original change raisePercent

per week $430 hrs/week 40 $10.75/hr. :job oldConvert per week. assuch -- basis same on the paysboth calculateFirst

=×

=÷=÷=

=×

Best Deals Problem 13: The correct answer is C. What is the problem asking? Which store costs less for the order? What are the facts?

Paper Factory: $27.50 per ream Papers, Inc.: 5 reams for $38 each, $37 for each ream after 5 Sales Warehouse: 750 sheets for $55.50 Need 20,000 sheets.


$1,480

$1,485$1,500

20,000 $0.074 sheetper $0.074 sheets 750 $55.50 : WarehouseSales

$37) reams (35 $38) reams (5 :Inc. Papers, $37.50 reams 40 :FactoryPaper

reams 40 m)sheets/rea (500 sheets 20,000 Need option.each for cost totalCalculate

=×=÷

=×+×=×

=÷



Best Deals Problem 14: The correct answer is B. What is the problem asking? Where can you get the calculator cheaper? What are the facts?

Store A: $65 marked down by 25%, employees get 15% off marked price Store B: $65 marked down by 40%


$26 - $65 0.40) ($65 - $65 discount 40% - price original :B Store

$7.31 - $48.75 0.15) (48.75 - $48.75 $48.75 $16.25 - $65 0.25) ($65 - $65

discount 15% employee -discount 25% - price original :A Store :storeeach for price theCalculate

$39.00

$41.44

==×

==×==×

KeyTrain Level 7 Applied Mathematics Multiple Unknowns - Answers


Multiple Unknowns – Answers Multiple Unknowns Problem 1: The correct answer is C. What is the problem asking? How many of each ticket were sold? What are the facts?

First floor tickets were $3.00, balcony were $2.00 Totals sales were $2,300 from 900 tickets.


500 400 - 900 Y - 900 Floor First (Balcony) 400 Y

2300- 2Y - 3X - 2,700 3Y 3X

:subtract and 3by equation 1st Multiply

$2,300 2Y 3X :amount sales Total 900 Y X :sold ticketsTotal

Y be cketsbalcony ti ofnumber Let Xby etsfloor tickfirst ofnumber Let

===

===+

=+=+

Multiple Unknowns Problem 2: The correct answer is A. What is the problem asking? What is the size of the rectangle? What are the facts?

Perimeter = 54 feet Twice the length is 3 ft. more than the width


10 L 20 2L 17 3 2L

17 W 51 3 - 54 3W

54 2W W) (3 :equationfirst into W 3 2L Substitute

W 3 2L width than themore 3 islength the twice:equation Second

54 2W 2L perimeter for Equation

===+=

====+++=

+=

=+=



Multiple Unknowns Problem 3: The correct answer is C. What is the problem asking? What does each person earn a day? What are the facts?

Engineer worked 6 days and Assistant worked 7 days for $4,500 Engineer worked 5 days and Assistant worked 3 days for $2,900.


$300 A $900 3A

$2,900 3A $2,000 $2,900 3A $400) (5

$2,900 3A 5E andday per $400 E $6,800 17E

$20,300 21A 35E - $2,900] 3A [5E 7 (-$13,500) 21A) (- (-18E) $4,500] 7A [6E 3-

: termslikeget tohaveyou n subtractio use To

$2,900 3A 5E $4,500 7A 6E

pay. seach week'for equation an Write

==

=+=+×

=+==

=+==+×=+==+×

=+=+

Multiple Unknowns Problem 4: The correct answer is D. What is the problem asking? How many gallons to each vehicle? What are the facts?

Total gallons = 60 One gets 12 gallons less than the other


gallons 24 X 12 - 36 X

gallons 36 Y 72; 2Y 12 60 2Y

60 12 - 2Y 60 Y 12) - (Y Y X

:1equation into 2equation Substitute

12 - Y X :other than theless 12 is One Y X gallons Total

carother in the gallons be YLet car onein gallons be XLet

==

==+==

=+=+

=+=



Multiple Unknowns Problem 5: The correct answer is C. What is the problem asking? What is the cost of each individual item? What are the facts?

4 quarts of oil and 40 gallons of gas cost $92.00. 6 quarts of oil and 52 gallons of gas cost $126.00.


oil ofquart afor $8.00 X $32; 4X60 - $92.00 4X

$92.00 $1.50) (40 4XThen

gas ofgallon afor $1.50 16 $24 Y so 24.00; $ 16Y $252.00- 104Y - 12X- $276.00 120Y 2X1

:equation1st thefromequation 2ndsubtract hen tand 12Xget to2by equation 2nd and 3,by equation 1st Multiply

$126.00 52Y 6X :receipt nd2 $92.00 40Y 4X :receipt1st

gas of gallons be Y and oil of quarts be XLet

===

=×+

=÷====+

=+=+

Multiple Unknowns Problem 6: The correct answer is D. What is the problem asking? How much was borrowed from each bank? What are the facts?

Fifth Bank charges 7%, Sixth Bank charges 8% Interest payments for one year are $1,250 Borrowed $300 more from Fifth Bank than Sixth Bank


$8,560 300 B A Then $8,260 B 1,239; 0.15B

1,260 0.08B 0.07B 21 1,260 0.08B B) 0.07(300

2nd. intoequation 1st Substitute $1,260 0.08B 0.07A Interest

300 B A :borrowedamount in Difference B beBank Sixth Let A beBank Fifth Let

=+===

=++=++

=+=+=

KeyTrain Level 7 Applied Mathematics Troubleshooting - Answers


Troubleshooting – Answers Troubleshooting Problem 1: The correct answer is C. What is the problem asking? Is the area actually 37.68 square yards? What are the facts?

The diameter is 12 yards. Set up and solve the problem:

did. what theyis thisYes, 37.68. 6 3.14 2 r 2 C instead? ncecircumfere calculate they Did

No 452. 12 12 3.14 d d 3.14 radius? of insteaddiameter use they Did

No. 36. 6 6 r r pi?forget they Did

correct.not isThat yd. sq. 37.68 figuresThey yd. sq. 113 yd. 6 yd. 6 3.14 Area

yards 6 2

12 2

diameter r Radius, r r 3.14 r is circle a of Area 2

=××==

=××=××

=×=×

=××=

===

××=

π

π

Troubleshooting Problem 2: The correct answer is D. What is the problem asking? Is 15 yards enough for 7 jackets? What are the facts?

Seven yards makes 3 jackets. Set up and solve the problem:

more yards 31

1 15 - 31

16 needYou yards. 15bought only You

yards. 31

16 3

49

37

7

:need jackets 7For

yards 31

2 37

jackets 3yards 7

:requiremust jacket each then jackets,3 makes material of yards 7 If

=

==×

==

KeyTrain Level 7 Applied Mathematics Troubleshooting - Answers


Troubleshooting Problem 3: The correct answer is A. What is the problem asking? Is $1,550 75% off from $2,067? What are the facts?

Original price $2,067. 75% off sale was $1,550. Set up and solve the problem:

price.regular thefrom price theof 75% subtracted have should then theyoff, 75% wassale theIf discount. 25% aor price, original theof 75% wasmarked price The

$517 $1,550 - $2,067 price Sale $1,550 0.75 $2,067 Discount 75% Discount - priceRegular price Sale

===×===

Troubleshooting Problem 4: The correct answer is C. What is the problem asking? Is 7% off of $72.95 really $44.95? What are the facts?

Four items totaled $72.95. They charged $44.95. She was supposed to get 7% off.


$44.95 $28 - $72.95 $7) (4 - $72.95 :instead itemeach off $7 gave that theyfindyou answers,different theBy trying

$44.95. of price a reached they how determine error to and trial Usecorrectnot wasprice The $67.84

$5.11 - $72.95 $72.95) (0.07 - $72.95

discount - priceregular Price Sale :be should charge what theCalculate

==×

==

×==

Troubleshooting Problem 5: The correct answer is D. What is the problem asking? Was the charge correct? What are the facts?

Charges $18 per hour for 2 hours and 15 minutes Drove 21 miles one way. Charge $0.27 per mile


correct. wascharge The $51.84 $11.34 $40.50 charge Total $11.34 mileper $0.27 miles 42

miles 42 2 21 miles totalso back, and both there is mileage(Remember mileage.for Charge $40.50 hour per $18 hours .252

:for time Charge hours 2.25 minutes 15 hours 2

:hours tominuts and hoursConvert

=+==×

=×==×

=

KeyTrain Level 7 Applied Mathematics Nonlinear Functions - Answers


Nonlinear Functions – Answers Nonlinear Functions Problem 1: The correct answer is B. What is the problem asking? What is the braking distance at 60 mph? What are the facts?

The formula is: b = 0.074v2


feet 266 b 3,600 0.074 b60 0.074 b

:sovleandformula in the for v 60 Substitute 2

=×=×= =

Nonlinear Functions Problem 2: The correct answer is B. What is the problem asking? What distance will give twice the light intensity as it does at 5 meters? What are the facts?

2d 72 l :is formula The ÷=


meters 3.5 d root, square thefind tocalculator a Using12.4 d d 72 5.8

lumens/m 5.8 2 2.9 :intensitylight thedoublefor distance thefind oyou want t Now lumens/m 2.9 25 72 5 72 l

:dfor solveandintensity thedoublethen meters, 5at intensity light theFind

2

2

2

==÷=

=×=÷=÷=

Nonlinear Functions Problem 3: The correct answer is D. What is the problem asking? What is the average age of the staff? What are the facts? The ages of the staff members are shown in the table. Set up and solve the problem:

years 41 torounded 40.84 25 1,021 people ofnumber total age Average25 3 2 6 4 3 2 3 2 :is people ofnumber Total

1,021 156 100 270 164 117 72 96 46 :numbers totalAdd156 3 52 :52 Age 100 people 2 50 :50 Age

270 people 6 45 :45 Age 164 people 4 41 :41 Age 117 people 3 39 :39 Age 72 people 2 36 :36 Age 96 people 3 32 :32 Age 46 people 2 23 :23 Age

:people ofnumber by the divide and members staff theof age total theCalculate

=÷=÷==+++++++

=+++++++=×=×

=×=×=×=×=×=×

KeyTrain Level 7 Applied Mathematics Nonlinear Functions - Answers


Nonlinear Functions Problem 4: The correct answer is A. What is the problem asking? What is the balance starting with $1,500 after 5 years at 5% annual interest? What are the facts? The formula is: B = P (1 + i )2


$1,905.96 B math.) (powers) lexponentia performcan which calculator a use should(You

0.004) (1 $1,500 B

60 12 5 months ofNumber 0.004 12 0.05 interest Monthly

:monthsofnumber theandmonth per interest theneed formula, theuse To

60

=

+×=

=×==÷=

Nonlinear Functions Problem 5: The correct answer is D. What is the problem asking? What is the cost of electricity per 1,000 cubic feet of gas when running at full speed? What are the facts? Currently run at 50% speed, electricity costs $0.08 per 1,000 cubic feet. Electrical use at different speeds shown on graph. Set up and solve the problem:

gas offeet cubic 1,000every for $0.12 $X Solving, $XKwh 6

$0.08Kwh 4

:proportion a use Now gas. offeet cubic 1,000 everyfor Kwh 6 use wouldcompressor theload, 100%At gas. offeet cubic 1,000

every for elctricity ofKwh 4 useyou load 50%at graph, the toAccording

=

=

Documents

Applied Mathematics Level 7 - Weeblyslccareerreadiness.weebly.com/uploads/5/4/8/8/54881327/...Problems in Level 7 are the most difficult in the Work Keys system. The mathematics used