Applied Mathematics Level 6 - School Of Social Justice ...sj.lvlhs.org/ourpages/auto/2012/4/8/49927094... · Applied Mathematics Problem Solving ... As before, start with the basic

KeyTrain Level 6 Applied Mathematics Introduction

Copyright © 2000, SAI Interactive, Inc. For use only by Chicago Public Schools. Page 1

For

Applied Mathematics

Level 6

Published by SAI Interactive, Inc., 340 Frazier Avenue, Chattanooga, TN 37405.

Copyright © 2000 by SAI Interactive, Inc. KeyTrain is a registered trademark of SAI Interactive, Inc. WorkKeys 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 6

Applied Mathematics

Introduction Welcome to Level 6 of Applied Mathematics. At Level 6 the tasks are more complex. Problems will require several steps and calculations to solve. The wording and organization of the problems may also be more difficult. Although the problems are more difficult than in previous levels, the math involved is not. The key to understand these new problems is to see them as a series of smaller, easier problems. By breaking larger problems down into smaller ones, you will be able to solve these also.



The Problems Will Include:

- Solving complicated multiple-step problems that may require manipulation of the original information

- Calculations using negative numbers, fractions, ratios, percentages and mixed numbers

- Calculating multiple rates and then compare the ratios or use them to perform other calculations

- Finding areas of rectangles and volumes of rectangular solids

- Determining the best deal using the result in another problem, and

- Finding mistakes in calculations.

Types of Numbers and Quantities The problems in this level deal with the same types of numbers and quantities you have used before: fractions, decimals, percentages and common units of measurement ( for weight, length, time, volume and temperature). You will also work with mixed units of measurement. You may have to convert units in order to solve other problems. In addition, you will also learn to find the area and volume of basic shapes using a formula. You may have to rearrange the formula to solve for the size of the shape from the area or volume.



This Level Is Divided Into Seven Lessons:

• Problem Solving Techniques

• Multiple Step Problems

• Fractions and Decimals

• Percentages

• Area and Volume

• Rates, and • Best Deals.

KeyTrain Level 6 Applied Mathematics Problem Solving


Level 6

Applied Mathematics

Problem Solving There are many different techniques that can be used to solve problems. Sometimes there may be more than one way to solve a problem. In these cases there may be no right or wrong method to use. Some of these problem-solving methods are especially useful when the problems are more difficult. If several steps are involved, or the equations are very difficult, there may be an easier way to figure out the answer. This section will review some of these methods.



Problem Solving Strategies Some of the problem solving strategies you can use are:

• Making a Drawing or Diagram

• Guessing and Testing • Working Backwards • Solving a Similar but Simpler Problem

Examples of each of these methods will be shown. In all cases, it is not how you get the answer that is important. The important thing is if the answer is right. You can and should check your answer to be sure.

Making a Drawing or Diagram If the solution to a problem is not immediately obvious, then a drawing or diagram may help. This is most helpful if you can picture the problem in your mind, but you don't know how to write an equation for the problem. The drawing or diagram helps you to organize information and solve the problem.

1 2 3 4 5 $4 $8 $12 $16 $25

You can use charts or tables. or

You can draw a diagram.



Using a Chart to Solve a Problem Here is an example of using a diagram or chart to solve a problem:

The facts and the question are fairly easy to understand. But it is not so easy to see how to write an equation to solve the problem. It may be easier to use a chart or table to solve the problem. Make a table of the pay for each year by adding the raises to the initial pay: Job 1 Salary Job 2 Salary Year 1 $22,000 $26,000 Year 2 $26,000 ($22,000 + $4,000) $28,000 ($26,000 + $2,000) Year 3 $30,000 ($26,000 + $4,000) $30,000 ($28,000 + $2,000) Answer: You will make the same money in the two jobs in the third year.

There are two jobs you can apply for. The first job pays $22,000 the first year, with raises of $4,000 each year after. The second job pays $26,000 the first year with raises of $2,000 each year after. When would you make as much money in the first job as in the second?



Using a Diagram to Solve a Problem Here is another example where a diagram is useful:

Again, the facts and questions are easy. But setting up the problem is not. This problem become much easier if you make a drawing: 1st 2nd 3rd 4th 5th row row row row row

You are planting a garden in the corner of your backyard. You begin by planting one plant in the corner. Then you plant 3 plants in a diagonal on the second row. Next you plant 5 plants in the third diagonal row. How many plants will you need in the fifth row?

The fifth row will have 9 plants.



Guessing and Testing In some situations, guessing and testing is a very effective problem-solving method. It is especially useful when the answer must be selected from a fixed number of choices. (For example, if you know the answer is a whole number and not a decimal or fraction.) These are the common steps for solving a problem with guessing and testing:

1) Guess an answer to the problem. 2) Test to see if the answer is correct. 3) If the answer is correct, you are done.

If not, then adjust your guess and try again. Note that your second (and third) guesses should be better as you learn from your first guess.



Using Guessing and Testing to Solve a Problem Here is a problem that can be solved with guessing and testing:

As before, start with the basic steps: What is the problem asking? How many videos and cassettes to buy. What are the facts? Must buy 10 items, and comes as close as possible to $220. Videos cost $24.95, cassettes cost $12.95 Use guess and test: First Guess: 5 Videos 5 x $24.95 = $124.95 5 Cassettes 5 x $12.95 = $ 64.75 TOTAL: $189.70

(Not enough spent, so order more videos since they are more expensive.)

Second Guess: 7 Videos 7 x $24.95 = $174.65 3 Cassettes 3 x $12.95 = $ 38.85 TOTAL: $213.50 (Pretty close to $220. If you try 8 videos and 2 cassettes, you will

see that it is over $220.) Therefore the answer is 7 videos and 3 cassettes.

A manager for Tapes “R” Us has budgeted $220 this week for new merchandise for her store. Video cost $24.95 including tax. Cassettes cost $12.95 including tax. She wants to purchase exactly 10 items. How many videos and how many cassettes should she buy to use the most of her budget?



Here is another example of how to guess answers even when you have an equation:

As before, start with the basic steps: What is the problem asking? What is the weight of each barrel? What are the facts? The weight of the first barrel, A, is twice the weight of the

second barrel, B. Together they weight 21 lbs.,

so A + B = 21

Use guess and test: First Guess: The second barrel, B weighs 10 lbs. Therefore the first barrel, A, must weigh 2 x 10 = 20 lbs. The total weight would then be 10 + 20 = 30 lbs.

(Too heavy – they are only supposed to be 21 lbs. Try a lower number.)

Second Guess: The second barrel, B, weighs 8 lbs. Therefore the first barrel, A, must weight 2 x 8 = 16 lbs. The total weight would then be 8 + 16 = 24 lbs. (Still too heavy – they are only supposed to be 21 lbs. Try a lower

number.)

Third Guess: The second barrel, B, weighs 7 lbs. Therefore the first barrel, A, must weight 2 x 7 = 14 lbs. The total weight would then be 7 + 14 = 21 lbs. Correct!

You produced two barrels of a chemical. The first barrel weighed twice as much as the second barrel. Together the two barrels weighed 21 pounds. What was the weight of each barrel?



Working Backwards Sometimes you may know the final result of a problem or math calculation. You may be asked to find the beginning numbers or data. To solve these problems you can work the problem backwards. Simply reverse the order of the math operations. Here is an example of a problem that can be worked backwards:

As before, start with the basic steps: What is the problem asking? Find the original cost of the rod and reel (without the case, markup or sales tax). What are the facts?

Mr. Lund charged $162, which he found by taking the cost of the rod and reel, adding $10, then multiplying by 2 for markup, then adding $12 for tax.

To solve the problem: Perform the math steps in reverse order using the reverse operations (when he

added, you subtract and when he multiplied you divide).

Check your answer by performing the calculations forward again.

Mr. Lund runs a sporting goods store. He sells a rod and reel in a case for $162. To determine the selling price, he added $10 for the case to the cost of the rod and reel. Then he doubled the total for his markup. Finally, he added a $12 sales tax. How much did the rod and reel cost Mr. Lund?

reel. and rod for the $65 paid He$65 $10 - $75 :case theadding Reverse

$75 2 $150 :markup theReverse$150 $12 - $162 : taxsales theReverse

==÷=



Solve a Similar but More Simple Problem This strategy works great when you are working with larger numbers, fractions or decimals. These numbers can sometimes make the problem confusing. To decide how to solve the problem, make up a similar problem situation where the numbers are smaller and easier to understand. Then use the same strategy to solve your problem. Here is a problem where fractions and unit conversions make the problem appear more difficult:

As before, start with the basic steps: What is the problem asking? Find the number of full packages of raisins you can make. What are the facts? You have 5 ½ pounds of raisins. Each package must have 9 ounces of raisins. Solve the problem:

If the numbers are confusing, imagine a more simple problem using different numbers. For instance:

How many packages will 10 pounds of raisins fill if each package holds 2 pounds? Here the answer is easy. You could fill 5 packages. You find this by dividing 10 lbs. by 2 lbs. (the total amount of raisins by the amount in each bag).

How many packages will 5 ½ pounds of raisins fill if each package holds 9 ounces?



Therefore you know that to solve the original problem, you must divide the total amount of raisins (5 ½lbs.) by the amount in each bag (9 ounces). To do this, you must first convert the different weights to the same unit of measurement.

package.) fullanother for enought notbut raisins, extra some be will(There packages. full 9 isanswer The

9.7678... ounces 9 ounces 88 :answer thefind todivideThen

ounces 88 pound 1ounces 16 lbs. 5.5 lbs.

215

:ounces topounds econvert th solve, To

=÷

=×=

ounces? 9 holdspackage each if fill raisins of pounds

215 willpackagesmany How



Summary -- Problem Solving Strategies This section has reviewed four strategies for solving more difficult problems:

• Using a Drawing or Diagram

• Guess and Test • Working Backwards • Solving a More Simple Problem

You may want to use these methods in problems later in this level. If you find a problem to be particularly difficult, then see if one of these methods can help you.

KeyTrain Level 6 Applied Mathematics Multiple Steps


Level 6

Applied Mathematics

Multiple Steps Solving problems in level 6 may require several steps. Often this may involve converting measurement units before performing other calculations. For example, say you need to add two lengths. However one length is given in inches and the other is given in yards. You must then convert the lengths to the same units before adding. If you need unit conversion factors during any of these problems, you can refer to the page of Formulas on the next page.



Formulas and Conversions

MEASUREMENT Distance Electricity 1 foot (ft.) = 12 inches (in.) 1 kilowatt-hour = 1,000 watt-hours 1 yard (yd.) = 3 feet amps = watts / volts 1 mile (mi.) = 5,280 feet 1 mile ≈ 1.61 kilometers (km.) FORMULAS 1 inch = 2.540 centimeters (cm.) (x is used to indicate multiply 1 foot = 0.3048 meters (m.) pi is equal to 3.14) 1 meter = 1,000 millimeters (mm.) 1 meter = 100 centimeters Rectangle 1 kilometer = 1,000 meters perimeter = 2(length + width) 1 kilometer ≈ 0.62 miles area = length x width Area Rectangular Solid (Box) 1 square foot (sq. ft.) = 144 square inches (sq. in.) volume = length x width x height

1 square yard (sq. yd.) = 9 square feet 1 acre = 43,560 square feet Cube volume = (length of side)3 Volume 1 cup (C.) = 8 fluid ounces Triangle 1 quart (qt.) = 2 pints (pt.) = 4 cups sum of angles = 180° 1 gallon (gal.) = 4 quarts area = ½ (base x height) 1 gallon (gal.) = 231 cubic inches (cu. in.) 1 liter (l.) ≈ 0.264 gallons = 1.056 quarts Circle 1 cubic foot (cu. ft.) = 1,728 cubic inches number of degrees in a circle = 360°

1 cubic foot = 7.48 gallons circumference ≈ 3.14 x diameter or 1 cubic yard (cu. yd.) = 27 cubic feet pi x diameter 1 board foot = 1 inch by 12 inches by 12 inches area ≈ 3.14 x (radius)2 or pi x (radius)2 Weight 1 ounce (oz.) ≈ 28.350 grams (g.) Cylinder 1 pound (lb.) = 16 ounces volume ≈ 3.14 x (radius)2 x height or 1 pound ≈ 453.592 grams pi x (radius)2 x height 1 milligram (mg.) = 0.001 grams 1 kilogram (kg.) = 1,000 grams Cone

1 kilogram ≈ 2.2 pounds volume ≈ 3.14 x (radius)2 x height 1 ton = 2,000 pounds 3 Temperature Sphere (Ball) °C = .56(°F – 32) or 5/9(°F – 32) volume ≈ 4 x 3.14 x (radius)3 ° F = 1.8(°C) + 32 or (9/5 x °C) + 32 3



Basic Method for Solving Word Problems Remember that longer or complicated word problems are not really any more difficult than shorter problems. The math operations are the same. Just break the problem down into smaller parts. Solve the smaller parts, and then you can find the answer easily! As before, remember to use the basic method for solving word problems:

1. Read the problem. Find what it is asking.

2. Write down the facts you have.

3. Set up and solve the problem.

4. Check your answer.

Example of a Multiple-step Problem

Actually, this is just a subtraction problem. But before subtracting the temperature, you need to know how many days passed from Sunday to Saturday. By counting the days, you can find that 6 days have passed. If this is not clear, you can use a diagram to help: 1 2 3 4 5 6

Sunday Monday Tuesday Wednesday Thursday Friday Saturday Then you can determine the temperature of Saturday: 35°F - (6 x 7°F) = 35°F - 42°F = -7°F

During one winter day the temperature on Sunday was 35°F. During the rest of the week, the temperature dropped 7 degrees each day. What was the temperature on Saturday?



Types of Problems This section will deal with the following types of problems:

• Quantities • Positive and Negative Numbers, and

• Money.

These topics have been covered in general in earlier sections. If you need some review of these topics, go back to the appropriate sections in Levels 3, 4 or 5.



Multiple Steps Problem 1 A person earned $100 a week for 15 weeks. He puts $35.75 into a savings account each of these weeks and spends the rest. How much does he spend during the 15 weeks? Check the correct answer. _____ A. $536.25 _____ B. $963.75 _____ C. $1,063.75 _____ D. $1,500.00



Multiple Steps Problem 2 Refer to the table below to answer this question.

Calcium and Sodium in Breakfast Foods

FOOD SERVING CALCIUM SODIUM Bacon Butter Coffee

Corn Flakes Egg

Syrup Milk

Orange Juice Pancake

Toast Tomato Juice

2 pieces 1 pat

250 ml 25 g

1 20 ml 250 ml 250 ml 27 g

1 slice 250 ml

2 mg 1 mg

0 4 mg 28 mg 25 mg

291 mg 25 mg 27 mg 21 mg 17 mg

325 mg 41 mg

0 251 mg 50 mg 4 mg

120 mg 4 mg

115 mg 170 mg 740 mg

As a dietitian, your patient is on a low-sodium diet (less than 1,100 mg of sodium per meal). She has a breakfast of orange juice, corn flakes, milk, 2 slices of toast with one pat of butter each and coffee. How much sodium did the patient have for breakfast? Check the correct answer. _____ A. 364 mg _____ B. 586 mg _____ C. 756 mg _____ D. 797 mg



Multiple Steps Problem 3 Refer to the table below to answer this question.

Calcium and Sodium in Breakfast Foods

FOOD SERVING CALCIUM SODIUM Bacon Butter Coffee

Corn Flakes Egg

Syrup Milk

Orange Juice Pancake

Toast Tomato Juice

2 pieces 1 pat

250 ml 25 g

1 20 ml 250 ml 250 ml 27 g

1 slice 250 ml

2 mg 1 mg

0 4 mg 28 mg 25 mg

291 mg 25 mg 27 mg 21 mg 17 mg

325 mg 41 mg

0 251 mg 50 mg 4 mg

120 mg 4 mg

115 mg 170 mg 740 mg

As a dietitian, your patient is on a low-sodium diet (less than 1,100 mg of sodium per meal). On another day, your patient has orange juice, 2 slices of bacon, 3 eggs, 2 slices of toast with butter and milk. Is she still following her diet plan of 1,100 mg of sodium per meal? Check the correct answer. _____ A. Yes _____ B. No _____ C. Cannot tell



Multiple Steps Problem 4 On Monday Memorial Hospital had 100 patients. Tuesday it received 15 new patients and discharged 3. Wednesday it received 9 and discharged 12. Thursday it received 5 and discharged 2, and Friday it received 13 and discharged 5. How many patients were in the hospital at the end of the day Friday? Check the correct answer. _____ A. 36 _____ B. 80 ______ C. 120 _____ D. 164

H HOSPITAL



Multiple Steps Problem 5 A checking account has $500 to pay installment payments with. The payments were $35 a month for 4 months and then $25 a month for 3 months. How much was left in the account after the payments were made? Check the correct answer. _____ A. $285 _____ B. $360 _____ C. $425 _____ D. $440



Positive and Negative Numbers

The next couple of problems involve both positive and negative numbers. As a review, remember these rules for doing math with positive and negative numbers:

Adding If the numbers have the same sign, add the numbers and use the same sign:

If the numbers have different signs, subtract and use the sign of the larger number:

Subtracting

Change the sign of the number being subtracted, then add as shown above: 2 - 5 = 2 + (-5) = -3

-7 - 6 = -7 + (-6) = -13

-3 - (-8) = -3 + 8 = 5

9 - (-4) = 9 + 4 = 13 Multiplying or Dividing If both numbers are the same sign, then the answer is positive.

If the numbers have different signs, then the answer is negative.

negative.) both were sincesign negative abut with 11, 6 (5 (-11) (-6) (-5)

13) 9 4as(Same 13 9 4=+=+

=+=+

negative) isanswer theso negative is 15 9,n larger tha is 15 and 6, 9 - (15 6- (-15) 9

positive) isanswer theso positive, is 5 3,n larger thais5and2, 3 - (5 2 5 (-3)

==+

==+

9 (-6) 54- 32 4- 8-6 7 42 21 3 7=÷=×

=÷=×

7- (-7) 49 7- 8 56- 54- 6 9- =÷=÷=×



Multiple Steps Problem 6 One day the temperature rose 5 degrees in the morning, then it dropped 9 degrees in the afternoon. The temperature at dawn was 3 degrees below. What was the temperature at the end of the day? Check the correct answer. _____ A. -7 degrees _____ B. -1 degree _____ C. 1 degree _____ D. 7 degrees



Multiple Steps Problem 7 The highest and lowest temperatures recorded in New York one year were 38 degrees Celsius and –21 degrees Celsius. The next year the highest and lowest temperatures were 36°C and -25°C. What was the difference in the lowest and highest temperatures over the two years? Check the correct answer. _____ A. 13°C _____ B. 17°C _____ C. 61°C _____ D. 63°C



Multiple Steps Problem 8 When you last balanced your checkbook, you had $463.76. You then wrote checks for $12.56, $52.11 and $26.03. You deposited $101.32 and $10.98. Your bank says your balance is $485.36, but your checkbook says $511.39. What, if anything, did you do wrong in tracking your checking account? Check the correct answer. _____ A. Nothing, the bank is wrong. _____ B. Carried wrong when adding $52.11. _____ C. Forgot to subtract $26.03 check. _____ D. Forgot to add $10.98 deposit.



Summary – Multiple Step Problems The problems in this section were not highly complex. They just required more than one calculation or comparison. The key to solving longer problems is to see them as a series of smaller, easier problems. If you can see this in the problems you face, you will be able to solve much more difficult problems than these were. You will see problems like this in the next sections.

KeyTrain Level 6 Applied Mathematics Fractions and Decimals


Level 6

Applied Mathematics

Fractions and Decimals This section has problems dealing with fractions and decimals. These problems use the same mathematical operations that were covered in the Level 5 section on fractions and decimals. These include: addition, subtraction, multiplication and division. The difference here is that there may be several steps required to solve the problem. Again, break the problem down into smaller steps. By solving each smaller, easy step, the larger problem can be solved.



Review Basic Math Operations with Fractions 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 the 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 fractions, 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 ==×

531

58

12

54

21

54 ==×=÷

1518

15121

311

511

323

512 ==×=×



Fractions and Decimals Problem 1

How much does a one-foot section of steel weight? (use decimals) Check the correct answer. _____ A. 3.2 lbs. _____ B. 6.4 lbs. _____ C. 6.6 lbs. _____ D. 7.2 lbs.

pounds. 43105 weighslongfeet

2116 measures that steel ofbar A



Fractions and Decimals Problem 2

32369 D. _____

32963 C. _____

60 B. _____

3274 A. _____

answer.correct Check the drills? 15 make toneeded are rod of inchesmany How

made. drilleach for inch waste 325 allowmust You

long. inches 1614 isbit Each rod. drill from made are bits Drill



Fractions and Decimals Problem 3 You must drill 5 equally spaced holes in a line along a board. The holes will measure 4 3/8 inches apart, center to center. The holes are 3/4" in diameter. What is the total length of the holes (i.e. the distance between the end holes, including the holes)? Check the correct answer.

inches 2120 D. _____

inches 4118 C. _____

inches 8717 B. _____

inches 2117 A. _____



Fractions and Decimals Problem 4 A customer wants to carpet a family room measuring 14' 6" by 22' 9" and a hallway that is 4' by 9'8". Ignoring any waste, about how much carpet is needed for this job? Check the correct answer. _____ A. 37 sq. yd. _____ B. 41 sq. yd. _____ C. 330 sq. yd. _____ D. 396 sq. yd.



Fractions and Decimals Problem 5 Three electrical appliances have power ratings of 1 7/8 watts, and 2 other appliances have power ratings of 4 3/4 watts. What is the total power used by these appliances? Check the correct answer.

watts8115 D. _____

watts4110 C. _____

watts856 B. _____

watts855 A. _____



Fractions and Decimals Problem 6 A jeweler is making a copy of a 15-inch chain to sell in his store. The clasp is 1/2 inch long. Each link in the chain is 1/4 inch long. How many links must be used to make the chain? Check the correct answer. _____ A. 4 _____ B. 30 _____ C. 58 _____ D. 60



Fractions and Decimals Problem 7 Your co-worker packed a case of 12 gears for shipment. Each gear weighs 1 lb. 3 oz. The box and packing weigh 2 lbs. He marked the shipping weight of the box as 15.6 lbs. Is the shipping weight correct? If not, why? _____ A. Yes, it is correct. _____ B. No, he forgot to add the box weight. _____ C. No, he converted the ounces wrong. _____ D. No, he only counted 10 gears.



Summary – Fractions and Decimals These problems have used fractions in more complicated calculations. These may include finding common denominators, converting mixed numbers, or several math operations. Simply focus on what math operations are required to solve the problem. Then you can convert denominators or mixed numbers as you need to solve the equations.

KeyTrain Level 6 Applied Mathematics Percentages


Level 6

Applied Mathematics

Percentages In Level 5, you saw how percentages can be used to describe portions of a larger amount. Level 5 problems usually asked to find a portion, or percentage, of a larger number. An example of this is finding the sale price of a $10 shirt that is on sale for 30% off. The answer would be $7. ($10 - 30% of $10.) In Level 6, some of the problems may work the opposite way. The problem may give an amount that is a certain percentage of a larger number. The answer will be to find the larger amount. An example of this would be finding the regular price of a shirt that has been marked down by 30% and is now on sale for $7. The answer is $10.



Percentages As a review, percent means the number of parts out of 100 total parts:

100% of the circle 50% of the circle 25% of the circle is green. is green. is green.

Finding the Percentage of a Number As a review, you can find the percentage of a number using a ratio, or by multiplying by the percentage as a fraction or decimal. For example, what is 75% of 80? Using a ratio:

You know that 75% mans 75 parts of 100. Set up equal fractions to find how many parts out of 80 is equal to 75 parts of 100. Then cross multiply.

By multiplying by the percentage as a fraction or decimal:

Convert the percentage to a decimal and multiply.

As you become familiar with fractions, you will probably find the second method to be faster. You will know that 75% is the same as multiplying by 0.75.

0.25 41

10025

25% 0.5 21

10050

50% 1.0 100100

100% ========

60 100 80 75 X 80 75 X 100 80X

10075

=÷×=×=×=

60 80 0.75 0.75 10075

75% =×==



Find the Percentage One Number is of Another Some problems may give two numbers, and ask how many percent one is of another. For instance, how many percent is 8 of 24? Here is how this might be asked: You can use a proportion or ratio to solve this problem. (You know a day has 24 hours.)

Most businesses have an eight-hour workday. What percent of the day are most workers at their jobs?

33% 0.333 0.333 24 8 :percent convert to then

248 dividingby decimal equivalent theFind :2 Method

33% 24 100 8 X 100 8 24 X :multiply Cross100X

248

⎯→⎯=÷

=÷×=×=×

=



Percentages Problem 1 Find the percentage described to the nearest whole number. 163 is what percent of 921.5? Answer:

Percentages Problem 2 Find the percentage described to the nearest whole number. 516.5 is what percent of 675.7? Answer:




Percentages Problem 4 Find the percentage described to the nearest whole number. 157.8 is what percent of 273? Answer:















Finding a Number when a Percent of it is Known In Level 6 problems may ask to find a number by giving a known percentage of that number. For instance:

From this statement, you know that 160 teenagers must have been the 80% of all the teenagers that were surveyed. So you must determine what number 160 is 80% of. In other words, what number times 80% gives 160?

In a recent survey, 160 teenagers said that television influences them to buy advertised products. The survey summary states that 80% of teenagers said that TV influences their buying. How many teenagers were surveyed?

200 80 100 160 X 100; 160 80 X :multiply Cross

10080

X

160

:ratio a use also couldYou

200 0.80 160 X so 160; 0.80 X 80% X

=÷×=×=×

=

=÷==×=×



Percentages Problem 11 Find the number described to the nearest whole number. 124.44 is 34 percent of what number? Answer:








Percentages Problem 15 Find the number described to the nearest whole number. 18 is 25 percent of what number? Answer:












Percentages Word Problems Here is an example of a word problem that finds a number knowing a percentage of that number:

1. First, read the problem carefully. What is the problem asking?

How many fuses were inspected?

2. What are the facts? 5% of the fuses were defective. 32 were found to be defective.

3. Set up and solve the problem.

4. Check that the answer is reasonable.

Five percent of a batch of fuses was found to be defective. If 32 fuses were defective, how many fuses were inspected?

640 0.05 32 X32 X 0.05

32? equalsnumber what of 5%

=÷==×

ok. checks this,calculator a Using32 0.05 640 5% 640defectivenumber thefindingby Check

=×=×



Percentages Problem 21 Twenty-five percent (25%) of the homes in a neighborhood have computers. If 50 homes in the neighborhood have computers, how many homes are there? _____ A. 12.5 _____ B. 75 _____ C. 100 _____ D. 200



Finding the Percent Increase or Decrease Changes are often described in terms of a percentage increase or decrease. The percentage change is always the percentage that the change is of the original amount.

100% amount Original

change ofAmount change Percentage ×=

Consider a salary increase from $72 per day to $90 per day. What is the percent increase?

25% 100% 0.25 100% 7218

increasePercent

18 72 - 90 change ofAmount

=×=×=

==



Percentages Problem 22 Find the percent described to the nearest whole number. What is the percent increase from 614 to 1,031.52? Answer:

Percentages Problem 23 Find the percent described to the nearest whole number. What is the percent increase from 345 to 451.95? Answer:








Percentages Problem 27 Find the percent described to the nearest whole number. What is the percent decrease from 255 to 140.25? Answer:



Percentages Problem 28 Find the percent described to the nearest whole number. What is the percent decrease from 592 to 444? Answer:





Percentages Problem 31 Find the percent described to the nearest whole number. What is the percent decrease from 423 to 16.92? Answer:



Discounts Discounts is just another word for saying a percent decrease.

Green Thumb Nursery ordered $175 worth of seeds. When they ordered, they got a 15% discount because the order was over $100. How much did they pay for the seeds?

$148.75 $26.25 - $175 - paidactually they So$26.25 0.15 $175 15% $175 then is change The

100% $175

change 15%

So $175. isamount original theand 15%, is changepercent The

100% amount Original

change ofAmount change Percentage

:hatRemember t 15%.by reducedwasprice themeansdiscount 15% The

===×=×

×=

×=

change theoriginal the



Percentages Problem 32 One day in March, 75% of the stores' customers paid with credit cards. If there were 50 customers that day, how many used a credit card? Check the correct answer. _____ A. 13 _____ B. 25 _____ C. 38 _____ D. 40



Percentages Problem 33 Shares of stock were sold and a profit of $1,320 was made. The profit was 15% over a 30-day period. How much were the shares worth when they were originally purchased? Check the correct answer. _____ A. $198 _____ B. $1,122 _____ C. $8,000 _____ D. $8,800



Percentages Problem 34 Recently at a theater, they were evaluating attendance at several events. From the first event to the second event the attendance dropped from 250 to 230. Find the percent decrease to the nearest percent? Check the correct answer. _____ A. 8% _____ B. 9% _____ C. 20% _____ D. 92%



Percentages Problem 35 You have determined that you saved $12.50 when you bought a pair of jeans on sale. They were on sale for 25% off. What was the original price and what was the sale price? Check the correct answer. _____ A. $25.00, $12.50 _____ B. $50.00, $12.50 _____ C. $37.50, $50.00 _____ D. $50.00, $37.50



Percentages Problem 36 A hardware store owner bought a used refrigerator for $155 and marked it to sell for a profit of 30% on the cost. He then sold it for 10% less than the marked price. What was the selling price? Check the correct answer. _____ A. $170.50 _____ B. $181.35 _____ C. $186.00 _____ D. $201.50



Percentages Problem 37 Using the same refrigerator sale: A hardware store owner bought a used refrigerator for $155 and marked it to sell for a profit of 30% on the cost. He then sold it for 10% less than the marked price. What is the percent profit made by the owner? Check the correct answer. _____ A. 15% _____ B. 17% _____ C. 19% _____ D. 30%



Percentages Problem 38 A sales person receives $250 per week and a 3% commission on all sales over $5,000. What are her total earnings if her weekly sales totaled $9,525? Check the correct answer. _____ A. $135.75 _____ B. $285.75 _____ C. $385.75 _____ D. $535.75



Percentages Problem 39 A dealer sells air compressors at 20% off of the manufacturer's suggested retail price (MSRP). Then the dealer had a sale where she took an additional 30% off of hes normal price. If the MSRP was $180 and the dealer charged $90, did she charge you right? If not, why? Is the price correct? If not, why? Check the correct answer. _____ A. Yes, it is correct. _____ B. No, she took only the 30% off MSRP _____ C. No, she forgot to take the 30% off. _____ D. No, she took 30% off of the MSRP, not the normal price.



Percentages Summary This section has included several different types of problems using percentages. These included finding the percentage increase or decrease, determining the total from a percentage of the total, and finding discounts and markups. As you can see, there are many different words and terms used to describe percentages. The key to working with more complicated problems is to read them carefully. You can always use the same ratio to solve the problem. However you need to be careful to put the right numbers in the right place! This is another place where practice can save you money. When you are buying items on sale, make sure you are getting the full discount that was advertised!

KeyTrain Level 6 Applied Mathematics Area and Volume


Level 6

Applied Mathematics

Area and Volume Many jobs require calculations of area and volume. Construction, engineering, landscaping, decorating, surveying and sewing are all examples where this type of math is essential to making the most of your resources. In Level 6, problems will involve manipulating the area of squares, rectangles, circles and triangles. Other problems will deal with the volume of rectangular solids.



Perimeter, Area and Volume As a review, recall the difference between perimeter, area and volume: 8 ft. 6 ft. 6 ft.

8 ft. Perimeter The total length around the outside of an object. Perimeter = 6 ft. + 8 ft. + 6 ft. + 8 ft. = 28 ft. Area The surface area of a flat shape. Area = 6 ft. x 8 ft. = 24 sq. ft. Volume

The total size of a 3-dimensional object. This is like the amount an object could hold if it were a container.



Area of a Rectangle As was mentioned in Level 5, the area of a region is the number of square units of space needed to cover the region. For instance, the size of rooms in a house can be measured in square feet. The area of a rectangle or square can be found by multiplying the length by the width: Area = Length x Width Suppose you wanted to cover the area shown below with one-foot square tiles. To determine the number of tiles, you would multiply 7 x 8 = 56 tiles. This is the same as finding the area in square feet. 8 ft.

Length x Width = Area 8 ft. x 7 ft. = 56 sq. ft.

7 ft.



Rearranging a Formula for Area Sometimes the formula needs to be altered or reversed to solve a problem. If the area and width of a rectangle are given, then rearrange the formula to find the length. 22 in. ? in. Example: If the area of a rectangle is 352 square inches, and the

length is 22 inches, what is its width? Area = Length x Width

Rearrange the equation to find the width: (Remember that division is the opposite of multiplication.)

in. 16 in. 22 in. sq. 352 Length Area

=÷=÷=

WidthWidth

352 sq. in.



Using Area to Solve a Word Problem Here is an example of using area to solve a word problem: 1. First, read the problem carefully. What is the problem asking?

What is the length (depth) of the store? 2. What are the facts?

The store is 8,800 sq. ft. in area and 110 ft. wide. 3. Set up and solve the problem.


Area = 80 ft. x 110 ft. = 8,800 sq. ft.

You are laying out a new retail store and need to arrange some displays. You know that the store is 8,800 sq. ft. in size, and that it is 110 ft. wide. How deep (from front to back) is the store?

ft. 80 110 8,800 Length Width Area Length WidthLength Area

=÷=÷=×=



Area of a Triangle For a triangle, the area is always one half of the base times the height.

ft. sq. 28 8ft. ft. 7 21

height base 21

Area

triangle,aFor

=××

××=

base – 7 ft.

height - 8 ft.



Circumference of Circles The perimeter of a circle is called the circumference. The circumference (C) is equal to 3.14 times the diameter. This is normally written as:

C = π x d

Where π is pi (said like “pie”), and is equal to 3.14159… Recall that the diameter (d) is the total width of the circle. The radius (r) is the distance from the center to a point on the circle. The circumference can also be written as:

radius diameter

r 2 d C ××=×= ππ



Area of Circles The area of a circle (A) is equal to 3.14 times the radius times the radius. This is normally written as:

Since the radius is equal to half the diameter,

3.14159... toequal is and ),pie"" like (said pi is where

r r r A 2

π

ππ =××=

4 d 2) (d r A 222 ÷×=÷×=×= πππ

radius diameter



Area and Volume Problem 1 Please answer the question below. Round the answer to two decimal places. Use π = 3.14 when needed. What is the area of a square that is 7.1 yards on a side? Answer:

Area and Volume Problem 2 Please answer the question below. Round the answer to two decimal places. Use π = 3.14 when needed. What is the perimeter of a square that is 16.8 ft. on a side? Answer:



Area and Volume Problem 3 Please answer the question below. Round the answer to two decimal places. Use π = 3.14 when needed. What is the area of a square that is 19.8 meters on a side? Answer:

Area and Volume Problem 4 Please calculate the area as described below. Please answer the question below. . Use π = 3.14 when needed. What is the area of a circle that is 12.6 inches in diameter? Answer:



Area and Volume Problem 5 Please answer the question below. Round the answer to two decimal places. Use π = 3.14 when needed. What is the area of a circle that is 2.2 feet in diameter? Answer:

Area and Volume Problem 6 Please answer the question below. Round the answer to two decimal places. Use π = 3.14 when needed. What is the perimeter (circumference) of a circle that has a radius of 19.3 yards? Answer:



Area and Volume Problem 7 Please answer the question below. Round the answer to two decimal places. Use π = 3.14 when needed. What is the perimeter (circumference) of a circle that has a radius of 6.5 feet? Answer:

Area and Volume Problem 8 Please answer the question below. Round the answer to two decimal places. Use π = 3.14 when needed. What is the perimeter of a rectangle that is 19.1 inches on one side and 4 inches on the other? Answer:



Area and Volume Problem 9 Please answer the question below. Round the answer to two decimal places. Use π = 3.14 when needed. What is the area of a rectangle that is 3.4 inches on one side and 4 inches on the other? Answer:

Area and Volume Problem 10 Please answer the question below. Round the answer to two decimal places. Use π = 3.14 when needed. What is the perimeter of a rectangle that is 5.5 ft. by 9.8 ft.? Answer:



height = 3

width = 4 depth = 3

Volume The volume of a container tells the size of the inside of the container. Common measures of volume are the gallon, cup, and cubic foot. To determine the volume of a rectangular solid, multiply the width times the depth times the height:

Volume = 4 x 3 x 3 = 36



Figuring Volume Here is a word problem that involves volume: 1. First, read the problem carefully. What is the problem asking?

How many 1 cubic meter boxes will it hold? In other words, what is the volume of the container in cubic meters?

2. What are the facts?

The container is 7 meters long by 5 meters wide by 3 meters high 3. Set up and solve the problem:


One layer of the container holds: 7 x 5 = 35 boxes Three layers high holds: 35 x 3 = 105 boxes.

A shipping container has the following dimensions: length – 7 m., width – 5 m., height – 3 m. You must ship square boxes 1 meter long on each side inside the shipping container. How many boxes will it hold?

boxes. 105 hold it will meters, wholeof dimensions hascontainer theSincemeterscubic105 3m 5m 7m height width length Volume =××=××=



Area and Volume Problem 11 You bought a toolbox that is 1 ft. long, 6 inches wide and 8 inches high. What is the volume of this toolbox? _____ A. 48 cubic inches _____ B. 128 cubic inches _____ C. 480 cubic inches _____ D. 576 cubic inches



Area and Volume Problem 12 A parcel of land has an area of 2 square miles and a width of 6,500 feet. How long is the land? Check the best answer. _____ A. 1.23 miles _____ B. 8,580 feet _____ C. 12,989 feet _____ D. 10,560 feet



Area and Volume Problem 13 You are carpeting a room 16 feet by 14 feet 6 inches with a rug that costs $18.50 per square foot of finished space. How much would the flooring cost? Check the correct answer. _____ A. $232 _____ B. $4,144 _____ C. $4,292 _____ D. $4,322



Area and Volume Problem 14 You have a space in front of your house, 10 feet long and 3 feet wide. You need to put a concrete sidewalk 6 inches thick in this space. How many cubic feet of concrete are needed for this sidewalk? Check the correct answer. _____ A. 12.5 cubic feet _____ B. 15 cubic feet _____ C. 30 cubic feet _____ D. 180 cubic feet



Area and Volume Problem 15 A 9 foot by 12 foot long rug is placed in a room that has an area of 238 square feet with a length of 17 feet. How much of the room is left uncovered? Check the correct answer. _____ A. 14 square feet _____ B. 108 square feet _____ C. 130 square feet _____ D. 217 square feet



Area and Volume Problem 16 In square feet, how large a tablecloth will be needed to cover a table that measures 41 inches long and 25 inches wide? How large must the tablecloth be? Check the correct answer. _____ A. 7.1 square feet _____ B. 10.25 square feet _____ C. 14.76 square feet _____ D. 1025 square feet



Area and Volume Problem 17 A large aquarium is 4 m. long, 2 m. wide and 3 m. deep? How many cubic meters of water will it hold? Check the correct answer. _____ A. 9 cubic meters _____ B. 24 cubic meters _____ C. 36 cubic meters _____ D. 48 cubic meters



Area and Volume Problem 18 In a new home, the front entry and living room are going to be partially carpeted. The entry measures 10 feet by 9 feet and the living room measures 24 feet by 18 feet. If they carpet 90% of the area, how many square feet of carpet will they need to purchase? Check the correct answer. _____ A. 342 square feet _____ B. 432 square feet _____ C. 470 square feet _____ D. 522 square feet



Area and Volume Problem 19 One pound of grass seed covers 135 square feet of lawn and costs $4.65. What is the cost of seeding a lawn that measures 15 yards by 12 yards? Check the correct answer. _____ A. $6.20 _____ B. $55.80 _____ C. $120.00 _____ D. $7,533.00



Area and Volume Problem 20 A chemical storage bin measures 6 meters by 3 meters by 5.4 meters. How many bins will be needed to store 120 cubic meters of chemical? Check the correct answer. _____ A. 1 _____ B. 2 _____ C. 32 _____ D. 97



Area and Volume Problem 21 You want to add a cellar to your store. The construction company told you it would cost $5.90 per cubic yard to dig the cellar, plus $20 per square foot to finish it. How much will it cost to add a cellar 36 feet long, 14 feet wide and 8 feet deep? Check the best answer. _____ A. $881.07 _____ B. $10,080.00 _____ C. $10,960.87 _____ D. $23,788.80



Area and Volume Problem 22 How many cubic yards of concrete are needed for a driveway that measures 12 feet wide, 81 feet long and 6 inches deep? How many cubic feet of concrete are needed for this driveway? Check the correct answer. _____ A. 97 cubic feet _____ B. 486 cubic feet _____ C. 972 cubic feet _____ D. 5,832 cubic feet



Area and Volume Problem 23 You are carving a block of ice for a banquet. It measures 1½ feet long, 1½ feet wide and 1½ feet thick? Ice weighs 57 pounds per cubic foot. What is the weight of the ice block? Check the correct answer. _____ A. 3.375 pounds _____ B. 16.89 pounds _____ C. 57 pounds _____ D. 192.375 pounds



Area and Volume Problem 24 Which has the greater volume:

a 6 cm. cube or a rectangle solid with a 3 cm. square base and a height of 12 cm.?

Which one is larger? Check the correct answer. _____ A. The cube _____ B. The rectangle _____ C. They are the same _____ D. There is not enough information



Area and Volume Problem 25 A moving crate measures 5’3” wide by 3’6” high by 4’9” long. The box is marked 87.3 cubic feet. Is the size marked correct? If not, why? Check the correct answer. _____ A. Yes, it is correct. _____ B. No, forgot to multiply by the height _____ C. No, they added instead of multiplying _____ D. No, converted in. to ft. wrong.



Summary – Area and Volume These problems have shown how area and volume can be used to plan many different types of jobs. By calculating ahead, you can save money on materials and estimate the total cost of the job. Be aware that sometimes people do not say the units correctly on area and volume measurements. For instance, carpet is normally sold by the square yard. However the salesperson may simply say "$10 per yard". Similarly, dirt is usually sold by the cubic yard, even though people may say "per yard" for short.

KeyTrain Level 6 Applied Mathematics Rate Problems


Level 6

Applied Mathematics

Rate Problems A rate is a comparison of two quantities with different units. This is commonly used to describe how fast or how often something occurs. Say you drive 200 miles in 4 hours. What is the rate of travel? In other words, what is your speed?

hoursper miles 50 hours 4

miles 200=

.in as ,per"" work with thespoken usually is rate The .

4200fraction the toequal is hours 4in miles 200 of rate

theTherefore dividing. like is in"" wordThe fractions. as expressed becan Rates

hour per miles



Rates Problem 1 The next 10 problems will give you practice with rates. Please calculate the rate as described below. Round the answer to two decimal places.

Rates Problem 2 This problem will give you practice with rates. Please calculate the rate as described below. Round the answer to two decimal places.

68.9 cases in 45 hours equals how many cases per hour? Answer:

14 inches in 20 seconds equals how many inches per second? Answer:





8.5 yards in 22 minutes equals how many yards per minute? Answer:

92.5 meters in 54 seconds equals how many meters per second? Answer:





66.2 sections in 49 meters equals how many sections per meter? Answer:

69.2 parts in 63 shifts equals how many parts per shift? Answer:





86.9 miles in 48 hours equals how many miles per hour? Answer:

68.8 dollars in 45 tons equals how many dollars per ton? Answer:





5.5 kilometers in 90 hours equals how many kilometers per hour? Answer:

96.3 dollars in 63 hours equals how many dollars per hour? Answer:



Predictions and Comparisons In industry, rates are used to make predictions and comparisons. One common example is calculating time rates. If a filter can process 10 gallons of water per minute, then how many gallons of water can it process in one day? To find this, multiply the rate by the number of minutes in a day:

day

gallons 14,400 day

hours 24 hour

minutes 60 minutegallons 10 =××

When you do this, you can see that the same units of measurement on the top and bottom cancel each other out. The minutes in gallons per minute and in minutes per hour cancel, and the same with the hours. Therefore the final rate has the units of gallons per day.



Multi-Rates Many problems in the workplace involve more than one rate. In Level 6, problems may use several rates or other calculations at once. An example of a multi-rate problem would be:

You can calculate this using the rates given:

You will need to order 220 elbows.

It takes 11 pipe elbows to assemble one chlorine pump. Your team can assemble four pumps in one day. If you need to order parts for next week (5 working days), how many pipe elbows should you order?

weekelbows 220

weekdays 5

daypumps 4

pumpselbows 11 =××



Rates and Word Problems Here is an example of using rates to solve a word problem:

1) First, read the problem carefully. What is the problem asking? How long does it take for each shoe?

2) What are the facts? Worked from 11:45 a.m. to 4:30 p.m. Took half hour for lunch Completed 15 shoes

3) Set up and solve the problem.

1. Find how long he worked: 11:45 to 12:00 is 15 min.; 12:00 to 4:30 is 4 hrs. 30 min.; subtract 30 minutes for lunch Total = 15 min. + (4 hrs. 30 min.) - 30 min.

= 15 + (4 x 60 + 30 min.) - 30 min. = 255 minutes

2. Find the rate:

4) Check that the answer is reasonable. 17 minutes x 15 shoes = 255 min. = 4 hrs. 15 min.

An employee in a shoe repair store must schedule his day so that he can complete all of his work. To do that, he must know how long it takes to complete each job. One day he worked on 15 shoes from 11:45 a.m. until 4:30 p.m., taking a half hour for lunch. Approximately how long did it take to complete each shoe?

shoeper minutes 17 shoes 15

255min.=



Rates Problem 11 A secretary can type 79 words per minutes. If she works 5 eight-hour days each week, how many words does she type in a week? Check the correct answer. _____ A. 30 words _____ B. 3,160 words _____ C. 189,600 words _____ D. 198,600 words



Rates Problem 12 Two pumps are used to empty a sewage tank. One pump pumps 60 gallons per minute (60 gpm) and the second pumps 50 gpm. They pumped for 30 minutes. How many total gallons are pumped? Check the correct answer. _____ A. 330 _____ B. 2,500 _____ C. 3,000 _____ D. 3,300



Rates Problem 13 It takes 2.5 yards of material to make a dress. Harley’s Clothing Design estimates that they can produce 52 dresses each week. How much material will they need to purchase to make dresses for a year? Check the correct answer. _____ A. 130 yards _____ B. 910 yards _____ C. 2,704 yards _____ D. 6,760 yards



Rates Problem 14 One pump pumps 125 gpm into a mixing tank while another pumps 100 gpm out of the tank. The tank starts with 1,000 gallons. How many gallons are in the tank after 10 minutes? Check the correct answer. _____ A. 330 _____ B. 1,250 _____ C. 2,500 _____ D. 3,300



Rates Problem 15 An assembly line produces 2 toasters per minute. It is 2:00 p.m., and the line has finished 560 out of its goal of 900 for the day. The line stops at 5:00 p.m. A co-worker is urging you to stop the line for a half hour break, saying that you will make the goal anyway. Should you stop the line for a half hour break? Check the correct answer. _____ A. Yes, you will beat the goal anyway. _____ B. No, you will be 40 toasters short if you do. _____ C. OK, you will have just enough. _____ D. Can’t tell from this information.



Summary – Rate Problems Understanding rates is one of the keys to efficient planning. Using rates, you can predict how much product a company can produce and how much materials will be needed. You can determine if a person or team will be able to accomplish its job in time. Rates can be adjusted for different time periods. If you know how many parts can be made in an hour, then you can tell how many parts can be made in a week, month or year. Use this in your job to see how efficient your business is.

KeyTrain Level 6 Applied Mathematics Best Deals


Level 6

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.

Here is an example:

Compute one company at a time: Company A: $0.04/kwhr x 4,000 kwhr = $160 Company B: ($0.03/kwhr x 4,000 kwhr) + $100 = $120 + $100 = $220 Company A will supply the required electricity for less.

Power company A sells electricity for $0.04/kwhr (kilowatt hour). Company B sells for $0.03/kwhr plus a $100/month charge. If your business uses 4,000 kilowatt hours per month, which company should you use?



Best Deals Problem 1 The next 10 problems will give you practice at determining the best deal. Each problem will show you two different prices for the same goods. Determine which is cheaper, or if they are the same.

Best Deals Problem 2 This problem will give you practice at determining the best deal. 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. 1 liter of acetone for $9.75 _____ B. 32 liters of acetone for $184.64 _____ C. They are the same.

Which is cheaper? Check the correct answer. _____ A. 58 copies for $5.80 _____ B. 12 copies for $1.32 _____ C. They are the same.





Which is cheaper? Check the correct answer. _____ A. 54 quarts of oil for $67.50 _____ B. 39 quarts of oil for $31.20 _____ C. They are the same.

Which is cheaper? Check the correct answer. _____ A. 93 lbs. of hamburger for $267.84 _____ B. 67 lbs. of hamburger for $145.39 _____ C. They are the same.





Which is cheaper? Check the correct answer. _____ A. 8 boxes of pens for $30.00 _____ B. 36 boxes of pens for $86.40 _____ C. They are the same.

Which is cheaper? Check the correct answer. _____ A. 89 boxes of labels for $1,566.40 _____ B. 14 boxes of labels for $385.00 _____ C. They are the same.





Which is cheaper? Check the correct answer. _____ A. 36 cans of tuna for $18.72 _____ B. 2 can of tuna for $1.86 _____ C. They are the same.

Which is cheaper? Check the correct answer. _____ A. 9 gallons of gas for $12.42 _____ B. 67 gallons of gas for $64.32 _____ C. They are the same.





Which is cheaper? Check the correct answer. _____ A. 65 cases of paper for $1,267.50 _____ B. 92 cases of paper for $1,794.00 _____ C. They are the same.

Which is cheaper? Check the correct answer. _____ A. 92 cases of soda for $330.28 _____ B. 40 cases of soda for $177.20 _____ C. They are the same.



Best Deals Problem 11 A printing business employs 35 people. The employer offers his employees an insurance package that costs him $2,170. He has been investigating various plans. A new plan would cost him $59 per employee plus a $70 sign up fee. Is the new plan a better deal? Check the correct answer. _____ A. Yes _____ B. No _____ C. Not enough information



Best Deals Problem 12 Your business is studying phone companies. Company A charges $12.50 per month plus $0.15 per minute for any call. Company B charges $14.95 per month plus $0.12 per minute for any call. You average 2 1/2 hours of calls each month. Which company is the better deal and how much will you save? Check the correct answer. _____ A. Company A by $2.05 per month _____ B. Company B by $2.05 per month _____ C. Company A by $2.45 per month _____ D. Company B by $2.45 per month



Best Deals Problem 13 You are fencing in a garden area for a client. She is unsure of what shape she wants but is debating between a circle or a square. She has purchased 50 feet of fencing to go around the garden. Your choices would be a circular garden with a diameter of 15.5 feet or a 12 1/2 foot square. Which shape most effectively uses the fencing already purchased? Check the correct answer. _____ A. Square _____ B. Circle _____ C. Neither is better _____ D. Not enough information



Best Deals Problem 14 A graduate is looking for a job. She has been offered two different jobs that she must decide between. One job pays $6.50 per hour plus an 8% commission on sales over $3,000. Her perspective employer guarantees that she will work 40 hours per week and should easily sell $6,000 worth of merchandise each week. The second job pays $7.25 per hour for a 40-hour week and a 5% commission on all sales. (Assume $5,000 worth of sales each week). Where would your annual salary be best? Check the correct answer. _____ A. Job 1, $6.50 per hour _____ B. Job 2, $7.25 per hour _____ C. They have the same annual salary _____ D. There is not enough information



Best Deals Problem 15 You inherit $5,000 from your uncle and want to invest the money. You go to two banks to find the best deal. Sunshine Bank suggests you invest the money in a Certificate of Deposit (CD) that earns 5% interest every six months. Moonlight Bank suggests that you purchase a stock that is currently paying 9% in annual dividends. You plan on investing the money for 5 years. Where will you get the best deal? Check the correct answer. _____ A. Sunshine Bank _____ B. Moonlight Bank _____ C. They give the same return _____ D. There is not enough information



Best Deals Problem 16 You are catering an anniversary party that is expecting 25 people. You plan on serving cake and punch. You may purchase these items from two different distributors. The first company will sell you a cake that serves 25 people for $25.95 and ingredients to make punch totaling $6.50. The second company charges $1.05 per person for cake and $0.25 per person for punch. Where will you get the best deal? Check the correct answer. _____ A. First Company _____ B. Second Company _____ C. They are the same price _____ D. There is not enough information



Summary – Best Deals Comparing different options to find the best deal can save you or your company money. Don't be afraid to shop around for a better deal! In each situation, compare the options:

• Determine the options • Break the options into smaller problems • Compute the different options • Compare each option and determine the best one.

KeyTrain Level 6 Applied Mathematics Answers


Level 6

Applied Mathematics

Answers

KeyTrain Level 6 Applied Mathematics Multiple Steps - Answers


Multiple Steps – Answers Multiple Steps Problem 1: The correct answer is B. What is the problem asking? How much is spent? What are the facts? Earned $100 per week for 15 weeks Saved $35.75 per week Set up and solve the problem:

$963.75 $536.25 - $1,500 Spent Saved - Earned

so earned, he everythingspent or savedeither He $536.25 15 $35.75 saved Total

$1,500 15 $100 earned Total

==

=×==×=

Check your answer: $963.75 + $536.26 = $1,500.00 Multiple Steps Problem 2: The correct answer is D. What is the problem asking? How much sodium is in the meal? What are the facts? Using the table, she had: orange juice 4 mg corn flakes 251 mg milk 120 mg 2 toasts (170 mg x 2) 340 mg 2 butters (41 mg x 2) 82 mg coffee 0 mg Set up and solve the problem:

mg 797 82 340 120 251 4 :above numbers theAdd

=++++

Check your answer: First, check your list to make sure everything is included, and then estimate.



Multiple Steps Problem 3: The correct answer is A. What is the problem asking? How much sodium is in the meal? What are the facts? Using the table, she had: orange juice 4 mg bacon (2 slices/serving) 325 mg 3 eggs (50 x 3) 150 mg milk 120 mg 2 toasts (170 mg x 2) 340 mg 2 butters (41 mg x 2) 82 mg Set up and solve the problem:

Yes 1,100? than less totalIs mg 1,021 82 340 120 150 325 4

:above numbers theAdd =+++++

Check your answer: First, check your list to make sure everything is included, and then estimate. Multiple Steps Problem 4: The correct answer is C. What is the problem asking? How many patients were left? What are the facts? Patients received (added) and discharged (subtracted) as shown. Set up and solve the problem:

120 5 - 13 112 :Friday 112 2 - 5 109 :Thursday 109 12 - 9 112 :Wednesday 112 3 - 15 100 :Tuesday

day.by day it do weIf or totals.day by day it doCan

=+=+=+=+

Check your answer: Check by totaling received and discharged: 100 + 42 – 22 = 120.



Multiple Steps Problem 5: The correct answer is A. What is the problem asking? How much is left in the checking account? What are the facts? How much is left in the checking account: Original balance - $500 Payment 4 months - $35 Payment 3 months - $25 Set up and solve the problem:

$285 $215 - $500 Balance Payments - Original

$215 $25) (3 $35) (4 Payments

==

=×+×=

Multiple Steps Problem 6: The correct answer is A. What is the problem asking? The temperature at the end of the day What are the facts? Temperature rose 5 degrees Then it dropped 9 degrees The temperature in the morning was 3 degrees below zero Set up and solve the problem:

degrees 7- 9 - 5 3- re temperatuFinal degrees 9 dropped eTemperatur

degrees 5 rose eTemperatur degrees 3-

:re temperatuoriginal Start with

=+=

Multiple Steps Problem 7: The correct answer is D. What is the problem asking? What is the difference in the highest and lowest temperatures? What are the facts? Highest temperature in the 2 years was 38 Celsius. Lowest temperature in the 2 years was –25 degrees Celsius. Set up and solve the problem:

C. 63 25 38 (-25) - 38 :is res temperatu twoin the Difference =+= Check your answer. Estimate as: 40 + 20 = 60 Similar



Multiple Steps Problem 8: The correct answer is C. What is the problem asking? Is your balance correct, and if not, what did you do wrong? What are the facts? Last balance of $463.76 Checks of $12.56, $52.11, $26.03 Deposits of $101.32, $10.98 Set up and solve the problem:

check. $26.03 hesubtract t forgot toYou $26.03 $485.36 - $511.39

:balanceyour andbank in Difference OK.bank So

$485.36 $10.98 $101.32 $26.03 - $52.11 - $12.56 - $463.76 :again balanceCheck

=

=++

KeyTrain Level 6 Applied Mathematics Fractions and Decimals - Answers


Fractions and Decimals – Answers Fractions and Decimals Problem 1: The correct answer is B. What is the problem asking? What is the weight of a one-foot section? What are the facts?

pounds 43

105 Bar weighs andfeet 21

16 measuresBar

Set up and solve the problem:

6.4 332

4423

233

4432

21 2) (16

43 4) (105

:fractionsimproper use Or, 6.4 16.5 105.75

:decimal convert toCan

foot 1?

feet 2116

pounds 43105

=×=÷=+×

÷+×

=÷

=

Fractions and Decimals Problem 2: The correct answer is C. What is the problem asking? How much rod is needed for 15 drills? What are the facts?

inch waste

325

requires drillEach

longinch 161

4 is drillEach


inches 32963

322,025

32135 15

:drills 15by Multiply

32135

325

32130

325

1665

:add andr denominatocommon Find 1665

161 16) (4

1614 :fraction to

1614Convert

325

1614 needs drillEach

==×

=+=+

=+×

=

+



Fractions and Decimals Problem 3: The correct answer is C. What is the problem asking? Total length including holes. What are the facts?

diameterin inches

43 are Holes

apart inches 834 are Centers


diameter of hole centers of holes is 3/4 inch are 4 3/8 inches apart

inches 4118

8146

86 4

835

:add andMultiply

835 to

834Convert

centers)last theof endeach on hole of half onefor isinch

43( inches

43 inches

834 4 is Distance

==+×

+×

Fractions and Decimals Problem 4: The correct answer is B. What is the problem asking? What is the number of square yards of carpet needed for a room and hallway? What are the facts? Room: 14 feet 6 inches by 22 feet 9 inches Hallway: 4 feet by 9 feet 8 inches Set up and solve the problem:

yd. sq. 41 9 368.7 :9by dividingby yards square Convert to

ft. sq. 368.7 38.7 330 Total ft. sq. 38.7 ft. 9.67 ft. 4 :Hall

ft. sq. 330 ft. 22.75 ft. 14.5 :Room feet.) decimal convert to example,(For unit. single a convert to toNeed

area. thefind tolengthsmultiply Must

=÷

=+==×

=×



Fractions and Decimals Problem 5: The correct answer is D. What is the problem asking? What is the total power used by appliances? What are the facts?

watts434 with appliances 2 and watts

871 with appliances 3


watts8115

8121

876 45

r)denominatocommon (need 4

38 845

)4

19 (2 )8

15 (3

)434 (2 )

871 (3 power Total

==+

=

+=

×+×=

×+×=

Fractions and Decimals Problem 6: The correct answer is C. What is the problem asking? How many links must be used to make the chain? What are the facts?

inch

21 clasp inch;

41 --link Each

inches 15 --length Total


links. 58 4 14.5 4 2114

multiply) then andinvert divide, (to ? 41 inches

2114

:linkeach by total thedivide toneedyou links, ofnumber toal thedetermine order toIn inches

2114 inch

21 - inches 15

:linkschain theoflength thedetermine toclasp hesubtract t First,

=×=×

=÷

=



Fractions and Decimals Problem 7: The correct answer is C. What is the problem asking? Is the shipping weight wrong, and if so, why? What are the facts? 12 gears at 1 lb. 3 oz. each Box weighs 2 lbs. Marked as 15.6 lbs. Set up and solve the problem:

lbs.) 1.3 equaled oz. 3 lb. 1 assumedactually (He wrongbemust conversion The lbs. 13.9 be uld weight wo thegears, 10 countedonly had he If

lbs. 14.3been have uld weight wo thebox, themissed he If lbs. 16.3 lbs. 2 lbs. 1.19 12 t Weigh

pound) ain ounces (16 1.19 lb.) 163( lb. 1 oz. 3 lb. 1

=+×=

=+=

KeyTrain Level 6 Applied Mathematics Percentages - Answers


Percentages – Answers Percentages Problem 1: The correct answer is 18%.

18% become which 0.18) to(round 0.176 921.5 163 =÷ Percentages Problem 2: The correct answer is 76%.

76% become which 0.76) to(round 0.764 675.7 5.516 =÷ Percentages Problem 3: The correct answer is 48%.


58% become which 0.58) to(round 0.578 273 8.157 =÷ Percentages Problem 5: The correct answer is 36%.





59% become which 0.59) to(round 0.587 990.2 1.582 =÷



Percentages Problem 10: The correct answer is 51%.

51% become which 0.51) to(round 0.508 593.2 5.301 =÷ Percentages Problem 11: The correct answer is 366.

366 0.34 124.44 so 124.44; 0.34 X 34% X =÷=×=× Percentages Problem 12: The correct answer is 283.




72 0.25 18 so 18; 0.25 X 25% X =÷=×=× Percentages Problem 16: The correct answer is 190.




927 0.43 398.61 so 398.61; 0.43 X 43% X =÷=×=×



Percentages Problem 20: The correct answer is 715.

715 0.62 443.3 so 443.3; 0.62 X 62% X =÷=×=× Percentages Problem 21: The correct answer is D. What is the problem asking? How many homes are there in total? What are the facts? 25% have computers 50 homes have computers Set up and solve the problem:

200 0.25 50 ? :percent by thenumber divide OR

200 25 50 100 ?

100) ofout e(percentag 10025

?50

:ratio a Use

=÷=

=÷×=

=


68% 100% 0.68 100% 614

417.52 increasePercent 417.52 614 - 1,031.52 change ofAmount

=×=×=

==


31% 100% 0.31 100% 345

106.95 increasePercent 106.95 345 - 451.95 change ofAmount

=×=×=

==


39% 100% 0.39 100% 508


=×=×=

==




64% 100% 0.64 100% 843


=×=×=

==


94% 100% 0.94 100% 839


=×=×=

==


45% 100% 0.45 100% 255

114.75 decreasePercent 114.75 140.25 - 255 change ofAmount

=×=×=

==


25% 100% 0.25 100% 592148 decreasePercent

148 444 - 592 change ofAmount =×=×=

==


90% 100% 0.90 100% 852


=×=×=

==


4% 100% 0.04 100% 11


=×=×=

==


96% 100% 0.96 100% 423

406.98 decreasePercent 406.98 16.92 - 423 change ofAmount

=×=×=

==



Percentages Problem 32: The correct answer is C. What is the problem asking? How many customers used a credit card? What are the facts? 75% of customers paid with credit cards There were a total of 50 customers that day Write and solve the problem: Need to find 75% of 50 customers 75% x 50 = 0.75 x 50 = 37.5 (round up to 38) Check your answer:

50. of (75%) 43about is 38

Percentages Problem 33: The correct answer is D. What is the problem asking? What was the original purchase price of the shares of stock? What are the facts? Profit = $1,320 Profit was 15% (days does not matter) Write and solve the problem:

$8,800 0.151320

OR$8,800 15 132000 X

100 1320 15X

X$1,320

10015

:ratio a useCan ? 100%

$1,320 15%

=

=÷=

×=

=

==

Check your answer: $1,320 0.15 $8,800 15% $8,800 =×=×



Percentages Problem 34: The correct answer is A. What is the problem asking? What is the percent of decrease in attendance? What are the facts? 1st event attendance was 250 2nd event attendance was 230 Write and solve the problem:

8% 100% 0.08 100% 250 20

0.08 25020

amount originaldecreaseamount decreasePercent

20 230 - 250 decreaseAmount

=×=×÷=

===

==

Check your answer: amount decrease 20 8% 250 ==× Percentages Problem 35: The correct answer is D. What is the problem asking? What is the original price and sale price? What are the facts? Amount saved -- $12.50 Jeans were 25% off original price Write and solve the problem:

$37.50 $12.50 - $50 price Sale

price) (original $50 n

$1250.00 25n

price) (originaln $12.50

10025

:ratio Use$12.50 %25

==

=

=

=

=

Check your answer: OK $12.50 25% 50$ =×



Percentages Problem 36: The correct answer is B. What is the problem asking? What is the selling price? What are the facts? Cost -- $155 Marked price was 30% above cost Sold for 10% less than marked price Write and solve the problem:

$181.35 $201.50) (0.10 - $201.50 $201.50) (10% - $201.50 for Sold

$201.50 $155) (0.30 $155 $155) (30% $155 Price Marked

$155 Cost Original

=×=×=

=×+=×+=

=

Check your answer: again.math Check the Percentages Problem 37: The correct answer is B. What is the problem asking? What is the profit made by the owner? What are the facts? Sold for -- $181.35 Original cost -- $155 Write and solve the problem:

17% 100 .17 increasePercent

0.17 $155

$26.35 cost original

increaseamount Ratio$26.35 $155 - $181.35 increaseAmount

=×=

===

==

Check your answer: OK $26.35 17% $155 =×



Percentages Problem 38: The correct answer is C. What is the problem asking? Total earnings for the week What are the facts? Base pay -- $250 per week 3% commission on sales over $5,000 Total weekly sales -- $9,525 Write and solve the problem:

$385.75 $135.75 $250 Pay Total$135.75 $4,525 0.03 $4,525 3%

$5,000) - ($9,525 3% $5,000over sales of %3 Commission

Commission ase Pay Total

=+==×=×=

×==

+= B

Check your answer: again.math Check Percentages Problem 39: The correct answer is D. What is the problem asking? Is the price right? If not, why? What are the facts? MSRP = $180 Normal price 20% off Sale 30% off normal price Write and solve the problem:

$90 0.30) ($180 - $144 MSRP of 30% subtractedyou ifBut

$90not $126, MSRP off 30% error, and By trial$100.80

0.30) ($144 - $144 price Sale$144.00

0.20) ($180 - $180 price Normal

=×

=

=×=

=×=

KeyTrain Level 6 Applied Mathematics Area and Volume - Answers


Area and Volume – Answers Area and Volume Problem 1: The correct answer is 50.41 square yards. Area = length x width = 7.1 yds. x 7.1 yds. = 50.41 sq. yds. Area and Volume Problem 2: The correct answer is 67.2 feet. Perimeter = (2 x length) + (2 x width) = (2 x 16.8 ft.) + (2 x 16.8 ft.) = 67.2 ft. Area and Volume Problem 3: The correct answer is 392.04 square meters. Area = length x width = 19.8 m. x 19.8 m. = 392.04 sq. m. Area and Volume Problem 4: The correct answer is 124.63 square inches.

in.) sq. 124.63 to(round in. sq. 124.626 in. 6.3 ft. 6.3 3.14 Areain. 6.3 2 ft. 12.6 diameter 2

1 r r Area 2

=××==÷=== π

Area and Volume Problem 5: The correct answer is 3.80 square inches.

ft.) sq. 3.80 to(rounded 3.7994 ft. 1.1 ft. .11 3.14 r Area

ft. 1.1 2

2.2 diameter 21 r ft. 2.2 d

2 =××==

====

π

Area and Volume Problem 6: The correct answer is 121.2 yards.

yards) 121.20 to(rounded yards 121.204 38.6 3.14 nceCircumfere. yds 38.6 yds. 19.3 2 r 2 diameter so diameter, 2

1 r d nceCircumfere=×=

=×=×==×= π

Area and Volume Problem 7: The correct answer is 40.82 feet.

feet 40.82 13 3.14 ncreCircumfereft. 13 ft. 6.5 2 r 2 diameter so diameter, 2

1 r d nceCircumfere=×=

=×=×==×= π

Area and Volume Problem 8: The correct answer is 46.2 inches. Perimeter = (2 x length) + (2 x width) = (2 x 19.1 in.) + (2 x 4 in.) = 46.2 inches



Area and Volume Problem 9: The correct answer is 13.6 square inches. Area = length x width = 3.4 in. x 4 in. = 13.6 sq. in. Area and Volume Problem 10: The correct answer is 30.6 feet. Perimeter = (2 x length) + (2 x width) = (2 x 5.5 ft.) + (2 x 9.8 ft.) = 30.6 feet Area and Volume Problem 11: The correct answer is D. What is the problem asking? What is the volume of the toolbox? What are the facts? Length = 1 foot Width = 6 inches Height = 8 inches Set up and solve the problem:

inches cubic 576 inches 8 inches 6 inches 12 volume height width length volume

:same thearet measuremen of units thesure Make

=××=××=

Area and Volume Problem 12: The correct answer is B. What is the problem asking? What is the length of the parcel? What are the facts? Area = 2 square miles Width = 6,500 feet Set up and solve the problem:

feet 8,580 5,280 1.625 :feet back tolength Convert

mi. 1.625 mi. 1.231 mi. sq. 2

width area Length

miles 1.231 ft./mile 5,280

ft. 6,500 width feet) 5,280 mile (1 miles dth toConvert wi

:same thearet measuremen of units thesure Make

=×

=÷=

÷=

==

=



Area and Volume Problem 13: The correct answer is C. What is the problem asking? What is the cost of the flooring? What are the facts? Floor: 16 feet by 14 feet 6 inches Cost: $18.50 per sq. ft. Set up and solve the problem:

$4,292 ft. sq.per $18.50 ft. sq. 232 foot squareper cost area Cost Total

ft. sq. 232 units)common o(convert t ft. 14.5 ft. 16 in. 6 ft. 14 ft. 16 width length Area

=×=×=

=×=×=×=

Area and Volume Problem 14: The correct answer is B. What is the problem asking? How many cubic feet of concrete are needed for a sidewalk? What are the facts? Space: length – 10 feet; width – 3 feet; thickness – 6 inches Set up and solve the problem:

feet cubic 15 feet) .5 toinches 6 (converted ft. .5 ft. 3 ft. 10

height width length volume unit same the to valueseconvert th toneedyou and volume;impliesfeet Cubic

=××=

××=

Area and Volume Problem 15: The correct answer is C. What is the problem asking? What is the square feet remaining uncovered? What are the facts? Floor: Area = 238 sq. ft. Rug: 9 ft. by 12 ft. Set up and solve the problem:

area remaining of ft. sq 130 ft sq. 108 - ft. sq. 238 12ft.) ft. (9 - ft. sq. 238 Area Rug - AreaFloor Area Remaining

==

×==



Area and Volume Problem 16: The correct answer is A. What is the problem asking? What is the area of the tablecloth? What are the facts? Length = 41 inches Width = 25 inches 1 square foot = 144 square inches (from the conversion table) Set up and solve the problem:

ft. sq. 7.1 ft. in./sq. sq. 144

in. sq. 1,025 Area inchessquare1,025 in.25 in.41 r tableRectangula of Area

==

=×=

Area and Volume Problem 17: The correct answer is B. What is the problem asking? How many cubic meters are in the aquarium? What are the facts? Facts: length = 4 meters; width = 2 meters; height = 3 meters Set up and solve the problem:

meters cubic 24 m. 3 m. 2 m. 4

height width length Volume

=××=

××=

Area and Volume Problem 18: The correct answer is C. What is the problem asking? How many square feet of carpet are needed? What are the facts? Entry: 10 ft. by 9 ft. Living Room: 24 ft. by 18 ft. Carpet 90% of the total area Set up and solve the problem:

470) toup (round 469.8 0.90 ft. sq. 522 90% ft. sq. 522 :carpeted be theis area totalof 90%

ft. sq. 522 ft. sq. 432 ft. sq. 90 area room living areaentry Area Total ft. sq. 432 ft. 18 ft. 24 width length area Room Living

ft. sq. 90 ft. 9 ft. 10 width length areaEntry :roomeach of area Compute

=×=×

=+=+==×=×=

=×=×=



Area and Volume Problem 19: The correct answer is B. What is the problem asking? What is the cost of seeding the lawn? What are the facts? Lawn measures 15 yards by 12 yards 1 pound of seed costs $4.65 1 pound of seed covers 135 square feet 1 yard = 3 feet Set up and solve the problem:

$55.80 $4.65 12 lb.per price lbs. ofnumber Cost

lbs. 12 135 1,620 n

ft. sq. 1,620

lbs.n ft. sq. 135

lb. 1 :ft.) sq. 135 covers lb. (1 coveragefor ratio Use

ft. sq. 1,620 ft. 36 ft. 45 Area ft. 36 ft. 3 12 yards 12 ft. 45 ft. 3 15 yards 51

=×=×=

=÷=

=

=×==×==×=

Area and Volume Problem 20: The correct answer is B. What is the problem asking? How many bins are needed? What are the facts? Dimensions: 6 meters by 3 meters by 5.4 meters There are 100 cubic meters to be stored Set up and solve the problem:

meters) cubic 194.4 2 (97.2 meters cubic 100 hold willbins 2 So meters cubic 97.2 m. 5.4 m. 3 m. 6 height width length bin one of Volume

=×=××=××=



Area and Volume Problem 21: The correct answer is C. What is the problem asking? How much does the cellar cost? What are the facts? 36 feet long, 14 feet wide, 8 ft. deep $5.90 per cubic yard digging $20 per square foot finishing Set up and solve the problem:

$10,960.87 ft.) sq.per $20 ft. sq. (504 yd.) cu.per $5.90 yd. cu. (149.3 Cost

ft. sq. 504 ft. 14 ft. 36 width length Area 27)by (divided yd. cu. 149.3 ft. cu. 4,032

ft. 8 ft. 14 ft. 36 depth)(or height width length Volume

area.on based is Finishing on volume. based is Digging

=×+×=

=×=×===××=

××=

Area and Volume Problem 22: The correct answer is B. What is the problem asking? How much concrete is needed for the driveway? What are the facts? Facts: length – 81 ft.; width – 12 ft.; height (depth) – 6 inches Set up and solve the problem:

ft. cu. 486 ft.) toinches (converted ft. 0.5 ft. 12 ft. 81

depth)(or height width length Volume

=××=

××=

Area and Volume Problem 23: The correct answer is D. What is the problem asking? Weight of an ice block What are the facts? Facts: length – 1.5 ft.; width – 1.5 ft.; height (depth) – 1.5 ft. Weight: 57 pounds per cubic foot Set up and solve the problem:

lbs. 192.375 foot cubicper lbs. 57 feet cubic ofnumber t Weigh ft. cu. 3.375

ft. 1.5 ft. 1.5 ft. 1.5 depth)(or height width length Volume

=×==

××=××=



Area and Volume Problem 24: The correct answer is A. What is the problem asking? Which figure has the greater volume? What are the facts? Cube: 6 centimeters on each side Rectangle: 3 centimeters square base and 12 centimeter height Set up and solve the problem:

larger is cube The scentimeter cubic 108 cm. 12 cm. 3 cm. 3 height width length Volume Rectangle

scentimeter cubic 216 cm. 6 cm. 6 cm. 6

height width length Volume Cube

=××=

××==

××=××=

Area and Volume Problem 25: The correct answer is A. What is the problem asking? Is the volume right? If not, why? What are the facts? Size is 5 feet 3 inches by 3 feet 6 inches by 4 feet 9 inches Marked as 87.3 cubic feet Set up and solve the problem:

marked.correctly is volumeThe feet) cubic 87.3 to(rounded 87.28 ft. 4.75 ft. 3.5 ft. 5.25 Volume

ft. 4.75 129 4 4'9"

ft. 3.5 126 3 3'6"

ft. 5.25 123 5 5'3"

feet)toinches(convert height width length Volume Calcuate

=××=

=+=

=+=

=+=

××=

KeyTrain Level 6 Applied Mathematics Rate Problems - Answers


Rate Problems – Answers Rates Problem 1: The correct answer is 1.53 cases per hour.

hourper cases 1.53 hours 45 cases 68.9 hours 45cases 68.9

=÷=

Rates Problem 2: The correct answer is 0.7 inches per second.

secondper inches 0.7 seconds 20 inches 14 seconds 20inches 14

=÷=

Rates Problem 3: The correct answer is 0.39 yards per minute.

minute)per yards 0.39 to(rounded 0.386 minutes 22 yards 8.5 minutes 22

yards 8.5=÷=

Rates Problem 4: The correct answer is 1.71 meters per second.

secondper meters 1.71 seconds 54 meters 92.5 seconds 54

meters 92.5=÷=

Rates Problem 5: The correct answer is 1.35 sections per meter.

meterper sections 1.35 meters 49 sections 66.2 meters 49sections 66.2

=÷=

Rates Problem 6: The correct answer is 1.10 parts per shift

shift)per parts 1.10 to(rounded 1.098 shifts 63 parts 69.2 shifts 63

parts 69.2=÷=

Rates Problem 7: The correct answer is 1.81 miles per hour.

hourper miles 1.81 hours 48 miles 86.9 hours 48miles 86.9

=÷=



Rates Problem 8: The correct answer is 1.53 dollars per ton.

per ton) dollars 1.53 to(rounded 1.528 tons45 dollars 68.8 tons45dollars 68.8

=÷=

Rates Problem 9: The correct answer is 0.06 kilometers per hour.

hourper kilometers 0.06 hours 90 kilometers 5.5 hours 90

kilometers 5.5=÷=

Rates Problem 10: The correct answer is 1.53 dollars per hour.

hour)per dollars 1.53 to(rounded 1.528 hours 63 dollars 96.3 hours 63dollars 96.3

=÷=

Rates Problem 11: The correct answer is C. What is the problem asking? How many words are typed in a week? What are the facts? 79 word per minute 5 days 8 hours each day Set up and solve the problem:

words189,600 te word/minu79 minutes 2,400 WordsTotalminutes 2,400 ur minutes/ho 60 hours/day 8 days 5 Time Total

=×==××=

Rates Problem 12: The correct answer is D. What is the problem asking? How many total gallons are pumped? What are the facts? Pump 1 -- 60 gallons per minute Pump 2 -- 50 gallons per minute Two pumps for 30 minutes Set up and solve the problem:

gallons 3,300 1,500 1,800 Gallons Total

gallons 1,500 minutes 30 gpm 50 2 Pumpgallons 1,800 minutes 30 gpm 60 1 Pump

=+==×==×=



Rates Problem 13: The correct answer is D. What is the problem asking? How much material is needed for a year? What are the facts? 1 dress – 2.5 yards of material Can produce 52 dresses each week; 1 year = 52 weeks Set up and solve the problem:

material of yards 6,760 syards/dres 2.5 ar dresses/ye 2,704 Material ofAmount ardresses/ye 2,704 r weeks/yea52 ek dresses/we 52 Dresses Total

=×==×=

Rates Problem 14: The correct answer is B. What is the problem asking? How many gallons are left in the tank? What are the facts? Pump 1 -- 125 gallons per minute into the tank Pump 2 -- 100 gallons per minute out of the tank Time = 10 minutes Initial Volume = 1,000 gallons Set up and solve the problem:

gallons 1,250 1,000 - 1,250 1,000

out 2gallons pump -in gallons 1 pump volumeinitial Volume Finalsubtract) means(out out gallons 1,000 minutes 10 gpm 100 2 Pump

add) means(in in gallons 1,250 minutes 10 gpm 125 1 Pump

=+=+==×==×=

Rates Problem 15: The correct answer is B. What is the problem asking? Can you make the goal and still take a break? What are the facts? Make 2 toasters per minute Goal: 900 toasters; already made 560 toasters Time: 3 hours (2:00 p.m. – 5:00 p.m.) Set up and solve the problem:

short. toasters40 be You will break. a takestill and goal themakecannot you No, toasters.340 make toneedYou

toasters300 hours 2.5 hour toasters/120 more hours 2.5for run willproduction break,hour half a you take If

toasters340 560 - 900 NeededAmount hourper toasters120 hoursper minutes 60 minuteper toasters2 Rate

=×

===×=

KeyTrain Level 6 Applied Mathematics Best Deals - Answers


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

acetone ofliter per $5.77 liters 32 $184.64 acetone ofliter per $9.75 liter 1 $9.75

=÷=÷

Best Deals Problem 2: The correct answer is A.

copyper $0.11 copies 11 $1.32 copyper $0.10 copies 58 $5.80

=÷=÷

Best Deals Problem 3: The correct answer is B.

oil ofquart per $0.80 quarts 39 $31.20 oil ofquart per $1.25 quarts 54 $67.50

=÷=÷


hamburger of poundper $2.17 lbs. 67 $145.39 hamburger of poundper $2.88 lbs. 93 $267.84

=÷=÷


pens ofbox per $2.40 boxes 36 $86.40 pens ofbox per $3.75 boxes 8 $30.00

=÷=÷


labels ofbox per $27.50 boxes 14 $385.00 labels ofbox per $17.60 boxes 89 $1,566.40

=÷=÷


tunaofcan per $0.93 cans 2 $1.86 tunaofcan per $0.52 cans 36 $18.72

=÷=÷


gas ofgallon per $0.96 gallons 67 $64.32 gas ofgallon per $1.38 gallons 9 $12.42

=÷=÷



Best Deals Problem 9: The correct answer is C.

paper of caseper $19.50 cases 92 $1,794.00 paper of caseper $19.50 cases 65 $1,267.50

=÷=÷


soda of caseper $4.43 cases 40 $177.20 soda of caseper $3.59 cases 92 $330.28

=÷=÷

Best Deals Problem 11: The correct answer is A. What is the problem asking? Which insurance plan is a better deal? What are the facts? Employees – 35 Current Plan cost -- $2,170 New Plan cost == $59/employee + $70 Write and solve the problem: New Plan: ($59 x 35) + $70 = $2,135 Compare this to the current cost of $2,170. The new plan is the best deal. Best Deals Problem 12: The correct answer is B. What is the problem asking? What is the total cost for each company? What are the facts? Company A -- $12.50/month + $0.15/minute Company B -- $14.95/month + $0.12/minute You use 2.5 hours per month Write and solve the problem: 2.5 hours = (60 x 2) + 30 = 150 minutes

Company A: $12.50 + (150 x $0.15) = $35.00 Company B: $14.95 + (150 x $0.12) = $32.95 Company B is a better deal. The difference in cost is: $35.00 - $32.95 = $2.05 per month



Best Deals Problem 13: The correct answer is A. What is the problem asking? Which shape uses the purchased fencing best? What are the facts? Purchased 50 feet fencing Circular garden -- 15.5 ft. diameter Square garden – 12.4 ft. length per side Write and solve the problem:

fencing. purchased theall use willshape square Theft. 50 12.5 4

sideeach 4 length :Square ft. 48.87 ft. 15.5 3.14

diameter) d (where d ncecircumfere length :Circle :optioneach aroundlength theCompute

=×=×=

=×==×== π

Best Deals Problem 14: The correct answer is B. What is the problem asking? Which job would give the best annual salary? What are the facts? Job 1 -- $6.50 per hour; sales of $6,000 per week with an 8% commission

on sales over $3,000 Job 2 -- $7.25 per hour; sales of $5,000 per week with a 5% on all sales 40 hours per week Write and solve the problem:

$540.00 $250 $290 $5,000) (0.05 $290

$5,000) (5% 40) ($7.25 2 Job$500.00 $240 $260 $3,000) (0.08 $260

$3,000) (8% 40) ($6.50 1 Job

=+=×+=×+×=

=+=×+=×+×=

Best Deals Problem 15: The correct answer is D. What is the problem asking? Which investment is better What are the facts? Sunshine Bank – 5% interest every 6 months Moonlight Bank – 9% annual dividends Write and solve the problem: This problem cannot be evaluated because interest rates fluctuate with time. Best Deals Problem 16: The correct answer is A. What is the problem asking? Which company is giving the better deal?



What are the facts? First company -- cake $25.95 for 25; punch $6.50 for 25 Second company – cake $1.05 per person; punch $0.25 per person Serving 25 people Write and solve the problem:

$0.05.by dealbetter aoffer company first The$32.50 $6.25 $26.25 25) ($0.25 25) ($1.05

:company 2nd $32.45 $6.50 $25.95

:company1st :companyeach for cost total theCompare

=+=×+×

=+

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Applied Mathematics Level 6 - School Of Social Justice ...sj.lvlhs.org/ourpages/auto/2012/4/8/49927094... · Applied Mathematics Problem Solving ... As before, start with the basic