MY MATH PATH 4 DECIMAL NUMBERS

Transcription

1 MY MATH PATH 4 DECIMAL NUMBERS Contents Learning outcomes...2 Glossary... Introduction...4 Section : Place value and rounding numbers...0 Exercise...2 Section 2: Comparing numbers...4 Exercise Section : Rounding numbers...7 Exercise...20 Section 4: Operations with decimals...22 Exercise Section 5: Multiplying whole numbers and decimal fractions...26 Exercise Section 6: Dividing whole numbers and decimal fractions...29 Exercise Section 7: Order of operations...5 Exercise Practice Test...8 Answers...40 MY MATH PATH 4 Decimal Fractions

2 Learning outcomes Being able to work with whole numbers, decimals and fractions is important in everyday life. Estimating an answer to a problem helps put the problem in context. This process should be part of any problem solving activity. Checking answers and solutions to problems is equally important. When you have completed this Unit, you should be able to: round numbers calculate and solve problems involving adding, subtracting, multiplying and dividing whole numbers, and decimals fractions check that answers and solutions to problems are reasonable in the context of the given questions MY MATH PATH 4 Decimal Fractions 2

3 Glossary Difference The answer to a subtraction question is called the difference. Equivalent Equivalent quantities are equal. Factor A number that divides evenly into another number. E.g., 2 and are factors of 6. Prime number A prime number is divisible only by itself and one. The number one is not a prime number. The first few prime numbers are 2,, 5, 7,,... Product The answer to a multiplication question is called the product. Quotient The answer to a division question is called the quotient. Sum The answer to an addition question is called the sum. MY MATH PATH 4 Decimal Fractions

4 Introduction to Fractions This is the beginning of an adventure with numbers that represent part of the whole thing. These numbers are fractions. The idea of fractions may be very comfortable to most of us because our minds are used to dealing with parts in our everyday life. Look at the pictures and use a fraction to answer the questions. one quarter 4 one third one half 2 two thirds 2 three quarters 4 How much gas is left? a) of a tank b) of a tank MY MATH PATH 4 Decimal Fractions 4

5 This full bubble gum machine is the whole thing. Do we need to fill up the bubble gum machine? c) No, it s full. d) Yes, it s full MY MATH PATH 4 Decimal Fractions 5

6 Common Fractions Common Fractions are written with two numbers, one above the other, with a line in between. The line may be straight or on an angle / 2 2/ (it also means 2 ) The denominator is the bottom number. The denominator tells how many equal parts there are in the whole thing. 2 numerator denominator The numerator is the top part in a common fraction. The numerator tells how many of the equal parts we are actually describing or talking about. This pizza has been cut into eight pieces, all the same size (equal). The denominator to use while talking about this pizza is 8. The 4 numerator will be the exact number of pieces of the pizza you are describing. This is pizza, and that is the whole thing. If someone ate all 8 pieces or 8 8 (eight-eights) that person ate pizza. of the pieces would be 8 of the pizza. You will learn more about common fractions in unit. MY MATH PATH 4 Decimal Fractions 6

7 Decimal Fractions What is a decimal fraction? Decimal Fractions are another way of considering parts of the whole thing and the whole thing is one. You use decimal fractions everytime you think about money! The dollars are written as whole numbers; the cents are written as a decimal fraction of a dollar. A decimal fraction does not have a denominator written underneath it. Instead a decimal point (.) separates the whole number from the fraction. Our number system is called a decimal system because it is based on the number ten ( deci is the Latin word for ten). So in decimal fractions the whole thing is divided into tenths; the tenths are divided by ten to make hundredths; the hundredths are divided by ten to make thousandths and so on. Decimal fractions are uses a great deal in our everyday lives, especially in money and measurement. $ kilometres to the store 2.6 metres of material.8 kilogram of pasta As you know, fractions describe part of the whole thing a fraction is smaller than. And as you know, (the whole thing) can be many things. For example, it can be one dollar one city one school one paycheck one year one second one pregnancy... So a decimal fraction may represent part of a pregnancy, part of a second or part of anything you want. Here are the characteristics of a decimal fraction: A decimal point. separates whole numbers from fractions. A decimal fraction starts with a decimal point MY MATH PATH 4 Decimal Fractions 7

8 In a decimal fraction, the denominator is not written. Remember that the denominator in a common fraction is the bottom number and tells how many equal parts there are in the whole thing. 2 denominator But in a decimal fraction the denominator is understood. We tell the size of the denominator by looking at how many numerals are placed after the decimal point. Decimal denominators are always ten or ten multiplied by tens. 0.4 has a denominator of has a denominator of has a denominator of has a denominator of has a denominator of has a denominator of A whole number and a decimal fraction can be written together. This is called a mixed decimal fraction $2.9 Every whole number has a decimal point after it, even though we usually do not bother to write the decimal point unless a decimal fraction follows the number. We can also put zeros to the right of the decimal point of any whole number without changing its value. =. = = 275. = MY MATH PATH 4 Decimal Fractions 8

9 Reading and Writing Decimal Fractions Common fractions with a denominator of 0 are written as a decimal fraction with one place to the right of the decimal point. This is the tenth place. 2 =.2 = two tenths 0 6 =.6 = six tenths 0 Introduction Exercise One Write each fraction as a decimal fraction and in words. a. b. c. 4 =.4 = four tenths 0 = = 0 9 = = 0 When we write decimal fractions, a zero is usually placed to the left of the decimal point to show there is no whole number. This zero keeps the decimal point from being lost or not noticed. Introduction Exercise Two Write each decimal fraction as a common fraction and in words. a. 0. = b. 0.5 = 0 three tenths c. 0.7 = Solutions are on p.7 MY MATH PATH 4 Decimal Fractions 9

10 Section : Place value Place value Words are formed with letters. Numbers are formed with digits. The ten digits of our number system are: 0,, 2,, 4, 5, 6, 7, 8 and 9 By stringing these digits together we can form numbers. The position that a digit occupies in a number tells us how many ones, tens, hundreds, thousands (and so on) that the number contains. If the number has a decimal point, then the position of the digits after the decimal point indicates how many tenths, hundredths, thousandths (and so on) the number contains. The place values or position values in a number are shown in Figure. Example Figure Place values Write the place value for each underlined digit. Solution Example 2 a b c a. tens b. thousandths c. hundred thousands Write in expanded form. Solution = MY MATH PATH 4 Decimal Fractions 0

11 Example Write three million in standard notation. Solution To write a number in standard notation means to write it using only digits. Three million in standard notation is Now complete Exercise and check your answers. MY MATH PATH 4 Decimal Fractions

12 Exercise. Write the place value for each underlined digit. a. 852 b c d e Underline the digit for the place value named. a. tens b. tenths c. hundredths d. millions e. ten thousandths Write in expanded form. a. 82 b. 8.2 c d Luna had 6 one hundred dollar bills, tens and 8 loonies. How much money did she have? MY MATH PATH 4 Decimal Fractions 2

13 5. Bill had 5 ten dollar bills, dimes and 2 pennies. How much money did Bill have? 6. Ida found 4 pennies and loonie. Together with her ten dollar bill, how much money does Ida have? 7. Dan wants $20.4. How many hundred dollar bills, ten dollar bills, loonies, dimes and pennies should he receive? 8. A certain company lost 8. million dollars last year. Write this number in standard notation. Answers are on page 40. MY MATH PATH 4 Decimal Fractions

14 Section 2: Comparing numbers There are only three possibilities when comparing numbers. Either one number is equal to (=) another number, greater than (>) another number or less than (<) another number. When comparing decimal fractions, it is often a good idea to write both numbers with the same number of digits after the decimal point. This is possible because placing zeros after the last decimal digit will not change the value of the number. For example: 6.5 = = = and so on. We can do the same with whole numbers, but only after placing a decimal point at the end of the whole number. For example: Example 804 = = = = and so on. Use >, < or = to compare each pair of numbers. a b c Solution a. Since has three decimal digits, write 0.0 with three decimal digits, by putting another 0 after the. Comparing: < 0.00 b. 69 and have the same value. They are said to be equivalent: 69 = c. 05 is greater than 0.5. Notice that a decimal could be placed at the end of > 0.5 Notice how greater than or less than signs always point to the smaller number, such as > 5 or 5 <. MY MATH PATH 4 Decimal Fractions 4

15 Example 2 Find the smallest of the three numbers below: Solution The equivalent numbers with three decimal digits are: The smallest number is Now complete Exercise 2 and check your answers. MY MATH PATH 4 Decimal Fractions 5

16 Exercise 2. Use >, < or = to compare the numbers. a b c d e Find the smallest number in each set of numbers. a b c d e Use an is equal to (=) or an is not equal to ( ) symbol to compare the numbers. a d b e c f Jill bought gas for 49.9 cents per litre. Jack said he bought gas for $0.499 per litre. Who paid the most for gas? Answers are on page 40. MY MATH PATH 4 Decimal Fractions 6

17 Section : Rounding numbers Estimation can be used to calculate the approximate cost of something. Example You and a friend are having dinner at a restaurant. As you look at the menu, you try to estimate the cost of your meal. Main Course Roast Beef $.95 Chicken $.50 Spaghetti $ 9.75 Vegetarian Lasagna $0.99 Desserts Cheesecake $ 4.25 Apple Pie $.90 Ice Cream $ 2.75 Beverages Milk $.50 Coffee $ 0.99 Tea $ 0.99 Soft Drinks $.55 You choose chicken, apple pie and coffee. Your friend chooses lasagna, ice cream and a soft drink. To estimate the cost of each person s meal, you can round each price up or down. In this example, if you round up, your estimate will be higher than the actual cost. Your meal Your friend Chicken $2 Lasagna $ Apple pie $ 4 Ice cream $ Coffee $ Soft drink $ 2 Total estimated cost $7 $6 Note that estimates are not exact answers and may vary depending on whether you round up or down. MY MATH PATH 4 Decimal Fractions 7

18 No one would ever report his or her age as being years old. Knowing that the person is approximately 0 years old is usually sufficient. Most values like age can never be known exactly. To make things simple, we often round numbers to a convenient place value. Example The population of Canada is people. Round to the nearest million. Solution Step : Underline the millions digit Step 2: If the next digit after the underlined digit is a 0,, 2,, or 4, then do not change the underlined digit. If the next digit is a 5, 6, 7, 8, or 9, then add to the underlined digit. Since the next digit is a 4, the number is still Step : Change all the digits after the underlined digit to zeros. Then remove the underline. The population of Canada is The steps required to round any decimal fraction are as follows: Step : Underline the digit to which the number is being rounded. Step 2: If the next digit after the underlined digit is a 0,, 2,, or 4, then do not change the underlined digit. If the next digit is a 5, 6, 7, 8, or 9, then add to the underlined digit. Step : Change all the digits after the underlined digit to zeros. If any of the digits after the underlined digit follow a decimal point, then drop them from the number. Step 4: Remove the underline from your answer. Example 2 We use the sign, which means is approximately equal to to compare a rounded number. For example: Round the following to the nearest tenth. a..57 b c MY MATH PATH 4 Decimal Fractions 8

19 Solution a. Step : the digit in the tenths place is a 5. Underline it..57 Step 2: the digit after the 5 is a 7. Add to the Step : change the 7 to a zero. Since this zero comes after the decimal point, drop it..60 =.6 Step 4:.57.6 b. Step : the digit in the tenths place is a 0. Underline it Step 2: the digit after the 0 is a. The 0 remains unchanged Step : replace the and 4 with zeros and drop them = 0.0 Step 4: c. Step : the digit in the tenths place is a 9. Underline it Step 2: the digit after the 9 is a 6. Adding to 9 = 0, which is carried to the ones place Step : replace all the numbers after the underlined number with zeros Step 4: Now complete Exercise and check your answers MY MATH PATH 4 Decimal Fractions 9

20 Exercise. Round to the nearest hundred. a d. 47 b. 20 e c. 85 f Round to the nearest one (also called rounding to the nearest whole number). a. 6. b. 8.7 c d e Round as indicated. a to the nearest ten thousand b to the nearest tenth c to the nearest hundredth d to the nearest ten thousandth e. 0.8 to the nearest whole number f to the nearest ten MY MATH PATH 4 Decimal Fractions 20

21 4. Round to the nearest hundredth. a b c d e Mike had $9.50, Mary had $84.05 and Sandra had $4.88. Round their amounts to the nearest dollar. Answers are on page 40. MY MATH PATH 4 Decimal Fractions 2

22 Section 4: Operations with decimals When adding or subtracting whole numbers of decimal fractions it is important that we add only ones digits to ones digits, tens digits to tens digits, hundreds digits to hundreds digits, tenths digits to tenths digits and so on. To accomplish this, always make sure that the decimal points are lined in a column and each decimal fraction has the same number of digits after the decimal point. Example Add the following: Solution Arrange the numbers as shown. Note that a decimal point is included in the number 8 and every number has been written with three digits after the decimal point Example 2 Subtract the following: Solution Align the decimal points. Rewrite 54 as Notice that we must borrow from the 4 and again from the When numbers are added, the answer is called a sum. The sum of 8 and 5 is. When numbers are subtracted, the answer is called a difference. The difference of 0 and 4 is 6. Now complete Exercise 4 and check your answers. MY MATH PATH 4 Decimal Fractions 22

23 Exercise 4. Find the sum. a. 846 b. c Find the sum. a b c Find the sum. a b c Find the sum. a b MY MATH PATH 4 Decimal Fractions 2

24 5. Find the difference. a b c d e f g h i Find the difference. a b c d Mom found three dimes, three quarters, one nickel, seven pennies and two loonies in Billy s pocket. How much money did Billy have? 8. Nadine had $50 before she bought a calculator for $9.95 and a geometry set for $2.98. How much did she have after these purchases? MY MATH PATH 4 Decimal Fractions 24

25 9. Calculate the balance values. Date Item Withdrawal Deposit Balance June June 4 cash withdrawal a June 8 salary deposit b June 2 BC Tel.69 c June 20 car repair cheque 46.5 d June 0 rent cheque e 0. Round each of the numbers to the nearest ten and estimate the sum or difference. a. 462 b Answers are on page 4. MY MATH PATH 4 Decimal Fractions 25

26 Section 5: Multiplying whole numbers and decimal fractions When two numbers are multiplied together, the number is called a product. The numbers being multiplied together are called factors. For example, when 2 = 6, we say that 2 and are factors while 6 is the product. Example Find the product of 57 and 274. Solution Multiply 57 first by 4, then by 7 and then by 2 as shown. Add these products When multiplying decimal fractions, great care must be taken in where to place the decimal point in the product. Example 2 Multiply.5 by Solution As above, multiply.5 first by 9, then by 0 and then by 4. Arrange these products as shown. Since the factors have a total of five decimal places, the product must also have five decimal places..5 2 decimal places decimal places decimal places Now complete Exercise 5 and check your answers. MY MATH PATH 4 Decimal Fractions 26

27 Exercise 5. Complete the times table chart Use the times table chart to help answer the following. a. the product of 0 and any number is always. b. the product of and any number is always. c. The product of 5 and any number always ends in a or a. d. Look at the products in the 9 s row or column. Except for the 0 product, the sum of the digits in each product is always.. Multiply a. 4 b. 96 c. 944 d MY MATH PATH 4 Decimal Fractions 27

28 e. 8 f. 52 g. 602 h Find the products. a b c d e Tim earns $ bi-weekly. There are 26 bi-weekly paydays in one year. How much will he earn in a year? 6. Holly drives an average speed of 84 km/hr in her car. How far will she travel in 4.5 hours? Answers are on page 4. MY MATH PATH 4 Decimal Fractions 28

29 Section 6: Dividing whole numbers and decimal fractions When one number is divided by another, the answer is called a quotient. The number being divided is called a dividend, while the dividing number is called a divisor. There is one number that can never be divided into another. It is impossible to divide zero into any number. For example, 5 0 is impossible. We say that division by zero is undefined. There are three ways of indicating division: quotient divisor dividend dividend divisor = quotient dividend divisor = quotient For example, the problem of dividing 0 by 2 can be written as: 2 0 or 0 2 or 0 2 We read 2 0 as 2 into 0, 0 2 as 0 divided by 2 and 0 2 as 0 over 2. Example Find by long division. Solution Write the question as 2 into 860. Now, 2 into 8? There are no 2 s in 8. 2 into 86 goes 7 times. Place 7 above the 6 and 7 2 = 84. Place 84 below 86 and subtract. Bring down the 0. Now 2 goes into 20 once. Place the above the 0 and 2 = 2. Place 2 below 20 and subtract. Since there is nothing to bring down, 8 is the remainder. We write the quotient as 7 R = 7 R8 MY MATH PATH 4 Decimal Fractions 29

30 Whenever we divide whole numbers we often get a remainder. We use the letter R to indicate the remainder. When dividing decimal fractions, we must take special care in dealing with the decimal points. We can make the task simpler by rearranging the question so that we always divide by a whole number. Example 2 Find by long division. Solution Write as 0.2 into Multiply both the divisor and dividend by 00 and rewrite as 2 into This allows us to divide by a whole number. Place the decimal point in the quotient directly above the decimal point in the dividend. Divide as in Example. Notice that 2 goes into 20 zero times = 40.9 The process of long division can sometimes produce endless non-zero remainders. We can deal with this by rounding the quotient to an appropriate place value. Example Find 0.7 by long division. Round the answer to the nearest hundredth. Solution Write as 0.7 into. Multiply both the divisor and dividend by 0. Rewrite as 7 into 0. Now we can divide by a whole number. Since we must round the quotient to two places after the decimal point, write three zeros after the 0. Divide as before. MY MATH PATH 4 Decimal Fractions 0

31 Rounding the quotient, 0.7 = Now complete Exercise 6 and check your answers. MY MATH PATH 4 Decimal Fractions

32 Exercise 6. Divide. a b c d. 8 e. 0 6 f. 6 0 g. 6 9 h. 4 4 MY MATH PATH 4 Decimal Fractions 2

33 i j k l Divide. a b c d e f g h i MY MATH PATH 4 Decimal Fractions

34 . Divide. Round the answers to the nearest hundredths. a. 6 b c d. 6 e f g h Three children decide to share the cost of their mother s $40 gift. How much should each pay? Round the answer to the nearest cent. 5. Kim drove 0 km on 28.5 litres of gas. How far does she travel on one litre of gas? Round the answer to the nearest tenth. 6. Tracy bought 9 apples for $.7. How much did each apple cost? Answers are on page 42. MY MATH PATH 4 Decimal Fractions 4

35 Section 7: Order of operations Consider the following calculation: Is it 5 4 = 20 or = 4? In other words, do we add first or multiply first? Mathematicians have decided upon the following order of operations:. Brackets, such as ( ), { } or [ ] First, look for brackets and perform all operations within the brackets. 2. Multiplying and dividing Second, perform all multiplying and dividing going from left to right.. Adding and subtracting The last operations to be performed are adding and subtracting, from left to right. An easier way of remembering the order of operations is using the acronym BEDMAS. It will help you to remember the order of operations. The letters stand for: Brackets or parenthesis Exponents Division Multiplication }left to right Addition Subtraction }left to right This is an internationally agreed-upon system for dealing with multiple operations. We can now answer the above question: = = 4 Example Find 50-2( + 8) MY MATH PATH 4 Decimal Fractions 5

36 Solution First, add + 8 inside the brackets ( + 8) = 50-2() Next, multiply 2(). This expression means 2. The operation of multiplication is always indicated when a number is placed in front of brackets. 50-2() = Last, subtract = 28 When working towards a solution, it is safest to perform just one operation at a time. Example 2 Calculate Solution First the division, then the multiplication and finally the adding. Example = = = 67 Calculate 2[ ( )] Solution First, the innermost brackets. Then, within brackets do multiplying or dividing going from left to right. Finally, multiply by 2. 2[ ( )] = 2[ (0.25)] = 2[2 0.5 (0.25)] = 2[.0 (0.25)] = 2[4] = 8 Now complete Exercise 7 and check your answers. MY MATH PATH 4 Decimal Fractions 6

37 Exercise 7. Calculate a b c d e. 2( ) f. 99 (0-0.) 2. Calculate a. 6 - (2 + - ) 2 b c. [0. + ( + 0.)] d. 4-7 ( ) 0.2 e. (9-7)(9 + 7) f. 0.5(.0-0.)( ) g (8. -.) - 25 h (0.8-0.). Donna has saved $80. She plans to save $55 every week for the next 8 weeks. How much will she have saved altogether after 8 weeks? 4. Four people went to dinner. Three people had the $2.95 dinner special while the fourth person had a $4.95 dinner. The wine totaled $ If they split the bill four ways, how much should each pay? Answers are on page 42. MY MATH PATH 4 Decimal Fractions 7

38 Practice Test. Write the place value for each underlined digit. a b Underline the tenths digit in Use >, < or = to compare the numbers below. a. 9.0 _ 90. b. 84 _ c _ 0.02 d _ Round each of the following as indicated. a. 452 to the nearest hundred b to the nearest tenth c to the nearest ten 5. Add Subtract from Multiply the following. a b MY MATH PATH 4 Decimal Fractions 8

39 8. Divide the following. a b Divide 7 by 0. and round the answer to the nearest hundredth. 0. Calculate the following. a b. 0.4[0-0.2( )]. Donna won $5 000 and decided to share half of it among her three children. How much should each child receive? MY MATH PATH 4 Decimal Fractions 9

40 Answers Introductory Exercise One - page 9 b.. one tenth c..9 nine tenth Introductory Exercise Two - page 9 b. 5/0 five tenth c..7 seven tenth Exercise - page 2. a. ones b. thousandths c. hundredths d. ten thousand e. hundred thousandths f. millions g. tenths h. ones i. hundredths j. thousands 2. a b c d e a. 82 = b. 8.2 = c = d = $68 5. $ $ hundred dollar bills, 0 ten dollar bills, loonies, 4 dimes and penny dollars Exercise 2 - page 6. a. < b. = c. > d. < e. = 2. a..6 b c d. 265 e a. b. c. = d. e. f. 4. Neither cents = $0.499 Exercise - page 20. a b. 200 c. 00 d. 0 e f a. 6 b. 9 c. 50 d. 56 e.. a b. 0.0 c d e. f a. 0.9 b c. 0.2 d e Mike $20, Mary $84 and Sandra $44 MY MATH PATH 4 Decimal Fractions 40

41 Exercise 4 - pages a. 94 b. 094 c a. 044 b c a b c a. 9.6 b a. 4 b. 9 c d. 2.9 e f g..82 h i a b c d $.7 8. $ a. $280.5 b. $ 7.96 c. $ 8.27 d. $99.92 e. $ a. 460 b Exercise 5 - pages a. 0 b. that number c. 0 or 5 d. 9. a. 205 b. 248 c d e. 9. f g h a b c d e $ km MY MATH PATH 4 Decimal Fractions 4

42 Exercise 6 - pages 2-4. a. 8 b. 6 c. 9 d. 8 e. 0 f. Undefined. It is impossible to divide any number by 0. g. 7 h. i. 85 j. 0.6 k l a. 24 R4 b. 092 R6 c. 50 d. 2.4 e..06 f. 40 g. 600 h..7 i..0. a. 20. b c. 8. d e. 4.5 f g h $ km 6. $0.9 Exercise 7 - page 7. a. 4 b. 8 c. 8 d. 7 e. 5 f a. 4 b. 9 c. 0.6 d. 2 e. 2 f g. 2 h. 2. $80 + 8($55) = $ [2.95() ] 4 = $9.08 Practice Test - pages 8-. a. thousands b. hundredths a. > b. = c. < d. > 4. a. 500 b. 7.0 c a b a. 400 R5 b a. 20 b..68. $8. MY MATH PATH 4 Decimal Fractions 42

43 Downloading Information MyMathPath.com Resources: Copyright 20 Province of British Columbia Ministry of Advanced Education and Labour Market Development Permission granted to make copies for educational purposes. Based on the Resources: Adult Literacy Fundamental Mathematics, Instructor s Manual and Test-Bank Copyright 200 Province of British Columbia Ministry of Advanced Education and Labour Market Development Acknowledgments: Textbook: Sylvie Trudel Moodle Course Developers - Faculty: Don Bentley Robert Kim Sylvie Trudel Moodle Course Developers Capilano University Students in the College and University Preparation Program: Crystalina Alcazar Ivan Arzave Jasmine Blanchard Bill Brown Derek Brownbridge Michael Drummond Kristofer Dukic Kameron Fakhari Mackenzie Grant David Hart Abbid Jaffer Alexander Kay Michael Kulch Nicholas Schopp Bobbi-dee Stillas Ksenia Teder Daniel Thomas Michael Villareal Ashley Wong MY MATH PATH 4 Decimal Fractions 4

44 MY MATH PATH 5 COMMON FRACTIONS Contents Learning outcomes...2 Glossary...2 Unit : Operations with fractions... Exercise...5 Unit 2: Equivalent fractions...7 Exercise Unit : Adding and subtracting common fractions... Exercise... Unit 4: Multiplying common fractions...6 Exercise Unit 5: Dividing common fractions...9 Exercise Unit 6: Conversions between common fractions and decimal fractions...22 Exercise Unit 7: Conversions between decimal fractions and common fractions...25 Exercise Unit 8: Estimation...28 Exercise Practice Test...0 Answers...2 MY MATH PATH 5 Common Fractions

45 Learning outcomes Being able to work with whole numbers, decimals and fractions is important in everyday life. Estimating an answer to a problem helps put the problem in context. This process should be part of any problem solving activity. Checking answers and solutions to problems is equally important. The calculator is a useful tool, not only for obtaining accurate results, but also for checking answers. When you have completed this Module, you should be able to: calculate and solve problems involving adding, subtracting, multiplying and dividing fractions estimate answers to problems check that answers and solutions to problems are reasonable in the context of the given questions Glossary Factor A number that divides evenly into another number. E.g., 2 and are factors of 6. Improper fraction An improper fraction has a numerator larger than the denominator. Mixed number A mixed number is a combination of a whole number and a proper fraction. Product The answer to a multiplication question is called the product. Proper fraction A proper fraction has a numerator smaller than the denominator. Quotient The answer to a division question is called the quotient. Sum The answer to an addition question is called the sum. MY MATH PATH 5 Common Fractions 2

46 Unit : Operations with fractions Common fractions are numbers that express parts of a whole. Suppose a pie is cut into five equal pieces and we eat two of them. Then we can say that we ate two-fifths or 2 5 of the pie. A common fraction is always made of two numbers. The top number is called the numerator and the bottom number is called the denominator. 2 numerator 5 denominator When working with common fractions it is important to remember that the whole is the same as the number one. If we ate the whole pie, it is the same as eating 5 / 5 of the pie. Fractions can be used to represent numbers less than one, more than one or equal to one. Proper fractions are less than one. Their numerators (top) are less than their denominators. For example, 2 <, 4 <, 9 0 < and 40 <. Improper fractions are equal to or greater than one. For example, 4 and 2 >. >, 0 0 =, 9 4 > Mixed numbers are composed of a whole number and a proper fraction. Mixed numbers are always greater than one. For example, 4 >, 5 8 >, 0 2 > and >. The rectangles shown below have been divided into four equal parts. The shaded part can be described by the improper fraction as 9 4 or by the mixed number 2 4. MY MATH PATH 5 Common Fractions

47 Example Write 5 as a mixed number. Solution The idea is to find the number of 5 5 in 5 or 5 = = = 2 5 The quickest way to write an improper fraction as a mixed fraction is to divide the numerator by the denominator Example 2 Write 4 8 as an improper fraction. Solution The idea is to find the number of eighths in 4 / = = = 5 8 The quickest way to change a mixed number to an improper fraction is to multiply the whole number by the denominator and add this product to the numerator. 4 8 = MY MATH PATH 5 Common Fractions 4

48 Now complete Exercise and check your answers. Exercise. Write the fraction that describes the shaded regions. a. b. c. d. 2. Write as mixed numbers. a. 7 5 b. 9 2 c. 4 9 d. 2 4 e. 8 8 f. 64 g h. 0 9 i MY MATH PATH 5 Common Fractions 5

49 . Write as improper fractions. a. 2 2 b. 0 4 c. 4 5 d. 8 e. f g h i Bill has quarters. How much money, in dollars and quarters is this? Answers are on page 2. MY MATH PATH 5 Common Fractions 6

50 Unit 2: Equivalent fractions In each of the following, one half of the pie is shaded The common fractions 2, 2 4, and 4 8 are called equivalent fractions because they represent the same value. Here, 2 = 2 4 = 4 8. There are many, actually infinitely many, ways of expressing the value ½. 2 = 2 4 = 6 = 4 8 = 5 0 = 6 2 = 7 4 =... For the ease of understanding, it is always best to express fractions in the lowest terms. A fraction is said to be in the lowest terms when neither the numerator or the denominator is divisible by the same prime number. For example, 0 is not in the lowest terms since both 5 0 and 5 are divisible by 5. Prime numbers are divisible only by one or themselves: Example 2,, 5, 7,,, 7, 9, 2,... Write 40 as a product of primes. Solution The process is to keep dividing by the prime numbers 2,, 5, 7,,... Divide 40 by = 2 70 Divide 70 by = MY MATH PATH 5 Common Fractions 7

51 5 is not divisible by 2 or, so divide 5 by = The quotient 7 is a prime so we are done. The method used in Example is not the only way to write 40 as a product of primes. We could easily have obtained the same answer as follows: 40 = 4 0 = = Factoring the numerator and denominator of a fraction is one way of helping to reduce the fraction to lowest terms. Prime numbers that occur in both the numerator and denominator can then be cancelled and replaced with a one. For example: Example 2 Reduce: a. 0 6 b c Solution Use the method in Example to factor the numerator and denominator. Replace crossed out factors with ones. Then multiply the remaining factors. a b c There are many short cuts to reducing fractions and you may already be aware of some. If you have a method that works, keep using it. Check with your instructor on the validity of your method. Now complete Exercise 2 and check your answers. MY MATH PATH 5 Common Fractions 8

52 Exercise 2. Factor (write as a product of primes). Example: 60 = 2 0 = = a. 80 b. 8 c. 5 d. 49 e. 77 f. 00 g. 44 h. 20 i. 200 j List the prime numbers that are greater than 20 but less than 40.. Factor the numerator. Factor the denominator. Then reduce to lowest terms by cancelling. a b. 6 c. 8 2 d. 0 5 e. 2 2 f MY MATH PATH 5 Common Fractions 9

53 g h i j k l m n. 5 7 o p Is 200 a prime number? Answers are on page 2. MY MATH PATH 5 Common Fractions 0

54 Unit : Adding and subtracting common fractions Notice how we can add the shaded parts of the whole The fractions 2 8 and have a common denominator. Before adding or subtracting fractions, 8 the fractions must be written with a common denominator. The idea behind adding fractions that do not have a common denominator, like and is shown below. Notice that the method for writing as 4 is the reverse method of reducing fractions to 2 lowest terms. Example Add Solution We need a common denominator. Think as follows. The larger denominator is 8. Will 6 divide evenly into 8? No. Now double the = 6. Will 6 divide evenly into 6? No. Triple 8. 8 = 24. Will 6 divide evenly into 24? Yes. The common denominator is 24. Multiply 5 6 number. 4 and 4 8 to get equivalent fractions. Rewrite the answer as a mixed MY MATH PATH 5 Common Fractions

55 Example Subtract Solution The common denominator is 0. Subtract the fractions and the whole numbers. Reduce the answer to the lowest terms Example 2 Subtract 4 2 Solution The common denominator is 6. Notice that we cannot subtract 6 from 2. We must borrow 6 a whole 6 6 from the 4. Then add this 6 6 to 2. Now we can subtract. 6 MY MATH PATH 5 Common Fractions 2

56 MY MATH PATH 5 Common Fractions Now complete Exercise and check your answers. Exercise. Add or subtract as indicated. Reduce your answers to the lowest terms. a. 4 4 b c d e f. 2 6

57 g h i Add, then reduce your answers to the lowest terms. a. 4 2 b. 2 4 c d e f g. 2 4 h MY MATH PATH 5 Common Fractions 4

58 . Subtract, then reduce your answers to lowest terms. a. 5 6 b c d. 4 4 e f g h If Jack ate half a pie and Jill ate a third of the same pie, how much was left over? 5. Roy worked 4 hours on Monday, 5 2 hours on Wednesday and 4 Friday. How many hours altogether did he work? hours on 6. Joan cut 7 4 inches off a 2 inch board. How much of the board was left? Answers are on page. MY MATH PATH 5 Common Fractions 5

59 Unit 4: Multiplying common fractions The pie below is cut into 8 parts. 4 of the pie is shaded. Suppose we want to find 2 of 4 or 2. Notice that the word of means times in arithmetic of 4 = 2 4 = 8 When multiplying fractions, multiply the numerator times the numerator and the denominator times the denominator. Example Multiply Solution Simply multiply numerators and denominators. Example = Multiply Solution Write both numbers as improper fractions, then multiply numerators and denominators. Rewrite the answer as a mixed number It is sometimes possible to cancel common factors from the question before multiplying. MY MATH PATH 5 Common Fractions 6

60 Example Multiply Solution Write 2 as 7. Factor all numerators and denominators. Cancel common factors. If we 2 did not cancel first, then and we would have to reduce 28 to lowest terms 42 regardless The steps to multiplying fractions are: Step : Rewrite any mixed numbers as improper fractions. Step 2: Factor, if possible, all numerators and denominators. Step : Cancel factors common to both the numerator and denominator. Step 4: Multiply the remaining factors. Now complete Exercise 4 and check your answers. Exercise 4. Multiply, then write the products in the lowest terms. a. 2 7 b c d MY MATH PATH 5 Common Fractions 7

61 e. 4 5 f g. 2 h i j Shelly exercised 4 exercise? of an hour every day. How many hours in a week did she. Bill did 2 of the 69 math questions. How many questions did he not do? 4. Carol spent three quarters of the hour long English class staring out the window. 2 How much time did she spend doing this? Answers are on page. MY MATH PATH 5 Common Fractions 8

62 Unit 5: Dividing common fractions How many quarters are there in one half? The drawing suggests that there are two quarters in the shaded half of the pie. Mathematically, we are asking, what is one half divided by one quarter, or Notice also that The number 4 is the reciprocal of 4. The reciprocal of a fraction is found by interchanging the numerator and the denominator. Note that your calculator has a reciprocal key x Division questions can be rewritten as multiplication questions. Dividing by a number is the same as multiplying by the reciprocal of that number. Example Divide 2 Solution 2 Rewrite as a multiplication question by changing the to a and the 2, then multiply. 2 to its reciprocal MY MATH PATH 5 Common Fractions 9

63 Example 2 Divide Solution Rewrite as improper fractions. Then rewrite as a multiplication question. Then multiply and rewrite the answer as a mixed number Now complete Exercise 5 and check your answers.. Divide Exercise 5 a. 2 b c. 2 d e. 2 f g h MY MATH PATH 5 Common Fractions 20

64 i. j Kim made 9 plates in 2 4 hours. How much time did it take her to make one plate?. Greg cut two thirds of the cake into five equal pieces and then ate one piece. How much of the whole cake did Greg eat? Answers are on page. MY MATH PATH 5 Common Fractions 2

65 Unit 6: Conversions between common fractions and decimal fractions It is often convenient to write a common fraction as a decimal fraction. For example, we think of three quarters as 75 cents or $0.75. In other words To write a common fraction as a decimal fraction, divide the numerator by the denominator. Example 4 or 4 4 Write 8 as a decimal fraction. Solution Write 8 needed. as a long division question. Write as.00 and add more decimal place zeros as 8 = Example 2 MY MATH PATH 5 Common Fractions 22

66 Write 5 as a decimal fraction. Solution Use long division to divide the fraction. Note that the quotient has repeating threes. Use a bar over the repeating digit. Place the whole number 5 before the decimal = 5. Now complete Exercise 6 and check your answers. Exercise 6. Rewrite as decimal fractions and then memorize the results. a. 2 b. c. 2 d. 4 e. 2 4 f. 4 g. 5 h. 2 5 i. 5 j Rewrite as decimal fractions. MY MATH PATH 5 Common Fractions 2

67 a. 6 b. 5 6 c. 8 d. 5 8 e. 2 4 f. 2 g. 0 h. 00 i. 5 2 j Rewrite as decimal fractions and note the pattern. a. 9 b. 9 2 c. 9 d. 9 4 e. 5 9 f. 9 6 g. 9 7 h. 9 8 i Write as decimal fractions and notice the pattern. a b c. 99 d. 99 e f Answers are on page 4. MY MATH PATH 5 Common Fractions 24

68

69 Unit 7: Conversions between decimal fractions and common fractions There are three types of decimal fractions:. Terminating decimals have a fixed number of digits. For example, 0.5,.625 and Repeating decimals have one digit or a group of digits which repeat forever. A bar is placed over the repeating digit or group of repeating digits. For example: = = = Non-repeating decimals have an infinite number of digits but no repeating digit or group of digits. For example: Both terminating and repeating decimals can be written as common fractions. Non-repeating decimals cannot be written as common fractions. Study the following: 0.7 = = = = = = When a decimal is written as a fraction, the decimal digits appear in the numerator of the fraction and the denominator has the same number of zeros as there are decimal digits. Example - Write as a common fraction. Reduce to the lowest term. Solution Write as a mixed number. The decimal part of has four digits, so the denominator is = MY MATH PATH 5 Common Fractions 28

70 Now complete Exercise 7 and check your answers. Exercise 7. Write as common fractions but do not reduce to the lowest terms. a. 0.5 b. 0.6 c d.. e f g h i j Write as common fractions and reduce to the lowest terms. a. 0.5 b c d. 0.2 e f g. 0.6 h. 0.4 MY MATH PATH 5 Common Fractions 29

71 . Write as decimal fractions. a. 7 9 b c. 2 9 d Answers are on page 4 MY MATH PATH 5 Common Fractions 0

72 Section 8: Estimation Estimation is a useful tool in mathematical problem solving. Estimating gives you an approximate or ballpark answer. People often estimate differently. There is no one correct way to estimate. It takes practice to get good at choosing numbers that make estimating easier. Here are some estimating strategies:. Round to numbers that are easy for you to compute. For example, about how much is 29 4? Some possible estimates are: 25 6 = = = Use numbers that make sense to you. When you round, sometimes the result will be an underestimate (less than the exact answer) and sometimes it will be an overestimate (more than the exact answer). problem estimate the exact answer is = 20 a little more than = 250 a little less than = 400 about 400. Choose numbers that are compatible and easy to work with for the problem. For example, how much is 2775 divided by 6? Some possible estimates are: = = Use whole numbers to estimate fractions and decimals. For example: problem estimate = = = 29 2 = 7 5. Group numbers to make estimation easier. For example, is approximately = 00. This is because is about 00, is about 00 and 98 is about 00. Now complete Exercise 8 and check your answers. MY MATH PATH 5 Common Fractions

73 Exercise 8. Decide whether these will total more or less than $20. a. $ $8.99 b. $ $8.9 + $7.9 c. $ $6.99 d. $ $.29 + $6.89 e. $ $.99 f. $ $ $ Estimate the following a. $ $.29 + $6.89 b c d e f g h i j Answers are on page 4. Now complete the Practice Test and check your answers. MY MATH PATH 5 Common Fractions 2

74 Practice Test. Write as mixed numbers. a. 25 b Write as improper fractions. a. 4 5 b Reduce to lowest terms. a b Add or subtract as indicated, then reduce the answers to lowest terms. a. 2 5 b c d e MY MATH PATH 5 Common Fractions

75 5. Multiply or divide, then reduce the answers to the lowest terms. a. 6 4 b. 4 2 c. 9 5 d Write as decimal fractions. a. 5 8 b Write as common fractions. a b Audrey claims to have read two-thirds of a 96 page report. How many pages did she read? 9. Bill did 4 of his assignment the first night, then would he have left to do on the third night? the second night. How much 0. The recipe calls for cup of sugar if you want to serve 6 people. How much sugar 4 should you use to serve 2 people?. Estimate the following. a b In driving a distance of 489 km, Paul s car consumes 5.2 litres of gasoline. Estimate how many kilometres he can travel on litre of gasoline. Answers are on pages 5. MY MATH PATH 5 Common Fractions 4

76 Answers Exercise - page 5. a a. 2 5 g a. 2 b. 0 or b. 9 2 h. 9 4 b. 4 c i. 2 c. 2 5 c. 0 5 or 0 d. 2 4 or d. e. f. 5 9 d. 8 e. 00 f g h i = 7 4 or 7 dollars and quarters Exercise 2 - page 9. a. 80 = 2 40 = = = b. 8 = 2 9 = 2 c. 5 = 5 7 d. 49 = 7 7 e. 77 = 7 f. 00 = 2 50 = = g. 44 = 2 72 = = = = h. 20 = 2 0 = i. 200 = 2 00 = j. 50 = 2 75 = 2 25 = , 29,, 7. a. b. 8 2 g. 6 2 h. 7 2 m. 0 c. i n. o. 4. No. 200 = d. j. 0 p e. f. 5 4 k. 4 l MY MATH PATH 5 Common Fractions 5

77 Exercise - pages. a. f a = 2 f = 7 2. a. 6 = 2 b. 8 8 = c. 8 5 or 5 h g. 2 2 b. 2 g. 2 = c. 8 or h d. 4 6 = 2 i. 24 d. = b. c. 5 d f. 4 5 g h = = of the pie was left over = 2 2 = 2 hours = 4 4 inches 25 0 = 5 6 e = 5 6 e = e Exercise 4 - page 7. a. 4 f b. 4 g. 9 h c. d. 2 6 = 5 6 i e. 4 = j = = 5 hours in a week 4 69 = = 2 questions or 4 2 = 4 2 = 9 8 or 8 hours 2 69 = 69 = 2 questions Exercise 5 - page 20. a. 5 6 g. 5 h. b. c d. 48 i. 6 j. 5 e. 2 f. 8 MY MATH PATH 5 Common Fractions 6

78 = = hour per plate = 2 of the cake 5 Exercise 6 - page 2. a. 2 = 0.5 b. = 0. c. 2 = 0. 6 d. 4 = 0.25 e. 2 4 = 0.5 f. 4 = 0.75 g. 5 = 0.2 h. 2 5 = 0.4 i. 5 = 0.6 j. 4 5 = a. 06. b c d e f g. 0. h. 0.0 i. 46. j a. 9 = 0. b. 2 9 = 0. 2 c. 9 = 0. d. 4 9 = 0. 4 e. 5 9 = 0. 5 f. 6 9 = 0. 6 g. 7 9 = 0. 7 h. 8 9 = 0. 8 i. 9 9 = 4. a b d e Exercise 7 - page a. 0 g b h c i d. 0 j a. b. c. d g. h a b c d e e. 25 f. f Exercise 8 - page 29. a. less b. more c. less d. less e. less f. more 2. (answers may vary slightly) a. $5 b. 70 c. 9 d e. 500 f. 25 g. 40 h. 200 i. 50 j. 200 MY MATH PATH 5 Common Fractions 7

79 Practice Test - pages 0. a. 8 b a. 2 5 b. 50. a. 5 b a. b a. 2 or b. or a b a. 000 b. 2 c c. 8 d d. or 2 2 e = 264 pages 9. - = = 5 2 of the assignment = 4 cup. (answers may vary slightly) a. 45 b kilometres MY MATH PATH 5 Common Fractions 8

80 Moodle Courses and Downloading Information MyMathPath.com Resources: Copyright 20 Province of British Columbia Ministry of Advanced Education and Labour Market Development Permission granted to make copies for educational purposes. Based on the Resources: Adult Literacy Fundamental Mathematics, Instructor s Manual and Test-Bank Copyright 200 Province of British Columbia Ministry of Advanced Education and Labour Market Development Acknowledgments: Textbook: Sylvie Trudel Moodle Course Developers - Faculty: Don Bentley Robert Kim Sylvie Trudel Moodle Course Developers Capilano University Students in the College and University Preparation Program: Crystalina Alcazar Ivan Arzave Jasmine Blanchard Bill Brown Derek Brownbridge Michael Drummond Kristofer Dukic Kameron Fakhari Mackenzie Grant David Hart Abbid Jaffer Alexander Kay Michael Kulch Nicholas Schopp Bobbi-dee Stillas Ksenia Teder Daniel Thomas Michael Villareal Ashley Wong MY MATH PATH 5 Common Fractions 9

81 MY MATH PATH 6 - MODULE RATIOS AND PROPORTIONS Contents Learning outcomes...2 Glossary...2 Introduction... Section : Ratios and rates...4 Exercise...7 Section 2: Proportions...9 Exercise 2... Section : Solving proportions...2 Exercise...4 Section 4: Applications...5 Exercise Practice Test...7 Answers...9 MY MATH PATH 6 MODULE Ratios and Proportions

82 Learning outcomes Proportions can be used to solve a wide variety of problems, from consumer problems to trade-related problems. The method for solving proportions will be used in later topics such as percent, algebra, trigonometry and geometry. When you have completed this Module, you should be able to: read, write, interpret, and compare ratios read, write and identify proportions and use them to solve problems Glossary Cross-products The products you get when you cross-multiply ratios are called cross-products. Proportion If two ratios are equal, they form a proportion. Rate A comparison of different Sections. Ratio A ratio is a comparison of similar Sections. Variable The letter used to represent an unknown quantity is a variable. MY MATH PATH 6 MODULE Ratios and Proportions 2

83 Introduction Proportions are very useful in solving problems in everyday life. A proportion is made up of two equal fractions or ratios. You can use ratios to compare how much chocolate mix is required for a quantity of milk. You can find out how much chocolate mix should be used with litre (000ml) of milk, if you know that it takes 25 g of chocolate mix to 250 ml of milk for serving. 25g x 250ml 000ml Since, we can see that the x in the ratio is 00 grams So, you would need 00 g of chocolate mix for litre of milk. In this Module, you will work with ratios that compare 2 or quantities. You will also learn a method for finding out if two ratios form a proportion and how to find a missing part of a proportion. You will then use proportions to solve problems. MY MATH PATH 6 MODULE Ratios and Proportions

84 Section : Ratios and rates Ratios are used frequently in everyday life to compare two or more quantities having the same Sections. For example, when you prepare a tin of orange juice at a mix of to, that means you use one tin of concentrate to three tins of water. A ratio is simply a comparison of two or more quantities. A ratio can be written in any of three ways. The example of the orange juice can be written: in words: to with a colon: : or as a fraction: In all cases, the ratio is read as: one to three. Ratios are easy to use, but there are a few simple rules: The quantity that is given first must be written first or must be given as the numerator of the fraction. Example Write three to two as a ratio. Solution to 2, :2, or 2 The ratio should, like fractions, be reduced to its lowest terms. Example 2 Reduce the ratio of 2:2. Solution 2:2 = 6: or 6 [Note: the in the ratio 6 must be included.] MY MATH PATH 6 MODULE Ratios and Proportions 4

85 Example Reduce the ratio Solution Example Unlike fractions, however, there may be more than two terms in a ratio. Fertilizer bags contain numbers like 6:20:24 that identify the ratio of phosphorus to nitrogen to potash. If the relative amount of each quantity was cut in half, the new ratio would be 8:0:2. Even though the amounts are less, the ratio between them remains the same. This can even be reduced further by halving each number again. Solution 6:20:24 = 8:0:2 = 4:5:6 When comparing measurements, both quantities should normally use the same Section of measure. Example 5 Compare 0 m to 00 cm. You must first convert both measurements to either metres or centimetres. (Remember, m = 00 cm.) Solution Converting both to centimetres: 0 m to 00 cm becomes 000 cm to 00 cm or 000:00 reducing to 0: Converting both to metres: 0 m to 00 cm becomes 0 m to m or 0: MY MATH PATH 6 MODULE Ratios and Proportions 5

86 Notice that, regardless of which Section we choose, metres or centimetres, the ratio is the same and is written without the Sections. A ratio may also be used to express a rate such as kilometres per hour (km/h) or revolutions per minute (rpm). In these cases, since the quantities cannot be converted to similar Sections, the Sections must be written with the rate. Example 6 The batter hit 4 pitches out of 8. Write this as a rate. Solution 4 hits 8 pitches hit 2 pitches Now complete Exercise and check your answers. MY MATH PATH 6 MODULE Ratios and Proportions 6

87 Exercise. Write the following as ratios. Use both : notation and fraction notation. For example: 8 km to 5 km = 8:5 or 8 5 a. $7 to $0 b. 2 g to 46 g c. coffee to 4 coffees d. 0 oranges to oranges 2. Write the following as ratios: (Remember to reduce them to lowest terms.) a. cups of flour to 2 cups of sugar b. 6 cm to cm c. 0 mm to 2 mm d. 20 cm to m e. minutes to hour f. parts blue to 5 parts green to 7 parts white g. 5% phosphorus to 20% nitrogen to 5% potash h. $9.00 to $2.00 to $6.00 MY MATH PATH 6 MODULE Ratios and Proportions 7

88 . Write the following as rates. Reduce them to lowest terms. a. Joanne drove 250 km in 5 hours. What was her average speed in km per hour? b. Martin was paid $480 for working a 40 hour week. How much was he paid per hour? c. A baseball player has 24 hits out of 75 times at bat. Write his batting average as a rate. Bonus question: Batting averages are usually expressed as a decimal. Convert the above rate into a decimal. Answers are on page 9. MY MATH PATH 6 MODULE Ratios and Proportions 8

89 Section 2: Proportions A statement that two ratios are equal is called a proportion. For example, This statement is a proportion because both ratios reduce to We can also determine if two ratios are equal by using a method called cross-multiplying. The products are called cross-products. If the cross-products of two ratios are equal, then the ratios form a proportion. Example Is 4 6 equal to 8 2? Solution 4 2 = 48 and 6 8 = 48 so 4 8 This is a proportion. 6 2 MY MATH PATH 6 MODULE Ratios and Proportions 9

90 Example 2 Is 2:5 equal to 0:25? Solution Write the ratios as 2 5 and 0 25 and cross-multiply = 50 and 5 0 = 50 Example So 2 0. This is a proportion Is 7 8 equal to 5 7? Solution = 9 and 8 5 = 20 The ratios are not equal and so this is not a proportion. 7 5 ( means not equal to ) 8 7 Now complete Exercise 2 and check your answers. MY MATH PATH 6 MODULE Ratios and Proportions 0

91 Exercise 2. Determine if the following ratios are equal. a. 2:, 6:9 b. 4 5, 2 5 c. 20:, 40:6 d. 5 8, 25 4 e. :4, 8:2 f , g. 7:2, 42:2 h. 0., i. 4 5, 4 j. 00:8, 25:2 Answers are on page 9. MY MATH PATH 6 MODULE Ratios and Proportions

92 Section : Solving proportions Whenever we know three quantities in a proportion we can always find the fourth. The most common method of doing this is called cross-multiplying. Example Solution 2 Find n. 6 n Cross-multiply. 2 n = 6 2n = 6 n 6 2 (to find n divide 6 by 2) n = (missing term) In the example above, we write 2 n as 2n. Whenever a letter is multiplied by a number, there is no need to write a times sign between them. For example, y means times y. The letters we use to represent unknown quantities in mathematics are called variables. It is also important to understand the step we took after cross-multiplying in the previous example. If we know that 2 times n equals 6, or 2n=6, then we can find n by dividing both 2n and 6 by 2. Example 2 Solution y 4 Find y y = 4 2 7y = 84 y 84 7 y = 2 (missing term) MY MATH PATH 6 MODULE Ratios and Proportions 2

93 Example 6 Find x. x 9 Solution 6 x = 9 6x = 9 6x = x = 6 x = 2 (missing term) Example 4 Solution 20 is to as x is to 5 In this problem we need to first write the proportion and then solve for the unknown variable. 20 x 5 x = 20 5 x = x x = 00 (missing term) Now complete Exercise and check your answers. MY MATH PATH 6 MODULE Ratios and Proportions

94 Exercise. Solve each proportion for the letter given. a. 2 N g. 0: = 4:c 2 b y h. 7:d = :6 c. 9 5 a i. x: = 2 :9 d. x 8 j. 5 is to as 45 is to y 5 2 e. x:5 = 8:2 k. 2 is to as N is to 2 f. 2: = r:7 l. 9 is to a as 5 is to Answers are on page 9. MY MATH PATH 6 MODULE Ratios and Proportions 4

95 Section 4: Applications Proportions are easily used to solve word problems. Example If a dozen apples cost $.80, how much will 8 apples cost? Solution Convert the problem to a proportion: a dozen apples is to $.80 is the same as 8 apples is to y, where y is the cost of 8 apples y 2 y = y = 4.40 y Example y = $.20 Therefore, 8 apples would cost $.20. If a car travels 400 km on 52 litres of gas, how many litres will it take to travel 250 km? Solution apples price Make sure similar Sections are aligned on the horizontal like in this example OR aligned on the verticle like in example 2A. Either way, you will get the same answer. 400km 52litres 250km xlitres 400 x = x = 000 x OR similar Sections are aligned on the VERTICLE. 400km 52litres 250km xlitres similar Sections are aligned on the HORIZONTAL. x = 2.5 L It will take 2.5 L to travel 250 km. Now complete Exercise 4 and check your answers. MY MATH PATH 6 MODULE Ratios and Proportions 5

96 Exercise 4 Solve each problem. Round answers to the nearest hundredth where appropriate.. 4 litres of paint are needed to paint 64 fence posts. How many posts can be painted with 5 litres of paint? 2. On a road map cm is used to represent 0 km. How far apart are towns that are 6 cm apart on the map?. A. m gold wire costs $200. How much will 0.8 m of the wire cost? 4. An engine uses a 40: mixture of gas to oil. How much oil should be added to 5 litres of gasoline? 5. kg of dates cost $27. How much will 0.5 kg of dates cost? 6. The label on a medicine bottle indicates that 0 cm of liquid contains 250 mg of medication. A prescription requires 40 mg of medication. How much liquid is required? 7. A pharmacist pays $2.65 for 60 medicine capsules. At this rate how much would 50 capsules cost? 8. If it takes 4 hours to drive 80 km, how long will it take to drive 700 km? Answers are on page 20. MY MATH PATH 6 MODULE Ratios and Proportions 6

97 Practice Test. Write as ratios and reduce to lowest terms: a. dimes to three quarters b. 6 minutes to one hour 2. Reduce these ratios to lowest terms. a. 2:4 b. 5:8:24 c Determine whether the following ratios are equal: a. :6 and 6:26 b. 4. 5, MY MATH PATH 6 MODULE Ratios and Proportions 7

98 4. Solve the proportions for the given variable: a. 9:2 = x:6 b. b: = 20: c. 0 8 c 45 d. 6 d 8 e. 7.2:6 = 8:e f. f If 2 kg of grass seed will cover 5 square metres of lawn, how many kg of seed are required for a lawn of square metres? 6. In a poll of 2000 people, 650 said that they preferred chocolate ice cream, and the rest said they preferred vanilla. If you were ordering ice cream for 5000 people, how many would you expect to want vanilla? Answers are on page 20. MY MATH PATH 6 MODULE Ratios and Proportions 8

99 Answers Exercise - pages 7-8. a. 7:0 or 7 0 b. 2:46 or 2 46 c. :4 or 4 d. 0: or 0 2. a. :2 b. 6: c. 5:6 d. :5 e. :20. :5:7 g. :4: h. :4:2. a. 50 km/hr. b. $2/hr. c. 8/25 = 0.2 Exercise 2 - page. a. yes b. yes c. yes d. no e. no f. yes g. yes h. no i. yes j. yes Exercise - page 4. a. N = 8 b. y = c. a = 27 5 or 52 5 or 5.4 d. x = 20 e. x = 20 f. r = 4 or or. g. c = or.2 h. d = 42 5 i. x = 6 j. y = k. N = 8 l. a = 27 5 or 52 or MY MATH PATH 6 MODULE Ratios and Proportions 9

100 Exercise 4 - page x = 80 posts x 0 x x = 60 km.. m 0. 8 c = $ c 40 5 x x = 0.25 L or 25 ml c = $ ċ 0 x x =.6 cm c c = $ x x = 7.7 h Practice Test - pages a. or 2:5 5 b. or : a. 6:7 b. 5:6:8 c. :. a. yes b. yes 4. a. x = 2 b. b = 20 c. c = 25 d. d = e. e = 5 f. f = kg people MY MATH PATH 6 MODULE Ratios and Proportions 20

101 Downloading Information MyMathPath.com Resources: Copyright 20 Province of British Columbia Ministry of Advanced Education and Labour Market Development Permission granted to make copies for educational purposes. Based on the Resources: Adult Literacy Fundamental Mathematics, Instructor s Manual and Test-Bank Copyright 200 Province of British Columbia Ministry of Advanced Education and Labour Market Development Acknowledgments: Textbook: Sylvie Trudel Moodle Course Developers - Faculty: Don Bentley Robert Kim Sylvie Trudel Moodle Course Developers Capilano University Students in the College and University Preparation Program: Crystalina Alcazar Ivan Arzave Jasmine Blanchard Bill Brown Derek Brownbridge Michael Drummond Kristofer Dukic Kameron Fakhari Mackenzie Grant David Hart Abbid Jaffer Alexander Kay Michael Kulch Nicholas Schopp Bobbi-dee Stillas Ksenia Teder Daniel Thomas Michael Villareal Ashley Wong MY MATH PATH 6 MODULE Ratios and Proportions 2

102 MY MATH PATH 6 - MODULE 2 PERCENT Contents Learning outcomes...2 Glossary...2 Introduction...2 Unit : Converting between decimals and percents... Exercise...5 Unit 2: Converting between fractions and percents...7 Exercise Unit : Adding and subtracting percents... Exercise...2 Unit 4: Percent increase and decrease... Exercise Unit 5: Percent applications - sales tax...5 Exercise Practice Test...6 Answers...7 MY MATH PATH 6 Module 2 Percent

103 Learning outcomes Percent is often used in everyday life. Percents are used to calculate taxes (PST and GST), markups, discounts, interest on loans, credit cards and bank accounts and nutrient values in foods. When you have completed this Module you should be able to: convert between decimals, fractions and percent solve a variety of percent problems Percent Glossary An amount out of 00. It is indicated by the symbol %. Introduction Percent, or the symbol %, is very common in everyday life. Sales are described as 25% or 50% off, mortgages are lent at % interest and sales tax is 7%. The word percent means per hundred. The word cent comes from the Latin word centum which means hundred. The following are several common statements involving percents: I got 80% on my math test G.S.T. is 7% sales tax is 7% credit card interest rates vary from 5-20% Percents (or percentages as they are commonly called) are closely related to fractions and decimals. 7% means 7 per hundred 7 means 7 per hundred means 7 per hundred, so 7 7% = = When we change from percent notation to fraction or decimal notation, we are simply replacing the percent sign with or MY MATH PATH 6 Module 2 Percent 2

104 Unit : Converting between decimals and percents Percent to decimal To convert from a percent to a decimal:. Write the numeral without the percent sign. 2. Multiply the numeral by 0.0; which means you move the decimal point two places to the left. Example Convert 7.25%, 6% and 6¾% to decimals. Solutions. 7.25% = = (move the decimal two places to the left) 7.25% = % = =.6 (move the decimal two places to the left). 6 4 % = = (convert the fraction 4 to the decimal 0.75, then move the decimal two places to the left) MY MATH PATH 6 Module 2 Percent

105 Decimals to percent To convert from a decimal to a percent: Multiply the decimal number by 00%. This means you move the decimal point two places to the right and write the percent sign, %, after the number. Leave out the decimal point if appropriate. Example 2 Convert 0.86, 2.45, to percent. Solutions = % = 8.6% (move the decimal 2 places to the right) = % = 245% (move the decimal 2 places to the right) = % = 0.8% (move the decimal 2 places to the right) Now complete Exercise and check your answers. MY MATH PATH 6 Module 2 Percent 4

106 Exercise. Convert to a decimal. a. 7% b. 2% c. 47.8% d % e. 2% f. 200% g. 46% h. 0.% i. 5.5% j. 0.% k. 20% l. 4 % m. 4 2 % n % o. 22% p. 0% q. 0.8% r. 2.6% s % t % 2. Convert to a percent. a b. 0.8 c d e f. 0.5 g. 0.6 h i. j..00 k. 2 l..47 m n o. 0.8 p MY MATH PATH 6 Module 2 Percent 5

107 q.. r. 2.4 s t A recent survey indicated that 0.25 of all adults chose chicken as their favorite meal. Find percent notation for % of hockey fans are supporting Vancouver. Find decimal notation for 92.5%. 5. Lemonade contains 2.5% real lemon juice. Find decimal notation for 2.5%. 6. On a recent math test 0.8 of the students received marks of 80% or more. What percent of the students got 80% or more? 7. A steep hill has a grade of 8%. What is the grade in decimal notation? 8. The present G.S.T. rate is 7%. Express this rate in decimal notation. 9. Sales tax in Springville is 0.2 of the purchase price. Express this rate using percent notation. Answers are on page 7. MY MATH PATH 6 Module 2 Percent 6

108 Unit 2: Converting between fractions and percents Percent to fractions To convert from a percent to a fraction:. Write the numeral without a percent sign. 2. Multiply the numeral by. 00. Reduce the fraction to its lowest terms. Example Convert 2%, 5 2 %, 6 2 % and 2.5% to fractions. Solutions 2. 2% = 2 = = % 5 = 2 2 = = % = % = = = % = = = MY MATH PATH 6 Module 2 Percent 7

109 Fractions to percent To convert from a fraction to a percent:. Multiply the fraction by 2. Reduce to the lowest terms. 00 %. Example 2 Convert 2, 8 and 2 to percents. Solutions = % % 66 % = % % 7 % % % % 50% Now complete Exercise 2 and check your answers. MY MATH PATH 6 Module 2 Percent 8

110 Exercise 2. Convert to a fraction. a. 45% b. 20% c. 7% d. 00% e. 65% f. 75% g. 40% h. 0% i. 2% j. 250% k. 7 2 % l % m. 2 2 % n. 25% o. 87.5% p. % q. 6 2 % r. 8 % s. 8 % t. 6.25% MY MATH PATH 6 Module 2 Percent 9

111 2. Convert to a percent. a. 4 b. 5 2 c. 4 d. 0 7 e. 5 4 f. 0 6 g. 2 h. 5 i j. 00 k. l. 0 m. 6 n. 2 2 o. 8 p. 9 q r s. 80 t. Answers are on page 7. MY MATH PATH 6 Module 2 Percent 0

112 Unit : Adding and subtracting percents There are many cases where percents must be added or subtracted. Example The deductions to Frank s paycheque amounted to 6.2%. What percent of the paycheck did Frank take home? The total paycheck must be represented as 00% of the paycheck 00% - 6.2% = 6.8% Example 2 Sulfuric acid by weight contains 2% hydrogen and 65% oxygen. The rest is sulfur. What percent of sulfuric acid is sulfur? The total weight must be represented by 00% 00% - (2% + 65%) = 00% - 67% = % Now complete Exercise and check your answers. MY MATH PATH 6 Module 2 Percent

113 Exercise. Calculate the following. a. 90% - 5% b. 5% + % 4 c. 00% % d. 2% + % 2 e. 00% - (2% + 6%) f. 00% + 7% g. 54% + 9% + 7.9% + 2.8% + 9 % Mary spends % of her time sleeping and 2 % 5 percent of her time is spent working and sleeping? of her time working. What. If % of Canadians are not employed, what percent are employed? 4. If a shopkeeper discounts (reduces) an item by %, what percent of the old price will be the new selling price? 5. Sales tax is 6%. What percent of the selling price must you actually pay when the sales tax is included? % of carbon dioxide is carbon and the rest is oxygen. What percent of carbon dioxide is oxygen? Answers are on page 7. MY MATH PATH 6 Module 2 Percent 2

114 Unit 4: Percent increase and decrease Percent is often used to state increases or decreases. Statements like the cost of living rose 4% or he had to take a 0% decrease to save his job are common examples. To find a percentage increase or decrease:. Calculate: Largest Number - Smallest Number Original amount 2. Translate the decimal number obtain in step to percent.. Example The cost of tuition for a full-time college student recently increased from $ to $64.50 per semester. What was the percent increase?. The original tuition for the student was $ Using the following formula:: Largest Number - Smallest Number Original amount = = (rounding) 2. Change from decimal to percent: 0. = 0.x 00% = 0% Example 2 By changing furnace filters and maintaining the furnace properly, it is estimated that a monthly heating bill of $84 can be reduced to $7.92. What is the percent of decrease?. The original heating bill was $ Using the following formula:: Largest Number - Smallest Number Original amount = = Change from decimal to percent: 0.2 = 0.2 x 00% = 2% MY MATH PATH 6 Module 2 Percent

115 Now complete Exercise 4 and check your answers. Exercise 4. Jack s cheque increased from $800 to $850 a month. What is the percent increase? 2. Mona will receive $0.65 more per hour next month. What is the percent increase if she receives $2.05 an hour now? Round your answer to one decimal place.. Sam bought a tape deck for $60 and later sold it for $50. What is the percent decrease? Round your answer to one decimal place. 4. Last year Sally made $2,484.2 and paid $, in taxes. This year she made exactly the same, but paid $4,89.50 in taxes. a. What was the percent increase in her salary? b. What was the percent increase in her taxes? Round your answers to one decimal place. 5. Two years ago, calves were $0.89 per pound. Last year they were $0.75 per pound. This year they are $0.96 per pound. a. What is the percent decrease from two years ago to one year ago? b. What is the percent increase from last year to this year? Round your answers to one decimal place. Answers are on page 7. MY MATH PATH 6 Module 2 Percent 4

116 Unit 5: Percent applications - sales tax Percent is widely used in sales tax computations. The provincial sales tax rate in British Columbia (PST) is currently 7%. This means that the tax is 7% of the purchase price. Another sales tax, the goods and services tax (GST) is collected for the federal government and is also 7% of the purchase price. The total price of an article is the price plus PST and GST. Example A refrigerator is priced at $ plus PST and GST. How much is the PST, the GST and the total price? First, find the PST, 7% of $699.95: In mathematics, the word of usually means multiplied. To calculate 7% of $699.95, change the of to x. 7% x $ = 0.07 x $ = $ Round to the nearest cent = $49.00 GST is also 7% of $ = $49.00 The total price is $ $ $49.00 = $ Now complete Exercise 5 and check your answers. Exercise 5. The sales tax in a certain city is 8.5%. How much tax is charged on a purchase of $84.00? What is the total price? 2. A stove is priced at $ PST is 7% and GST is 7%. Find the amount of the PST, the GST and the total price of the stove. Answers are on page 7. MY MATH PATH 6 Module 2 Percent 5

117 Practice Test. Convert to decimals: a. % b. 52.% 2. Convert to fractions: a. 25% b. 8 %. Convert to percentage: a b c. 4 d. 4. The Smith s rent went from $75 a month to $400 a month. What is the percent increase? Round your answer to one decimal place. 5. Sodium sulfate is 2.4% sodium, 22.5% sulfur and the rest is oxygen. What percent is oxygen? 6. In one year, the price of silver fell from $9.20 an ounce to $6.5. What is the percent decrease? Round your answer to one decimal place. Answers are on page 8. MY MATH PATH 6 Module 2 Percent 6

118 Answers Exercise - pages 5-6. a. 0.7 b c d e..2 f or 2 g..46 h i j k. 0.2 l m n o. 2.2 p. 0. q r..26 s t a. 47% b. 8% c. 7% d. 0.4% e..6% f. 50% g. 60% h. 27.8% i. 00% j. 00% k. 200% l. 47% m % n. 0.07% o. 8% p. 46% q. 0% r. 240% s % t. 2%. 25% % % Exercise 2 - page 0 b. 5 f a. 20 g. 5 2 m. 8 h. 7 c. 00 i o. 8 n. 4 d. e. 20 j. 2 k. 2 8 q. 6 p. l. 2 r. 2 s. 6 5 t a. 25% b. 40% c. 75% d. 70% e. 80% f. 60% g. 50% h. 60% i. 80% j. 7% k. / % l. 0% m. 6 2 / % n / % o. 7½% p. / 9 % q. 62½% r. 5% s. / 9 % t. 27 / % Exercise - page 2. a. 55% b/ 5¾% c. 77.5% d. 2¼% e. 72% f. 07% g. 8.7% / 5 %. 89% / % 5. 06% % MY MATH PATH 6 Module 2 Percent 7

119 Exercise 4 - pages % %. 58.% 4. a. 0% b. 6.% 5. a. 5.7% b. 28% Exercise 5 - page 5. $2.64 tax, $46.64 total 2. $.50 PST, $.50 GST, $52.95 total. $ $4,6 Saskatchewan, $4,592 BC 5. 8% 6. 4% 7. $90 8. $,7.4 Practice Test - pages 6. a. 0. b a. ¼ b. 5 / 6. a. 57% b. 0.0% c. 25% d. / % or.% % or 6.7% % 0..0% MY MATH PATH 6 Module 2 Percent 8

120 Moodle Courses and Downloading Information MyMathPath.com Resources: Copyright 20 Province of British Columbia Ministry of Advanced Education and Labour Market Development Permission granted to make copies for educational purposes. Based on the Resources: Adult Literacy Fundamental Mathematics, Instructor s Manual and Test-Bank Copyright 200 Province of British Columbia Ministry of Advanced Education and Labour Market Development Acknowledgments: Textbook: Sylvie Trudel Moodle Course Developers - Faculty: Don Bentley Robert Kim Sylvie Trudel Moodle Course Developers Capilano University Students in the College and University Preparation Program: Crystalina Alcazar Ivan Arzave Jasmine Blanchard Bill Brown Derek Brownbridge Michael Drummond Kristofer Dukic Kameron Fakhari Mackenzie Grant David Hart Abbid Jaffer Alexander Kay Michael Kulch Nicholas Schopp Bobbi-dee Stillas Ksenia Teder Daniel Thomas Michael Villareal Ashley Wong MY MATH PATH 6 Module 2 Percent 9

121 MY MATH PATH 6 MODULE POWERS AND ROOTS Contents Learning outcomes...2 Glossary... Introduction...4 Unit : Powers - repeated multiplication...5 Exercise...8 Unit : Square roots...0 Exercise... Practice Test... Answers...4 MY MATH PATH 6 Module Powers and Roots

122 Learning outcomes Exponents and roots are used throughout algebra, and in other sciences as well. A scientific calculator can be used to find any required root or power of a number. When you have completed this Module, you should be able to: understand and use roots and powers find a root or a power using a scientific calculator MY MATH PATH 6 Module Powers and Roots 2

123 Glossary Base The number that is multiplied by itself a certain number of times, depending on the exponent. Cube To cube a number means to raise it to the third power. Exponent The small number placed above and to the right of a base number. It indicates how many times that base is to be multiplied by itself. Power A base number raised to a given exponent. Radical An expression involving the symbol. Radicand In the expression 5, 5 is called the radicand. Root A number that, multiplied by itself a given number of times, equals a radicand. Square To square a number means to raise it to the second power. Square root A radical with an index of two (often omitted). MY MATH PATH 6 Module Powers and Roots

124 Introduction There is a story about the mathematician who invented the game of chess. He presented this new game to the King, who was very impressed. The King announced that the mathematician could name his reward for inventing such a wonderful game. The mathematician thought for a moment, and then asked that he be given some rice for his family - but that the amount of rice be counted as follows: On the first square of the chess board place one grain of rice, on the second square 2, on the rd square 4, on the 4 th square 8, and so on, doubling the number each time. Now, a chessboard has 64 squares, so the number of rice grains on the last square of the chessboard will be 2 6 or about 9,22,72,000,000,000,000. This amount of rice would be greater than all of the rice ever harvested in the history of the world. This module deals with ways to handle such giant numbers, plus other numbers that use exponents. MY MATH PATH 6 Module Powers and Roots 4

125 Unit : Powers - repeated multiplication As you may recall, multiplying is a way of indicating that we should add the same number so many times. For example, 5 = The power operation is a way of indicating that we should multiply the same number so many times. For example, 5 = The power operation is indicated by placing a small number, called an exponent, to the top right of some number, or base, that we then multiply so many times together. 5 is called a power. 5 is called the base. is called the exponent. The power, 5, is read five to the third or more commonly, five cubed. Powers with exponents of 2 are said to be squared, while those with exponents of are said to be cubed. Below are some examples of how we read powers. 7 2 is read seven squared 7 is read seven cubed 9 4 is read nine to the fourth 2 8 is read twelve to the eighth Notice that the exponent is not really a number - it is really an instruction: it says multiply the number below me by itself this many times. So 9 4 = 656, not 6. We can use exponents to find powers of integers, decimals and fractions: (-) 5 = (-)(-)(-)(-)(-) = -24 (.25) = (.25)(.25)(.25) = = = 656 You probably realize now why exponents are used - they save a lot of time writing down repeated multiplication. Scientific calculators have a raise to the power key, usually labelled y x (sometimes x y or a x is used). To use this function, press the y x key in between the base number and the exponent, then press =. MY MATH PATH 6 Module Powers and Roots 5

126 Example To find 5 =? press: y x 5 = Did you get 24? Example 2 To find (.25) =? press:.25 y x = Did you get.9525? Example To find (2.57) 0 =? press: 2.57 y x 0 = Did you get ? Take a look at the following patterns: 2 4 = = = = = = 8 0 = = = 2 2 = = 0 0 = 00 2 = 2 0 = = 0 0 = Notice that each line is equal to one factor (2 or 0) divided into the line above it. It seems reasonable the 2 = 2 or 0 = 0. After all, the exponent gives the number of factors that are multiplied together. But why should 2 0 = or 0 0 =? A zero exponent does look odd, but it has to follow the same pattern: 2 2 = 2 2 = = 0 2 = = 2 0 = 0 0 = 0 0 = Note that your calculator can use zero as an exponent, except in the case 0 0, which is undefined. Otherwise, any base number raised to the 0 power equals. Finally, here is a shortcut you can use on your calculator: The most common exponent used in algebra is 2. Your calculator will have a key labelled x 2. You can use this key instead of the y x key, and save yourself some time: x 2 or x 2 = Did you get 69? MY MATH PATH 6 Module Powers and Roots 6

127 Fractions can also be squared as follows: To square the 2 and separately: 2 x 2 is 4 and x 2 is 9 and rewrite as Now complete Exercise and check your answers. MY MATH PATH 6 Module Powers and Roots 7

128 Exercise. How would you read the following? a. 2 b. 0 c. x d. 4 2 e Write the following as powers. a b c. 4 4 d. (9.7)(9.7)(9.7)(9.7) e. 5x5x5x5x5x5x5x5x5 f. 26. Evaluate the following: a. 2 = b. 4 2 = c. 5 2 = d = e = MY MATH PATH 6 Module Powers and Roots 8

129 f. 2 = g. 2 2 = h. 4 2 = i. (0.5) = 2 j. = k. (5.) = l. 9 0 = 5 m. 2 8 = n. (.2) 2 = o. (0.) 5 = 4. The following are called powers of ten. Evaluate them. a. 0 0 = b. 0 = c. 0 2 = d. 0 = e. 0 4 = f. 0 5 = g. 0 6 = h. 0 7 = i. Get the idea? = Answers are on page. MY MATH PATH 6 Module Powers and Roots 9

130 Section 2: Square roots A root is the reverse of a power. Since 5 2 = 25, we say that 25 is the square of 5. We could reverse this statement, and say that 5 is the square root of 25. In symbols, this would be written: The symbol 25 = 5 call the symbol means find the number, that when squared, equals the number inside. We a radical and the number inside it is the radicand. The radicand in the expression 25 is 25. The expression 25 reads the square root of 25. Take a look at the following: 9 = because = 9 49 = 7 because 7 7 = = 0 because 0 0 = = 0.5 because = 0.25 = because = What about numbers that do not have whole number square roots? For example, what is the square root of 0? Look for the button on your calculator. Use this button to find the above square roots. Did you get the right answers? Try the following, using your button. 2 = Did you get ? 95 = Did you get ? 0.05 = Did you get ? 2 = Did you get ? MY MATH PATH 6 Module Powers and Roots 0

131 A = b because b b = A. Now complete Exercise 2 and check your answers. Exercise 2. a. 9 = because = 9 b. 8 = because = 8 c. 25 = because = 25 d. 00 = because = 00 e. 4 = because = 4 f. 0 = because = 0 g. = because = h = because = 4900 i. 600 = because = 600 j. = because = 4 4 MY MATH PATH 6 Module Powers and Roots

132 k. = because = l. m. 4 4 = because = = because = n.. 2 = because =.2 o = because = 0.64 p = because = 0.0 Answers are on page. MY MATH PATH 6 Module Powers and Roots 2

133 . Write the following as powers. Practice Test a. b. 9 9 c. (0.7)(0.7)(0.7)(0.7) d Evaluate the following: a. 0 = b. 9 2 = c. 5 2 = d = e. 2 = f. 5 = 2 g. (2.) =. Evaluate the following:. a. 4 = b. 64 = c. 0 = d. = e. 792 = f. = 6 Answers are on page. MY MATH PATH 6 Module Powers and Roots

134 Answers Exercise - pages 7-. a. three squared b. ten cubed c. x cubed d. one half to the fourth e. six to the sixth 2. a. 2 b. 8 5 c. 4 d e. 5 0 f. 26. a. 69 b. 96 c. 225 d. 400 e. 625 f. 4 2 g. 8 h. 6 8 i j. 27 k. 5. l. 25 m. 64 n..44 o a. b. 0 c. 00 d. 000 e f g h i. number of zeros = exponent for powers of ten Exercise 2 - pages. a. b. 9 c. 5 d. 0 e. 2 f. 0 g. h. 70 i. 40 j. 4 2 m. n.. o. 0.8 p k. 5 l. 2 Practice Test - pages 2-2. a. b. 9 c. (0.7) 4 d a. b. 8 c. 225 d e. f. 2 g a. 2 b. 8 c. 0 d. e. 89 f. 4 MY MATH PATH 6 Module Powers and Roots 4

135 Moodle Courses and Downloading Information MyMathPath.com Resources: Copyright 20 Province of British Columbia Ministry of Advanced Education and Labour Market Development Permission granted to make copies for educational purposes. Based on the Resources: Adult Literacy Fundamental Mathematics, Instructor s Manual and Test-Bank Copyright 200 Province of British Columbia Ministry of Advanced Education and Labour Market Development Acknowledgments: Textbook: Sylvie Trudel Moodle Course Developers - Faculty: Don Bentley Robert Kim Sylvie Trudel Moodle Course Developers Capilano University Students in the College and University Preparation Program: Crystalina Alcazar Ivan Arzave Jasmine Blanchard Bill Brown Derek Brownbridge Michael Drummond Kristofer Dukic Kameron Fakhari Mackenzie Grant David Hart Abbid Jaffer Alexander Kay Michael Kulch Nicholas Schopp Bobbi-dee Stillas Ksenia Teder Daniel Thomas Michael Villareal Ashley Wong MY MATH PATH 6 Module Powers and Roots 5

136 MY MATH PATH 6 MODULE 4 STATISTICS Contents Learning outcomes...2 Glossary... Introduction...4 Unit : Interpreting graphs...5 Exercise...8 Unit 2: Mean, median, mode and range...5 Exercise Practice Test...2 Answers...25 MY MATH PATH 6 Module 4 Statistics

137 Learning outcomes Statistics is the branch of mathematics that helps you collect, organize and find meaning in numerical facts. The word statistics also refers to the numerical facts themselves. When you have completed this Module, you should be able to: calculate median, mean, mode and range interpret bar, line, and circle graphs MY MATH PATH 6 Module 4 Statistics 2

138 Glossary Bar graph A type of graph that uses side-by-side horizontal bars or vertical bars of different lengths to compare different items of information. Circle graph A type of graph that is circular in shape. The circle graph compares different information by assigning pie Units of different sizes to the various categories of information. Data Factual information you collect to solve a problem. Graph A diagram that pictures data or information in a way that makes it easier to understand. Common graphs are bar graphs, circle graphs and line graphs. Interpret The way you explain or arrive at a conclusion based on a study of data. Line graph A series of points, plotted between the horizontal and vertical axes of a graph, that form a line when connected to one another. Each point on the line represents specific values of the quantities labelled along the horizontal and vertical axes of the graph. Mean The sum of all the values in a set divided by the number of values in the set. Median The value in the middle of a data set when the data are arranged in numerical order. There are an equal number of data values greater than the median value and an equal number of data values less than the median value. Mode The number that appears most often in a set of data. Range The difference in the maximum and minimum values for a set of data. MY MATH PATH 6 Module 4 Statistics

139 Introduction Statistics is the branch of mathematics that helps you collect, organize and find meaning in numerical facts or data. Data can be organized into a table and then used to construct a graph. Figure shows data on level of education and employment in 996. Education Percentage of employment Grade 8 or less 20% Some secondary 40% Graduated from high school 62% Some college/university 64% Post-secondary diploma 69% University degree 77% Figure This type of statistical information frequently appears in newspapers and magazines. Note how easy it is to read and interpret the data. For example, what conclusion or conclusions would you draw about higher education and employment? MY MATH PATH 6 Module 4 Statistics 4

140 Unit : Interpreting graphs Graphs are convenient devices for showing data or illustrating information. Since they are drawn, rather than written, a graph makes it possible to get a quick impression of a great deal of data and to easily make comparisons and draw conclusions. We can see various kinds of graphs in newspapers and magazines as well as on the television. Graphs are particularly easy to produce using a computer. There are three types of common graphs: bar line circle Although these graphs all appear different, they have properties in common that you must know if you are to correctly construct your own graphs. All graphs have to be titled and labelled so that everyone looking at or analyzing the graph will get the same information. As you go through the graphing Units, pay particular attention to how the graphs are all titled and labelled. How to read any graph You can get information from a graph quickly if you follow these steps:. Read the title of the graph. 2. Read the labels and range of numbers (scale) along the sides.. Determine what units the graph uses. 4. Look for groups, patterns and differences. Bar graphs Favourite Television Shows The bar graph is used to show comparisons and trends. Bar graphs get their name from the thick bars with which they are drawn. The bars may be vertical or horizontal. A small space usually separates each bar. The bars are often different colours. For example, the graph in Figure 2 can be easily used to answer the question: Which kinds of television programs were chosen as a favorite by 50 or more college students? Detective and western shows were chosen as favorites by 50 or more students. Number of students Detective Comedy Variety Western Type of TV show Figure 2 Bar graph MY MATH PATH 6 Module 4 Statistics 5

141 Line graphs Line graphs plotted from sets of data are good for showing trends and developments over a period of time. A line graph can be drawn with either straight or curved lines that extend across the graph in a horizontal direction. The graph in Figure shows how the number of farms has dropped steadily over a thirty-year period up to 970. Often it is necessary to find a particular quantity from a line graph. For example, to Figure Line graph answer the question How many millions of farms were there in 960? locate 960 on the horizontal scale. Move up to the point on the graph. Then move left to the number on the vertical scale. The vertical scale shows that there were 4 million farms in 960. Circle graphs A circle graph shows an entire quantity divided into various parts. Each part of the circle is called a Unit (or sector) and has its own name and value. Circle graphs are also known as pie graphs because they can look like a pie cut into pieces. In most cases, values on circle graphs consist of either parts of a dollar or a fraction or a percent of a whole. An example of a circle graph showing the expenses of a student going to a community college is shown in Figure 4. Figure 4 Circle graph MY MATH PATH 6 Module 4 Statistics 6

142 This circle graph clearly shows which college expense is the greatest and which is the least. The Unit of the circle graph showing tuition is largest; so tuition is the greatest expense. The circle graph can be used to calculate certain different data. For example, you can use the circle graph in the example above to find how much money would be spent on tuition if the total college expenses were $2400 for a year. Find 2% of $ = 552 Write 2% as a decimal and multiply. Tuition would be $552 per year. Now complete Exercise and check your answers. MY MATH PATH 6 Module 4 Statistics 7

143 Exercise Use this graph to answer questions to.. Which material makes up the greatest part of the human body? 2. Which materials make up 20% or less of the human body?. If a student has a mass of 64 kg, about how much of the body mass is: a. fat b. bone c. muscle d. other MY MATH PATH 6 Module 4 Statistics 8

144 Use the following graph to answer questions 4 to. Suppose you looked back at your old bills and graphed the amount of money paid each bimonthly billing for electricity in 995 and 996. Which billings in 995 were: 4. more than $50? 5. less than $00? 6. $50 or less? Which billings in 996 were: 7. less than $00? 8. $50 or more? 9. $65 or less? Find the total amount of the hydro bill for: Hydro users can choose to make equal monthly payments. Each payment is the average of last year s bills. Find the calculated equal monthly payment for each of these years. Round answers to the nearest cent. MY MATH PATH 6 Module 4 Statistics 9

145 Use the following graph for questions 4 to What are the units of measurement for altitude and for temperature in the graph? 5. How much has the temperature decreased from sea level to an altitude of,000 feet? 6. Think about extending the line to estimate the temperature at 0,000 feet and at 5,000. How reliable do you think these estimates are? Why? 7. Think about extending the line to estimate the temperature at 500 feet below sea level. How reliable do you think this estimate would be? Why? 8. Why would early aviators have been interested in the graph? MY MATH PATH 6 Module 4 Statistics 0

146 Refer to the graph to answer questions 9 to 22. You work as a quality control supervisor at the Compu-Rite Parts factory. A technician prepares a report on the rejected data boards with a graph, as shown. 9. What dates does the defect report cover? How many days of production data are shown? 20. What is the highest level of defects reported on the graph? 2. What is the lowest level of defects reported on the graph? 22. On what day did the highest level of defects occur? MY MATH PATH 6 Module 4 Statistics

147 Based on the bar graph, answer questions 2 to 27. As the purchasing agent for the electrical department, you are responsible for keeping a supply of dry-cell batteries for hand-held equipment used by the technicians. A trade magazine shows that some types of batteries perform better than others. The report displays a bar graph, as shown here. 2. Which type of battery lasts the longest under either high-rate or low-rate use? 24. The department is using heavy-duty batteries now. How do they compare to the other types of batteries available? 25. The department needs batteries in flashlights to last through a minimum 2-hour shift of high-rate use. Which battery types would you select to meet this requirement? 26. Under conditions requiring only occasional, low-rate use such as portable radio gear batteries should last at least 48 hours. Which battery types would you select to meet this requirement? 27. Cost is an important factor. What does the graph say about the cost of each type of battery? MY MATH PATH 6 Module 4 Statistics 2

148 Use the circle graph to answer questions 28 to 0. The circle graph shown here is about types of treatment at the Cedar Hill Walk-in Clinic. 28. What kind activity was provided most often at the Clinic during the year? 29. Compare the number of minor surgery treatments to the number of routine examinations. There are about times as many routine examinations as there are minor surgery treatments. 0. Rank the types of treatment given at the clinic this year, starting with the most frequent, down to the least frequent. The bar graph below is about electronic appliances. Study the bar graph and then answer questions to 5. MY MATH PATH 6 Module 4 Statistics

149 . Should automobiles be included in the list of appliances? Why or why not? 2. How many kinds of appliances are shown?. What appliances are used in over half of the households? 4. What appliances are used in less than half of the households? 5. What other information do you need to find how many households have home computers? The following bar graph shows the average braking distance required for a car with antilock brakes travelling at different speeds on dry pavement. The braking distance is measured from the time the brakes are applied until the car comes to a complete stop. Refer to this graph to answer questions 6 and What is the unit of measurement used to indicate the braking distance? 7. What is the braking distance shown for a car traveling at a speed of 80 kilometres per hour? Answers are on pages MY MATH PATH 6 Module 4 Statistics 4

150 Unit 2: Mean, median, mode and range The mean, median and mode are three different measures of the middle of a set of data. Finding the mean The average of a set of data is called the mean. Suppose six people are weighed and their mass in kilograms each recorded as follows and you want to find the mean (average) mass in kilograms for this set of six people To find the mean, first add the items = is the sum of the items. Then divide the sum by the number of items. It is convenient to round up the answer to the nearest tenth = 6.7 The mean mass is 6.7 kg. the mean sum of the items = number of items Finding the median and the mode The mean gives you some information about a set of numbers or data. The median and the mode are two other different measures of a set of data. Suppose 5 families have the following numbers of children:,, 5,, 7,, 0,, 2, 8,, 2, 0,, 2 The mean of these values is 2.6, yet no family has 2.6 children and this is not a meaningful value in this situation. Instead, re-arrange the data in order. The median is the middle number. It has as many numbers above as it has below. The median family has 2 children. MY MATH PATH 6 Module 4 Statistics 5

151 In the following set of data, with an even number of quantities, there are two middle numbers. The median is halfway between these two numbers is the median The median is a quantity where half the numbers in the set of data are above it and half are below. Remenber that the data must be arranged in order first. The mode is the most common number that appears in a data set. Using the same 5 families, select the number that appears most often. The mode is child per family. The mode is the number that appears most often in a set of data. Note that some data sets could have more than one mode, while others may not have a mode. For example, if in the 5 families there was one less family with one child, there would actually be three modes;, 2, and. If all the families had different numbers of children there would be no mode. The mean, median and mode in a set of data may be the same number or they may all be different. MY MATH PATH 6 Module 4 Statistics 6

152 Finding the range To find the range in a set of numbers, find the difference between the largest and smallest number in the set. The difference is the range. To determine range consider the above data: largest number = 8 smallest number = 0 range = 8-0 = 8 The range is the difference between the largest and smallest number in a data set. Now complete Exercise 2 and check your answers. MY MATH PATH 6 Module 4 Statistics 7