Decimal Numbers and Percent Unit NSSAL (Draft) C. David Pilmer 2013 (Last Updated: June 2014) NSSAL ©2012 i Draft C. D. Pilmer This resource is the intellectual property of the Adult Education Division of the Nova Scotia Department of Labour and Advanced Education. The following are permitted to use and reproduce this resource for classroom purposes. • Nova Scotia instructors delivering the Nova Scotia Adult Learning Program • Canadian public school teachers delivering public school curriculum • Canadian non-profit tuition-free adult basic education programs The following are not permitted to use or reproduce this resource without the written authorization of the Adult Education Division of the Nova Scotia Department of Labour and Advanced Education. • Upgrading programs at post-secondary institutions (exception NSCC's ACC) • Core programs at post-secondary institutions • Public or private schools outside of Canada • Basic adult education programs outside of Canada Individuals, not including teachers or instructors, are permitted to use this resource for their own learning. They are not permitted to make multiple copies of the resource for distribution. Nor are they permitted to use this resource under the direction of a teacher or instructor at a learning institution. Acknowledgments The Adult Education Division would like to thank Dr. Genevieve Boulet (MSVU) for reviewing this resource and providing valuable feedback. The Adult Education Division would also like to thank the following ALP instructors for piloting this resource and offering suggestions during its development. Eileen Burchill (IT Campus) Lynn Cuzner (Marconi Campus) Carissa Dulong (Truro Campus) Krys Galvin (Truro Campus) Barbara Gillis (Burridge Campus) Nancy Harvey (Akerley Campus) Barbara Leck (Pictou Campus) Suzette Lowe (Lunenburg Campus) Shelly Meisner (IT Campus) Alice Veenema (Kingstec Campus) NSSAL ©2012 ii Draft C. D. Pilmer Table of Contents Introduction (for Learners) ……………………………………………………………….. Prerequisite Knowledge …………………………………………………………………… Introduction (for Instructors) ……………………………………………………………… iv v vi Introduction to Decimal Numbers ………………………………………………………… Comparing Decimals ……………………………………………………………………… Rounding Decimals ………………………………………………………………………. Equivalent Fractions and Decimals ……………………………………………………….. Introduction to Percent ……………………………………………………………………. Comparing Fractions, Decimals and Percentages ………………………………………… Adding and Subtracting Decimal Numbers ………………………………………………. Multiplying Decimal Numbers ……………………………………………………………. Dividing Decimal Numbers ……………………………………………………………….. Estimation Questions Involving Percentages ………………………..……………………. Calculator Questions ………………………………………………………………………. 1 6 13 14 18 26 30 37 43 54 58 Appendix …………………………………………………………………………………… Connect Four Fraction Decimal Equivalency Game ………………………………….. Connect Four Fraction Percent Equivalency Game …………………………………… Connect Four Percentage Game ………………………………………………………. Additional Practice: Ordering Decimals ………………………………………………. Additional Practice: Ordering Decimals and Fractions ……………………………….. Additional Practice: Ordering Decimals, Fractions and Percentages …………………. Post-Unit Reflection …………………………………………………………………… Soft Skills Rubric ……………………………………………………………………… Answers ………………………………………………………………………………... 62 63 64 65 66 68 69 72 73 74 NSSAL ©2012 iii Draft C. D. Pilmer Introduction (for Learners) Welcome to the Decimal Numbers and Percent Unit. Like the Fractions Unit, we will initially spending a bit of time understanding what a decimal and percent are and how to order decimals and percentages from smallest to largest. This understanding is very important before we try to introduce operations (i.e. addition, subtraction, multiplication, and division) with decimals and percents. Prerequisite Knowledge This unit was written under the assumption that learners understand the concepts covered in the Level III Whole Number Operations Unit and Level III Fractions Unit. We will be revisiting many of the concepts addressed in those units. The expectation for this unit is that learners are comfortable with: • The addition, subtraction, and multiplication of multi-digit numbers. e.g. 198 + 35 • e.g. 928 − 294 e.g. 251× 58 Divide a multi-digit number by a single digit number. e.g. 8456 ÷ 7 • Ordering fractions from smallest to largest without using a calculator. e.g. Order 2 1 12 3 6 1 1 4 8 , 1 , , 2 , , , 2 , and from smallest to largest. 10 8 16 5 9 7 12 16 Introduction (for Instructors) This unit is similar to the Fraction Unit in that learners initially spend a significant amount of time understanding the magnitude of decimals and percentages, before ever completing operations with decimals and percentages. We actually tap into much of the information and understanding that the learners acquired in the Fractions Unit; hence, it is a prerequisite for this unit. Please ensure that learners arrange fractions, decimals and percentages from smallest to largest before they proceed to the sections involving operations. NSSAL ©2012 iv Draft C. D. Pilmer Introduction to Decimal Numbers We see decimal numbers everywhere. • Money: "Yoshi deposited $210.75 into his bank account." • Measurement: "The distance from my house to work is 14.3 kilometres." "The container holds 1.57 litres of fluid." "The package weighs 6.8 kilograms." "The property is 0.85 acres in size." "The winning time in the 100 metre dash was 9.72 seconds." "When we started the experiment, the fluid was at 18.2oC." • Probability: "The probability of obtaining a head when flipping a fair coin is 0.5." • Statistics: "The mean weight (i.e. average weight) of males in the class is 85.2 kg." Decimals are just another way of writing fractions, and vice versa. 1 ("one-tenth"), which is represented by the 10 area model on the right, can also be written in its decimal form as 0.1 ("zero decimal one"). For example, the fraction 1 ("one-hundredth"), which is represented 100 by the area model on the right, can also be written in its decimal form as 0.01 ("zero decimal zero one"). For example, the fraction Fractional Form 7 ("seven tenths") 10 81 ("eighty-one hundredths') 100 3 ("three hundredths") 100 417 ("four hundred seventeen thousandths") 1000 79 ("seventy-nine thousandths") 1000 NSSAL ©2012 Decimal Form 0.7 ("zero decimal seven") 0.81 ("zero decimal eight one") 0.03 ("zero decimal zero three") 0.417 ("zero decimal four one seven") 0.079 ("zero decimal zero seven nine") 1 Draft C. D. Pilmer Please note that many math resources and math teachers will say that decimal numbers should be read in the same manner as their fractional counterparts. e.g. 0.7 should be read as "seven tenths." e.g. 0.81 should be read as "eighty-one hundredths." e.g. 0.417 should be read as "four hundred seventeen thousandths." The rationale for this approach from people in the education community is that it forces learners to understand place value, and therefore conveys a deeper level of understanding. However, mathematicians disagree with this approach stating that decimals like 0.7 should be read as "zero decimal seven" because it clearly conveys to the listener that we are dealing with a decimal, rather than its equivalent fraction. In this resource, we are going to follow the practices of the mathematicians. Place Value and Decimals Ten-Thousandths Thousandths Hundredths Tenths • Ones Tens Ones Period Hundreds Thousands Ten Thousands Hundred Thousands Thousands Period Millions Ten Millions Hundred Millions Millions Period Some decimals are larger than 1. 28.93 "twenty-eight decimal nine three" 93 Fractional Form: 28 ("twenty-eight and ninety-three hundredths") 100 Expanded Form: 20 + 8 + 0.9 + 0.03 or 1 1 ( 2 ×10 ) + (8 ×1) + 9 × + 3 × 10 100 4319.2 "four thousand, three hundred nine decimal two" 2 Fractional Form: 4309 ("four thousand three hundred nine and two tenths") 10 Expanded Form: 4000 + 300 + 9 + 0.2 or 1 ( 4 ×1000 ) + ( 3 ×100 ) + ( 9 ×1) + 2 × 10 NSSAL ©2012 2 Draft C. D. Pilmer 7.065 "seven decimal zero six five" 65 Fractional Form: 7 ("seven and sixty-five thousandths') 1000 Expanded Form: 7 + 0.06 + 0.005 or 1 1 ( 7 ×1) + 6 × + 5 × 100 1000 25.304 "twenty-five decimal three zero four" 304 Fractional Form: 25 ("twenty-five and three hundred four thousandths") 1000 Expanded Form: 20 + 5 + 0.3 + 0.004 or 1 1 ( 2 ×10 ) + ( 5 ×1) + 3 × + 4 × 10 1000 Questions 1. What decimal numbers are represented by each of these area models? (a) (b) (c) Answer: ________ (d) (e) Answer: ________ NSSAL ©2012 Answer: ________ Answer: ________ (d) Answer: ________ 3 Answer: ________ Draft C. D. Pilmer 2. Circle the adjoining numbers that are equivalent decimals and fractions. 9 10 0.05 5 10 0.5 46 1000 0.46 1 0.07 7 100 0.087 87 100 0.046 1 8 100 1.8 527 1000 0.0527 37 1000 0.037 67 100 0.13 8 10 0.009 9 100 0.09 37 100 6.7 13 1000 1.26 64 1000 2.07 2 7 10 0.27 3 9 100 1.3 7 1000 2.7 9 1000 3.09 56 1000 0.126 0.39 3 9 10 0.056 56 100 2 3 3.64 2 604 1000 3.604 3 3 3 406 1000 1 8 10 26 100 3. Express each fraction in its decimal form. (a) 256 = 1000 (d) 1 97 = 1000 (b) 6 = 100 (e) 2 (c) 3 7 = 1000 9 = 10 (f) 13 58 = 100 4. Express each decimal in its fractional form. Do not put the fraction in its simplest form. (a) 0.95 = (b) 0.4 = (c) 4.508 = (d) 1.08 = (e) 2.003= (f) 6.059 = 5. Write the decimal equivalent to each of the following. (a) thirty-five and six tenths _____________ (b) seven and nine hundredths _____________ (c) fifty-eight thousandths _____________ (d) one thousand and fifteen hundredths _____________ (e) two hundred six and three hundred nine thousandths _____________ (f) seventy and one tenth _____________ NSSAL ©2012 4 Draft C. D. Pilmer (g) five and thirty-seven thousandths _____________ (h) four hundred and twenty-nine thousandths _____________ 6. Write each decimal as a fraction, using both numerals and words. A completed example has been provided. Fraction Decimal Numerals Words 4 2 two and four hundredths e.g. 2.04 100 (a) 32.8 (b) 0.472 (c) 13.067 (d) 7.59 (e) 327.09 7. Write each decimal number in both expanded forms. (a) 42.8 (b) 9.31 (c) 302.429 (d) 18.034 (e) 4209.07 NSSAL ©2012 5 Draft C. D. Pilmer Comparing Decimals In this section we will be comparing decimal numbers, and in a few instances comparing decimal numbers to fractions. You will have to remember the strategies we used in the Fraction Unit that you completed earlier. There are two techniques that we would like you to learn for comparing decimals. 1. Benchmarks When we were working with fractions, we used the benchmarks 0, 1 , and 1 to gauge the 2 size of fractions. We will do the same for decimals. Examples of Decimals that are Close to Zero 1 0.1 i.e. 10 8 0.08 i.e. 100 17 0.017 i.e. 1000 Examples of Decimals that are Close to One Half 6 0.6 i.e. 10 46 0.46 i.e. 100 519 0.519 i.e. 1000 Examples of Decimals that are Close to One 9 0.9 i.e. 10 94 0.94 i.e. 100 4 1.004 i.e. 1 1000 Example 1 Order the numbers 0.53, 1.1 and 0.091 from smallest to largest. Answer: 53 0.53 or is close to one half. 100 1 1.1 or 1 is close to one. 10 91 0.091 or is close to zero. 1000 Order from Smallest to Largest: 0.091, 0.53, 1.1 Example 2 Order the numbers 0.952, 0.489, 9 9 , 0.07, and from smallest to largest. 8 16 Answer: The question involves decimals and fractions, but we can use the benchmark strategy with all of these numbers. NSSAL ©2012 6 Draft C. D. Pilmer 952 0.952 or is close to one (slightly less than one). 1000 489 0.489 or is close to one half (slightly less than one half). 1000 9 is close to one (slightly more than one). 8 7 0.07 or is close to zero. 100 9 is close to one half (slightly more than one half). 16 9 9 Order from Smallest to Largest: 0.07, 0.489, , 0.952, 8 16 2. Comparing Digits Start on the left of both numbers and compare corresponding digits. If the digit of one number is larger, then this is the larger decimal number. If the digits are the same, move one place to the right and repeat the procedure. In some cases, you might want to add additional zeros to the decimal number for comparison purposes (e.g. 0.54 = 0.540). Example 3 Which is larger? (a) 1.6 or 1.4 (b) 0.576 or 0.582 (c) 2.95 or 2.9 Answers: (a) Step 1: Start on the left and compare the unit digits Same 1.6 1.4 Step 2: Move one place to the right (to the tenths) and compare the digits Different 1.6 1.4 Step 3: Since the 6 is bigger than the 4, we can conclude that 1.6 is larger than 1.4 NSSAL ©2012 7 Draft C. D. Pilmer (b) Step 1: Unit digits are the same. Same 0.576 0.582 Step 2: Tenths digits are the same. Same 0.576 0.582 Step 3: Hundredths digits are different. Different 0.576 0.582 Step 4: Since the 8 is bigger than the 7, we can conclude that 0.582 is larger than 0.576. (c) Step 1: Add a zero to 2.9 such that both numbers have three digits Step 2: Unit digits are the same. Same 2.95 2.90 Step 3: Tenths digits are the same. Same 2.95 2.90 Step 4: Hundredths digits are different. Different 2.95 2.90 Step 5: Since the 5 is bigger than the 0, we can conclude that 2.95 is larger than 2.9. NSSAL ©2012 8 Draft C. D. Pilmer Questions Do not use a calculator for any of these questions. 1 , or 1. 2 Decimal Closest to: 1. For each of the following decimal numbers, indicate whether it is closer to 0, Decimal Closest to: Decimal Closest to: (a) 0.6 (b) 0.1 (c) 0.9 (d) 0.08 (e) 0.45 (f) 1.01 (g) 0.502 (h) 0.89 (i) 0.12 (j) 0.901 (k) 0.005 (l) 0.486 (m) 0.892 (n) 0.59 (o) 0.092 2. In each case, you are given two numbers. Circle the larger number. You will have to use the benchmark strategy because every question deals with both decimals and fractions. (a) 0.89 1 12 (b) 0.56 7 8 (c) 4 8 0.1 (d) 3 32 0.907 (e) 0.451 3 3 (f) 0.879 5 12 (g) 13 12 0.009 (h) 7 16 0.58 3. In each case, you are given two numbers. Circle the larger number. (a) 0.7 0.3 (b) 0.47 0.52 (c) 0.198 0.192 (d) 1.24 1.04 (e) 2.04 2.4 (f) 0.09 0.078 (g) 3.1 3.098 (h) 0.57 0.507 NSSAL ©2012 9 Draft C. D. Pilmer (i) 5.61 5.618 (j) 2.09 1.98 (k) 7.029 7.08 (l) 12.899 12.988 (m) 0.4 0.409 (n) 31.29 31.3005 (o) 3.01 2.999 (p) 7.5 7.809 (q) 15.35 15.2 (r) 0.75 0.739 (s) 15 16 0.44 (t) 1 8 0.81 (u) 0.51 1 10 (v) 2.04 2 (w) 4 1 2 4.7 (x) 3 11 12 5 6 3.009 4. Place the following numbers by the appropriate arrow on the number line below. 1.4, 1.9, 0.6, 2.7, 0.1, 3.1, 0.8, 2.2, 1.3, 2.5 0 1 2 3 5. Place the following numbers by the appropriate arrow on the number line below. 0.54, 2.89, 2.4, 0.097, 1.46, 1.039, 2.62, 1.75, 3.05, 1.95 0 NSSAL ©2012 1 2 10 3 Draft C. D. Pilmer 6. Place the following numbers by the appropriate arrow on the number line below. 7 1 4 3 2.78, , 3.3, 1.57, 2 , 0.07, 3 , , 1.2, 2.44 10 16 100 8 0 1 2 3 7. Order the following numbers from smallest to largest. (a) 0.9, 1.3, 0.4, 1.6, 0.2 _________, _________, _________, _________, _________ (b) 0.59, 1.23, 0.08, 0.55, 1.14 _________, _________, _________, _________, _________ (c) 0.8, 0.09, 0.52, 1.01, 0.83, 1.1 _________, _________, _________, _________, _________, _________ (d) 0.26, 0.19, 1, 0.98, 0.3, 0.2 _________, _________, _________, _________, _________, _________ (e) 0.2, 0.08, 0.72, 0.006, 0.24, 0.209 _________, _________, _________, _________, _________, _________ (f) 0.64, 0.7, 0.05, 0.78, 0.619, 0.092, 0.4 _________, _________, _________, _________, _________, _________, _________ (g) 0.542, 0.9, 7 1 , 0.85, , 0.862, 0.3 100 2 _________, _________, _________, _________, _________, _________, _________ NSSAL ©2012 11 Draft C. D. Pilmer (h) 1 8 43 , 0.16, , 0.201, 0.6, 1.3, 0.649 8 1000 _________, _________, _________, _________, _________, _________, _________ (i) 0.48, 1.002, 1 5 , 0.509, 1 , 0.4, 1.1 16 8 _________, _________, _________, _________, _________, _________, _________ NSSAL ©2012 12 Draft C. D. Pilmer Rounding Decimals Let's revisit the rounding rules for whole numbers. • Decide which is the last digit to keep • Leave that digit the same if the next digit is less than 5 (this is called rounding down) • However, increase it by 1 if the next digit is 5 or more (this is called rounding up) e.g. e.g. e.g. e.g. When 83 is rounded to the nearest ten, we obtain 80. When 86 is rounded to the nearest ten, we obtain 90. When 1452 is rounded to the nearest hundred, we obtain 1500. When 23 805 is rounded to the nearest thousand, we obtain 24 000. For this course, we will be rounding decimal numbers to: • the nearest tenth (i.e. first decimal place) • the nearest hundredth (i.e. second decimal place) • the nearest thousandth (i.e. third decimal place) The rules for rounding decimal numbers are the same as those for rounding whole numbers; round down if the next digit is less than 5 and round up if the next digit is 5 or more. e.g. 7.36 rounded to the nearest tenth (i.e. first decimal place) is 7.4. e.g. 7.34 rounded to the nearest tenth is 7.3. e.g. 0.5293 rounded to the nearest tenth is 0.5. e.g. 0.5293 rounded to the nearest hundredth (i.e. second decimal place) is 0.53. e.g. 0.5293 rounded to the nearest thousandth (i.e. third decimal place) is 0.529. e.g. 4.196 rounded to the nearest hundredth is 4.20 (not 4.2 ← rounded to nearest tenth) e.g. 2.63 rounded to the nearest whole number is 3. Questions 1. (a) When 14.547 is rounded to the nearest hundredth, we obtain __________. (b) When 14.547 is rounded to the nearest whole number, we obtain __________. (c) When 14.547 is rounded to the nearest tenth, we obtain __________. 2. (a) When 251.93 is rounded to the nearest tenth, we obtain __________. (b) When 251.93 is rounded to the nearest ten, we obtain __________. (c) When 251.93 is rounded to the nearest whole number, we obtain __________. 3. (a) (b) (c) (d) When 7.0648 is rounded to the nearest thousandth, we obtain __________. When 7.0648 is rounded to the nearest whole number, we obtain __________. When 7.0648 is rounded to the nearest tenth, we obtain __________. When 7.0648 is rounded to the nearest hundredth, we obtain __________. 4. (a) (b) (c) (d) (e) When 437.3295 is rounded to the nearest whole number, we obtain __________. When 437.3295 is rounded to the nearest tenth, we obtain __________. When 437.3295 is rounded to the nearest thousandth, we obtain __________. When 437.3295 is rounded to the nearest hundred, we obtain __________. When 437.3295 is rounded to the nearest hundredth, we obtain __________. NSSAL ©2012 13 Draft C. D. Pilmer Equivalent Fractions and Decimals We are aware that decimals are just another way of expressing fractions, and vice versa. However, there are some common equivalent fractions and decimals that we should all know offthe-top-of-our-heads. Fractions with a denominator of 2 1 2 = 0.5 =1 2 2 3 1 = 1= 1.5 2 2 4 =2 2 Notice the resulting sequence: 0.5, 1, 1.5, 2,… 1 (As we go up by with the fractions, the corresponding decimals go up by 0.5) 2 Fractions with a denominator of 3 2 1 = 0.666... = 0.333... 3 3 = 0.6 = 0.3 3 =1 3 4 1 = 1= 1.333... 3 3 = 1.3 Many of these result in repeating decimals. The line above a digit indicates that digit repeats indefinitely. Notice the resulting sequence: 0.3 , 0.6 , 1, 1.3 ,… 1 (As we go up by with the fractions, the corresponding decimals go up by 0.333… or 3 0.3 .) Fractions with a denominator of 4 1 2 = 0.25 = 0.5 4 4 3 = 0.75 4 4 =1 4 Notice the resulting sequence: 0.25, 0.5, 0.75, 1, 1.25,… 1 (As we go up by with the fractions, the corresponding decimals go up by 0.25) 4 Fractions with a denominator of 5 1 2 = 0.4 = 0.2 5 5 3 = 0.6 5 4 = 0.8 5 Notice the resulting sequence: 0.2, 0.4, 0.6, 0.8, 1,… 1 (As we go up by with the fractions, the corresponding decimals go up by 0.2) 5 NSSAL ©2012 14 Draft C. D. Pilmer Example 1: Order the numbers 3 2 1 5 , 0.58, , 0.46, 1.3, , 1 0.962 from smallest to largest. 5 3 20 9 Answer: It is important not to convert all the fractions to their decimals equivalents. Such conversions work in some cases, but not in all. Do not forget to use benchmarks. 3 can be converted to 0.6, its decimal equivalent. 5 2 can be converted to 0.666.. or 0.6 , its decimal equivalent. 3 1 is close to the benchmark 0. 20 5 1 1 is close to (and slightly more than) the benchmark 1 . 2 9 1 5 3 2 , 0.46, 0.58, , , 0.962, 1.3, 1 Appropriate Order: 20 9 5 3 Questions Do not use a calculator to complete any of these questions. 1. For each of the following decimals, state the equivalent fraction or mixed number. (a) 0.75 = (b) 3.5 = (c) 1.8 = (d) 2.3 = (e) 7.2 = (f) 5.6 = 2. For each of the following decimals, state the equivalent fraction or mixed number. 1 2 1 (a) 9 = (b) = (c) 6 = 4 3 5 (d) 7 3 = 4 1 (e) 9 = 2 2 (f) 8 = 3 3. In each case, you are given two numbers. Circle the larger number. (a) 3 5 (c) 1 (e) 4 NSSAL ©2012 0.65 (b) 3 4 1 3 1.09 (d) 2 1 2 2.6 2 3 4.59 (f) 6 4 5 6.78 15 0.875 Draft C. D. Pilmer (g) 2 1 4 2.304 (h) 7 1 5 7.19 4. Place the following numbers by the appropriate arrow on the number line below. 7 1 3 2 1 1 1 , 2.069, , 0.9, 3 1.44, 2 , , 3.37, 2.539, 8 3 5 25 4 5 0 1 2 3 5. Place the following numbers by the appropriate arrow on the number line below. 95 9 2 2 1 , 0.098, , 2.2, 3 , 2.43, 1 , 2.71 1 , 1.87, 3.12, 100 16 5 3 4 0 1 2 3 6. Order the following numbers from smallest to largest. 3 1 11 5 93 (a) 2 , 1 , 1.6, 0.09, , 2.4, 1 , 5 4 100 12 8 _________, _________, _________, _________, _________, _________, _________, _________ 1 9 3 1 41 (b) 1 , 2 , 0.587, 1 , 2 , 1.58, 2.7, 4 16 3 10 100 _________, _________, _________, _________, _________, _________, _________, _________ NSSAL ©2012 16 Draft C. D. Pilmer 1 9 1 19 4 (c) 2 , , 2.011, 0.892, 1 , 2.6, 1 , 4 16 10 20 5 _________, _________, _________, _________, _________, _________, _________, ________ 7. Open-ended Questions (i.e. more than one acceptable answer) Your Answer (a) Provide a decimal number that is between 3.4 and 3.5. (b) Provide a mixed number that is between 2.5 and 2.8. (c) (d) 1 1 and 1 5 3 2 9 Provide a decimal number that is between and 10 3 Provide a decimal number that is between 1 (e) Provide a mixed number that is between 3.1 and 3.3. (f) Provide a decimal number that is between 2 4 and 3.01 5 (Have your instructor check your answers to question 7.) 8. With a classmate, friend, or family member, play at least two rounds of the Connect Four Fraction Decimal Equivalency Game found in the appendix of this resource. Record in the chart below whom you played and who won. Opponent Winner Round #1 Round #2 NSSAL ©2012 17 Draft C. D. Pilmer Introduction to Percent We see percentages everywhere. • The union negotiated a 2% wage increase for this year. • The dress is marked 30% off. • Approximately 70% of the class is female. • Babe Ruth, who played professional baseball from 1914 to 1935, hit a homerun 11.76% of the time at bat. • Candice left a tip of 20% for the exceptional service she received at the restaurant. • The mortgage rate on Lei's condominium is 5.25% per annum. The word percent comes from the Latin phrase per centum, which means "per 100." For example, when one says 13%, it means 13 per 100 and 13 , or by the decimal 0.13 . The can be represented by the fraction 100 area model for this particular percentage is shown on the right; 13 of the 100 equal parts are shaded. Percentages are just another way of expressing fractions or decimals; they all mean the same thing but look slightly different. Percent 7% 23% 89% 109% 16.7% 0.1% 25% 80% 150% NSSAL ©2012 Fraction (or Mixed Number) 7 100 23 100 89 100 109 9 =1 100 100 16.7 167 = 100 1000 0.1 1 = 100 1000 25 1 = 100 4 80 4 = 100 5 150 50 1 = 1= 1 100 100 2 Decimal 0.07 0.23 0.89 1.09 0.167 0.001 0.25 0.80 1.50 18 Draft C. D. Pilmer Converting Percentages to Decimals Simply remove the percent sign and slide the decimal point two places to the left (i.e. divide by 100). Example 1 Convert the following percentages to decimals. (a) 68% (b) 135% (c) 15.9% Answers: Slide decimal point two places to the left Remove percent sign (a) 68% 68. 135. 1.35 Therefore: 135% = 1.35 0.159 Therefore: 15.9% = 0.159 Slide decimal point two places to the left Remove percent sign (c) 15.9% Therefore: 68% = 0.68 Slide decimal point two places to the left Remove percent sign (b) 135% 0.68 15.9 Converting Decimals to Percentages Simply slide the decimal point two places to the right (i.e. multiply by 100) and add the percent sign. Example 2 Convert the following percentages to decimals. (a) 0.52 (b) 2.68 (c) 0.743 Answers: Slide decimal point two places to the right (a) 0.52 Add a percent sign 52. Slide decimal point two places to the right (b) 2.68 NSSAL ©2012 Therefore: 0.52 = 52% 268% Therefore: 2.68 = 268% 74.3% Therefore: 0.743 = 74.3% Add a percent sign 268. Slide decimal point two places to the right (c) 0.743 52% Add a percent sign 74.3 19 Draft C. D. Pilmer Converting Percentages to Fractions Simply drop the percent sign, express as fraction with a denominator of 100, and simplify the fraction if necessary. Example 3 Convert the following percentages to fractions (or mixed numbers). (a) 43% (b) 65% (c) 108% (d) 7.9% (e) 0.6% (f) 216.4% Answers: With questions (d) through (f), we initially have fractions with decimals in them. We do not leave the number in this form. If we multiply the numerator and denominator by 10, we can rectify this problem. 43 (a) 43% = 100 (b) 65% = 65 65 ÷ 5 13 = = 100 100 ÷ 5 20 108 8 8÷4 2 (c) 108% = = 1= 1 = 1 100 100 100 ÷ 4 25 (d) 7.9% = 7.9 7.9 ×10 79 = = 100 100 ×10 1000 (e) 0.6% = 0.6 0.6 ×10 6 6÷2 3 = = = = 100 100 ×10 1000 1000 ÷ 2 500 216.4 16.4 16.4 ×10 164 164 ÷ 4 41 (f) 216.4% = = 2= 2 = 2= 2 = 2 100 1000 100 ×10 1000 1000 ÷ 4 250 Equivalent Fractions, Decimals, and Percentages In the previous section, we examined equivalent fractions and decimals; we are going to expand on this slightly by also including equivalent percentages. Fractions with a denominator of 2 1 2 = 1= 100% = 0.5 = 50% 2 2 3 = 1.5 = 150% 2 4 = 2= 200% 2 Fractions with a denominator of 3 1 2 = 0.3 = 33.3% = 0.6 = 66.6% 3 3 3 = 1= 100% 3 4 = 1.3 = 133.3% 3 Fractions with a denominator of 4 1 2 = 0.5 = 50% = 0.25 = 25% 4 4 NSSAL ©2012 3 = 0.75 = 75% 4 20 4 = 1= 100% 4 Draft C. D. Pilmer Fractions with a denominator of 5 1 2 = 0.4 = 40% = 0.2 = 20% 5 5 3 = 0.6 = 60% 5 4 = 0.8 = 80% 5 Questions Calculators are not permitted for any of these questions. 1. For each of the area models below, supply the corresponding percent, decimal, and fraction. (a) (b) (c) Percent: Percent: Percent: Decimal: Decimal: Decimal: Fraction: Fraction: Fraction: 2. When you are downloading program or application for your digital device, you will often see a bar on your screen indicating what portion of that program or application has been downloaded at that instant. Below, you have been supplied with download bars with shaded portions. In each case, estimate the percentage of the program or application that has been downloaded at that time (i.e. There is a range of acceptable answers.). (a) Percent: _______ (b) Percent: _______ (c) Percent: _______ (d) Percent: _______ (e) Percent: _______ NSSAL ©2012 21 Draft C. D. Pilmer 3. Convert the following percentages to decimals. No work needs to be shown. (a) 79% = _______ (b) 16% = _______ (c) 9% = _______ (d) 145% = _______ (e) 29.4% = _______ (f) 7% = _______ (g) 208% = _______ (h) 81.7% = _______ (i) 4.5% = _______ (j) 0.8% = _______ 4. Convert the following decimals to percentages. No work needs to be shown. (a) 0.19 = _______ (b) 0.48 = _______ (c) 1.73 = _______ (d) 0.692 = _______ (e) 0.06 = _______ (f) 2.09 = _______ (g) 0.073 = _______ (h) 1.548 = _______ (i) 0.002 = _______ (j) 1.7 = _______ 5. Convert the following percentages to fractions (or mixed number). In some cases, the fraction will have to be simplified. (a) 39% (b) 91% (c) 16% (d) 129% (e) 235% (f) 5.1% (g) 4.6% (h) 48.2% (i) 0.4% (j) 320.6% NSSAL ©2012 22 Draft C. D. Pilmer 6. Convert the following fractions or mixed numbers to percentages. No work needs to be shown; it all comes down to remembering equivalent fractions and percentages. 3 2 (a) = (b) 1 = 4 5 1 (c) 3 = 2 (e) 1 (d) 2 = 3 4 = 5 1 (f) 3 = 4 2 (g) 1 = 3 1 (h) 2 = 5 7. Complete the following table of equivalent fractions, decimals and percentages. Percent (a) (c) (e) (g) Fraction 83 100 3 4 5 Decimal Percent (b) Fraction Decimal 67% (d) 5% 0.39 1 (f) 0.719 (h) 3 4 216.3% 8. Of the three percentages supplied, which one makes the most sense in the context of the given situation. Situation It was a fantastic sale item. The price of the item had been reduced by _______. 120% 3% 40% (b) When Jacob renewed his mortgage, he was pleased that the rate of interest had dropped by _______ per annum. 13% 28% 1% (c) With a few more men in the course than woman, we were not surprised when we were told that _______ of the class was comprised of women. 45% 62% 18% Maxine's science teacher was very pleased with Maxine's performance on the test. Her mark was _______, which was the highest mark in the class. 72% 96% 58% Montez was satisfied with the service at the restaurant and therefore left a tip of _______ for the waiter. 15% 45% 2% (a) (d) (e) NSSAL ©2012 23 Available Percentages Draft C. D. Pilmer 9. Below you have been supplied with diagrams of cylindrical containers filled with fluid. Match each of the numbers below with the most appropriate diagram. Place your answers in the boxes below each diagram. This is an estimation activity; no calculations are required. Do not assume that equivalent fractions, decimals, and percentages will be going in the same 4 box. For example 82%, 0.81 and might all go in the same box even though they are not 5 equivalent; they are, however, very close to each other. 2.16 140% 0.8 1 4 7 9 0.52 69% 194% 0.72 2.57 218% 1 3 7 0.91 2 1 6 89% 5 8 51% 7 10 1.93 81% 7 8 0.24 260% 19 20 26% 1.43 1 2 2 1 (a) (b) (c) (d) (e) (f) (g) (h) (i) NSSAL ©2012 24 Draft C. D. Pilmer 10. Circle the two adjoining numbers that are equivalent decimals, fractions, or percentages. 93 100 9.3% 17 1000 1.7% 0.053 93% 0.8 0.17 5.3% 53 100 1 8 100 0.8% 0.008 0.28 0.53 2.75% 13.1% 1.31 131 10 280% 5.03% 503 1000 275% 131 1000 0.131% 2.75 47% 3.3 1 3 34.7% 0.347 0.275% 4.4 4 4.6 25% 47 100 347% 0.3% 3 10 30% 3 100 3 3 2 4 5 2 3 1.387 138.7% 387 100 0.275 2 3 4 11. Use the numbers in the chart below to correctly complete the following statement. There are _____ people in the Sampson family. Of those, ____ are female. That means that percentage of females in this family is _____%, which can also be represented by the fraction _____. The percentage of males in this family is _____%, which can also be represented by the fraction _____. 75 6 3 4 1 4 8 25 12. With a classmate, friend, or family member, play at least two rounds of the Connect Four Fraction Percent Equivalency Game found in the appendix of this resource. Record in the chart below whom you played and who won. Opponent Winner Round #1 Round #2 NSSAL ©2012 25 Draft C. D. Pilmer Comparing Fractions, Decimals and Percentages In this section we will be comparing fractions, decimals and percentages for the purpose of ordering them from smallest to largest. Most learners will tend to leave the decimals as 1 decimals, convert percentages to decimals mentally, and use the benchmarks 0, , and 1 for 2 fractions. This strategy works well in most, but not all, cases. Example Order the following from smallest to largest. 8 1 7 3 107%, , 1.546, 9.8%, , , 0.51, 9 100 16 4 Answer: 107% = 1.07 (slightly larger than the benchmark 1) 8 is slightly less than the benchmark 1. 9 1 1.546 is slightly larger than 1.5 or 1 . 2 9.8% = 0.098 (close to the benchmark 0) 1 = 0.01 (close to the benchmark 0, and smaller than 9.8% or 0.098) 100 7 1 is slightly less than the benchmark . 16 2 1 0.51 is slightly more than the benchmark . 2 3 1 = 0.75, which is half way between the benchmarks and 1. 4 2 Proper Order: 1 7 3 8 , 9.8%, , 0.51, , , 107%, 1.546 100 16 4 9 Questions Do not use a calculator to complete any of these questions. 1. For each of the following, indicate whether it is closer to 0, Closest to: 1 , or 1. 2 Closest to: Closest to: (a) 98% (b) 11% (c) 45% (d) 0.02 (e) 0.899 (f) 0.6 NSSAL ©2012 26 Draft C. D. Pilmer Closest to: Closest to: Closest to: (g) 8 9 (h) 11 20 (i) 1 16 (j) 0.3% (k) 1.05 (l) 102% (m) 13 24 (n) 56.2% (o) 16 15 2. In each case, circle the larger number. (a) 39% 83% (b) 14.7% 14.2% (c) 136% 98% (d) 3.1% 2.99% (e) 9% 0.48 (f) 83% 0.78 (g) 2.3% 0.005 (h) 1 12 0.65 (i) 0.99 105% (j) 1.45 89% (k) 7 8 29% (l) 47% 9 16 (m) 135% 14 15 (n) 1 10 2% (o) 0.198 7 16 (p) 93.5% 3 4 (q) 2 3 30% (r) 1 (s) 0.546 21.5% 4 5 (t) 7.59% 0.48 1 4 (u) 5 8 0.19 81.2% (v) 8.3% 1 100 0.009 1 3 215% 3. Place the following by the appropriate arrow on the number line below. 5 1 1 1 200%, 0.93, 2 , 54.7%, 1.099, 3 , 125%, , 0.422, 2 , 180% 20 4 8 16 0 NSSAL ©2012 1 2 27 3 Draft C. D. Pilmer 4. Place the following by the appropriate arrow on the number line below. 19 3 3 1 , 2.85, 9.7%, 2 , 0.713, 155%, 3.2, , 1.389, 96%, 215% 20 6 4 0 1 2 3 5. Order the following from smallest to largest. (a) 32%, 124%, 0.7%, 91.2%, 5.8% _________, _________, _________, _________, _________ (b) 0.82, 14%, 0.1, 64%, 0.745 _________, _________, _________, _________, _________ (c) 123%, 1.45, 8.2%, 0.61, 57.2% _________, _________, _________, _________, _________ (d) 5 9 , 0.792, 3.8%, 86%, 12 10 _________, _________, _________, _________, _________ (e) 1.96, 68.5%, 0.4, 1 1 , 1 , 20% 32 10 _________, _________, _________, _________, _________, _________ (f) 0.276, 57.6%, 8 31 , 1.1, , 30.2% 16 32 _________, _________, _________, _________, _________, _________ 1 5 (g) 1 , 209%, 0.89, , 0.096, 64.5% 4 8 _________, _________, _________, _________, _________, _________ NSSAL ©2012 28 Draft C. D. Pilmer (h) 0.956, 91%, 6 1 , 50.3%, 0.08, 14 100 _________, _________, _________, _________, _________, _________ 3 7 1 (i) 1 , 28%, 0.9, , 194%, 1.02, 5 7 3 _________, _________, _________, _________, _________, _________, _________ (j) 214%, 0.34, 1 7 7 , 2 , 94.5%, 1 , 1.092 5 8 12 _________, _________, _________, _________, _________, _________, _________ 6. Open-ended Questions (i.e. more than one acceptable answer) Your Answer (a) Provide a percent that is between 34% and 35%. (b) Provide a percent that is between (c) Provide a percent that is between 0.78 and 0.81. (d) Provide a decimal number that is between 7% and 9.3%. (e) Provide a decimal number that is between 8.2 and 8.3. (f) Provide a decimal number that is between 1 (g) Provide a mixed number that is between 145% and 156%. (h) Provide a mixed number that is between 2 (i) Provide a mixed number that is between 3.01 and 3.25 3 2 and . 5 5 1 2 and 1 . 10 10 3 and 3. 4 (Have your instructor check your answers to question 6.) NSSAL ©2012 29 Draft C. D. Pilmer Adding and Subtracting Decimal Numbers In the Whole Number Operations Bridging Unit, we learned how to add multi-digit whole numbers. To accomplish this, we start by stacking the numbers vertically such that corresponding place values line up (e.g. units with units, tens with tens) and add from right to left. If the sum in any corresponding place value is 10 or greater, we regroup (i.e. carry the excess to the next larger place value). e.g. 158 + 265 Answer: Add the Units Add the Tens ↓ ↓ 1 1 Add the Hundreds ↓ 1 1 1 1 5 8 + 2 6 5 1 5 8 + 2 6 5 1 5 8 + 2 6 5 3 2 3 4 2 3 8 units plus 5 units is 13 units. Regroup the 13 to 1 ten and 3 units. 1 ten plus 5 tens plus 6 tens 1 hundred plus 1 hundred is 12 tens. Regroup the 12 plus 2 hundreds is 4 to 1 hundred and 2 tens. hundreds Adding Decimals We follow the same procedure when adding decimal numbers. We start by stacking the numbers vertically such that the corresponding place values line up (i.e. tenths with tenths, hundredths with hundredths, etc.). Again we add from right to left. If the sum in any corresponding place value is 10 or greater, we regroup (i.e. carry the excess to the next larger place value). e.g. 0.67 + 2.84 Answer: Add the Hundredths ↓ 1 6 7 8 5 0 + 2 1 0 + 2 . . 2 7 hundredths plus 5 hundredths is 12 hundredths. Regroup the 12 to 1 tenth and 2 hundredths. NSSAL ©2012 Add the Tenths ↓ 1 . . 6 7 8 5 . 5 2 1 tenth plus 6 tenths plus 8 tenths is 15 tenths. Regroup the 15 to 1 unit and 5 tenths. Transfer down the decimal point. 30 Add the Units ↓ 1 0 + 2 . . 6 7 8 5 3 . 5 2 1 unit plus 2 units is 3 units. The final answer is 3.52. Draft C. D. Pilmer e.g. 0.471 + 4.89 + 0.055 Answer: Add the Thousandths 0 4 + 0 Add the Hundredths ↓ 4 7 1 8 9 0 5 5 . . . 2 0 4 + 0 . . . 4 7 1 8 9 0 5 5 1 6 6 1 thousandth plus 5 thousandths is 6 thousandths. 7 hundredths plus 9 hundredths plus 5 hundredths is 21 hundredths. Regroup the 21 to 2 tenths and 1 hundredth. Add the Tenths ↓ 1 Add the Units ↓ 2 0 4 + 0 . . . ↓ 1 4 7 1 8 9 0 5 5 2 0 4 + 0 . . . 4 7 1 8 9 0 5 5 . 4 1 6 5 . 4 1 6 2 tenths plus 4 tenths plus 8 tenths is 1 unit plus 4 units is 5 units. The final 14 tenths. Regroup the 14 to 1 unit and answer is 5.416. 4 tenths. Transfer down the decimal point. e.g. 2.95 + 14.86 + 0.7 Answer: Add the Hundredths ↓ 9 5 8 6 7 1 2 1 4 + 0 . . . 1 NSSAL ©2012 Add the Tenths ↓ 2 2 1 4 + 0 Add the Units ↓ 1 . . . 2 9 5 8 6 7 . 5 1 31 2 1 4 + 0 1 . . . 9 5 8 6 7 8 . 5 1 Add the Tens ↓ 2 2 1 4 + 0 1 . . . 9 5 8 6 7 1 8 . 5 1 Draft C. D. Pilmer In the Whole Number Operations Bridging Unit, we learned how to subtract multi-digit whole numbers. To accomplish this, we start by stacking the numbers vertically such that corresponding place values line up (e.g. units with units, tens with tens) and subtract from right to left. If the digit being subtracted is larger than the digit from which it is being subtracted, regroup (i.e. borrow) one from the digit in the next larger place value. e.g. 392 - 145 Answer: Subtract the Units 3 − 1 Subtract the Tens ↓ ↓ 8 12 8 12 9 4 2 5 9 4 2 5 4 7 3 − 1 7 We cannot take 5 units from 2 units. Therefore we regroup (i.e. borrow) 1 from the tens. Subtract the Hundreds ↓ 8 tens minus 4 tens is 4 tens. 8 12 9 4 2 5 2 4 7 3 − 1 3 hundreds minus 1 hundred is 2 hundreds. Subtracting Decimals We follow the same procedure when subtracting decimal numbers. We start by stacking the numbers vertically such that the corresponding place values line up (i.e. tenths with tenths, hundredths with hundredths, etc.). Again we work from right to left. If the digit being subtracted is larger than the digit from which it is being subtracted, regroup (i.e. borrow) one from the digit in the next larger place value. e.g. 5.96 - 3.45 Answer: Subtract the Hundredths ↓ 5 − 3 . . 9 6 4 5 1 6 hundredths minus 5 hundredths is 1 hundredth. NSSAL ©2012 Subtract the Tenths ↓ 5 − 3 . . 9 6 4 5 . 5 1 9 tenths minus 4 tenths is 5 tenths. Transfer down the decimal point. 32 Subtract the Units ↓ 5 − 3 . . 9 6 4 5 2 . 5 1 5 units minus 3 units is 2 units. The final answer is 2.51. Draft C. D. Pilmer e.g. 7.63 - 2.18 Answer: Subtract the Hundredths ↓ 7 − 2 . . 5 13 6 1 3 8 Subtract the Tenths ↓ 7 − 2 5 5 13 . . 6 1 3 8 . 4 5 Subtract the Units ↓ 5 13 . . 6 1 3 8 5 . 4 5 7 − 2 We cannot take 8 5 tenths minus 1 tenth is 4 hundredths from 3 tenths. Transfer down the hundredths. Therefore we decimal point. regroup (i.e. borrow) 1 from the tenths. 13 hundredths minus 8 hundredths is 5 hundredths. 7 units minus 2 units is 5 units. The final answer is 5.45. e.g. 40.59 - 12.7 Answer: This question is a little more challenging because in the second step (i.e. subtracting the tenths), we cannot initially regroup (i.e. borrow) from the units because there are zero units. That means we have to regroup from the tens to the units, and then the units to the tenths. Subtract the Hundredths 4 0 − 1 2 . . Subtract the Tenths ↓ 9 ↓ 3 10 5 9 7 4 1 0 2 − 15 . . 9 Subtract the Tens 9 − NSSAL ©2012 9 . 8 9 Subtract the Units ↓ 3 10 4 1 0 2 . . 7 . 8 9 15 5 7 5 7 9 − 33 ↓ 9 3 10 4 1 0 2 . . 5 7 2 7 . 8 9 15 9 Draft C. D. Pilmer e.g. 6.35 - 0.728 Answer: Initially change 6.35 to 6.350; they are equivalent decimals. Subtract the Thousandths ↓ 6 − 0 . . 3 7 4 10 5 2 0 8 Subtract the Hundredths ↓ 6 − 0 . . 3 7 4 10 5 2 0 8 2 2 2 Subtract the Tenths ↓ Subtract the Units ↓ 5 − 6 0 . . 13 4 10 5 3 7 5 2 0 8 6 0 . . 5 . 6 2 2 − . 6 2 2 13 4 10 3 7 5 2 0 8 Questions Do not use a calculator for any of these questions. 1. Complete the indicated operation. Show all your work. (a) 12.72 + 34.16 (b) 62.53 + 7.31 (c) 38.6 + 50.27 (d) 6.423 + 0.39 NSSAL ©2012 34 Draft C. D. Pilmer (e) 6.39 + 35.572 (f) 142.8 + 87.53 (g) 0.265 + 6.81 + 38.7 (h) 7.46 + 0.085 + 0.93 2. Complete the indicated operation. Show all your work. (a) 64.87 - 21.52 (b) 6.95 - 2.91 (c) 3.547 - 1.819 (d) 13.52 - 7.8 (e) 9.28 - 0.415 (f) 7.042 - 0.36 NSSAL ©2012 35 Draft C. D. Pilmer (g) 28.049 - 6.27 (h) 7.406 - 0.85 3. A container filled with water weighs 4.56 kilograms. Once the water is removed, the container weighs 0.89 kilograms. What was the weight of the water that was removed? 4. Jack gained 1.36 kilograms in the first week and 2.06 kilograms in the second week. How much weight did he gain over that two week period? 5. The odometer on Akira's car initially read 23 467.4 kilometers. After driving 825.7 kilometres, what would be the new odometer reading? 6. Montez's time on the 100 metre dash was 10.54 seconds. Hinto's time was 10.92 seconds. How many seconds earlier did Montez arrive at the finishing line as compared to Hinto? NSSAL ©2012 36 Draft C. D. Pilmer Multiplying Decimal Numbers In the Level III Whole Number Operations Unit, we learned three techniques for multiplying whole numbers: traditional algorithm, multiplying using the expanded form, and the lattice method. You chose the method you preferred; the same will apply here in this section. e.g. 67 × 49 Traditional Algorithm 6 7 × 4 9 6 0 3 2 6 8 0 3 2 8 3 Using Expanded Form Lattice Method 60 + 7 × 40 + 9 6 5 4 2 8 2 4 0 carry 1 6 carry 7 3 0 0 0 4 3 9 2 2 2 4 5 8 6 4 8 3 2 8 3 3 3 Multiplying Decimals 1. Multiply the decimals as though they were whole numbers (i.e. initially ignore the decimal points) 2. The decimal point in the product is placed so that the number of decimal places in the product is equal to the sum of the number of decimal places in the factors. Note: When multiplying decimals, you do not need to line up the decimal points, unlike question involving addition and subtraction of decimal numbers. Example 1 Complete the indicated operation. (a) 67 × 4.9 (b) 6.7 × 4.9 (c) 0.67 × 4.9 (d) 0.67 × 0.49 Answers: These questions were chosen because all of their solutions rely on knowing that 67 × 49 = 3283 , which was calculated above. (a) 67 - zero decimal places 4.9 - one decimal place The final answer should have one decimal place (0 + 1 = 1) Therefore: 67 × 4.9 = 328.3 (b) 6.7 - one decimal place 4.9 - one decimal place The final answer should have two decimal places (1 + 1 = 2) Therefore: 6.7 × 4.9 = 32.83 NSSAL ©2012 37 Draft C. D. Pilmer (c) 0.67 - two decimal places 4.9 - one decimal place The final answer should have three decimal places (2 + 1 = 3) Therefore: 0.67 × 4.9 = 3.283 (d) 0.67 - two decimal places 0.49 - two decimal places The final answer should have four decimal places (2 + 2 = 4) Therefore: 0.67 × 0.49 = 0.3283 Example 2 Complete the following operation. Show all your work. 49.7 × 0.53 Answer: Change the question to 497 × 53 ; we will deal with the decimals points in a later step. Again, you can choose one of the three multiplication techniques that you prefer. Traditional Algorithm 4 9 7 × 5 3 1 4 9 1 2 4 8 5 0 2 6 3 4 1 Using Expanded Form Lattice Method 400 + 90 + 7 × 50 + 3 2 1 2 3 4 5 2 0 0 2 7 0 5 0 0 4 9 7 5 1 0 0 0 0 0 2 6 3 4 1 3 1 2 2 6 5 2 2 3 3 4 0 1 1 5 2 7 4 1 1 We will now consider the decimal points. 49.7 - one decimal place 0.53 - two decimal places The final answer should have three decimal places (1 + 2 = 3) Therefore: 49.7 × 0.53 = 26.341 To determine whether the answer is reasonable, round the decimal numbers to numbers that are more manageable. We could round 49.7 to 50, and round 0.53 to 0.5. Since 50 × 0.5 = 25 , then we can assume that the answer of 26.341 is reasonable. NSSAL ©2012 38 Draft C. D. Pilmer Questions Do not use a calculator for any of these questions. 1. Complete the indicated operation. Show all your work. Please note that we have addition, subtraction and multiplication questions in here. (b) 7.43 × 2.6 (a) 3.7 × 6.5 (c) 0.45 × 0.52 (d) 3.4 + 18.92 (e) 1.73 × 4 (f) 0.49 × 9.1 NSSAL ©2012 39 Draft C. D. Pilmer (g) 83.91 - 7.28 (h) 63 × 0.29 (i) 0.453 × 0.78 (j) 90.36 - 12.5 2. For each question, we have provided three possible solutions. Use your estimation skills to determine which of three the correct answer is. We do not want you to work these out on paper or use a calculator. Instead, we want you to round the decimals to numbers that are more manageable, and estimate the final answer in your head. For example, 9.13 × 4.9 could be changed to 9 × 5 , which has a product of 45. You would then look for the answer that is close to 45. (a) 7.08 × 3.2 10.28 22.656 2.2125 (b) 82.53 − 6.89 75.64 89.42 568.6317 (c) 153.6 + 23.7 129.9 194.3 177.3 (d) 4.3 × 89.5 384.85 512.75 319.65 (e) 452.5 + 351.7 924.2 734.2 804.2 (f) 51.3 × 49.68 2548.584 3264.374 1956.924 NSSAL ©2012 40 Draft C. D. Pilmer 3. Six questions are supplied below. You must use your estimation skills to determine which arrow on the number line below best represents the solution to each of the six questions. Question 0.98 × 2.1 5.23 − 4.37 1.1 + 1.97 a 0 Arrow b Question 0.326 + 2.21 2.34 − 1.98 0.49 × 2.88 Arrow d c 1 e 2 f 3 4. Use your estimation skills to match up each question with each answer. Questions (a) 2.93 + 3.208 Answers 32.185 (b) 16.08 - 5.239 311.74 (c) 7.85 × 4.1 21.78 (d) 47.9 + 32.7 6.138 (e) 29.58 - 7.8 126.7 (f) 39.8 × 6.1 80.6 (g) 98.3 + 28.4 1110.9 (h) 409.8 - 98.06 10.841 (i) 52.9 × 21 242.78 5. John's car holds 48.7 litres of gas. If his vehicle can travel 15 kilometres on a litre of gas, how far can it travel on a full tank? 6. Kadeer had $14.15 in his iTunes account. If he purchases a song for $1.29 from iTunes, how much will be left in his account? NSSAL ©2012 41 Draft C. D. Pilmer 7. If Meera makes $12.65 per hour, how much will she make, before deductions, in a 38 hour work week? 8. The shrub is 38.7 cm tall. They expect that it will grow an additional 3.5 cm over the year. What is the expected height of the tree in a year's time? 9. If each plastic pellet weighs 0.58 grams, how much does 45 pellets weigh? NSSAL ©2012 42 Draft C. D. Pilmer Dividing Decimal Numbers In this section, we will show you two ways to solve these types of division question. One technique uses the traditional algorithm; the other uses the partial quotient method. You chose the method you prefer. Dividing a Decimal Number by a Whole Number With a first few questions we are only going to be looking at questions where we are dividing a decimal number by a whole number (e.g. 165.2 ÷ 7 , 4.23 ÷ 9 ). Example 1 Complete the operation 165.2 ÷ 7 . Answer: In these explanations, and the ones that follow, we will be using the terms divisor, quotient, and dividend. These terms have been described in the diagram on the right. For this particular question, the dividend is 165.2, the divisor is 7, and the quotient is the final answer. Traditional Algorithm Do the long division as you would with whole numbers, then place the decimal point in the quotient directly above the decimal point in the dividend. 23.6 7 165.2 -14 25 - 21 42 - 42 0 Therefore: 165.2 ÷ 7 = 23.6 quotient divisor dividend Partial Quotient Method Initially ignore the decimal point and pretend that you are dividing two whole numbers. 236 7 1652 200 1400 252 30 210 42 6 42 0 Now move the decimal point in the quotient, the same number of places and in the same direction as the decimal point in the dividend. In this case, the dividend should be 165.2, where the decimal point is one place to the left. Therefore our quotient should be 23.6; notice that the decimal point is also one place to the left. Therefore: 165.2 ÷ 7 = 23.6 This answer looks reasonable because we know that 140 ÷ 7 = 20 , therefore we would expect that 165.2 ÷ 7 would be a little more than 20. NSSAL ©2012 43 Draft C. D. Pilmer Example 2 Complete the operation 4.23 ÷ 9 . Answer: Traditional Algorithm Do the long division as you would with whole numbers, then place the decimal point in the quotient directly above the decimal point in the dividend. 0.47 9 4.23 - 36 63 - 63 0 Therefore: 4.23 ÷ 9 = 0.47 Partial Quotient Method Initially ignore the decimal point and pretend that you are dividing two whole numbers. 47 9 423 40 360 63 7 63 0 In this case, the dividend should be 4.23, where the decimal point is two places to the left. Therefore our quotient should be 0.47; notice that the decimal point is also two places to the left. Therefore: 4.23 ÷ 9 = 0.47 4.5 1 is equal to or 0.5. 9 2 Therefore we would expect that 4.23 ÷ 9 is slightly less than 0.5. This answer looks reasonable because we know that 4.5 ÷ 9 or Example 3 Match each division question with the appropriate answer. We are not asking you to work these out using paper-and-pencil or a calculator; rather, we are asking you to use your estimation skills. Questions (a) 2090.2 ÷ 7 (b) 51.84 ÷ 6 (c) 315.84 ÷ 8 (d) 2.13 ÷ 4 (e) 467.37 ÷ 9 Answers 39.48 0.5325 298.6 51.93 8.64 Answers: • We know that 2100 ÷ 7 = 300 , therefore 2090.2 ÷ 7 is likely equal to 298.6. • We know that 48 ÷ 6 = 8 , therefore 51.84 ÷ 6 is likely equal to 8.64. • We know that 320 ÷ 8 = 40 , therefore 315.84 ÷ 8 is likely equal to 39.48. 2 • We know that 2 ÷ 4 or is equal to 0.5, therefore 2.13 ÷ 4 likely equals 0.5325. 4 • We know that 450 ÷ 9 = 50 , therefore 467.37 ÷ 9 is likely equal to 51.93. NSSAL ©2012 44 Draft C. D. Pilmer Briefly Revisiting Fractions We know that fractions are one way of expressing the operation of division. 3 = 3÷4 4 e.g. e.g. 9 = 9÷5 5 e.g. 27 = 27 ÷ 100 100 We also know that equivalent fractions can be created by multiplying or dividing the numerator and denominator of a fraction by the same number. 12 12 ÷ 3 4 e.g.= = 15 15 ÷ 3 5 35 35 ÷ 5 7 e.g.= = 20 20 ÷ 5 4 20 20 ÷ 10 2 e.g.= = 30 30 ÷ 10 3 7 7 × 2 14 e.g.= = 8 8 × 2 16 2 2×3 6 e.g. = = 5 5 × 3 15 9 9 ×10 90 e.g.= = 4 4 × 10 40 We will use both of these pieces of knowledge to help us understand the first step in dividing a decimal number by another decimal number. Consider the question 9 ÷ 0.6 . 9 . 0.6 • We could create an equivalent fraction by multiplying the numerator and denominator by 10. 9 9 ×10 90 = = 0.6 0.6 ×10 6 90 • The can be expressed as 90 ÷ 6 . 6 • We have shown that the answer (i.e. quotient) to 9 ÷ 0.6 (or 0.6 9 ) is equal to answer • The question 9 ÷ 0.6 can be expressed as to 90 ÷ 6 (or 6 90 ). Consider the question 8.1 ÷ 0.03 . 8.1 . 0.03 • We could create an equivalent fraction by multiplying the numerator and denominator by 100. 8.1 8.1×100 810 = = 0.03 0.03 ×100 3 810 • The can be expressed as 810 ÷ 3 . 3 • We have shown that the answer (i.e. quotient) to 8.1 ÷ 0.03 (or 0.03 8.1 ) is equal to • The question 8.1 ÷ 0.03 can be expressed as answer to 810 ÷ 3 (or 3 810 ). NSSAL ©2012 45 Draft C. D. Pilmer Consider the question 7.675 ÷ 0.5 . 7.675 . 0.5 • We could create an equivalent fraction by multiplying the numerator and denominator by 10. 7.675 7.675 ×10 76.75 = = 0.5 0.5 ×10 5 76.75 can be expressed as 76.75 ÷ 5 . • The 5 • We have shown that the answer (i.e. quotient) to 7.675 ÷ 0.5 (or 0.5 7.675 ) is equal • The question 7.675 ÷ 0.5 can be expressed as to answer to 76.75 ÷ 5 (or 5 76.75 ). We can take this and apply it to a variety of division questions. e.g. 0.7 3.01 = 3.01 ÷ 0.7 = 3.01 3.01×10 30.1 = = = 7 30.1 0.7 0.7 ×10 7 5 5 ×10 50 e.g. 0.8 5 =÷ 5 0.8 = = = = 8 50 0.8 0.8 ×10 8 e.g. 0.02 9.28 = 9.28 ÷ 0.02 = 9.28 9.28 ×100 928 = = = 2 928 0.02 0.02 ×100 2 e.g. 0.09 0.0387 = 0.0387 ÷ 0.09 = 0.0387 0.0387 ×100 3.87 = = = 9 3.87 0.09 0.09 ×100 9 e.g. 0.004 0.0224 =0.0224 ÷ 0.004 = 0.0224 0.0224 ×1000 22.4 = = =4 22.4 0.004 0.004 ×1000 4 Let's look at all the division questions with equivalent quotients that we have discussed in the last two pages. 0.6 9 = 6 90 0.03 8.1 = 3 810 0.5 7.675 = 5 76.75 0.7 3.01 = 7 30.1 0.8 5 = 8 50 0.02 9.28 = 2 928 0.09 0.0387 = 9 3.87 0.004 0.0224 = 4 22.4 Notice that every case we started with a question where we were dividing by a decimal number, but in the end we had changed the question to one where we were dividing by a whole number. We should be able to solve new question as we have already learned how to divide a decimal by a whole number. So how could we describe the process of changing a division question from one where we are dividing by a decimal number to one where we are dividing by a whole number? We obviously do not want to go through the lengthy process of converting the division question to fraction question, creating an equivalent fraction, and then converting from a fraction question to a NSSAL ©2012 46 Draft C. D. Pilmer division question. Instead, we use the following shortcut. Start by quotient moving the decimal point to the right in the divisor until the divisor is a divisor dividend whole number. Then move the decimal point to the right in the dividend the same number of places as was moved for the divisor. If you move both one place to the right, it is equivalent to multiplying the numerator and denominator of a fraction by 10. If you move both two places to the right, it is equivalent to multiplying the numerator and denominator of a fraction by 100. Dividing a Whole Number or Decimal Number by a Decimal Number Step 1: Move the decimal point to the right in the divisor until the divisor is a whole number. Step 2: Move the decimal point to the right in the dividend the same number of places as was done in Step 1. Step 3: Divide through using the procedure that you prefer for dividing a decimal number by a whole number (i.e. what we did in Examples 1 and 2) Example 4 Complete the operation 21.87 ÷ 0.9 . Answer: Regardless of whether you prefer the traditional algorithm or the partial quotient method, you must start by changing the divisor (0.9) to a whole number. This is accomplished by moving the decimal point one place to the right in the divisor. We must then move the decimal point one place to the right in the dividend. This means that the question changes from 21.87 ÷ 0.9 to 218.7 ÷ 9 Traditional Algorithm Partial Quotient Method 243 24.3 9 218.7 -18 38 - 36 27 - 27 0 If 218.7 ÷ 9 = 24.3 , then 21.87 ÷ 0.9 = 24.3 9 2187 200 1800 387 40 360 27 3 27 0 Now move the decimal point in the quotient, the same number of places and in the same direction as the decimal point in the dividend. If 218.7 ÷ 9 = 24.3 , then 21.87 ÷ 0.9 = 24.3 NSSAL ©2012 47 Draft C. D. Pilmer Example 5 Complete the operation 0.2922 ÷ 0.06 . Answer: Start by changing the divisor (0.06) to a whole number. This is accomplished by moving the decimal point two places to the right in the divisor. We must then move the decimal point two places to the right in the dividend. This means that the question changes from 0.2922 ÷ 0.06 to 29.22 ÷ 6 . Traditional Algorithm Partial Quotient Method 487 4.87 6 29.22 - 24 52 - 48 42 - 42 0 If 29.22 ÷ 6 = 4.87 , then 0.2922 ÷ 0.06 = 4.87 6 2922 400 2400 522 80 480 42 7 42 0 Now move the decimal point in the quotient, the same number of places and in the same direction as the decimal point in the dividend. If 29.22 ÷ 6 = 4.87 , then 0.2922 ÷ 0.06 = 4.87 Example 6 Match each division question with the appropriate answer. In many cases, you may wish to move the decimal points in both divisor and dividend to make the question more manageable. We are not asking you to work these out using paper-and-pencil or a calculator; rather, we are asking you to use your estimation skills. Questions (a) 58.08 ÷ 0.8 (b) 0.3474 ÷ 0.06 (c) 83.72 ÷ 0.4 (d) 880.2 ÷ 9 (e) 0.0336 ÷ 0.07 Answers 209.3 0.48 5.79 72.6 97.8 Answers: • Change 58.08 ÷ 0.8 to 580.8 ÷ 8 . We know that 560 ÷ 8 = 70 , therefore 580.8 ÷ 8 (or 58.08 ÷ 0.8 ) is likely equal to 72.6. • Change 0.3474 ÷ 0.06 to 34.74 ÷ 6 . We know that 36 ÷ 6 = 6 , therefore 34.74 ÷ 6 (or 0.3474 ÷ 0.06 ) is likely equal to 5.79. NSSAL ©2012 48 Draft C. D. Pilmer • • • Change 83.72 ÷ 0.4 to 837.2 ÷ 4 . We know that 800 ÷ 4 = 200 , therefore 837.2 ÷ 4 (or 83.72 ÷ 0.4 ) is likely equal to 209.3 We know that 900 ÷ 9 = 100 , therefore 880.2 ÷ 9 is likely equal to 97.8. 3.5 Change 0.0336 ÷ 0.07 to 3.36 ÷ 7 . We know that 3.5 ÷ 7 or equals 0.5. Therefore 7 it is likely that 3.36 ÷ 7 (or 0.0336 ÷ 0.07 ) is equal to 0.48. Questions Do not use a calculator to complete any of these questions. 1. Complete each of the operations. Show all your work. (a) 32.04 ÷ 6 (b) 240.3 ÷ 9 (c) 2.415 ÷ 5 NSSAL ©2012 (d) 0.651 ÷ 7 49 Draft C. D. Pilmer 2. Match each division question with the appropriate answer. We are not asking you to work these out using paper-and-pencil or a calculator; rather, we are asking you to use your estimation skills. Questions (a) 389.6 ÷ 8 (b) 49.02 ÷ 6 (c) 2.024 ÷ 4 (d) 257.8 ÷ 9 (e) 32.83 ÷ 7 Answers 0.506 48.7 4.69 8.17 28.62 3. In each case, four division questions have been provided. From the last three division questions, circle the one which has the same quotient (i.e. generates the same answer) as the first division question. You do not want you to work any of these out using paper-and-pencil or a calculator. (a) 0.6 45.36 6 4.536 6 453.6 6 4536 (b) 8 7.36 0.8 73.6 0.8 0.736 0.8 736 (c) 0.5 385.6 5 3856 5 38.56 5 3.856 (d) 8 27.345 0.8 273.45 0.8 2734.5 0.8 2.7345 (e) 0.5 49 5 4.9 5 490 5 0.49 (f) 0.06 8.2 6 820 6 82 6 0.082 (g) 3 182.6 0.03 18260 0.03 18.26 0.03 1.826 (h) 0.07 58.2 7 582 7 5820 7 0.582 4. Complete the following operations. Show all your work. Please note that we have also included a few addition, subtraction, and multiplication questions. (a) 4.48 ÷ 0.8 (b) 2.58 ÷ 0.03 NSSAL ©2012 50 Draft C. D. Pilmer (c) 7.59 + 12.8 (d) 66.01 ÷ 0.7 (e) 4.7 × 6.8 (f) 3.4 ÷ 0.05 (g) 0.2616 ÷ 0.3 (h) 183.2 − 59.16 NSSAL ©2012 51 Draft C. D. Pilmer 5. Match each question with the appropriate answer. Note that we have included addition, subtraction, multiplication, and division questions. With some of the division questions, you may wish to move the decimal points in both divisor and dividend to make the question more manageable. We are not asking you to work these out using paper-and-pencil or a calculator; rather, we are asking you to use your estimation skills. Questions (a) 639.1 ÷ 7 (b) 289.4 + 315.7 (c) 23.76 ÷ 0.6 (d) 20.5 × 61.8 (e) 0.24 ÷ 0.5 (f) 453.6 − 198.8 (g) 0.496 ÷ 0.08 (h) 0.9 ×135.6 (i) 32.68 ÷ 0.4 Answers 39.6 122.04 0.48 91.3 254.8 6.2 817 1266.9 605.1 6. Six questions are supplied below. You must use your estimation skills to determine which arrow on the number line below best represents the solution to each of the six questions. Question 6.65 ÷ 7 1.49 + 1.04 0.145 ÷ 0.05 a 0 Arrow b Question 3.05 − 2.97 0.31× 5.2 1.278 ÷ 0.6 Arrow c 1 d 2 e f 3 7. The prize money of $169.80 has to be shared equally by 6 people. How much money does each person get? NSSAL ©2012 52 Draft C. D. Pilmer 8. Hamid uses 0.2 kg of ground beef when making a single hamburger patty. How many patties can he make using 3.64 kg of ground beef? 9. Ryan makes $13.50 per hour. How much will he make, before deductions, if he works 6.5 hours? 10. Tylena paid $5.79 for 0.6 kg of meat. How much would one kilogram of the meat cost? 11. Jessie cycled 85.3 km on day one, 93.6 km on day two, and 78.8 km on day three. How far did she cycle in that three day period? NSSAL ©2012 53 Draft C. D. Pilmer Estimation Questions Involving Percentages We use percentages every day when we work out the price of an item after taxes, determine the sale price of an item, and calculate the tip for your waiter or waitress. In many cases, we use our estimation skills when addressing these real-life situations. • In Nova Scotia, when you purchase most items, you have to pay a 15% harmonized sales tax on those goods. For example, a bedroom suite advertised at $1395 will be subject to sales tax. It is important that you be able to estimate the tax on that purchase and the total cost of the purchase. • A discount is a reduction in a price. When a discount on an item is offered, the rate of discount is often advertised as a percent of the regular price. For example a sofa, regularly priced at $799, may be advertised as 25% off during a particular sale. It is important to be able to estimate the cost of the sofa after the discount so that you are not overcharged for that item. • When you go out to a restaurant for a meal, you are expected to tip the waiter or waitress for good service. Typically people tip between 15% (good service) and 20% (exceptional service). It is important that you be able to mentally calculate these tips so that the waiter or waiter receives appropriate amount for their level of service. Below we solve a variety of estimation questions involving percentages. As with any estimation question, there are a variety of ways of obtaining a reasonable estimate. In our solutions, we have only provided one reasonable estimate. We have tried to use the most common approach in each case, but we recognize that there are other perfectly acceptable techniques. Example 1 Your bill at a local restaurant is $68.95. The waitress offered exceptional service and you decide to give a tip of approximately 20%. How much money should she receive? Answer: • Round $68.95 to $70. • We know that 10% of $70 is $7. • Therefore 20% of $70 is $14. • The tip for the waitress should be approximately $14. Example 2 Jorell needs to purchase a new mattress for his bed. It costs $795 before taxes. He has to pay 15% tax. Approximately how much will he pay in taxes, and what will be the approximate cost of this purchase? Answer: • Round $795 to $800 • We know that 10% of $800 is $80, and that 20% of $800 is $160. • Therefore 15% of $800 would have to be half way between $80 and $160. That means the tax on the mattress would be approximately $120. • That means that the total cost who be slightly less than $920 ($800 + $120). NSSAL ©2012 54 Draft C. D. Pilmer Example 3 Lei is going to purchase a sofa that is regularly priced at $1195. Today the sofa is marked down by 30%. What is the approximate sales price of this item? Answer: • Round $1195 to $1200. • If the price is reduced by 30%, then 70% of the price is retained. • If 10% of $1200 is $120, then 70% of $1200 is found by multiplying $120 by 7. • Since 120 × 7 = 840 , then we can conclude that the sale price of the sofa is approximately $840. Example 4 The new jeans, Nasrin is interested in, regularly cost $79. Today they are marked down by 25%. How much will she pay approximately for these jeans including the 15% sales tax? Answer: • Round $79 to $80. • 25% off is the same as one-quarter off. 1 • One-quarter i.e. of $80 is $20. 4 • If the price is reduced by $20, then the approximate sale price of the jeans is $60. • Now we need to determine the 15% sales tax. We know that 10% of $60 is $6, and that 20% of $60 is $12. Therefore 15% of $60 will be halfway between $6 and $12. The sales tax will be approximately $15 on this item. • The total cost of the jeans will be approximately $75 ($60 + $15). Questions Do not use a calculator on any of these questions. 1. Solve each of the following. No work needs to be shown (i.e. Do it in your head.). (a) What is 10% of 40? _______ (b) What is 10% of 120? _______ (c) What is 10% of 500? _______ (d) What is 10% of 1400? _______ (e) What is 20% of 40? _______ (f) What is 20% of 120? _______ (g) What is 20% of 500? _______ (h) What is 20% of 1400? _______ (i) What is 15% of 40? _______ (j) What is 15% of 120? _______ (k) What is 15% of 500? _______ (l) What is 15% of 1400? _______ (m) What is 30% of 40? _______ (n) What is 30% of 120? _______ (o) What is 25% of 400? _______ (p) What is 25% of 1200? _______ NSSAL ©2012 55 Draft C. D. Pilmer 2. Manish's bill at a local restaurant is $49.45. The waiter offered exceptional service so Manish decides to give a tip of approximately 20%. How much money should the waiter receive? 3. Krys is purchasing a fall jacket for her son. It costs $39.95. Approximately how much will she have to pay after taxes (15%) for this item? 4. Ryan is purchasing a DVD boxed set of Season 13 of The Simpsons. It normally sells for $29.95 but today it is marked down by 30%. What is the approximate sale price of this item? 5. Alice received satisfactory service at the restaurant and therefore felt it was reasonable to leave a 15% tip on her $81.35 bill. Approximately how much should she leave? 6. All spring stock was marked down by 40% in a local clothing store. Approximately how much would one pay, after taxes (15%), for a spring dress regularly costing $59.95? 7. The $160 electronic device was marked down by 25% because a newer model of the same device was now on the market. Approximately how much will it cost for this discounted device after the paying sales tax (15%)? NSSAL ©2012 56 Draft C. D. Pilmer 8. The demand for a particular running shoe was much higher than the manufacturer expected. They were originally going to sell the shoes for $89.95 a pair. The manufacturer decides to increase the price by 20%. If they do this, what would be approximate new cost of the shoes before taxes? 9. The regular price of a season's pass for skiing is $295. If you purchase the pass early, you can save 30%. What is the approximate total cost, after taxes (15%), if you purchase this early-bird season's pass? 10. Eight questions are supplied below. You must use your estimation skills to determine which arrow on the number line below best represents the solution to each of the eight questions. Question 25% of 11.90 0.784 ÷ 0.4 1.43 − 1.316 30% of 4.90 a 0 Arrow c b Question 0.52 + 0.496 10% of 21.50 20% of 2.99 2.1×1.513 d Arrow e 1 f 2 g h 3 11. With a classmate, friend, or family member, play at least two rounds of the Connect Four Percent Game found in the appendix of this resource. Record in the chart below whom you played and who won. Opponent Winner Round #1 Round #2 NSSAL ©2012 57 Draft C. D. Pilmer Calculator Questions This is one of the few times in this course where we will allow you to use a calculator to solve problems. Our rationale is that you should know all the fundamentals concerning decimals and percentages at this point in time, and that we now want to expose you to multi-step problems with "messier" numbers that are better handled with a calculator, as opposed to using paper-and-pencil techniques. Example 1 John was born on July 8, 1955. In an attempt to get his twelve grandchildren to remember his birthday, John gives each child $78.55 cash at Christmas. How much money should he take out of his account to cover his grandchildren's gifts? Answer: • Simply multiply 12 by 78.55 on the calculator. • John needs to take out $942.60 to cover the gifts. Example 2 Suzzette has a 53.5 litre container of water. She wants to know how many 2.45 litre containers she can completely fill using the larger container of water. Answer: • Using a calculator: 53.5 ÷ 2.45 = 21.8 (rounded to one decimal place) • Normally we would round 21.8 up to 22, but this question is asking us how many "containers she can completely fill." For this reason, we will round down and say that she can completely fill 21 containers. Example 3 In week one, Carissa's expenses were $496.65, and her earnings were $757.50. The money not spent went into her savings. In week two she hopes to save twice as much money as week one. If she is able to do this, how much money will go into her savings in week two? Answer: • Her Savings on Week One: 757.50 - 496.65 = $260.85 • Her Desired Savings on Week Two: 2 × 260.85 = $521.70 Example 4 Barb is purchasing at shirt priced at $18.95, a pair of jeans at $46.95, and a knapsack at $39.95. What is the total cost after tax (15%)? Answer: • Find the total before tax. 18.95 + 46.95 + 39.95 = 105.85 • Determine the tax. 15% of 105.85 = 0.15 ×105.85 = 15.88 (rounded to the second decimal point) • Find the total after tax. 105.85 + 15.88 = $121.73 NSSAL ©2012 58 Draft C. D. Pilmer Example 5 At one particular store, customers can receive a 30% discount if their purchases before the discount total $150 or more. Shelly plans on purchasing a $89.45 set of bath towels and facecloths, a $49.95 set of blinds for her bedroom window, and a $17.45 toaster. What is her total cost after tax (15%)? Answer: • Find the total before taxes. 89.95 + 49.95 + 17.45 = 157.35 • Since their total purchase exceeds $150, they are able to receive the 30% discount. • If the price is reduced by 30%, then 70% of the price is retained. Take 70% of $157.35 to find the new total (before taxes). 70% of 157.35 = 0.70 ×157.35 = 110.15 (rounded off) • Determine the tax. 15% of 110.15 = 0.15 ×110.15 = 16.52 (rounded off) • Find the total after tax. 110.15 + 16.52 = $126.67 Example 6 Meera works 47 hours this week. She gets $15.60 per hour for the first 40 hours. She gets "time-and-a-half" for any hours after the 40 hours; this is considered overtime. How much will she earn, before deductions, for this work week? Answer: • Earnings for the First 40 Hours of Work: 40 ×15.60 = $624 • Hourly Earnings at Time-and-a-Half: 1.5 ×15.60 = $23.40 per hour • Earnings for the 7 Hours of Overtime: 7 × 23.40 = $163.80 • Total Earnings: 624 + 163.80 = $787.80 Example 7 Nashi makes $13.20 per hour plus a 2.5% commission on all her sales. If she works 36 hours and makes $6490 worth of sales, what will be her earnings before deductions? Answer: • Earnings from Hourly Wage: 36 ×13.20 = $475.20 • Commission Earnings: 2.5% of $6490 = 0.025 × 6490 = $162.25 • Total Earnings: 475.20 + 162.25 = $637.45 NSSAL ©2012 59 Draft C. D. Pilmer Questions Calculators are permitted with these questions. Show how you solved the each question. 1. If a long-distance phone provider offers a rate of $0.12 per minute, how long can you talk for $2.76? 2. Suppose it costs $27.50 per day plus $0.11 per kilometer for a rental car. What is the total bill if you have the car for three days and travel 657 kilometres? 3. Masato purchases a loft of bread ($2.65), a can of beans ($2.29) and hot dogs ($3.89). If he pays with a $10 bill, how much change will he receive? Please note that sales tax is not applied to food. 4. The garden center marked all plants down by 60% for their end-of-season sale. How much would you have to pay after taxes (15%) for a plant that normally cost $129.99? 5. Rana purchases a sweat top ($18.99), jeans ($34.99), and running shoes ($37.99) for each of her twin boys. What will be the total cost after paying the sales tax (15%)? 6. Last week Meera was making $13.40 per hour and working 36 works. This week her hours increased to 42 hours and her hourly wage increased to $14.70 per hour. How much will she make, before deductions, over this two week period? NSSAL ©2012 60 Draft C. D. Pilmer 7. Kendrick makes $12.50 per hour plus a 3% commission on all his sales. If he worked 38 hours and sold $5840 worth of merchandise this week, what would be his earnings before deductions? 8. Harris, a pipefitter working in oil project in northern Alberta, has to work a 12 hour shift on a statutory holiday. For doing so, his employer will pay him time-and-a-half for the first 8 hours and double-time for the remaining 4 hours. If his normal hourly rate is $32.60, how much will he make, before deductions, for this 12 hour shift? 9. The Boxing Day sale at a local clothing boutique advertised 40% off all purchases. Kimi wanted to purchase a blouse, regularly priced at $29.95, and a sweater, regularly priced at $37.59. If she purchases both during the sale, what is the total cost including sales tax (15%)? 10. Kevin travelled to friend's cottage using his car. The car's odometer initially read 33 407.2 kilometres. Upon arriving at the cottage, the odometer read 33 598.9 kilometres. If he used 13.5 litres of fuel during the trip, how many kilometres per litre did his car achieve on this trip? NSSAL ©2012 61 Draft C. D. Pilmer Appendix NSSAL ©2012 62 Draft C. D. Pilmer Connect Four Fraction Decimal Equivalency Game Number of Players: Two Objective: The winner is the first player to connect four of his/her pieces horizontally, vertically or diagonally. Instructions: 1. Roll a die to see which player will go first. 2. The first player looks at the board and decides which square he/she wishes to capture. The square with a specified decimal is captured by creating the equivalent fraction using the numerator and denominator strips at the bottom of the page. One paper clip is placed on each strip to do so. For example, if one chooses 3 on the numerator strip and 4 on the 3 denominator, then they can capture one square labeled 0.75 ( is equivalent to 0.75). They 4 either mark the square with an X or place a colored counter on the square. There may be other squares with that same difference but only one square can be captured at a time. 3. Now the second player is ready to capture a square but he/she can only move one of the paperclips. They then mark the square with the equivalent decimal using an O or a different colored marker. If a player cannot move a single paperclip to capture a square, a paperclip must still be moved in order to ensure that the game can continue. 4. Play alternates until one player connects four squares. Remember that only one player clip is moved at a time. If none of the players is able to connect four, then the winner is the individual who has captured the most squares. Game Board: 0.4 1 0.2 0.4 1 0.5 0.6 0.2 0.4 0.25 0.3 0.4 0.25 0.1 0.75 0.8 0.3 0.6 0.75 0.2 0.3 0.5 0.2 1 0.2 0.8 0.25 0.1 0.8 0.4 0.1 0.5 0.3 0.75 0.6 1 Numerator (Top) Strip: 1 NSSAL ©2012 2 3 Denominator (Bottom) Strip: 4 4 63 5 10 Draft C. D. Pilmer Connect Four Fraction Percent Equivalency Game Number of Players: Two Objective: The winner is the first player to connect four of his/her pieces horizontally, vertically or diagonally. Instructions: 1. Roll a die to see which player will go first. 2. The first player looks at the board and decides which square he/she wishes to capture. The square with a specified percent is captured by creating the equivalent fraction using the numerator and denominator strips at the bottom of the page. One paper clip is placed on each strip to do so. For example, if one chooses 3 on the numerator strip and 4 on the 3 denominator, then they can capture one square labeled 75% ( is equivalent to 75%). They 4 either mark the square with an X or place a colored counter on the square. There may be other squares with that same difference but only one square can be captured at a time. 3. Now the second player is ready to capture a square but he/she can only move one of the paperclips. They then mark the square with the equivalent decimal using an O or a different colored marker. If a player cannot move a single paperclip to capture a square, a paperclip must still be moved in order to ensure that the game can continue. 4. Play alternates until one player connects four squares. Remember that only one player clip is moved at a time. If none of the players is able to connect four, then the winner is the individual who has captured the most squares. Game Board: 40% 10% 20% 100% 40% 50% 25% 100% 25% 80% 60% 20% 30% 60% 40% 50% 30% 40% 75% 20% 30% 25% 80% 100% 20% 80% 75% 10% 20% 40% 10% 50% 100% 60% 30% 75% Numerator (Top) Strip: 1 NSSAL ©2012 2 3 Denominator (Bottom) Strip: 4 4 64 5 10 Draft C. D. Pilmer Connect Four Percentage Game Number of Players: Two Objective: The winner is the first player to connect four of his/her pieces horizontally, vertically or diagonally. Instructions: 1. Roll a die to see which player will go first. 2. The first player looks at the board and decides which square he/she wishes to capture. They place two paperclips on two strips below; one on the "Percentage" strip and one on the "Of" strip. Take the percentage of that number and capture the appropriate square (e.g. 20% of 40 allows one to capture an "8" square). They either mark the square with an X or place a colored counter on the square. There may be other squares with that same value but only one square can be captured at a time. 3. Now the second player is ready to capture a square but he/she can only move one of the paperclips. They then mark the square with that value using an O or a different colored marker. If a player cannot move a single paperclip to capture a square, a paperclip must still be moved in order to ensure that the game can continue. 4. Play alternates until one player connects four squares. Remember that only one paperclip is moved at a time. If none of the players is able to connect four, then the winner is the individual who has captured the most squares. Game Board: 10 16 10 12 8 20 30 8 3 24 15 10 2 5 18 4 25 30 25 20 10 6 16 8 6 4 12 2 3 5 18 24 15 20 12 4 Percentage: Of: 10% 15% 20% 25% NSSAL ©2012 20 65 40 80 100 120 Draft C. D. Pilmer Additional Practice; Ordering Decimals This is an optional section of material. 1 , or 1. 2 Decimal Closest to: 1. For each of the following decimal numbers, indicate whether it is closer to 0, Decimal Closest to: Decimal Closest to: (a) 0.56 (b) 0.93 (c) 0.07 (d) 1.008 (e) 0.4 (f) 0.19 (g) 0.897 (h) 0.054 (i) 0.089 (j) 0.106 (k) 0.61 (l) 0.403 2. In each case, you are given two numbers. Circle the larger number. (a) 0.2 0.8 (b) 0.38 0.92 (c) 0.867 0.263 (d) 3.14 3.64 (e) 2.7 2.5 (f) 4.07 4.12 (g) 3.459 3.624 (h) 1.908 1.98 (i) 3.001 2.85 (j) 6.13 6.112 (k) 8.3 8.408 (l) 1.89 1.982 (m) 0.52 0.509 (n) 3.59 3.505 (o) 4.01 4.099 (p) 9.056 9.06 (q) 17.45 17.2 (r) 5.856 5.9 (s) 3.298 3.41 (t) 4.61 4.3 (u) 8.5 8.601 (v) 3.04 3.009 3. In each case, you are given three numbers. Circle the largest number. (a) 0.408 0.4 0.431 (b) 2.6 2.58 2.098 (c) 7.4 7.59 7.329 (d) 4.02 4.2 4.002 NSSAL ©2012 66 Draft C. D. Pilmer 4. Place the following numbers by the appropriate arrow on the number line below. 2.16, 1.36, 1.63, 0.45, 3.08, 0.72, 2.82, 1.91, 0.14 2.67, 0.87 0 1 2 3 5. Place the following numbers by the appropriate arrow on the number line below. 1.183, 0.91, 2.462, 1.45, 0.076, 0.33, 3.278, 0.6, 2.8, 3.07, 1.782, 2.2 0 1 2 3 6. Order the following numbers from smallest to largest. (a) 1.5, 1.3, 0.8, 2.1, 0.6, 1.9 _________, _________, _________, _________, _________, _________ (b) 2.06, 1.45, 1.39, 0.73, 2.41, 0.09 _________, _________, _________, _________, _________, _________ (c) 0.907, 1.6, 0.34, 1.241, 0.7, 0.93, 1.78 _________, _________, _________, _________, _________, _________, _________ (d) 1.041, 0.4, 1.9, 0.28, 1.372, 1.83, 0.563 _________, _________, _________, _________, _________, _________, _________ NSSAL ©2012 67 Draft C. D. Pilmer Additional Practice; Ordering Decimals and Fractions This is an optional section of material. 1 , or 1. 2 Decimal Closest to: 1. For each of the following decimal numbers, indicate whether it is closer to 0, Decimal Closest to: Decimal Closest to: (a) 0.04 (b) 19 20 (c) 0.913 (d) 7 12 (e) 0.521 (f) 1 30 (g) 0.8 (h) 15 32 (i) 0.48 (j) 1.003 (k) 0.1 (l) 33 32 2. In each case, you are given two numbers. Circle the larger number. (a) 1 16 0.45 (b) 7 8 5 11 (c) 0.178 0.26 (d) 8 14 0.9 (e) 0.56 7 16 (f) 0.189 15 16 (g) 3 1 8 3.6 (h) 1 3 5 1.17 (i) 2.592 2 1 6 (j) 5 3 7 5.625 (k) 4 4.08 (l) 6.4 6 1 32 (m) 17 16 0.98 (n) 1.48 1 11 20 9 16 3. In each case, you are given three numbers. Circle the largest number. (a) 0.56 11 12 (c) 5.628 5 NSSAL ©2012 4 9 0.095 (b) 2.83 2 1 8 2 9 20 5.91 (d) 4.06 4.306 4 6 10 68 Draft C. D. Pilmer 4. Place the following numbers by the appropriate arrow on the number line below. 1 15 10 1 9 , 3.052, 1 , 2.8, 1.183, 3 , 0.1, 2 2 , 0.68, 1.4, 5 16 20 6 16 0 1 2 3 5. Order the following numbers from smallest to largest. 9 3 5 , 0.068, (a) 0.92, , 8 10 8 _________, _________, _________, _________, _________ 3 5 (b) 1.1, 0.427, 1.83, 1 , 7 9 _________, _________, _________, _________, _________ (c) 1 5 3 13 , 0.9, , 1 , 1.042, 0.78 12 50 16 _________, _________, _________, _________, _________, _________ (d) 0.17, 1 3 1 8 , , 1.7, , 0.905 25 100 15 _________, _________, _________, _________, _________, _________ (e) 0.8, 1.908, 1 7 1 , 0.91, 0.064, 1.15, 4 12 _________, _________, _________, _________, _________, _________, _________ (f) 3.171, 2 29 9 39 , 3 , 2.5, 3.26, 2 , 3.8 20 32 1000 _________, _________, _________, _________, _________, _________, _________ NSSAL ©2012 69 Draft C. D. Pilmer Additional Practice; Ordering Decimals, Fractions and Percentages This is an optional section of material. 1 , or 1. 2 Decimal Closest to: 1 (c) 18 1. For each of the following decimal numbers, indicate whether it is closer to 0, Decimal (a) Closest to: Decimal 16% 9 17 7 6 (d) (g) Closest to: (b) 0.842 (e) 98% (f) 47% (h) 8.3% (i) 0.05 (j) 52.6% (k) 27 1000 (l) 106% (m) 12.5% (n) 0.91 (o) 23 50 2. In each case, you are given two numbers. Circle the larger number. (a) 49.2% 53% (b) 13% 3 4 (c) 7 14 0.95 (d) 0.6 38% (e) 58.3% 1 8 (f) 0.09 13 16 (g) 11% 9 20 (h) 0.939 102% (i) 25% 1 16 (j) 2 6 7 2.57 (k) 78.9% 2.8 (l) 1 1 8 145% (m) 23 21 98.5% (n) 236% 2.1 3. In each case, you are given three numbers. Circle the largest number. (a) NSSAL ©2012 57% 1 12 0.357 (b) 70 1.3 1 7 9 119% Draft C. D. Pilmer 4. Place the following numbers by the appropriate arrow on the number line below. 91 1 3 11 28%, 2 , 95% , 1 , 28%, , 1.7, 140%, 0.08, 2.683, 1 , 216.5% 100 10 6 12 0 1 2 3 5. Order the following numbers from smallest to largest. 5 (a) 0.9, , 4%, 87.5%, 0.38 8 _________, _________, _________, _________, _________ (b) 0.529, 7 17 , 16%, 101%, 100 20 _________, _________, _________, _________, _________ (c) 61%, 0.003, 1 9 , 28.5%, , 0.96 10 6 _________, _________, _________, _________, _________, _________ (d) 153%, 1 1 , 0.72, 1.25, 209.5%, , 1.8 32 _________, _________, _________, _________, _________, _________ 4 47 3 , 1.1 , 2 , 250%, 63%, 187% (e) 1 , 10 7 100 _________, _________, _________, _________, _________, _________, _________ (f) 109%, 173%, 86.9%, 1.6, 1 31 13 , , 0.378 100 25 _________, _________, _________, _________, _________, _________, _________ NSSAL ©2012 71 Draft C. D. Pilmer Post-Unit Reflections What is the most valuable or important thing you learned in this unit? What part did you find most interesting or enjoyable? What was the most challenging part, and how did you respond to this challenge? How did you feel about this math topic when you started this unit? How do you feel about this math topic now? Of the skills you used in this unit, which is your strongest skill? What skill(s) do you feel you need to improve, and how will you improve them? How does what you learned in this unit fit with your personal goals? NSSAL ©2012 72 Draft C. D. Pilmer Soft Skills Rubric Look back over the module you have just completed and assess yourself using the following rubric. Use pencil or pen and put a checkmark in the column that you think best describes your competency for each description. I will look at how accurately you have done this and will discuss with you any areas for improvement. You will be better prepared for your next step, whether it is work or further education, if you are competent in these areas by the end of the course. Keep all of these rubrics in one place and check for improvement as you progress through the course. Date: Competent demonstrates the concept fully and consistently Throughout this module, I… Approaching Competency Developing Competency demonstrates the concept most of the time demonstrates the concept some of the time • Attended every class • Let my instructor know if not able to attend class • Arrived on time for class • Took necessary materials to class • Used appropriate language for class • Used class time effectively • Sustained commitment throughout the module • Persevered with tasks despite difficulties • Asked for help when needed • Offered support/help to others • Helped to maintain a positive classroom environment • Completed the module according to negotiated timeline • Worked effectively without close supervision Comments: (Created by Alice Veenema, Kingstec Campus) NSSAL ©2012 73 Draft C. D. Pilmer Answers Introduction to Decimals (pages 1 to 5) 1. (a) 0.5 (d) 0.41 (b) 0.3 (e) 0.7 (c) 0.13 (f) 0.29 2. 5 10 0.07 0.5 7 100 37 1000 9 100 604 1000 3. (a) 0.256 (d) 1.097 45 100 8 (d) 1 100 4. (a) 5. (a) (b) (c) (d) (e) (f) (g) (h) 0.046 1.8 0.037 1.26 7 10 3 2.7 3 1 0.09 2 8 10 46 1000 9 100 3.09 3.604 1 26 100 56 1000 0.056 (b) 0.06 (e) 2.007 (b) (c) 3.9 (f) 13.58 4 10 (e) 2 508 1000 59 (f) 6 1000 (c) 4 3 1000 35.6 7.09 0.58 1000.15 206.309 70.1 5.037 400.029 NSSAL ©2012 74 Draft C. D. Pilmer 6. Fraction Decimal Numerals Words 8 32 32.8 thirty-two and eight tenths 10 (a) (b) 0.472 472 1000 (c) 13.067 13 (d) 7.59 7 (e) 327.09 327 7. (a) 40 + 2 + 0.8 (b) 9 + 0.3 + 0.01 four hundred seventy-two thousandths 67 1000 59 100 seven and fifty-nine hundredths 9 100 and and thirteen and sixty-seven thousandths three hundred twenty-seven and nine hundredths ( 4 ×10 ) + ( 2 ×1) + 8 × 1 10 1 1 ( 9 ×1) + 3 × + 1× 10 100 1 1 1 + 2× + 9× 10 100 1000 1 1 (d) 10 + 8 + 0.03 + 0.004 or (1×10 ) + ( 8 ×1) + 3 × + 4× 100 1000 1 (e) 4000 + 200 + 9 + 0.07 or ( 4 ×1000 ) + ( 2 ×100 ) + ( 9 ×1) + 7 × 100 (c) 300 + 2 + 0.4 + 0.02 + 0.009 and ( 3 ×100 ) + ( 2 ×1) + 4 × Comparing Decimals (pages 6 to 12) 1. (a) 1 2 (b) 0 (c) 1 1 2 (f) 1 1 2 (h) 1 (i) 0 (j) 1 (k) 0 (l) (m) 1 (n) (d) 0 (g) NSSAL ©2012 (e) 1 2 1 2 (o) 0 75 Draft C. D. Pilmer 2. (a) (c) (e) (g) 3. (a) (c) (e) (g) (i) (k) (m) (o) (q) (s) (u) (b) 7 8 (d) 0.907 (f) 0.879 (h) 0.58 0.7 0.198 2.4 3.1 5.618 7.08 0.409 3.01 15.35 15 16 (b) (d) (f) (h) (j) (l) (n) (p) (r) 0.52 1.24 0.09 0.57 2.09 12.988 31.3005 7.809 0.75 (t) 0.81 0.51 (v) 0.89 4 8 3 3 13 12 (w) 4.7 (x) 5 6 11 3 12 2 4. The numbers should be placed along the number line in the following order. 0.1, 0.6, 0.8, 1.3, 1.4, 1.9, 2.2, 2.5, 2.7, 3.1 5. The numbers should be placed along the number line in the following order. 0.097, 0.54, 1.039, 1.46, 1.75, 1.95, 2.4, 2.62, 2.89, 3.05 6. The numbers should be placed along the number line in the following order. 3 7 1 4 0.07, , , 1.2, 1.57, 2 , 2.44 2.78, 3 , 3.3 8 10 16 100 7. (a) 0.2, 0.4, 0.9, 1.3, 1.6 (b) 0.08, 0.55, 0.59, 1.14, 1.23 (c) 0.09, 0.52, 0.8, 0.83, 1.01, 1.1 (d) 0.19, 0.2, 0.26, 0.3, 0.98, 1 (e) 0.006, 0.08, 0.2, 0.209, 0.24, 0.72 (f) 0.05, 0.092, 0.4, 0.619, 0.64, 0.7, 0.78 7 1 (g) , 0.3, , 0.542, 0.85, 0.862, 0.9 100 2 NSSAL ©2012 76 Draft C. D. Pilmer 8 43 ,1 , 1.3 8 1000 5 1 , 0.4, 0.48, 0.509, 1.002, 1.1, 1 (i) 8 16 (h) 0.16, 0.201, 0.6, 0.649, Rounding Decimals (page 13) 1. (a) 14.55 (b) 15 (c) 14.5 2. (a) 251.9 (b) 250 (c) 252 3. (a) (b) (c) (d) 7.065 7 7.1 7.06 4. (a) (b) (c) (d) (e) 437 437.3 437.330 440 437.33 Equivalent Fractions and Decimals (pages 14 to 17) 1. (a) 3 4 (d) 2 1 3 (b) 3 1 2 (c) 1 4 5 (e) 7 1 5 (f) 5 2 3 2. (a) 9.25 (b) 0.4 (c) 6.3 (d) 7.75 (e) 9.5 (f) 8.6 3. (a) (c) (e) NSSAL ©2012 0.65 1 1 3 2 4 3 (b) 0.875 (d) 2.6 (f) 6 77 4 5 Draft C. D. Pilmer (g) 2.304 (h) 7 1 5 4. The numbers should appear on the number line in the following order. 1 1 2 3 7 1 , , , 0.9, 1.44, 1 , 2.069, 2.539, 2 , 3 , 3.37 25 3 5 4 8 5 5. The numbers should appear on the number line in the following order. 9 95 2 2 1 0.098, , , 1 , 1 , 1.87, 2.2, 2.43, 2.71, 3.12, 3 16 100 5 3 4 5 93 3 1 11 , , 1 , 1.6, 1 , 2.4, 2 8 100 4 5 12 1 41 3 1 9 (b) , 0.587, 1 , 1.58, 1 , 2 , 2.7, 2 3 100 4 16 10 1 1 9 4 19 (c) , , 0.892, 1 , 1 , 2.011, 2 , 2.6 10 20 16 5 4 6. (a) 0.09, 7. In each case we have supplied three possible answers; there are many more than the ones we have listed. (a) 3.41, 3.459, 3.499 3 3 2 (b) 2 , 2 , 2 5 3 4 (c) 1.3, 1.25, 1.239 (d) 0.7, 0.74, 0.859 1 1 2 (e) 3 , 3 , 3 5 10 4 (f) 2.9, 2.83, 2.999, 3.005 Introduction to Percent (pages 18 to 25) 1. (a) 17%, 0.17, 17 100 (b) 43%, 0.43, 43 100 (c) 29%, 0.29, 29 100 2. There are a range of acceptable answers. (a) 40% to 47% (b) 11% to 19% (c) 85% to 94% (d) 63% to 73% (e) 4% to 7% 3. (a) 0.79 (c) 0.09 NSSAL ©2012 (b) 0.16 (d) 1.45 78 Draft C. D. Pilmer (e) 0.294 (g) 2.08 (i) 0.045 4. (a) (c) (e) (g) (i) 19% 173% 6% 7.3% 0.2% (b) (d) (f) (h) (j) 39 100 4 25 7 2 20 23 500 1 250 5. (a) (c) (e) (g) (i) 6. (a) (c) (e) (g) (f) 0.07 (h) 0.817 (j) 0.008 (b) (d) (f) (h) (j) 75% 350% 80% 166.6% 7. Percent (a) 83% (c) 460% (e) 5% (g) 71.9% 8. (a) (b) (c) (d) (e) (b) (d) (f) (h) Fraction 83 100 3 4 5 1 20 719 1000 48% 69.2% 209% 154.8% 170% 91 100 29 1 100 51 1000 241 500 103 3 500 140% 233.3% 325% 220% Decimal Percent 0.83 (b) 67% 4.6 (d) 139% 0.05 (f) 175% 0.719 (h) 216.3% Fraction 67 100 39 1 100 3 1 4 163 2 1000 Decimal 0.67 0.39 1.75 2.163 40% 1% 45% 96% 15% NSSAL ©2012 79 Draft C. D. Pilmer 9. Remember we are not grouping equivalent decimal, fractions, and percentages in the same boxes; rather, we are completing an estimation activity where we match the numbers to the most appropriate diagram. 7 1 1 (a) 51%, 0.52, (b) 26%, 0.24, (c) 69%, 0.72, 10 2 4 5 3 7 (e) 81%, 0.8, (f) 260%, 2.57, 2 (d) 140%, 1.43, 1 8 9 7 1 7 19 (g) 218%, 2.16, 2 (h) 89%, 0.91, (i) 194%, 1.93, 1 6 8 20 10. 93 100 17 1000 1.7% 1.387 53 100 93% 0.8% 13.1% 0.008 2 347% 3 4 275% 4 5 4 47 100 2 0.53 280% 131 1000 3 138.7% 3 10 2 3 4.6 30% 11. There are 8 people in the Sampson family. Of those, 6 are female. That means that percentage of females in this family is 75%, which can also be represented by the fraction 3 . 4 The percentage of males in this family is 25%, which can also be represented by the fraction 1 . 4 NSSAL ©2012 80 Draft C. D. Pilmer Comparing Fractions, Decimals, and Percentages (pages 26 to 29) 1. Closest to: Closest to: Closest to: 1 2 1 2 (a) 98% 1 (b) 11% 0 (c) 45% (d) 0.02 0 (e) 0.899 1 (f) 0.6 (g) 8 9 1 (h) 11 20 1 2 (i) 1 16 0 (j) 0.3% 0 (k) 1.05 1 (l) 102% 1 (m) 13 24 1 2 (n) 56.2% 1 2 (o) 16 15 1 2. (a) (c) (e) (g) (i) 83% 136% 0.48 2.3% 105% 7 8 (b) (d) (f) (h) (j) (m) 135% (n) (k) (o) (q) (s) (u) 7 16 2 3 4 5 81.2% (l) 14.7% 3.1% 83% 0.65 1.45 9 16 1 10 (p) 93.5% (r) 215% (t) 0.48 (v) 8.3% 3. The numbers should occur in this order along the number line (from left to right). 1 1 5 1 , 0.422, 54.7%, 0.93, 1.099, 125%, 180%, 200%, 2 , 2 , 3 20 4 8 16 4. The numbers should occur in this order along the number line (from left to right). 3 3 19 9.7%, , 0.713, 96%, 1.389, 155%, 1 , 215%, 2.85, 2 , 2.85, 3.2 6 4 20 5. (a) 0.7%, 5.8%, 32%, 91.2%, 124% (b) 0.1, 14%, 64%, 0.745, 0.82 (c) 8.2%, 57.2%, 0.61, 123%, 1.45 NSSAL ©2012 81 Draft C. D. Pilmer 5 9 , 0.792, 86%, 12 10 1 1 , 20%, 0.4, 68.5%, 1 , 1.96 32 10 8 31 0.276, 30.2%, , 57.6%, , 1.1 32 16 1 5 0.096, , 64.5%, 0.89, 1 , 209% 4 8 1 6 , 0.08, , 50.3%, 91%, 0.956, 100 14 3 7 1 28%, , 0.9, , 1.02, 1 , 194% 5 7 3 1 7 7 , 0.34, 94.5%, 1.092, 1 , 214%, 2 5 12 8 (d) 3.8%, (e) (f) (g) (h) (i) (j) Adding and Subtracting Decimal Numbers (pages 30 to 36) 1. (a) (c) (e) (g) 46.88 88.87 41.962 45.775 (b) (d) (f) (h) 69.84 6.813 230.33 8.475 2. (a) (c) (e) (g) 43.35 1.728 8.865 21.779 (b) (d) (f) (h) 4.04 5.72 6.682 6.556 3. 3.67 kg 4. 3.42 kg 5. 24 293.1 km 6. 0.38 seconds Multiplying Decimal Numbers (pages 37 to 42) 1. (a) (c) (e) (g) (i) 24.05 0.234 6.92 76.63 0.35334 NSSAL ©2012 (b) (d) (f) (h) (j) 82 19.318 22.32 4.459 18.27 77.86 Draft C. D. Pilmer 2. (a) (b) (c) (d) (e) (f) 22.656 75.64 177.3 384.85 804.2 2548.584 3. Question 0.98 × 2.1 5.23 − 4.37 1.1 + 1.97 Arrow d b f Question 0.326 + 2.21 2.34 − 1.98 0.49 × 2.88 Arrow e a c 4. Question (a) 2.93 + 3.208 Answers (c) 32.185 (b) 16.08 - 5.239 (h) 311.74 (c) 7.85 × 4.1 (e) 21.78 (d) 47.9 + 32.7 (a) 6.138 (e) 29.58 - 7.8 (g) 126.7 (f) 39.8 × 6.1 (d) 80.6 (g) 98.3 + 28.4 (i) 1110.9 (h) 409.8 - 98.06 (b) 10.841 (i) 52.9 × 21 (f) 242.78 5. 730.5 km 6. $12.86 7. $480.70 8. 42.2 cm 9. 26.1 grams Dividing Decimal Numbers (pages 43 to 53) 1. (a) 32.04 (c) 2.415 NSSAL ©2012 (b) 26.7 (d) 0.093 83 Draft C. D. Pilmer 2. Questions (a) 389.6 ÷ 8 (b) 49.02 ÷ 6 (c) 2.024 ÷ 4 (d) 257.8 ÷ 9 (e) 32.83 ÷ 7 Answers (c) 0.506 (a) 48.7 (e) 4.69 (b) 8.17 (d) 28.62 3. (a) 6 453.6 (b) 0.8 0.736 (c) 5 3856 (d) 0.8 2.7345 (e) 5 490 (f) 6 820 (g) 0.03 1.826 (h) 7 5820 4. (a) (c) (e) (g) 5.6 20.39 31.96 0.872 (b) (d) (f) (h) 639.1 ÷ 7 289.4 + 315.7 23.76 ÷ 0.6 20.5 × 61.8 0.24 ÷ 0.5 453.6 − 198.8 0.496 ÷ 0.08 0.9 ×135.6 32.68 ÷ 0.4 5. (a) (b) (c) (d) (e) (f) (g) (h) (i) 6. Question 6.65 ÷ 7 1.49 + 1.04 0.145 ÷ 0.05 86 94.3 68 124.04 (c) (h) (e) (a) (f) (g) (i) (d) (b) Arrow b e f Question 3.05 − 2.97 0.31× 5.2 1.278 ÷ 0.6 39.6 122.04 0.48 91.3 254.8 6.2 817 1266.9 605.1 Arrow a c d 7. $28.30 8. 18.2, but we round down such that the final answer is 18 hamburgers. 9. $87.75 NSSAL ©2012 84 Draft C. D. Pilmer 10. $9.65 11. 257.7 km Estimation Questions Involving Percentages (pages 54 to 57) 1. (a) 4 (c) 50 (e) 8 (g) 100 (i) 6 (k) 75 (m) 12 (o) 100 (b) (d) (f) (h) (j) (l) (n) (p) 12 140 24 280 18 210 36 300 Answers are likely to vary slightly from learner to learner on questions 2 through 9. This is to be expected with estimation questions. As long as your answer is close to our answer, assume that you estimation technique was perfectly valid. 2. Approximately $10 3. Approximately $46 4. Approximately $21 5. Approximately $12 6. Approximately $42 ($36 + $6) 7. Approximately $138 ($120 + $18) 8. Approximately $108 ($90 + $18) 9. Approximately $230 ($200 + $30) 10. Question 25% of 11.90 0.784 ÷ 0.4 1.43 − 1.316 30% of 4.90 NSSAL ©2012 Arrow g e a d Question 0.52 + 0.496 10% of 21.50 20% of 2.99 2.1×1.513 85 Arrow c f b h Draft C. D. Pilmer Calculator Questions (pages 58 to 61) 1. 23 minutes 2. $154.77 3. $1.17 4. $59.80 5. Hint: Remember that we are purchasing each of these items for her two boys. Answer: $211.54 6. $1099.80 7. $650.20 8. $652.00 9. $46.60 10. 14.2 kilometres per litre Additional Practice; Ordering Decimals (pages 66 and 67) 1. Decimal Closest to: 1 2 Decimal Closest to: (b) 0.93 1 (a) 0.56 (d) 1.008 1 (e) 0.4 (g) 0.897 1 (h) 0.054 (j) 0.106 0 (k) 0.61 (m) 12.5% 0 (n) 0.91 2. (a) (c) (e) (g) (i) (k) NSSAL ©2012 0.8 0.867 2.7 3.624 3.001 8.408 (b) (d) (f) (h) (j) (l) 86 1 2 1 2 1 2 1 Decimal Closest to: (c) 0.07 0 (f) 0.19 0 (i) 0.089 0 (l) 0.403 (o) 23 50 1 2 1 2 0.92 3.64 4.12 1.98 6.13 1.982 Draft C. D. Pilmer (m) (o) (q) (s) (u) 3. (a) (c) 0.52 4.099 17.45 3.41 8.601 (n) (p) (r) (t) (v) 3.59 9.06 5.9 4.61 3.04 0.431 7.59 (b) (d) 2.6 4.2 4. 0.14, 0.45, 0.72, 0.87, 1.36, 1.63, 1.91, 2.16, 2.67, 2.82, 3.08 5. 0.076, 0.33, 0.6, 0.91, 1.183, 1.45, 1.782, 2.2, 2.462, 2.8, 3.07, 3.278 6. (a) (b) (c) (d) 0.6, 0.8, 1.3, 1.5, 1.9, 2.1 0.09, 0.73, 1.39, 1.45, 2.06, 2.41 0.34, 0.7, 0.907, 0.93, 1.241, 1.6, 1.78 0.28, 0.4, 0.563, 1.041, 1.372, 1.83, 1.9 Additional Practice; Ordering Decimals and Fractions (pages 68 and 69) 1. Decimal Closest to: Decimal (a) 0.04 0 (b) 19 20 (d) 7 12 1 2 (e) 0.521 (g) 0.8 1 (h) 15 32 1 2 1 2 (j) 1.003 1 (k) 0.1 0 (j) (l) 6.4 (n) 1 0.45 (b) (c) 0.26 (d) (e) 0.56 (f) (g) 3.6 (h) 2.592 9 (k) 4 16 17 (m) 16 NSSAL ©2012 1 7 8 0.9 15 16 3 1 5 5.625 2. (a) (i) Closest to: 87 Decimal Closest to: (c) 0.913 1 (f) 1 30 0 (i) 0.48 1 2 (l) 33 32 1 11 20 Draft C. D. Pilmer 3. (a) (c) 11 12 (b) 2.83 5.91 (d) 4 4. 0.1, 0.68, 6 10 1 9 1 15 10 , 1.183, 1.4, 1 , 2 , 2 , 2.8 , 3.052, 3 5 16 6 16 20 9 3 5 , , 0.92, 8 10 8 5 3 0.427, , 1.1, 1 , 1.83 9 7 13 3 5 , 0.78, 0.9, 1.042, 1 , 1 50 12 16 1 8 3 , 0.17, , 0.905, 1 , 1.7 100 15 25 7 1 0.064, , 0.8, 0.91, 1.15, 1 , 1.908 12 4 29 39 9 , 2.5, 2 , 3.171, 3.26, 3 , 3.8 2 20 32 1000 5. (a) 0.068, (b) (c) (d) (e) (f) Additional Practice; Ordering Decimals, Fractions and Percentages (pages 70 and 71) 1. (a) (d) (g) (j) Decimal Closest to: 16% 0 9 17 7 6 52.6% Decimal Closest to: (b) 0.842 1 (c) 1 18 0 1 2 (e) 98% 1 (f) 47% 1 2 1 (h) 8.3% 0 (i) 0.05 0 1 2 (k) 27 1000 0 (l) 106% 1 2. (a) 53% (b) (c) 0.95 (d) (e) 58.3% (f) NSSAL ©2012 88 Decimal Closest to: 3 4 0.6 13 16 Draft C. D. Pilmer (g) 9 20 (h) (i) 25% (j) (k) 2.8 23 (m) 21 3. (a) 57% 4. 0.08, 28%, 102% (l) 6 7 145% (n) 236% (b) 1 2 7 9 3 1 11 91 , 95% , 1 , 140%, 1.7, 1 , 216.5%, 2.683, 2 6 10 100 12 5 , 87.5%, 0.9 8 7 17 , 16%, 0.529, , 101% 100 20 1 9 0.003, , 28.5%, 61%, , 0.96 10 6 1 0.72, 1 , 1.25, 153%, 1.8, 209.5% 32 47 4 3 , 63% , 1.1, 1 , 187%, 2 , 250% 100 7 10 13 31 0.378, , 86.9%, 109%, 1 , 1.6, 173% 25 100 5. (a) 4%, 0.38, (b) (c) (d) (e) (f) NSSAL ©2012 89 Draft C. D. Pilmer

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