Place Value of Whole Numbers

Transcription

1 Chapter 1 Review of Basic Arithmetic Place Value of Numbers and Rounding Numbers Place value is the value of a digit as determined by its position in a number, such as ones, tens, hundreds, etc. Place Value of Whole Numbers The position of each digit in a whole number determines the place value for the digit. Exhibit 1.1(a) illustrates the place value of the ten digits in the whole number: 3,867,254,129. In this example, 4 is in the 'thousands' place value and represents 4000, whereas 7 is in the 'millions' place value and represents 7,000,000. We read and write numbers, from left to right. A comma (or alternatively, a space) separates every three digits into groups, starting from the place value for 'ones', thereby making it easier to read a whole number. Exhibit 1.1(a): Place Value of a Ten-Digit Whole Number The place value of 'ones' is 10 0 ( = 1) and each place has a value 10 times the place value to its right, as shown in Table 1.1(a) below: Table 1.1(a) Place Value Chart of Whole Numbers ,000,000, ,000,000 10,000,000 1,000, ,000 10, Billions Hundred millions Ten millions Millions Hundred thousands Ten thousands Thousands Hundreds Tens Ones The vertical red lines denote the positions of the commas that separate the groups of three numbers, starting from the place value for 'ones'. The above 10-digit number written as 3,867,254,129 is called the standard form of writing numbers This can be written in expanded form as follows: 3,000,000, ,000, ,000, ,000, , , , Or 3 billion million + 60 million + 7 million thousand + 50 thousand + 4 thousand + 1 hundred + 2 tens + 9 ones The word and does not appear in the word form of whole numbers. This can also be written in word form as follows: Three billion, eight hundred sixty-seven million, two hundred fifty-four thousand, one hundred twenty-nine. Million: One thousand thousand (1000 x 1000). One million is a number represented by 1 followed by 6 zeros (1,000,000). In scientific notation: 1 x 10 6 For example, the population of Canada is about 35 million (35,000,000 or 35 x 10 6 ). Billion: One thousand million [1000 (1000 x 1000)]. One billion is a number represented by 1 followed by 9 zeros (1,000,000,000). In scientific notation : 1 x 10 9 For example, the population of the world is about 7 billion (7,000,000,000 or 7 x 10 9 ). In comparison, the current population of the world is about 200 times that of Canada.

2 4 Chapter 1 Review of Basic Arithmetic Place Value of Decimal Numbers The position of each digit in a decimal number determines the place value of the digit. Exhibit 1.1(b) illustrates the place value of the five-digit decimal number: The place value of each digit is found by decreasing powers of 10, as shown in Table 1.1(b) below: Exhibit 1.1(b): Place Value of a Five-Digit Decimal Number Table 1.1(b) Place Value Chart of Decimal Numbers 10-1 = = = = 1 10, = 1 100, Tenths Hundredths Thousandths Ten-thousandths Hundred-thousandths Example of the decimal number written in its standard form: Decimal numbers can be expressed in their word form in 3 ways. For example.85 can be expressed as: point eight five, point eighty-five, or eighty-five hundredths. Example 1.1(a) This can be written in expanded form as follows: Or 3 tenths + 5 hundredths + 7 thousandths + 9 ten-thousandths + 6 hundred-thousandths This can be written in word form as follows: Zero point three five seven nine six. Place Value of a Digit What is the place value of the digit 5 in each of the following and what amount does it represent? (i) 675,342 (ii) 35,721,890 (iii) 5,916,203,847 (iv) (i) 675,342 (ii) 35,721,890 (iii) 5,916,203,847 (iv) Place Value Thousands Millions Billions Thousandths Amount it Represents ,000,000 5,000,000, Example 1.1(b) Identifying the Digit of a Number Given its Place Value In 320, , identify the digit that occupies the following place values: (i) Hundred thousands (ii) Ten thousands (iii) Thousands (iv) Hundredths (i) 3 (ii) 2 (iii) 0 (iv) 5 Example 1.1(c) Writing Numbers in Expanded and Word Forms Write the following numbers in expanded and word forms: (i) 9,865,323 (ii) 43,583,621 (iii) 8,213,505,235 (iv)

3 Chapter 1 Review of Basic Arithmetic 5 (i) Number: 9,865,323 Expanded form: 9,000, , , , Word form: Nine million, eight hundred sixty-five thousand, three hundred twenty-three (ii) Number: 43,583,621 Expanded form: 40,000, ,000, , , , Word form: Forty-three million, five hundred eighty-three thousand, six hundred twenty-one (iii) Number: 8,213,505,235 Expanded form: 8,000,000, ,000, ,000, ,000, , , Word form: Eight billion, two hundred thirteen million, five hundred five thousand, two hundred thirty-five (iv) Number: Expanded form: Word form: Four hundred seventy-eight point two five or four hundred seventy-eight point twenty-five or four hundred seventy-eight and twenty-five hundredths Example 1.1(d) Writing Numbers in Standard Form Given Word Form Write the following in standard form: (i) Thirty-five thousand, eight hundred twenty-five (ii) Three million, three hundred forty-two thousand, six hundred seventeen (iii) Half of a million (iv) Three-quarters of a billion (i) 35,825 (ii) 3,342,617 1 (iii) Half of a million is # 1,000,000 = 500, (iv) Three-quarters of a billion is # 1,000,000,000 = 750,000,000 4 Rounding Whole Numbers and Decimals Rounding numbers makes them easier to work with and easier to remember. Rounding changes some of the digits in a number but keeps its value close to the original. It is used in reporting large quantities or values that change often, such as in population, income, expenses, etc. For example, the population of Canada is approximately 33 million or, Henry's car expense for this month is approximately $700. Rounding of numbers also makes arithmetic operations faster and easier when it is not required to find the exact answer. For example, if you are required to estimate the area of a rectangular plot of land that measures 114 m by 97 m, you would have to multiply 114 # 97 which will result in 11,058 m 2. However, you can get a quick estimate by rounding the measurements 114 to 110 and 97 to 100. This will result in an area of 110 # 100 = 11,000 m 2. Rounding Whole Numbers to the Nearest Ten, Hundred, Thousand, etc. Rounding whole numbers refers to changing the value of the whole number to the nearest ten, hundred, thousand, etc. It is also referred to as rounding whole numbers to multiples of 10, 100, 1000, etc.

4 6 Chapter 1 Review of Basic Arithmetic For example, Rounding a whole number to the nearest ten is the same as rounding it to a multiple of 10. Rounding a whole number to the nearest hundred is the same as rounding it to a multiple of 100. Rounding an amount to the nearest $10 refers to rounding the amount to a multiple of $10. Rounding Decimals to the Nearest Whole Number, Tenth, Hundredth, etc. Rounding decimals refers to changing the value of the decimal number to the nearest whole number, tenth, hundredth, thousandth, etc. It is also referred to as "rounding to a specific number of decimal places", indicating the number of decimal places that will be left when the rounding is complete. For example, Rounding to the nearest whole number is the same as rounding without any decimals. Rounding to the nearest tenth is the same as rounding to one decimal place. Rounding to the nearest hundredth is the same as rounding to two decimal places. Rounding to the nearest cent refers to rounding the amount to the nearest hundredth or to two decimal places. Rules for Rounding Whole Numbers and Decimals Step 1: Identify the digit to be rounded (this is the place value for which the rounding is required). Step 2: If the digit to the immediate right of the required rounding digit is less than 5 (0, 1, 2, 3, 4), do not change the value of the rounding digit. Step 3: If the digit to the immediate right of the required rounding digit is 5 or greater than 5 (5, 6, 7, 8, 9), increase the value of the rounding digit by one (round up by one number). For rounding whole numbers: after step 2, change the value of all the digits that are to the right of the rounding digit to 0. For rounding decimals: after step 2, drop all the digits that are to the right of the rounding digit. Example 1.1(e) Rounding Whole Numbers Round the following to the indicated place values: (i) 18,568 to the nearest ten (ii) $24,643 to the nearest $100 (i) Rounding 18,568 to the nearest ten. Identify the rounding digit in the tens place: 18,568 (6 is the digit in the tens place). The digit to the immediate right of the rounding digit is 8, which is greater than 5; therefore, increase the value of the rounding digit by one, from 6 to 7, and change the value of the digits that are to the right of the rounding digit to 0, which will result in 18,570. Therefore, 18,568 rounded to the nearest ten (or multiple of 10) is 18,570. (ii) Rounding $24,643 to the nearest $100. Identify the rounding digit in the hundreds place: 24,643 (6 is the digit in the hundreds place).

5 Chapter 1 Review of Basic Arithmetic 7 continued The digit to the immediate right of the rounding digit is 4, which is less than 5; therefore, do not change the value of the rounding digit, but change the value of the digits that are to the right of the rounding digit to 0, which will result in 24,600. Therefore, $24,643 rounded to the nearest $100 (or multiple of 100) is 24,600. Example 1.1(f) Rounding Numbers (Visual Method) Round the following to the indicated place value: (i) 627 to the nearest ten (multiples of 10) (ii) 16.5 to a whole number We can visualize these numbers on a number line to determine which number is nearest. (i) Rounding 627 to the nearest ten (multiples of 10) 627 is closer to 630, than to 620. Therefore, 627 rounded to the nearest ten is 630. (ii) Rounding 16.5 to a whole number 16.5 is exactly in between 16 and 17. By convention, if a number is exactly in the middle, we round it up. Therefore, 16.5 rounded to a whole number is 17. Example 1.1(g) Rounding Decimal Numbers Round the following decimal numbers to the indicated place value: (i) to the nearest hundredth (ii) $ to the nearest cent (iii) $ to the nearest cent (i) Rounding to the nearest hundredth Identify the rounding digit in the hundredths place: (4 is the digit in the hundredths place). The digit to the immediate right of the rounding digit is less than 5; therefore do not change the value of the rounding digit. Drop all the digits to the right of the rounding digit, which will result in Therefore, rounded to the nearest hundredth (or to two decimal places) is (ii) Rounding $ to the nearest cent Identify the rounding digit in the hundredths place: $ (7 is the digit in the hundredths place). The digit to the immediate right of the rounding digit is greater than 5; therefore, increase the value of the rounding digit by one, from 7 to 8, and drop all the digits that are to the right of the rounding digit, which will result in $ Therefore, $ rounded to the nearest cent (or to two decimal places) is $

6 8 Chapter 1 Review of Basic Arithmetic continued (iii) Rounding $ to the nearest cent Identify the rounding digit in the hundredths place: $ (9 is the digit in the hundredths place). The digit to the immediate right of the rounding digit is greater than 5; therefore, increase the value of the rounding digit by one, from 9 to 10, carrying the one to the tenths place, then to the ones, then to the tens to increase 3 to 4. Finally, drop all the digits that are to the right of the hundredths place. Therefore, $ rounded to the nearest cent (or to two decimal places) is $ Example 1.1(h) Rounding Percents Round the following percents to the number of decimal places as indicated: (i) % to one decimal place. (ii) % to two decimal places. (iii) % to two decimal places. (i) Rounding % to one decimal place Identify the rounding digit in the tenths place: % (8 is the digit in the tenths place). The digit to the immediate right of the rounding digit is less than 5; therefore, do not change the value of the rounding digit. Drop all the digits to the right of the rounding digit, which will result in 12.8%. Therefore, % rounded to one decimal place (or nearest tenth) is 12.8%. (ii) Rounding % to two decimal places Identify the rounding digit in the hundredths place: % (8 is the rounding digit in the hundredths place). The digit to the immediate right of the rounding digit is greater than 5; therefore, increase the value of the rounding digit by one, from 8 to 9, and drop all the digits that are to the right of the rounding digit, which will result in 67.39%. Therefore, % rounded to two decimal places (or nearest hundredth) is 67.39%. (iii) Rounding % to two decimal places Identify the rounding digit in the hundredths place: % (2 is the rounding digit in the hundredths place). The digit to the immediate right of the rounding digit is equal to 5; therefore, increase the value of the rounding digit by one, from 2 to 3, and drop all the digits that are to the right of the rounding digit, which will result in 25.33%. Therefore, % rounded to two decimal places (or nearest hundredth) is 25.33%. Example 1.1(i) Rounding a Set of Percents which Add Up to 100% Stephanie spends 40.18% of her salary on rent, 20.61% on food, 15.62% on travel, 10.15% on entertainment, 5.5% on pet food, and saves the remaining 7.94%. Express these percents rounded to the nearest percent without decimals. These percents rounded to the nearest percent without decimals (percent with whole numbers) would be: Rent: 40%, Food: 21%, Travel: 16%, Entertainment: 10%, Pet food: 6%, and Savings: 8%.

7 Chapter 1 Review of Basic Arithmetic 9 continued However, the sum of these percents would result in: 40% + 21% + 16% + 10% + 6% + 8% = 101% We have to round these numbers to give us a total of only 100%, so 1% has to be reduced from one of these rounded numbers such that it does not have a significant effect. Changing the largest percent will result in a smaller effect than changing the other percents. If 40% is reduced to 39%, the change is 39% - 40% = -2.5% but if 6% is reduced to 5%, the change is 40% 5% - 6% = %. Changing 40% has a smaller effect than changing the 6% value. 6% Therefore, the percents rounded to the nearest whole numbers are best represented as: Rent: 39%, Food: 21%, Travel: 16%, Entertainment: 10%, Pet food: 6%, and Savings: 8% 39% + 21% + 16% + 10% + 6% + 8% = 100% Note: When the final answer must be expressed as a percent or when using percents in calculations, rounding should not be done during intermediate calculations. Instead rounding to the required decimal places should only be done at the end of all calculations. In business and finance applications, final answers are generally rounded to the nearest hundredth (2 decimal places) for value of money (e.g., $ ) and percents (e.g., 4.86%). Estimation Estimation is finding a number that is close to the right answer. In situations when an exact value is not required, we use an estimated value by rounding some or all of the numbers, factors, divisors, etc., so that it is easy to arrive at an answer using mental calculations. In solving problems, particularly when using calculators, estimation helps to determine whether the calculated answer is reasonable and also helps to identify and prevent obvious mistakes. When using calculators, it is possible that numbers, operational keys, or decimal points may be incorrectly entered, leading to a wrong answer. Estimation helps to check the effective use of calculators. We use various methods of estimation to judge whether the answer from a calculation is reasonable. The method to perform an estimation depends on an individual s mathematical skill in multiplication because different numbers or different operations require different methods of estimation. The following are a few examples of estimation: To estimate the addition and subtraction of numbers, we may first round each number to the same place value and then add or subtract. Alternatively, we may round the numbers to their biggest place values first, and then add or subtract. For example, to estimate the value of , by rounding to the nearest hundred, we obtain, = 10,200 by rounding to the biggest place value, we obtain, = 10,300 Compare with the exact value: 10,206 To estimate the product of numbers, we may round each number to their biggest place value so that it has only one non-zero digit. Then, drop all the zeroes and mentally multiply the non-zero numbers. Finally, reinstate all the zeroes that were dropped earlier. Alternatively, we may round one or two factors that can be easily worked mentally with the other factors. For example, to estimate the value of , by rounding to the biggest place value, we obtain, = (3 5 2) 1,000,000 = 30 1,000,000 = 30,000,000 by rounding two factors, we obtain, = 3249 (5 2) 1000 = ,000 = 32,490,000 Compare with the exact value: 31,112,424

8 10 Chapter 1 Review of Basic Arithmetic To estimate the division of numbers, we may round the numbers to their biggest place value, drop the zeroes that are common to both the numerator and denominator, and divide mentally. Alternatively, we may approximate the numbers to fit in with the multiplication table. For example, to estimate the value of , by rounding to the biggest place value, we obtain, = 20 by rounding to multiples of 25, we obtain, = 23 Compare with the exact value rounded to the nearest two decimal places: Exercises Answers to the odd-numbered problems are available at the end of the textbook 1. Write the following numbers in (i) expanded form and (ii) word form: a. 5,249,346 b. 95,275,385 c. 1,146,045,787 d Write the following numbers in (i) expanded form and (ii) word form: a. 9,265,335 b. 30,673,984 c. 3,405,553,978 d Write the following in standard form: a. Sixty-five thousand, two hundred forty-four point thirty-four b. Twelve million, four hundred fifty-two thousand, eight hundred thirty-two c. One-eighth of a million d. Half of a billion 4. Write the following in standard form: a. Eight hundred thirty-three thousand, six hundred forty-one point eighty-two b. Thirty-two million, six hundred eighty-four thousand, two hundred fifty-six c. Three-quarters of a million d. One-tenth of a billion 5. Write the following in standard form: a. Eighty-nine million, six hundred thirteen thousand, five hundred twenty-two point one three b. Sixteen million, two hundred seventeen thousand, five hundred sixty-seven c. Half of a million d. Three-quarters of a trillion 6. Write the following in standard form: a. Nine hundred fifty thousand, six hundred fifty point zero five. b. Sixty-five thousand, eight hundred point fifty-six c. One-quarter of a billion d. Half of a thousand 7. Round the following to the indicated place values: a to the nearest ten b to the nearest thousand c. $25,972 to the nearest $1000 d. $895 to the nearest $10 8. Round the following to the indicated place values: a to the nearest hundred b. 53,562 to the nearest ten c. $7915 to the nearest $100 d. $1095 to the nearest $10 9. Round the following to the indicated place values: a to the nearest tenth b to the nearest whole number c. $ to the nearest cent d. $ to the nearest cent 10. Round the following to the indicated place values: a to the nearest thousandth b to the nearest whole number c. $ to the nearest cent d. $ to the nearest cent

9 Chapter 1 Review of Basic Arithmetic Round the following percents to the nearest whole percent (without decimals): a % b % c % d % 12. Round the following percents to the nearest whole percent (without decimals): a % b % c % d % 13. Round the percents in Problem 11 to the nearest percent with 1 decimal place. 14. Round the percents in Problem 12 to the nearest percent with 1 decimal place. 15. Round the percents in Problem 11 to the nearest percent with 2 decimal places. 16. Round the percents in Problem 12 to the nearest percent with 2 decimal places. For Problems 17 to 24, (i) estimate the values by first rounding the numbers to the nearest ten and (ii) calculate the exact answer. Express the final answer rounded to two decimal places. 17. a b c a b c a b c a b c a. 58 # 75 b. 472 # 48 c. 95 # a. 63 # 59 b. 35 # 97 c. 246 # a b c a b c Estimate the values in Problems 17 and 19 by first rounding the numbers to the nearest hundred. Express the final answer rounded to two decimal places. 26. Estimate the values in Problems 18 and 20 by first rounding the numbers to the nearest hundred. Express the final answer rounded to two decimal places. 27. A prepaid phone card to make calls to Singapore for 3 hours costs $5. Find the cost per minute rounded to the nearest cent. 28. A prepaid phone card to make calls to China for 1 hour costs $2.50. Find the cost per minute rounded to the nearest cent % of the employees of a large software company are engineers, 35.40% are project managers and team leads, 2.40% are senior managers, and the rest are administrative staff. Express these percents rounded to the nearest percent without decimals. 30. Amy invested 4.20% of her savings in bonds, 32.65% in stocks, 25.55% in mutual funds, and the balance in her brother's business. Express these percents rounded to the nearest percent without decimals. 1.2 Factors and Multiples Factors of a number are whole numbers that can divide the number with no remainder. For example, factors of 12 are 1, 2, 3, 4, 6, and 12. We can express factors of a number by showing how the product of two factors results in the number. 12 = 1 # = 2 # 6 12 = 2 # 2 # 3 12 = 3 # 4 Multiples of a number are whole numbers that can be divided by the number with no remainder. Multiples of a number can be expressed as the product of the number and a whole number. For example, multiples of 10: 10, (10+10), ( ), ( ), ( )... 10, 20, 30, 40, 50 10, (10#2), (10#3), (10#4), (10#5) Therefore, multiples of 10 are 10, 20, 30, 40, 50