m81293 Decimals By the end of this section, you will be able to: Name decimals Write decimals Convert decimals to fractions or mixed numbers Locate decimals on the number line Order decimals Round decimals b6e1863d-155b-4580-9d11-bb0d46db2d09 Before you get started, take this readiness quiz. Name the number 4,926,015 in words. If you missed this problem, review . Four million, nine hundred twenty-six thousand, fifteen Round 748 to the nearest ten. If you missed this problem, review . 750 Locate 310 on a number line. If you missed this problem, review .

You probably already know quite a bit about decimals based on your experience with money. Suppose you buy a sandwich and a bottle of water for lunch. If the sandwich costs $3.45, the bottle of water costs $1.25, and the total sales tax is $0.33, what is the total cost of your lunch? The total is $5.03. Suppose you pay with a $5 bill and 3 pennies. Should you wait for change? No, $5 and 3 pennies is the same as $5.03. Because 100 pennies=$1, each penny is worth 1100 of a dollar. We write the value of one penny as $0.01, since 0.01=1100. Writing a number with a decimal is known as decimal notation. It is a way of showing parts of a whole when the whole is a power of ten. In other words, decimals are another way of writing fractions whose denominators are powers of ten. Just as the counting numbers are based on powers of ten, decimals are based on powers of ten. shows the counting numbers. Counting number Name 1 One 10=10 Ten 10·10=100 One hundred 10·10·10=1000 One thousand 10·10·10·10=10,000 Ten thousand How are decimals related to fractions? shows the relation. Decimal Fraction Name 0.1 110 One tenth 0.01 1100 One hundredth 0.001 11,000 One thousandth 0.0001 110,000 One ten-thousandth When we name a whole number, the name corresponds to the place value based on the powers of ten. In Whole Numbers, we learned to read 10,000 as ten thousand. Likewise, the names of the decimal places correspond to their fraction values. Notice how the place value names in relate to the names of the fractions from .

This chart illustrates place values to the left and right of the decimal point.

Notice two important facts shown in . The “th” at the end of the name means the number is a fraction. “One thousand” is a number larger than one, but “one thousandth” is a number smaller than one. The tenths place is the first place to the right of the decimal, but the tens place is two places to the left of the decimal. Remember that $5.03 lunch? We read $5.03 as five dollars and three cents. Naming decimals (those that don’t represent money) is done in a similar way. We read the number 5.03 as five and three hundredths. We sometimes need to translate a number written in decimal notation into words. As shown in , we write the amount on a check in both words and numbers.

When we write a check, we write the amount as a decimal number as well as in words. The bank looks at the check to make sure both numbers match. This helps prevent errors.

Let’s try naming a decimal, such as 15.68. We start by naming the number to the left of the decimal. fifteen______ We use the word “and” to indicate the decimal point. fifteen and_____ Then we name the number to the right of the decimal point as if it were a whole number. fifteen and sixty-eight_____ Last, name the decimal place of the last digit. fifteen and sixty-eight hundredths The number 15.68 is read fifteen and sixty-eight hundredths. Name the number to the left of the decimal point. Write “and” for the decimal point. Name the “number” part to the right of the decimal point as if it were a whole number. Name the decimal place of the last digit. Name each decimal: ⓐ 4.3 ⓑ 2.45 ⓒ 0.009 ⓓ −15.571. ⓐ 4.3 Name the number to the left of the decimal point. four_____ Write "and" for the decimal point. four and_____ Name the number to the right of the decimal point as if it were a whole number. four and three_____ Name the decimal place of the last digit. four and three tenths ⓑ 2.45 Name the number to the left of the decimal point. two_____ Write "and" for the decimal point. two and_____ Name the number to the right of the decimal point as if it were a whole number. two and forty-five_____ Name the decimal place of the last digit. two and forty-five hundredths ⓒ 0.009 Name the number to the left of the decimal point. Zero is the number to the left of the decimal; it is not included in the name. Name the number to the right of the decimal point as if it were a whole number. nine_____ Name the decimal place of the last digit. nine thousandths ⓓ −15.571 Name the number to the left of the decimal point. negative fifteen Write "and" for the decimal point. negative fifteen and_____ Name the number to the right of the decimal point as if it were a whole number. negative fifteen and five hundred seventy-one_____ Name the decimal place of the last digit. negative fifteen and five hundred seventy-one thousandths Name each decimal: ⓐ 6.7 ⓑ 19.58 ⓒ 0.018 ⓓ −2.053 ⓐ six and seven tenths ⓑ nineteen and fifty-eight hundredths ⓒ eighteen thousandths ⓓ negative two and fifty-three thousandths Name each decimal: ⓐ 5.8 ⓑ 3.57 ⓒ 0.005 ⓓ −13.461 ⓐ five and eight tenths ⓑ three and fifty-seven hundredths ⓒ five thousandths ⓓ negative thirteen and four hundred sixty-one thousandths

Now we will translate the name of a decimal number into decimal notation. We will reverse the procedure we just used. Let’s start by writing the number six and seventeen hundredths: six and seventeen hundredths The word and tells us to place a decimal point. ___.___ The word before and is the whole number; write it to the left of the decimal point. 6._____ The decimal part is seventeen hundredths. Mark two places to the right of the decimal point for hundredths. 6._ _ Write the numerals for seventeen in the places marked. 6.17 Write fourteen and thirty-seven hundredths as a decimal. fourteen and thirty-seven hundredths Place a decimal point under the word ‘and’. ______. _________ Translate the words before ‘and’ into the whole number and place it to the left of the decimal point. 14. _________ Mark two places to the right of the decimal point for “hundredths”. 14.__ __ Translate the words after “and” and write the number to the right of the decimal point. 14.37 Fourteen and thirty-seven hundredths is written 14.37. Write as a decimal: thirteen and sixty-eight hundredths. 13.68 Write as a decimal: five and eight hundred ninety-four thousandths. 5.894 Look for the word “and”—it locates the decimal point. Mark the number of decimal places needed to the right of the decimal point by noting the place value indicated by the last word. Place a decimal point under the word “and.” Translate the words before “and” into the whole number and place it to the left of the decimal point. If there is no “and,” write a “0” with a decimal point to its right. Translate the words after “and” into the number to the right of the decimal point. Write the number in the spaces—putting the final digit in the last place. Fill in zeros for place holders as needed. The second bullet in Step 2 is needed for decimals that have no whole number part, like ‘nine thousandths’. We recognize them by the words that indicate the place value after the decimal – such as ‘tenths’ or ‘hundredths.’ Since there is no whole number, there is no ‘and.’ We start by placing a zero to the left of the decimal and continue by filling in the numbers to the right, as we did above. Write twenty-four thousandths as a decimal. twenty-four thousandths Look for the word "and". There is no "and" so start with 0 0. To the right of the decimal point, put three decimal places for thousandths. Write the number 24 with the 4 in the thousandths place. Put zeros as placeholders in the remaining decimal places. 0.024 So, twenty-four thousandths is written 0.024 Write as a decimal: fifty-eight thousandths. 0.058 Write as a decimal: sixty-seven thousandths. 0.067 Before we move on to our next objective, think about money again. We know that $1 is the same as $1.00. The way we write $1(or$1.00) depends on the context. In the same way, integers can be written as decimals with as many zeros as needed to the right of the decimal. 5=5.0−2=−2.05=5.00−2=−2.005=5.000−2=−2.000 and so on…

We often need to rewrite decimals as fractions or mixed numbers. Let’s go back to our lunch order to see how we can convert decimal numbers to fractions. We know that $5.03 means 5 dollars and 3 cents. Since there are 100 cents in one dollar, 3 cents means 3100 of a dollar, so 0.03=3100. We convert decimals to fractions by identifying the place value of the farthest right digit. In the decimal 0.03, the 3 is in the hundredths place, so 100 is the denominator of the fraction equivalent to 0.03. 0.03=3100 For our $5.03 lunch, we can write the decimal 5.03 as a mixed number. 5.03=53100 Notice that when the number to the left of the decimal is zero, we get a proper fraction. When the number to the left of the decimal is not zero, we get a mixed number. Look at the number to the left of the decimal. If it is zero, the decimal converts to a proper fraction. If it is not zero, the decimal converts to a mixed number. Write the whole number. Determine the place value of the final digit. Write the fraction. numerator—the ‘numbers’ to the right of the decimal point denominator—the place value corresponding to the final digit Simplify the fraction, if possible. Write each of the following decimal numbers as a fraction or a mixed number: ⓐ 4.09 ⓑ 3.7 ⓒ −0.286 ⓐ 4.09 There is a 4 to the left of the decimal point. Write "4" as the whole number part of the mixed number. Determine the place value of the final digit. Write the fraction. Write 9 in the numerator as it is the number to the right of the decimal point. Write 100 in the denominator as the place value of the final digit, 9, is hundredth. The fraction is in simplest form. Did you notice that the number of zeros in the denominator is the same as the number of decimal places? ⓑ 3.7 There is a 3 to the left of the decimal point. Write "3" as the whole number part of the mixed number. Determine the place value of the final digit. Write the fraction. Write 7 in the numerator as it is the number to the right of the decimal point. Write 10 in the denominator as the place value of the final digit, 7, is tenths. The fraction is in simplest form. ⓒ −0.286 There is a 0 to the left of the decimal point. Write a negative sign before the fraction. Determine the place value of the final digit and write it in the denominator. Write the fraction. Write 286 in the numerator as it is the number to the right of the decimal point.Write 1,000 in the denominator as the place value of the final digit, 6, is thousandths. We remove a common factor of 2 to simplify the fraction. Write as a fraction or mixed number. Simplify the answer if possible. ⓐ 5.3 ⓑ 6.07 ⓒ −0.234 ⓐ 5310ⓑ 67100 ⓒ −117500 Write as a fraction or mixed number. Simplify the answer if possible. ⓐ 8.7 ⓑ 1.03 ⓒ −0.024 ⓐ 8710ⓑ 13100ⓒ −3125

Since decimals are forms of fractions, locating decimals on the number line is similar to locating fractions on the number line. Locate 0.4 on a number line. The decimal 0.4 is equivalent to 410, so 0.4 is located between 0 and 1. On a number line, divide the interval between 0 and 1 into 10 equal parts and place marks to separate the parts. Label the marks 0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0. We write 0 as 0.0 and 1 as 1.0, so that the numbers are consistently in tenths. Finally, mark 0.4 on the number line. Locate 0.6 on a number line. Locate 0.9 on a number line. Locate −0.74 on a number line. The decimal −0.74 is equivalent to −74100, so it is located between 0 and −1. On a number line, mark off and label the multiples of -0.10 in the interval between 0 and −1 (−0.10, −0.20, etc.) and mark −0.74 between −0.70 and −0.80, a little closer to −0.70. Locate −0.63 on a number line. Locate −0.25 on a number line.

Which is larger, 0.04 or 0.40? If you think of this as money, you know that $0.40 (forty cents) is greater than $0.04 (four cents). So, 0.40>0.04 In previous chapters, we used the number line to order numbers. a<b‘ais less thanb’whenais to the left ofbon the number linea>b‘ais greater thanb’whenais to the right ofbon the number line Where are 0.04 and 0.40 located on the number line? We see that 0.40 is to the right of 0.04. So we know 0.40>0.04. How does 0.31 compare to 0.308? This doesn’t translate into money to make the comparison easy. But if we convert 0.31 and 0.308 to fractions, we can tell which is larger. 0.31 0.308 Convert to fractions. 31100 3081000 We need a common denominator to compare them. 3081000 3101000 3081000 Because 310>308, we know that 3101000>3081000. Therefore, 0.31>0.308. Notice what we did in converting 0.31 to a fraction—we started with the fraction 31100 and ended with the equivalent fraction 3101000. Converting 3101000 back to a decimal gives 0.310. So 0.31 is equivalent to 0.310. Writing zeros at the end of a decimal does not change its value. 31100=3101000and0.31=0.310 If two decimals have the same value, they are said to be equivalent decimals. 0.31=0.310 We say 0.31 and 0.310 are equivalent decimals. Two decimals are equivalent decimals if they convert to equivalent fractions. Remember, writing zeros at the end of a decimal does not change its value. Check to see if both numbers have the same number of decimal places. If not, write zeros at the end of the one with fewer digits to make them match. Compare the numbers to the right of the decimal point as if they were whole numbers. Order the numbers using the appropriate inequality sign. Order the following decimals using <or>: ⓐ 0.64__0.6 ⓑ 0.83__0.803 ⓐ 0.64__0.6 Check to see if both numbers have the same number of decimal places. They do not, so write one zero at the right of 0.6. 0.64__0.60 Compare the numbers to the right of the decimal point as if they were whole numbers. 64>60 Order the numbers using the appropriate inequality sign. 0.64>0.600.64>0.6 ⓑ 0.83__0.803 Check to see if both numbers have the same number of decimal places. They do not, so write one zero at the right of 0.83. 0.830__0.803 Compare the numbers to the right of the decimal point as if they were whole numbers. 830>803 Order the numbers using the appropriate inequality sign. 0.830>0.8030.83>0.803 Order each of the following pairs of numbers, using <or>: ⓐ 0.42__0.4 ⓑ 0.76__0.706 ⓐ >ⓑ > Order each of the following pairs of numbers, using <or>: ⓐ 0.1__0.18 ⓑ 0.305__0.35 ⓐ < ⓑ < When we order negative decimals, it is important to remember how to order negative integers. Recall that larger numbers are to the right on the number line. For example, because −2 lies to the right of −3 on the number line, we know that −2>−3. Similarly, smaller numbers lie to the left on the number line. For example, because −9 lies to the left of −6 on the number line, we know that −9<−6. If we zoomed in on the interval between 0 and −1, we would see in the same way that −0.2>−0.3and−0.9<−0.6. Use <or> to order. −0.1__−0.8. −0.1__−0.8 Write the numbers one under the other, lining up the decimal points. −0.1−0.8 They have the same number of digits. Since −1>−8,−1 tenth is greater than −8 tenths. −0.1>−0.8 Order each of the following pairs of numbers, using <or>: −0.3___−0.5 > Order each of the following pairs of numbers, using <or>: −0.6___−0.7 >

In the United States, gasoline prices are usually written with the decimal part as thousandths of a dollar. For example, a gas station might post the price of unleaded gas at $3.279 per gallon. But if you were to buy exactly one gallon of gas at this price, you would pay $3.28, because the final price would be rounded to the nearest cent. In Whole Numbers, we saw that we round numbers to get an approximate value when the exact value is not needed. Suppose we wanted to round $2.72 to the nearest dollar. Is it closer to $2 or to $3? What if we wanted to round $2.72 to the nearest ten cents; is it closer to $2.70 or to $2.80? The number lines in can help us answer those questions.

ⓐ We see that 2.72 is closer to 3 than to 2. So, 2.72 rounded to the nearest whole number is 3. ⓑ We see that 2.72 is closer to 2.70 than 2.80. So we say that 2.72 rounded to the nearest tenth is 2.7.

Can we round decimals without number lines? Yes! We use a method based on the one we used to round whole numbers. Locate the given place value and mark it with an arrow. Underline the digit to the right of the given place value. Is this digit greater than or equal to 5? Yes - add 1 to the digit in the given place value. No - do not change the digit in the given place value Rewrite the number, removing all digits to the right of the given place value. Round 18.379 to the nearest hundredth. Locate the hundredths place and mark it with an arrow. Underline the digit to the right of the 7. Because 9 is greater than or equal to 5, add 1 to the 7. Rewrite the number, deleting all digits to the right of the hundredths place. 18.38 is 18.379 rounded to the nearest hundredth. Round to the nearest hundredth: 1.047. 1.05 Round to the nearest hundredth: 9.173. 9.17 Round 18.379 to the nearest ⓐ tenth ⓑ whole number. ⓐ Round 18.379 to the nearest tenth. Locate the tenths place and mark it with an arrow. Underline the digit to the right of the tenths digit. Because 7 is greater than or equal to 5, add 1 to the 3. Rewrite the number, deleting all digits to the right of the tenths place. So, 18.379 rounded to the nearest tenth is 18.4. ⓑ Round 18.379 to the nearest whole number. Locate the ones place and mark it with an arrow. Underline the digit to the right of the ones place. Since 3 is not greater than or equal to 5, do not add 1 to the 8. Rewrite the number, deleting all digits to the right of the ones place. So 18.379 rounded to the nearest whole number is 18. Round 6.582 to the nearest ⓐ hundredth ⓑ tenth ⓒ whole number. ⓐ 6.58 ⓑ 6.6 ⓒ 7 Round 15.2175 to the nearest ⓐ thousandth ⓑ hundredth ⓒ tenth. ⓐ 15.218 ⓑ 15.22 ⓒ 15.2 Introduction to Decimal Notation Write a Number in Decimal Notation from Words Identify Decimals on the Number Line Rounding Decimals Writing a Decimal as a Simplified Fraction

Name a decimal number. Name the number to the left of the decimal point. Write “and” for the decimal point. Name the “number” part to the right of the decimal point as if it were a whole number. Name the decimal place of the last digit. Write a decimal number from its name. Look for the word “and”—it locates the decimal point. Place a decimal point under the word “and.” Translate the words before “and” into the whole number and place it to the left of the decimal point. If there is no “and,” write a “0” with a decimal point to its right. Mark the number of decimal places needed to the right of the decimal point by noting the place value indicated by the last word. Translate the words after “and” into the number to the right of the decimal point. Write the number in the spaces—putting the final digit in the last place. Fill in zeros for place holders as needed. Convert a decimal number to a fraction or mixed number. Look at the number to the left of the decimal. If it is zero, the decimal converts to a proper fraction. If it is not zero, the decimal converts to a mixed number. Write the whole number. Determine the place value of the final digit. Write the fraction. numerator—the ‘numbers’ to the right of the decimal point denominator—the place value corresponding to the final digit Simplify the fraction, if possible. Order decimals. Check to see if both numbers have the same number of decimal places. If not, write zeros at the end of the one with fewer digits to make them match. Compare the numbers to the right of the decimal point as if they were whole numbers. Order the numbers using the appropriate inequality sign. Round a decimal. Locate the given place value and mark it with an arrow. Underline the digit to the right of the given place value. Is this digit greater than or equal to 5? Yes - add 1 to the digit in the given place value. No - do not change the digit in the given place value Rewrite the number, removing all digits to the right of the given place value.

Name Decimals In the following exercises, name each decimal. 5.5 five and five tenths 7.8 5.01 five and one hundredth 14.02 8.71 eight and seventy-one hundredths 2.64 0.002 two thousandths 0.005 0.381 three hundred eighty-one thousandths 0.479 −17.9 negative seventeen and nine tenths −31.4 Write Decimals In the following exercises, translate the name into a decimal number. Eight and three hundredths 8.03 Nine and seven hundredths Twenty-nine and eighty-one hundredths 29.81 Sixty-one and seventy-four hundredths Seven tenths 0.7 Six tenths One thousandth 0.001 Nine thousandths Twenty-nine thousandths 0.029 Thirty-five thousandths Negative eleven and nine ten-thousandths −11.0009 Negative fifty-nine and two ten-thousandths Thirteen and three hundred ninety-five ten thousandths 13.0395 Thirty and two hundred seventy-nine thousandths Convert Decimals to Fractions or Mixed Numbers In the following exercises, convert each decimal to a fraction or mixed number. 1.99 199100 5.83 15.7 15710 18.1 0.239 2391000 0.373 0.13 13100 0.19 0.011 111000 0.049 −0.00007 −7100000 −0.00003 6.4 625 5.2 7.05 7120 9.04 4.006 43500 2.008 10.25 1014 12.75 1.324 181250 2.482 14.125 1418 20.375 Locate Decimals on the Number Line In the following exercises, locate each number on a number line. 0.8 0.3 −0.2 −0.9 3.1 2.7 −2.5 −1.6 Order Decimals In the following exercises, order each of the following pairs of numbers, using <or>. 0.9__0.6 > 0.7__0.8 0.37__0.63 < 0.86__0.69 0.6__0.59 > 0.27__0.3 0.91__0.901 > 0.415__0.41 −0.5__−0.3 < −0.1_−0.4 −0.62_−0.619 < −7.31_−7.3 Round Decimals In the following exercises, round each number to the nearest tenth. 0.67 0.7 0.49 2.84 2.8 4.63 In the following exercises, round each number to the nearest hundredth. 0.845 0.85 0.761 5.7932 5.79 3.6284 0.299 0.30 0.697 4.098 4.10 7.096 In the following exercises, round each number to the nearest ⓐ hundredth ⓑ tenth ⓒ whole number. 5.781 ⓐ 5.78 ⓑ 5.8 ⓒ 6 1.638 63.479 ⓐ 63.48 ⓑ 63.5 ⓒ 63 84.281

Salary Increase Danny got a raise and now makes $58,965.95 a year. Round this number to the nearest: ⓐ dollar ⓑ thousand dollars ⓒ ten thousand dollars. ⓐ $58,966 ⓑ $59,000 ⓒ $60,000 New Car Purchase Selena’s new car cost $23,795.95. Round this number to the nearest: ⓐ dollar ⓑ thousand dollars ⓒ ten thousand dollars. Sales Tax Hyo Jin lives in San Diego. She bought a refrigerator for $1624.99 and when the clerk calculated the sales tax it came out to exactly $142.186625. Round the sales tax to the nearest ⓐ penny ⓑ dollar. ⓐ $142.19 ⓑ $142 Sales Tax Jennifer bought a $1,038.99 dining room set for her home in Cincinnati. She calculated the sales tax to be exactly $67.53435. Round the sales tax to the nearest ⓐ penny ⓑ dollar.

How does your knowledge of money help you learn about decimals? Answers will vary. Explain how you write “three and nine hundredths” as a decimal. Jim ran a 100-meter race in 12.32 seconds. Tim ran the same race in 12.3 seconds. Who had the faster time, Jim or Tim? How do you know? Tim had the faster time. 12.3 is less than 12.32, so Tim had the faster time. Gerry saw a sign advertising postcards marked for sale at “10for0.99¢.” What is wrong with the advertised price?

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ If most of your checks were: …confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific. …with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved? …no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.

equivalent decimals Two decimals are equivalent decimals if they convert to equivalent fractions.