Free Pre-Algebra Lesson 31 ! page 1 Lesson 31 Decimal Fractions Expressing parts of a whole is a mathematical problem with several solutions. One way to do this is the ordinary fractions (classically called the “common” or “vulgar” fractions) we have been using in this book so far. Another way to express parts of a whole is with decimals. U.S. currency uses a decimal system, dollars and cents. Calculators default to decimals when displaying results. Decimals form the basis of the metric system. Many people feel more comfortable with decimal representations than with fractions. Decimals have many advantages – and some disadvantages we will also explore. Place Value Revisited The number system we use for whole numbers is called a Place Value system, because the placement of the digit in the number tells its value. It is a Base Ten system, because each place is worth ten times the next place to the right.1 For example, in the number 5,432,106 each digit has a place that defines its value. The rightmost digit is in the ones place, so we have 6 ones. The next digit, 0, holds the tens place – it tells us that there are no tens. We need the zero, because otherwise the 1 in the hundreds place would mean 1 ten rather than 1 hundred. Each place value is a power of 10, because it comes from multiplying the place to the right by 10. 108 107 millions 106 105 104 thousands 103 102 101 1 hundreds tens ones hundreds tens ones hundreds tens ones 5, 4 3 2, 1 0 6 The big idea for decimal fractions is to expand the place value system to the right, using the same principle that each place value is ten times the one to the right. Ten times the place to the right means the same as 1/10 the place to the left, so after the ones place we have the fractional places 1/10, 1/100, 1/1000, etc. We write a decimal point after the ones place to indicate that the digits to the right represent fractional parts of a whole. 103 102 101 1 thousands hundreds tens ones 2, 1 0 6. tenths hundredths thousandths 0 1 2 1/101 1/102 1/103 The ones place is highlighted above because it is the center of the number names. To the left of the ones place is the tens, to the right is the tenths. The number names proceed in the same order to the right as they did to the left, but the name is followed by the suffix “th” to indicate a fraction. Say It, Don’t Spray It Careful with those “th”s. When you’re writing a decimal number with no whole number part, you properly include a zero in the ones place. 0.012 In practice, however, many people leave the leading zero off: 0.012 Use your judgment about the requirements of a particular situation. The names also tell us that other systems are possible! Think about Roman numerals (not a place value system), or about the binary numbers used in computers (not a base ten system). 1 © 2010 Cheryl Wilcox Free Pre-Algebra Lesson 31 ! page 2 Reading and Writing Decimals To read a decimal number properly, say “and” for the decimal point, the regular name for the number after the decimal point, then the place value of the last digit on the right. The number, 2,106.012 is formally read “two thousand, one hundred six and twelve thousandths” The formal name is rarely used, (most people would say “point oh one two” rather than “and twelve thousandths”) but it is helpful because it reveals that the decimal is really a fraction. Let’s break down the decimal 0.012 into its parts to see how it is equal to twelve thousandths, that is 12/1000. 103 102 101 1 thousands hundreds tens ones 2, 1 0 6. tenths hundredths thousandths 0 1 2 1/101 1/102 1/103 Just as the 2 in the thousands place means two groups of a thousand, the 2 in the thousandths place means there are two parts of size 1/1000. In expanded form, this number is equal to ! 1$ ! 1 $ ! 1 $ 2 1000 + 1 100 + 0 10 + 6 1 + 0 # & + 1# + 2 #" 1000 &% " 10 % " 100 &% ( ) ( ) ( ) () The fraction part (after the decimal point) is ! 1$ ! 1 $ ! 1 $ 0 # & + 1# + 2# & " 10 % " 100 % " 1000 &% 1 2 1 10 2 + = • + 100 1000 100 10 1000 10 2 12 = + = 1000 1000 1000 = 0+ Example: Fill in the missing columns in the chart. Decimal Formal Name 0.97 Fraction “ninety-seven hundredths” 97 100 25.017 0.3 “twenty-five and seventeen thousandths” “three tenths” 25 17 1000 3 10 © 2010 Cheryl Wilcox Free Pre-Algebra Lesson 31 ! page 3 From Fraction to Decimal If a fraction has a denominator that is a power of ten, you can write it as a decimal just by filling in the numerator in the place value chart with the rightmost digit in the place value of the denominator. For example, the fraction 7/100 has a denominator of 100. Find the hundredths place on the chart and write 7 there: 1 ones tenths hundredths 1/101 1/102 0. thousandths tenthousandths hundredthousandths 1/103 1/104 1/105 7 Then flll in any zeros that come between the decimal point and the numerator. 1 ones tenths 0. hundredths 0 7 1/101 1/102 thousandths tenthousandths hundredthousandths 1/103 1/104 1/105 7 = 0.07 100 Zeros to the right of the last non-zero digit do not change the value of the number. The number 0.07 is different from 0.7, but 0.07 and 0.0700 are equivalent. Example: Write the fractions as decimals. Use the place value chart for reference. 1 ones tenths hundredths thousandths tenthousandths hundredthousandths 1/10 1/100 1/1000 1/10.000 1/100,000 . 19 = 0.19 100 19 = 0.019 1000 199 = 0.199 1000 199 = 0.00199 100,000 Suppose you have just any random fraction, without a special power of ten denominator. To change any fraction to a decimal, we read the fraction bar as a division symbol, and do long division or divide on a calculator. 1 = 1÷ 4 4 0.25 4 1.00 0.8 0.20 0.20 0.00 or Notice that if you change 1/4 to an equivalent fraction with denominator 100, you get the same result. 1 1 25 25 = • = = 0.25 4 4 25 100 © 2010 Cheryl Wilcox Free Pre-Algebra Lesson 31 ! page 4 Example: Write the fractions as decimals. 3 = 3 ÷ 5 = 0.6 5 21 = 21÷ 25 = 0.84 25 7 = 7 ÷ 20 = 0.35 20 27 = 27 ÷ 50 = 0.54 50 Sometimes the division does not work out evenly, resulting in what we call a repeating decimal. For example, 1 = 1÷ 3 3 0.333 3 1.000 0.9 0.100 0.090 0.010 You can see from the long division that it can never terminate. The calculator displays as many 3s as it can fit onto the screen, but it cannot show infinitely many 3s as it should. To write 3s on to infinity is impossible, so we have three choices: We can round the decimal to some predetermined place value. For example, your calculator rounds repeating decimals to the number of digits in its display. We can write a few 3s, followed by three dots (ellipses) that mean “and so forth.” 0.3333… This notation is considered casual, and somewhat unmathematical. It can be confusing in some circumstances because it is not specified exactly what the repeating part is. The correct mathematical notation is to put a bar over the repeating part of the decimal. This is difficult to type but easy to write by hand. 0.3 = 0.33333333333333333333333333333333... When you change a fraction to a decimal, it will either terminate, as 1/4 did, or repeat, as 1/3 did. However, sometimes the repeating part has one or more leading digits before it begins, and sometimes the repeating pattern has more than one digit. Example: Write the fractions as decimals. 2 = 2 ÷ 3 = 0.6 3 1 = 1÷ 6 = 0.16 6 5 = 5 ÷ 11= 0.45 11 41 = 41÷ 90 = 0.45 90 Did you notice that your calculator rounded up because the last digit is greater than 5? The one in the tenths place does not repeat.The repeating part of the decimal begins at the 6. Notice how your calculator rounds the last digit to 7. Here the two digit pattern 45 repeats forever. Compare this to the previous example. Here the four in the tenths place does not repeat. The repeating pattern of 5s begins in the hundredths place. © 2010 Cheryl Wilcox Free Pre-Algebra Lesson 31 ! page 5 Summary (and More) Decimals are an alternate way to write fractions of a whole amount. Decimal notation is equivalent to writing a fraction with a denominator that is a power of ten. To convert a decimal to a fraction, write the number after the decimal point as the numerator and use the place value of the rightmost digit as the denominator. 0.0047 = 0047 47 = 10000 10,000 A quick way to calculate the denominator is to write the numeral 1 followed by as many zeros as there are places after the decimal point. You can see in this example the decimal point is followed by four digits and the denominator of the fraction has four zeros. To convert a fraction to a decimal, interpret the fraction bar as a division symbol. If the denominator of the fraction has any factors other than 2 and 5 (the factors of 10), the division will not be even, and the decimal will have a repeating part. 5 5 = 22 2 • 11 5 = 5 ÷ 22 = 0.227 22 If the fraction can be converted to an equivalent fraction with denominator equal to a power of ten, then the decimal will terminate. (Convert by creating 2•5 combinations.) 3 3 5 • 5 • 5 375 = • = 8 2 • 2 • 2 5 • 5 • 5 1000 3 = 3 ÷ 8 = 0.375 8 If you are converting a mixed number, concentrate on the fraction part. You can append the whole number part before the decimal point once you’ve converted. 302 ( ) 43 = 302 + 43 ÷ 99 = 302 + 0.43 = 302.43 99 The disadvantage of decimal representations is the loss of precision. While the infinitely repeating form of a decimal fraction is precise, for practical purposes we must round repeating decimals before we can use them. The advantage of decimal representations lies in the ease of arithmetic operations, as we’ll see in the next section. Because we are working in our familiar place value system, the arithmetic operations are similar to those with whole numbers, rather than the special rules for fractions. ! © 2010 Cheryl Wilcox Free Pre-Algebra Lesson 31 ! page 6 Lesson 31: Decimal Fractions Worksheet Name _________________________________ 1 ones tenths hundredths thousandths tenthousandths hundredthousandths 1/10 1/100 1/1000 1/10.000 1/100,000 . Fill in the missing columns in the chart. Use the place value chart for reference. Decimal Formal Name Fraction 5.07 “three hundred eighty-one ten-thousandths” 13 Write the decimals as fractions. Simplify the fractions to lowest terms. 5.2 5.02 5.20 0.502 Write the fractions as decimals. Use a bar to indicate repeating decimals. 33 100 9 10 9 100 1 1000 8 25 39 40 49 90 6 11 1 9 2 9 1 3 7 9 © 2010 Cheryl Wilcox 33 100 Free Pre-Algebra Lesson 31 ! page 7 This ruler has been enlarged so you can see the marks clearly. The true size is shown at the bottom of the page. The upper part of the ruler, marked in inches, is divided into eighths. The lower part of the ruler, marked in centimeters, is divided into tenths. Write both the fraction and the decimal name for each point. The convention for centimeters is to use tenths rather than simplifying to lower terms (write 5/10 instead of 1/2 for cm). True Size © 2010 Cheryl Wilcox Free Pre-Algebra Lesson 31 ! page 8 Lesson 31: Decimal Fractions Homework 31A 1. a. Round 76,939 to the nearest thousand. b. Round 4,377,299 to the nearest million. 23a 2 207a Name __________________________________ 2. Write four fractions equivalent to 3. Find the prime factorization of 207. 9 25 " 7 19 % x! ' 5. Simplify 60 $ 20 & # 60 6. Evaluate 7. Find equivalent fractions with a common denominator. 8. Add, and write the answer as a mixed number. 9. Subtract. 3 8 3 5 + 8 6 7 4. Simplify a 8 when a = ! . 8 9 3 5 !2 8 6 5 6 10. Convert 50 inches to feet. (12 inches = 1 foot) 11. How many cups is 18 gallons? (1 gallon = 4 quarts, 1 quart = 4 cups) 12. The height (in feet, after t seconds) of a rock thrown down a deep well is given by the equation h = !16t 2 ! 24t . The bottom of the well is at –500 feet. 13. Convert feet to inches, then use the distance-rate-time formula. Has the rock already hit the bottom when t = 5 seconds? © 2010 Cheryl Wilcox A model train ran at the rate of 3 inches per second. How long does it take to travel a 6 foot track? Free Pre-Algebra Lesson 31 ! page 9 Solve the equations. 14. 4 x = 12 9 16. 3 5 9 x! = 7 7 7 15. !7 y + 4 = 11 17. x + 3 =8 4 18. Fill in the blanks. Decimal Formal Name Fraction 0.19 “two and four hundredths” 3 19. Write the decimals as fractions or mixed numbers. Simplify to lowest terms. 0.005 3.11 7.125 20. Write the fractions as decimals. Use a bar for repeating decimals. 1 2 1 3 2 3 1 4 1 5 1 6 1 7 1 8 1 9 © 2010 Cheryl Wilcox 4 10 Free Pre-Algebra Lesson 31 ! page 10 Lesson 31: Decimal Fractions Homework 31A Answers 1. a. Round 76,939 to the nearest thousand. 77,000 b. Round 4,377,299 to the nearest million. 2. Write four fractions equivalent to 3. Find the prime factorization of 207. 207 = 3 • 3 • 23 9 18 27 36 45 = = = = 25 50 75 100 125 4,000,000 " 7 23a 2 4. Simplify 207a = 19 % x! ' 5. Simplify 60 $ 20 & # 60 23 • a • a = 3 • 3 • 23 • a a 9 60 • 7 60 3 x ! 60 • 19 20 6. Evaluate = 7x ! 57 8 9 = !8 ÷8= ! 8 • 1 = !1 8 9 9 8 9 ! 7. Find equivalent fractions with a common denominator. 8. Add, and write the answer as a mixed number. 9. Subtract. 3 3 3 9 = • = 8 2 • 2 • 2 3 24 3 5 9 20 29 5 + = + = =1 8 6 24 24 24 24 7 3 5 9 20 !2 =7 !2 8 6 24 24 =6 5 5 2 • 2 20 = • = 6 2 • 3 2 • 2 24 10. Convert 50 inches to feet. (12 inches = 1 foot) 50 in 1 ft 50 2 1 • = ft = 4 ft = 4 ft 1 12 6 12 in 12 12. The height (in feet, after t seconds) of a rock thrown down a deep well is given by the equation h = !16t 2 ! 24t . The bottom of the well is at –500 feet. Has the rock already hit the bottom when t = 5 seconds? () 2 () h = !16 5 ! 24 5 = !16 • 25 ! 24 • 5 = !429 No, it is still above –500 feet. © 2010 Cheryl Wilcox a 8 when a = ! . 8 9 33 20 13 !2 =4 24 24 24 11. How many cups is 18 gallons? (1 gallon = 4 quarts, 1 quart = 4 cups) 18 gal 1 • 4 qts 1 gal • 4 cups 1 qt = 288 cups 13. Convert feet to inches, then use the distance-rate-time formula. A model train ran at the rate of 3 inches per second. How long does it take to travel a 6 foot track? 6 ft 12 in • = 72 in 1 1 ft 72 = 3t d = rt 72 = 3t 3t / 3 = 72 / 3 t = 24 seconds Free Pre-Algebra Lesson 31 ! page 11 Solve the equations. 14. 15. !7 y + 4 = 11 4 x = 12 9 4 x = 12 9 !7 y + 4 = 11 !7 y = 7 3 9 4 9 • x = 12 • 4 9 4 ! 7 y + 4 ! 4 = 11! 4 ! 7 y / !7 = 7 / !7 y = !1 x = 27 16. 3 5 9 x! = 7 7 7 17. x + 3x ! 5 = 9 3x = 14 x= 3x ! 5 + 5 = 9 + 5 3 =8 4 3 3 3 3 =8 x + ! = 8! 4 4 4 4 3 4 3 1 x = 8! =7 ! =7 4 4 4 4 x+ 3x / 3 = 14 / 3 14 3 18. Fill in the blanks. Decimal Formal Name 0.19 Fraction “nineteen hundredths” 19 100 2.04 3.4 “two and four hundredths” 2 “three and four tenths” 5 1 = 1000 200 3.11 = 3 11 100 7.125 = 7 20. Write the fractions as decimals. Use a bar for repeating decimals. 1 = 0.5 2 1 = 0.3 3 2 = 0.6 3 1 = 0.25 4 1 = 0.2 5 1 = 0.16 6 1 = 0.142857 7 1 = 0.125 8 1 = 0.1 9 © 2010 Cheryl Wilcox 100 3 19. Write the decimals as fractions or mixed numbers. Simplify to lowest terms. 0.005 = 4 125 1 =7 1000 8 4 10 Free Pre-Algebra Lesson 31 ! page 12 Lesson 31: Decimal Fractions Homework 31B 1. a. Round 509,887 to the nearest ten thousand. b. Round 84,087,361 to the nearest million. 4. Simplify 38a 2b 418a 3 7. Find equivalent fractions with a common denominator. 2 3 4 7 Name _________________________________________ 2. Write four fractions equivalent to 3. Find the prime factorization of 418. 5 18 " 7 9% y! ' 5. Simplify 50 $ 10 & # 25 6. Evaluate 8. Subtract. 9. Add. 2 4 ! 3 7 5 3 !1 when x = . x 5 2 4 +8 3 7 10. Convert 85 inches to feet. (12 inches = 1 foot) 11. How many minutes is 14 days? (1 day = 24 hours, 1 hour = 60 minutes) 12. The height (in feet, after t seconds) of a rock thrown down a deep well is given by the equation h = !16t 2 ! 24t . The bottom of the well is at –500 feet. 13. Convert 8 feet to inches, then use the distance-rate-time formula. Has the rock already hit the bottom when t = 6 seconds? © 2010 Cheryl Wilcox A model train ran at the rate of 3 inches per second. How long does it take to travel an 8 foot track? Free Pre-Algebra Lesson 31 ! page 13 Solve the equations. 14. 8n + 14 = 12 16. 11 9 =x! 4 2 15. 2 y = 16 3 17. 7 3 10 w+ = 15 15 15 18. Fill in the blanks. Decimal Formal Name Fraction 0.009 “three and seven tenths” 10 19. Write the decimals as fractions or mixed numbers. Simplify to lowest terms. 0.012 5.75 7.375 20. Write the fractions as decimals. Use a bar for repeating decimals. 1 11 10 11 1 12 11 12 1 40 39 40 7 9 7 10 77 100 © 2010 Cheryl Wilcox 23 1000