Free Pre-Algebra Lesson 31! page 1

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1 Free Pre-Algebra Lesson! page Lesson 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. For example, in the number,42,06 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 in the hundreds place would mean ten rather than hundred. Each place value is a power of 0, because it comes from multiplying the place to the right by millions thousands hundreds tens ones hundreds tens ones hundreds tens ones, 4 2, 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 /0 the place to the left, so after the ones place we have the fractional places /0, /00, /000, etc. We write a decimal point after the ones place to indicate that the digits to the right represent fractional parts of a whole thousands hundreds tens ones tenths hundredths thousandths 2, /0 /0 2 /0 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. When you re writing a decimal number with no whole number part, you properly include a zero in the ones place In practice, however, many people leave the leading zero off: 0.02 Use your judgment about the requirements of a particular situation. Say It, Don t Spray It Careful with those th s. 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). 200 Cheryl Wilcox

2 Free Pre-Algebra Lesson! 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, 2,06.02 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 0.02 into its parts to see how it is equal to twelve thousandths, that is 2/ thousands hundreds tens ones tenths hundredths thousandths 2, /0 /0 2 /0 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 /000. In expanded form, this number is equal to! 2( 000) + ( 00) + 0( 0) + 6( ) + 0 $ 0% & +! $ 00% & + 2! $ 000% & The fraction part (after the decimal point) is! 0 $ 0% & +! $ 00% & + 2! $ 000% & = = = = Example: Fill in the missing columns in the chart. Decimal Formal Name Fraction 0. ninety-seven hundredths twenty-five and seventeen thousandths three tenths Cheryl Wilcox

3 Free Pre-Algebra Lesson! page 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 /00 has a denominator of 00. Find the hundredths place on the chart and write there: ones tenths hundredths thousandths 0. /0 /0 2 /0 /0 4 /0 Then flll in any zeros that come between the decimal point and the numerator. ones tenths hundredths thousandths 0. 0 /0 /0 2 /0 /0 4 /0 00 = 0.0 Zeros to the right of the last non-zero digit do not change the value of the number. The number 0.0 is different from 0., but 0.0 and are equivalent. Example: Write the fractions as decimals. Use the place value chart for reference. ones. tenths hundredths thousandths tenthousandths hundredthousandths tenthousandths hundredthousandths tenthousandths hundredthousandths /0 /00 /000 /0.000 /00, = = = 0. 00,000 = 0.00 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. 4 = or Cheryl Wilcox Notice that if you change /4 to an equivalent fraction with denominator 00, you get the same result. 4 = = 2 00 = 0.2

4 Free Pre-Algebra Lesson! page 4 Example: Write the fractions as decimals. = = = 2 2 = = 20 = 0. 2 = 2 0 = Sometimes the division does not work out evenly, resulting in what we call a repeating decimal. For example, = You can see from the long division that it can never terminate. The calculator displays as many s as it can fit onto the screen, but it cannot show infinitely many s as it should. To write s 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 s, followed by three dots (ellipses) that mean and so forth. 0. 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. = 0... When you change a fraction to a decimal, it will either terminate, as /4 did, or repeat, as / 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 = 0.6 = 6 = = = = 4 0 = Did you notice that your calculator rounded up because the last digit is greater than? 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. Here the two digit pattern 4 repeats forever. Compare this to the previous example. Here the four in the tenths place does not repeat. The repeating pattern of s begins in the hundredths place. 200 Cheryl Wilcox

5 Free Pre-Algebra Lesson! page 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 = = 4 0,000 A quick way to calculate the denominator is to write the numeral 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 (the factors of 0), the division will not be even, and the decimal will have a repeating part. 22 = 2 = 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 combinations.) 8 = = = 8 = 0. 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 = 02 + ( 4 ) = = 02.4 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.! 200 Cheryl Wilcox

6 Free Pre-Algebra Lesson! page 6 Lesson : Decimal Fractions Worksheet Name ones. tenths hundredths thousandths tenthousandths hundredthousandths /0 /00 /000 /0.000 /00,000 Fill in the missing columns in the chart. Use the place value chart for reference. Decimal Formal Name Fraction.0 three hundred eighty-one ten-thousandths 00 Write the decimals as fractions. Simplify the fractions to lowest terms Write the fractions as decimals. Use a bar to indicate repeating decimals Cheryl Wilcox

7 Free Pre-Algebra Lesson! page 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 /0 instead of /2 for cm). True Size 200 Cheryl Wilcox

8 Free Pre-Algebra Lesson! page 8 Lesson : Decimal Fractions Homework A Name. a. Round 6, to the nearest thousand. b. Round 4,,2 to the nearest million. 2. Write four fractions equivalent to 2. Find the prime factorization of Simplify 2a2 20a. Simplify x! % 6. Evaluate a $ 20& ' 8 when a =! 8.. Find equivalent fractions with a common denominator Add, and write the answer as a mixed number Subtract. 8! Convert 0 inches to feet. (2 inches = foot). How many cups is 8 gallons? ( gallon = 4 quarts, quart = 4 cups) 2. The height (in feet, after t seconds) of a rock thrown down a deep well is given by the equation h =!6t 2! 24t. The bottom of the well is at 00 feet. Has the rock already hit the bottom when t = seconds?. Convert feet to inches, then use the distance-rate-time formula. A model train ran at the rate of inches per second. How long does it take to travel a 6 foot track? 200 Cheryl Wilcox

9 Free Pre-Algebra Lesson! page Solve the equations x = 2.!y + 4 = 6. x! =. x + 4 = 8 8. Fill in the blanks. Decimal Formal Name Fraction 0. two and four hundredths 4 0. Write the decimals as fractions or mixed numbers. Simplify to lowest terms Write the fractions as decimals. Use a bar for repeating decimals Cheryl Wilcox

10 Free Pre-Algebra Lesson! page 0 Lesson : Decimal Fractions Homework A Answers. a. Round 6, to the nearest thousand.,000 b. Round 4,,2 to the nearest million. 4,000, Write four fractions equivalent to 2 = 8 0 = 2 = 6 00 = 4 2. Find the prime factorization of = 2 4. Simplify 2a2 20a = 2 a a 2 a = a. Simplify x! % $ 20& ' x! = x! 6. Evaluate a 8 when a =! 8.! 8 8 =! 8 8 =! 8 8 =!. Find equivalent fractions with a common denominator. = = 24 6 = = Add, and write the answer as a mixed number = = 2 24 = 24. Subtract. 8! 2 6 = 24! = 6 24! = Convert 0 inches to feet. (2 inches = foot) 0 in ft 2 in = 0 2 ft = ft = 4 6 ft. How many cups is 8 gallons? ( gallon = 4 quarts, quart = 4 cups) 8 gal 4 qts gal 4 cups = 288 cups qt 2. The height (in feet, after t seconds) of a rock thrown down a deep well is given by the equation h =!6t 2! 24t. The bottom of the well is at 00 feet. Has the rock already hit the bottom when t = seconds? h =!6( ) 2! 24( ) =!6 2! 24 =!42 No, it is still above 00 feet.. Convert feet to inches, then use the distance-rate-time formula. A model train ran at the rate of inches per second. How long does it take to travel a 6 foot track? 6 ft 2 in ft = 2 in d = rt 2 = t 2 = t t / = 2 / t = 24 seconds 200 Cheryl Wilcox

11 Free Pre-Algebra Lesson! page Solve the equations x = 2 4 x = x = 2 x = 2 4.!y + 4 =!y + 4 =! y + 4! 4 =! 4!y =! y /! = /! y =! 6. x! = x! = x! + = + x = 4 x / = 4 /. x + 4 = 8 x + 4 = 8 x + 4! 4 = 8! 4 x = 4 x = 8! 4 = 4 4! 4 = 4 8. Fill in the blanks. Decimal Formal Name Fraction 0. nineteen hundredths two and four hundredths.4 three and four tenths Write the decimals as fractions or mixed numbers. Simplify to lowest terms = 000 = 200. = 00.2 = = Write the fractions as decimals. Use a bar for repeating decimals. 2 = 0. = 0. 2 = = 0.2 = = 0.6 = = 0.2 = Cheryl Wilcox

12 Free Pre-Algebra Lesson! page 2 Lesson : Decimal Fractions Homework B Name. a. Round 0,88 to the nearest ten thousand. b. Round 84,08,6 to the nearest million. 2. Write four fractions equivalent to 8. Find the prime factorization of Simplify 8a2 b. Simplify 0 48a 2 y! % 6. Evaluate $ 0& ' x when x =!.. Find equivalent fractions with a common denominator Subtract. 2! 4. Add Convert 8 inches to feet. (2 inches = foot). How many minutes is 4 days? ( day = 24 hours, hour = 60 minutes) 2. The height (in feet, after t seconds) of a rock thrown down a deep well is given by the equation h =!6t 2! 24t. The bottom of the well is at 00 feet. Has the rock already hit the bottom when t = 6 seconds?. Convert 8 feet to inches, then use the distance-rate-time formula. A model train ran at the rate of inches per second. How long does it take to travel an 8 foot track? 200 Cheryl Wilcox

13 Free Pre-Algebra Lesson! page Solve the equations. 4. 8n + 4 = 2. 2 y = = x! 2. w + = 0 8. Fill in the blanks. Decimal Formal Name Fraction 0.00 three and seven tenths Write the decimals as fractions or mixed numbers. Simplify to lowest terms Write the fractions as decimals. Use a bar for repeating decimals Cheryl Wilcox

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