Math:9

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Decimal Defined

Balancing your checkbook involves adding and subtracting decimals.

Decimals are very useful in daily life. Accountants, for example, deal with decimals all the time when adding and subtracting columns of dollars and cents and when calculating taxes. Engineers use decimals to calculate distances, angles, and dimensions. Retailers and clerks use decimals when marking or changing prices. Stockbrokers work with decimals to determine how much stocks and bonds have gone up or down. Bankers use decimals to calculate mortgage rates, car loans, and the interest earned on savings accounts. And any time you’re dealing with money, you, too, use decimals—to balance your checkbook, to calculate a tip at a restaurant, to determine how much you owe on a purchase, and many other instances.

A proper fraction is one way to express an amount less than a whole. For example, 1⁄3 is less than 3⁄3—that is, 1⁄3 is less than 1. Likewise, 2⁄5 is less than 5⁄5—that is, 2⁄5 is less than 1. A decimal is another way of writing a number that’s less than 1. In other words, a decimal is another way of writing a proper fraction. A decimal point (.) indicates the beginning of a decimal.

The prefix deci- in the word decimal means 10.

Place Values of Whole Numbers and Decimals

As you’ve already learned, each digit in a whole number represents a place value. Each place value in a whole number represents a quantity that’s ten times greater than the place value to its right. Decimals, too, have place values. As shown in the figure below, the digits to the left of the decimal point represent a whole number; the digits to the right of the decimal point represent a decimal, or a number less than 1.

Both whole numbers and decimals have place values.

Before you begin to study decimals, let’s review the place values of whole numbers. The digit at the far right in a whole number represents ones; the next digit to the left, tens; the next, hundreds; the fourth, thousands; and so on. Thus, the number 9,683,112 really represents the sum of these numbers:

2101003,00080,000600,000+9,000,000−−−−−−−−−−9,683,1122101003,00080,000600,000+9,000,000_9,683,112

To mark the original starting place in a whole number, add a decimal point to the far right. Thus, the whole number 9,683,112 can also be written as 9,683,112. and still have the same value. This type of notation is used for writing dollars and cents. Adding a decimal point at the end of a number doesn’t change the value of the number. Therefore, both $25 and $25.00 represent the same amount.

Now, look at the decimal part of the number in the previous figure. Just as with whole numbers, each digit in a decimal has a place value. If you start at the decimal point and move to the right, each place represents a quantity that’s 1⁄10 the value of the place to its left. The illustration shows the decimal .243576. In this number, the digit 2 represents 2⁄10; the digit 4 represents 4⁄100, the digit 3 represents 3⁄1,000, the digit 5 represents 5⁄10,000, the digit 7 represents 7⁄100,000, and the digit 6 represents 6⁄1,000,000. The place value of a digit in a decimal is determined by its distance from the decimal point. Thus, the decimal .243576 actually represents the sum of these numbers:

.2 .04 .003 .0005 .00007+.000006−−−−−−−− .243576 .2 .04 .003 .0005 .00007+.000006_ .243576

Example: For each of the digits in bold, write the decimal place name and then write the value of the equivalent fraction.

DigitDecimal Place NameFraction

.567tenths

510510

.865hundredths

61006100

.4378thousandths

71,00071,000

.9753ten-thousandths

310,000310,000

You may have noticed that although a decimal has no written denominator (as a fraction does), a decimal is read like a fraction. The denominator of the fraction form of a decimal is always a multiple of 10. The number of digits to the right of the decimal point tells you what the unwritten denominator is. The next section should help you to understand how to read a decimal properly.

Reading Decimals

To read a decimal, follow these steps:

Step 1: Ignore the decimal point and any zeros immediately to the right of the decimal point. (The importance of the zeros is explained later.) For example, in the decimal .003, ignore the decimal point and the two zeros. Instead, go immediately to the 3.

Step 2: Read the remaining digits as if they were a whole number. In the decimal .003, read the 3 as “three.”

Step 3: Say the place value of the digit on the far right. In the decimal .003, the digit on the far right is 3, which is in the thousandths place. Therefore, .003 should be read as “three thousandths.”

Example: Read each decimal aloud and then write it in words.

DecimalDecimal in words

.7seven tenths

.02two hundredths

.004four thousandths

.56fifty-six hundredths

.754seven hundred fifty-four thousandths

Writing Decimals as Fractions

To write a decimal as a fraction, follow these steps:

Step 1: Read the decimal aloud as you learned in the section above.

Step 2: Determine the place value of the decimal (tenths, hundredths, thousandths, and so on).

Step 3: Delete the decimal point and any zeros immediately following the decimal point.

Step 4: Write the decimal as a fraction, using the decimal as the numerator and the place value as the denominator.

The following examples illustrate this process for you.

Example: Write the decimal .23 as a fraction.

.23Read this decimal as “twenty-three hundredths.” Therefore, the place value is 100.

23Delete the decimal point from the numerator.

2310023100

Write the decimal as a fraction, using the decimal as the numerator and the place value as the denominator. Answer: 2310023100

Example: Write the decimal .016 as a fraction.

.016Read this decimal as “sixteen thousandths.” Therefore, the place value is 1,000.

16Delete the decimal point and any zeros immediately following it from the numerator.

161,000161,000

Write the decimal as a fraction using the decimal as the numerator and the place value as the denominator. Answer: 161,000161,000

Example: Write the following expressions as decimals. (Note the addition of zeros as necessary to hold place values.)

Decimal in WordsDecimal

four tenths.4

six tenths.6

six hundredths.06

forty-seven hundredths.47

six thousandths.006

fifteen thousandths.015

two hundred three thousandths.203

Moving the Decimal Point

The position of a decimal point in a number determines the value of that number. Therefore, if you move the decimal point, you change the value of the number. Moving the decimal point one place to the right is the same as multiplying the original number by 10. Moving the decimal point two places to the right multiplies the original number by 100. Moving the decimal point three places to the right multiplies the original number by 1,000. Study the following examples and notice the position of the decimal point in each one.

Rule: To multiply a decimal by any multiple of 10 (10, 100, 1,000, 10,000, 100,000, and so on), move the decimal point one place to the right for every zero in the multiplier.

Likewise, moving a decimal point one place to the left is the same as dividing the original number by 10. Moving the decimal point two places to the left divides the original number by 100. Moving the decimal point three places to the left divides the original number by 1,000. Study the following examples and notice the position of the decimal point in each one.

Rule: To divide a decimal by any power of 10 (10, 100, 1,000, 10,000, 100,000, and so on), move the decimal one place to the left for every zero in the divisor.

The Little Decimal Point

The location of a decimal point in a number is very important. Consider, for instance, the difference between $5.00 and $500.00, or between $.05 and $50.00. In each of these numbers, it’s the position of the decimal point that makes the difference!

When Zeros Mean Something

The use of zeros can alter the value of a decimal. For example, if you add a zero between the decimal point and the first digit of the number, it’s the same as dividing the decimal by 10, because you’ve actually moved the decimal one place to the left.

Rule: Adding one zero between the decimal point and the first digit of the number divides the number by 10. Adding two zeros divides the number by 100; three zeros by 1,000; and so on.

Likewise, if you remove a zero from between the decimal point and the first digit of the number, it’s the same as multiplying the decimal by 10, because you’ve actually moved the decimal one place to the right.

Rule: Removing a zero between the decimal point and the first digit of the number multiplies the number by 10. Removing two zeros multiplies the number by 100; three zeros by 1,000; and so on.

Table 1 below may help you understand how adding or deleting zeros between a decimal point and the first digit of the number changes the value of that decimal.

Let’s see how this works with whole numbers.

Before moving on, study the following examples to make sure you understand when zeros mean something in decimals.

When Zeros Mean Nothing

Adding or removing zeros from the far right of a decimal doesn’t change the value of the decimal. For example, adding a zero to .5 makes .50. Now, let’s compare the values of these two decimals:

.5 is read “five tenths” or 5⁄10.

.50 is read “fifty hundredths” or 50⁄100, which can be reduced to 5⁄10.

Let’s consider another example. One zero added to .12 is .120.

.12 is read “twelve hundredths” or 12⁄100.

.120 is read “one hundred and twenty thousandths” or 120⁄1000, which can be reduced to 12⁄100.

Rule: Removing or adding zeros at the right of a decimal doesn’t change the value of the decimal.

Mixed Decimals

A whole number plus a fraction is called a mixed number. A whole number plus a decimal is called a mixed decimal. Suppose you worked for 2.75 hours. The number of hours you worked is expressed as a mixed decimal. To read this number, follow these steps:

Step 1: Read the whole number (the number to the left of the decimal point). For the number 2.75, say “two.”

Step 2: Read the decimal point as “and.”

Step 3: Read the decimal (the number to the right of the decimal point). For the number 2.75, say “seventy-five hundredths.”

Therefore, the decimal 2.75 is read as “two and seventy-five hundredths.”

Example: Read each mixed decimal aloud and then write it in words.

Mixed DecimalDecimal in Words

21.07twenty-one and seven hundredths

5.13five and thirteen hundredths

14.26fourteen and twenty-six hundredths

300.007three hundred and seven thousandths

35.22thirty-five and twenty-two hundredths

496.325four hundred ninety-six and three hundred twenty-five thousandths

To write a mixed decimal when you’re given the decimal in words, first write the whole number, followed by a decimal point, then write the decimal part of the number as you learned earlier in this lesson.

Example: Write “twenty-two and fifteen hundredths” as a decimal.

22Write the whole number.

22.

↑Add a decimal point.

.15Write out the decimal part of the number, making sure that the last digit in the numerator (5) is in the proper place of the decimal (in this case, the hundredths place).

22.15Combine the whole number and the decimal. Answer: 22.15

Example: Read the words for each mixed decimal and then write it as a number.

Mixed Decimal in WordsMixed Decimal

two and twelve hundredths2.12

five and fifteen hundredths5.15

sixty-seven and twenty-nine hundredths67.29

one hundred forty and twenty-five thousandths140.025

fourteen and eighty-one thousandths14.081

twenty-one and fifty-six hundredths21.56

seven and six hundredths7.06

Let’s put to practical use what you’ve learned so far about decimals.

Rounding Decimals

To add, subtract, multiply, and divide decimals, it will help to learn more about them. Among the facts you’ll be learning in this assignment are how to round decimals, how to change mixed decimals to mixed numbers, and how to compare decimals. These facts will help you conquer the more complex tasks later in the lesson.

In the Whole Numbers lesson, you learned the steps for rounding whole numbers. These same steps are also used for rounding decimals. Step 4, however, is a new step. Be sure to read it carefully.

Step 1: Find the digit having the place value you want to round to. It will help if you circle this digit.

Step 2: Look at the digit immediately to the right of the circled digit.

Step 3: If the digit to the right is 5 or more, then round up by increasing the circled digit by 1. If the digit to the right is less than 5, round down—don’t change the circled digit.

Step 4: Drop all digits (including zeroes) to the right of the rounded digit.

Changing Mixed Decimals to Mixed Numbers

Earlier in this lesson, you learned how to change a decimal into a fraction. For example, you learned that .037 could be written as 37⁄1,000. Now, you’ll learn how to change a mixed decimal to a mixed number. Consider the following examples.

Example: Write 24.567 as a mixed number.

24Write the whole number part of the mixed decimal.

567Drop the decimal point from the decimal part, and make the decimal the numerator of your fraction.

567Draw a line under the numerator.

5671,0005671,000The place value of the decimal is the denominator of the fraction. In this case, the denominator is 1,000.

245671,000245671,000Combine the whole number and the fraction.

Answer: 245671,000245671,000

Example: Write 6.8 as a mixed number. Reduce to lowest terms.

6Write the whole number part of the mixed decimal.

8Drop the decimal point from the decimal part, and make the decimal the numerator of the fraction.

810810Draw a line under the numerator. The place value of the decimal is the denominator of the fraction.

810=8÷210÷2=45810=8÷210÷2=45To reduce this fraction to the lowest terms, divide both the numerator and the denominator by 2.

645645Combine the whole number and the fraction.

Answer: 645645

Example: Write 5.004 as a mixed number in its simplest form.

5Write the whole number part of the mixed decimal.

41,00041,000Drop the decimal point from the decimal part, and make the decimal the numerator of the fraction. The place value of the decimal is the denominator.

41,000=4÷41,000÷4=125041,000=4÷41,000÷4=1250Reduce the fraction by dividing the numerator and denominator by 4.

5125051250Combine the whole number and the fraction.

Answer: 5125051250

Comparing Decimals

Knowing how to compare decimals is important. Suppose you’re shopping for a car and you want to get the best deal. One car dealer is offering a finance package at 3.75 percent. Another is offering a finance package at 3.8 percent. Which is the better deal? To answer this question, you have to know how to compare decimals. You’ll remember that the symbol < means “less than” and the symbol > means “greater than.”

Suppose you wish to compare the decimals .69 and .83. Since both of these decimals have the same number of decimal places, follow these steps:

69 83Write the decimals omitting the decimal points.

69 < 83Compare the decimals just as you would whole numbers (69 is less than 83).

.69 < .83Reinsert the decimal points. Answer: .69 is less than .83.

Let’s say you need to know which is greater, .6 or .35. To compare these two decimals, follow these steps:

.60 .35Write the decimals so they have the same number of decimal places. To do this, add a zero or zeroes to the decimal with the smaller number of places until it has the same number as the decimal with the larger number of places. (Adding zeroes at the end of a decimal doesn’t change the value of the decimal.)

60 > 35Drop the decimal points and compare the decimals as you would whole numbers.

.60 > .35 or .6 > .35Reinsert the decimal points. Answer: .6 is greater than .35.

Example: Jeff completed a ski race in 1.02 minutes. Jerome completed the same race in 1.044 minutes. Who completed the race in less time? To answer this question, you must compare 1.02 and 1.044.

1.02 1.044Look at the whole numbers first. If they’re different, simply compare the whole numbers. If the whole numbers are the same, you must then compare the decimals.

.02 .044Temporarily drop the whole numbers and work with the decimals only.

.020 .044Write the decimals so they have the same number of decimal places. Add a zero after the 2 in .02.

20 < 44Drop the decimal points and compare the decimals as you would whole numbers. (Once a decimal point is dropped, you should ignore any zeroes at the beginning of the numbers before you compare the numbers.)

.02 < .044Reinsert the decimals. Be sure to reinsert the zeroes as well.

1.02 < 1.044Reinsert the whole numbers. Answer: Since 1.02 is less than 1.044, Jeff completed the ski race in less time.

Example: Juanita has a small business that she operates from her home. She is currently looking for a computer to add to her business. Juanita has examined four different computers with recall times of 0.0195 seconds, 0.01932 seconds, 0.01883 seconds, and 0.02159 seconds. Which of these times is the fastest? Write the decimals so they have the same number of decimal places. Drop the decimal points and compare the four numbers as you would whole numbers.

19501932188321591950193218832159Answer: Since 1883 is the smallest number, the computer with a recall time of 0.01883 seconds is the fastest.

Now that you know how to compare decimals, let’s go back to the car dealer’s finance package we discussed at the beginning of this section. One car dealer is offering a finance package at 3.75 percent and another is offering a package at 3.8 percent. Which is the better deal?

3.75 3.8The whole numbers are the same, so temporarily drop them and work with the decimals only.

.75 .80Write the decimals so they have the same number of decimal places. Add a zero after the 8 in .8.

75 < 80Drop the decimal points and compare the decimals as you would whole numbers. (75 is less than 80.)

3.75 < 3.80Reinsert the decimal points and the whole numbers. Answer: The 3.75 finance package is the better deal.