5.1 Decimals

Learning Objectives

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

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 10,00010,000 as ten thousand. Likewise, the names of the decimal places correspond to their fraction values. Notice how the place value names in Figure 5.2 relate to the names of the fractions from Table 5.2.

A chart is shown labeled “Place Value”. There are 12 columns. The columns are labeled, from left to right, Hundred thousands, Ten thousands, Thousands, Hundreds, Tens, Ones, Decimal Point, Tenths, Hundredths, Thousandths, Ten-thousandths, Hundred-thousandths.

Figure 5.2 This chart illustrates place values to the left and right of the decimal point.

Notice two important facts shown in Figure 5.2.

  • 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$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 Figure 5.3, we write the amount on a check in both words and numbers.

An image of a check is shown. The check is made out to Jane Doe. It shows the number $152.65 and says in words, “One hundred fifty two and 65 over 100 dollars.”

Figure 5.3 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.

Write Decimals

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

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.

Locate Decimals on the Number Line

Since decimals are forms of fractions, locating decimals on the number line is similar to locating fractions on the number line.

Round Decimals

In the United States, gasoline prices are usually written with the decimal part as thousandths of a dollar. For example, a gas stationmight 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 Figure 5.4 can help us answer those questions.

In part a, a number line is shown with 2, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9 and 3. There is a dot between 2.7 and 2.8 labeled as 2.72.  In part b, a number line is shown with 2.70, 2.71, 2.72, 2.73, 2.74, 2.75, 2.76, 2.77, 2.78, 2.79, and 2.80. There is a dot at 2.72.

Figure 5.4 ⓐ We see that 2.722.72 is closer to 33 than to 2.2. So, 2.722.72 rounded to the nearest whole number is 3.

ⓑ We see that 2.722.72 is closer to 2.702.70 than 2.80.2.80. So we say that 2.722.72 rounded to the nearest tenth is 2.7.2.7.

Can we round decimals without number lines? Yes! We use a method based on the one we used to round whole numbers.

Section 5.1 Exercises

Practice Makes Perfect

Name Decimals

In the following exercises, name each decimal.

  1. 5.5
  2. 7.8
  3. 5.01
  4. 14.02
  5. 8.71
  6. 2.64
  7. 0.002
  8. 0.005
  9. 0.381
  10. 0.479
  11. −17.9
  12. −31.4
    Write Decimals

In the following exercises, translate the name into a decimal number.

  1. Eight and three hundredths
  2. Nine and seven hundredths
  3. Twenty-nine and eighty-one hundredths
  4. Sixty-one and seventy-four hundredths
  5. Seven tenths
  6. Six tenths
  7. One thousandth
  8. Nine thousandths
  9. Twenty-nine thousandths
  10. Thirty-five thousandths
  11. Negative eleven and nine ten-thousandths
  12. Negative fifty-nine and two ten-thousandths
  13. Thirteen and three hundred ninety-five ten thousandths
  14. 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. 1.99
  2. 5.83
  3. 15.7
  4. 18.1
  5. 0.239
  6. 0.373
  7. 0.13
  8. 0.19
  9. 0.011
  10. 0.049
  11. −0.00007
  12. −0.00003
  13. 6.4
  14. 5.2
  15. 7.05
  16. 9.04
  17. 4.006
  18. 2.008
  19. 10.25
  20. 12.75
  21. 1.324
  22. 2.482
  23. 14.125
  24. 20.375
    Locate Decimals on the Number Line

In the following exercises, locate each number on a number line.

  1. 0.8
  2. 0.3
  3. −0.2
  4. −0.9
  5. 3.1
  6. 2.7
  7. −2.5
  8. −1.6
    Order Decimals

In the following exercises, order each of the following pairs of numbers, using .

  1. 0.9__0.6
  2. 0.7__0.8
  3. 0.37__0.63
  4. 0.86__0.69
  5. 0.6__0.59
  6. 0.27__0.3
  7. 0.91__0.901
  8. 0.415__0.41
  9. −0.5__−0.3
  10. −0.1_−0.4
  11. −0.62_−0.619
  12. −7.31_−7.3
    Round Decimals

In the following exercises, round each number to the nearest tenth.

  1. 0.67
  2. 0.49
  3. 2.84
  4. 4.63
    In the following exercises, round each number to the nearest hundredth.
  5. 0.845
  6. 0.761
  7. 5.7932
  8. 3.6284
  9. 0.299
  10. 0.697
  11. 4.098
  12. 7.096
    In the following exercises, round each number to the nearest ⓐ hundredth ⓑ tenth ⓒ whole number.
  13. 5.781
  14. 1.638
  15. 63.479
  16. 84.281
    Everyday Math
  17. 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.

  1. New Car Purchase Selena’s new car cost $23,795.95. Round this number to the nearest:
    ⓐ dollar

ⓑ thousand dollars

ⓒ ten thousand dollars.

  1. 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.
  2. 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.

Writing Exercises

  1. How does your knowledge of money help you learn about decimals?
  2. Explain how you write “three and nine hundredths” as a decimal.
  3. 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?
  4. Gerry saw a sign advertising postcards marked for sale at “10for0.99¢.” What is wrong with the advertised price?

Self Check

ⓐ 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.