Decimals

Name Decimals

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 \(\text{\$3.45}\), the bottle of water costs \(\text{\$1.25}\), and the total sales tax is \(\text{\$0.33}\), what is the total cost of your lunch?

The total is \(\text{\$5.03}.\) Suppose you pay with a \(\text{\$5}\) bill and \(3\) pennies. Should you wait for change? No, \(\text{\$5}\) and \(3\) pennies is the same as \(\text{\$5.03}.\)

Because \(\text{100 pennies}=\text{\$1},\) each penny is worth \(\frac{1}{100}\) of a dollar. We write the value of one penny as \(\$0.01,\) since \(0.01=\frac{1}{100}.\)

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
\(10\cdot 10=100\)	One hundred
\(10\cdot 10\cdot 10=1000\)	One thousand
\(10\cdot 10\cdot 10\cdot 10=10,000\)	Ten thousand

How are decimals related to fractions? shows the relation.

Decimal	Fraction	Name
\(0.1\)	\(\frac{1}{10}\)	One tenth
\(0.01\)	\(\frac{1}{100}\)	One hundredth
\(0.001\)	\(\frac{1}{1,000}\)	One thousandth
\(0.0001\)	\(\frac{1}{10,000}\)	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 .

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.

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

Condensed — the full section is in OpenStax Prealgebra 2e.

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

Example

Try it.

Write fourteen and thirty-seven hundredths as a decimal.

Solution

	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.

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.

\[\begin{array}{llll}5=5.0 & & & -2=-2.0 \\ 5=5.00 & & & -2=-2.00 \\ 5=5.000 & & & -2=-2.000\end{array}\]\[\text{and so on\ldots }\]

Condensed — the full section is in OpenStax Prealgebra 2e.

Convert Decimals to Fractions or Mixed Numbers

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 \(\text{\$5.03}\) means \(5\) dollars and \(3\) cents. Since there are \(100\) cents in one dollar, \(3\) cents means \(\frac{3}{100}\) of a dollar, so \(0.03=\frac{3}{100}.\)

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 \(0.03.\)

\[0.03=\frac{3}{100}\]

For our \(\text{\$5.03}\) lunch, we can write the decimal \(5.03\) as a mixed number.

\[5.03=5\frac{3}{100}\]

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.

Condensed — the full section is in OpenStax Prealgebra 2e.

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.

Example

Try it.

Locate \(0.4\) on a number line.

Solution

The decimal \(0.4\) is equivalent to \(\frac{4}{10},\) 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.

Example

Try it.

Locate \(-0.74\) on a number line.

Solution

The decimal \(-0.74\) is equivalent to \(-\frac{74}{100},\) 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 \(-0.70\).

Order Decimals

Which is larger, \(0.04\) or \(0.40?\)

If you think of this as money, you know that \(\text{\$0.40}\) (forty cents) is greater than \(\text{\$0.04}\) (four cents). So,

\[0.40>0.04\]

In previous chapters, we used the number line to order numbers.

\[\begin{array}{l} \\ ab\ ‘a\ \text{is greater than}\ b’\ \text{when}\ a\ \text{is to the right of}\ b\ \text{on the number line}\end{array}\]

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.	\(\frac{31}{100}\)	\(\frac{308}{1000}\)
We need a common denominator to compare them.		\(\frac{308}{1000}\)
	\(\frac{310}{1000}\)	\(\frac{308}{1000}\)

Because \(310>308,\) we know that \(\frac{310}{1000}>\frac{308}{1000}.\) Therefore, \(0.31>0.308.\)

\[\frac{31}{100}=\frac{310}{1000}\ \text{and}\ 0.31=0.310\]\[0.31=0.310\]

Example

Try it.

Order the following decimals using \(<\ \text{or}\ \text{>:}\)

1. ⓐ \(\ 0.64\ __0.6\)
2. ⓑ \(\ 0.83\ __0.803\)

Solution

ⓐ
	\(\ 0.64\ __0.6\)
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.	\(0.64>0.60\) \(0.64>0.6\)

ⓑ
	\(\ 0.83\ __0.803\)
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.	\(0.830>0.803\) \(0.83>0.803\)

Example

Try it.

Use \(<\ \text{or}\ >\) to order. \(-0.1__-0.8.\)

Solution

	\(\ -0.1\ __-0.8\)
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.	\(-0.1>-0.8\)

Condensed — the full section is in OpenStax Prealgebra 2e.

Round Decimals

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 \(\text{\$3.279}\) per gallon. But if you were to buy exactly one gallon of gas at this price, you would pay \(\text{\$3.28}\), 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 \(\text{\$2.72}\) to the nearest dollar. Is it closer to \(\text{\$2}\) or to \(\text{\$3}?\) What if we wanted to round \(\text{\$2.72}\) to the nearest ten cents; is it closer to \(\text{\$2.70}\) or to \(\text{\$2.80}?\) The number lines in can help us answer those questions.

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

Example

Try it.

Round \(18.379\) to the nearest hundredth.

Solution

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.

Example

Try it.

Round \(18.379\) to the nearest ⓐ tenth ⓑ whole number.

Solution

ⓐ 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.

Condensed — the full section is in OpenStax Prealgebra 2e.

Key Concepts

• Name a decimal number.
  1. Name the number to the left of the decimal point.
  2. Write “and” for the decimal point.
  3. Name the “number” part to the right of the decimal point as if it were a whole number.
  4. Name the decimal place of the last digit.
• Write a decimal number from its name.
  1. 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.
  2. Mark the number of decimal places needed to the right of the decimal point by noting the place value indicated by the last word.
  3. 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.
  4. Fill in zeros for place holders as needed.
• Convert a decimal number to a fraction or mixed number.
  1. 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.
  2. Determine the place value of the final digit.
  3. Write the fraction. numerator—the ‘numbers’ to the right of the decimal point denominator—the place value corresponding to the final digit
  4. Simplify the fraction, if possible.
• Order decimals.
  1. 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.
  2. Compare the numbers to the right of the decimal point as if they were whole numbers.
  3. Order the numbers using the appropriate inequality sign.
• Round a decimal.
  1. Locate the given place value and mark it with an arrow.
  2. Underline the digit to the right of the given place value.
  3. 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
  4. Rewrite the number, removing all digits to the right of the given place value.

Decimals

Name Decimals

In the following exercises, name each decimal.

Try it.

\(5.5\)

Solution

five and five tenths

Try it.

\(7.8\)

Try it.

\(5.01\)

Solution

five and one hundredth

Try it.

\(14.02\)

Try it.

\(8.71\)

Solution

eight and seventy-one hundredths

Try it.

\(2.64\)

Try it.

\(0.002\)

Solution

two thousandths

Try it.

\(0.005\)

Try it.

\(0.381\)

Solution

three hundred eighty-one thousandths

Try it.

\(0.479\)

Try it.

\(-17.9\)

Solution

negative seventeen and nine tenths

Try it.

\(-31.4\)

Write Decimals

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

Try it.

Eight and three hundredths

Solution

8.03

Try it.

Nine and seven hundredths

Try it.

Twenty-nine and eighty-one hundredths

Solution

29.81

Try it.

Sixty-one and seventy-four hundredths

Try it.

Seven tenths

Solution

0.7

Try it.

Six tenths

Try it.

One thousandth

Solution

0.001

Try it.

Nine thousandths

Try it.

Twenty-nine thousandths

Solution

0.029

Try it.

Thirty-five thousandths

Try it.

Negative eleven and nine ten-thousandths

Solution

−11.0009

Try it.

Negative fifty-nine and two ten-thousandths

Try it.

Thirteen and three hundred ninety-five ten thousandths

Solution

13.0395

Try it.

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.

Try it.

\(1.99\)

Solution

\(1\frac{99}{100}\)

Try it.

\(5.83\)

Try it.

\(15.7\)

Solution

\(15\frac{7}{10}\)

Try it.

\(18.1\)

Try it.

\(0.239\)

Solution

\(\frac{239}{1000}\)

Try it.

\(0.373\)

Try it.

\(0.13\)

Solution

\(\frac{13}{100}\)

Try it.

\(0.19\)

Try it.

\(0.011\)

Solution

\(\frac{11}{1000}\)

Try it.

\(0.049\)

Try it.

\(-0.00007\)

Solution

\(-\frac{7}{100000}\)

Try it.

\(-0.00003\)

Try it.

\(6.4\)

Solution

\(6\frac{2}{5}\)

Try it.

\(5.2\)

Try it.

\(7.05\)

Solution

\(7\frac{1}{20}\)

Try it.

\(9.04\)

Try it.

\(4.006\)

Solution

\(4\frac{3}{500}\)

Try it.

\(2.008\)

Try it.

\(10.25\)

Solution

\(10\frac{1}{4}\)

Try it.

\(12.75\)

Try it.

\(1.324\)

Solution

\(1\frac{81}{250}\)

Try it.

\(2.482\)

Try it.

\(14.125\)

Solution

\(14\frac{1}{8}\)

Try it.

\(20.375\)

Locate Decimals on the Number Line

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

Try it.

\(0.8\)

Solution

Try it.

\(0.3\)

Try it.

\(-0.2\)

Solution

Try it.

\(-0.9\)

Try it.

\(3.1\)

Solution

Try it.

\(2.7\)

Try it.

\(-2.5\)

Solution

Try it.

\(-1.6\)

Order Decimals

In the following exercises, order each of the following pairs of numbers, using \(<\ \text{or}\ >.\)

Try it.

\(0.9__0.6\)

Solution

>

Try it.

\(0.7__0.8\)

Try it.

\(0.37__0.63\)

Solution

<

Try it.

\(0.86__0.69\)

Try it.

\(0.6__0.59\)

Solution

>

Try it.

\(0.27__0.3\)

Try it.

\(0.91__0.901\)

Solution

>

Try it.

\(0.415__0.41\)

Try it.

\(-0.5__-0.3\)

Solution

<

Try it.

\(-0.1_-0.4\)

Try it.

\(-0.62_-0.619\)

Solution

<

Try it.

\(-7.31_-7.3\)

Round Decimals

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

Try it.

\(0.67\)

Solution

0.7

Try it.

\(0.49\)

Try it.

\(2.84\)

Solution

2.8

Try it.

\(4.63\)

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

Try it.

\(0.845\)

Solution

0.85

Try it.

\(0.761\)

Try it.

\(5.7932\)

Solution

5.79

Try it.

\(3.6284\)

Try it.

\(0.299\)

Solution

0.30

Try it.

\(0.697\)

Try it.

\(4.098\)

Solution

4.10

Try it.

\(7.096\)

In the following exercises, round each number to the nearest ⓐ hundredth ⓑ tenth ⓒ whole number.

Try it.

\(5.781\)

Solution

1. ⓐ 5.78
2. ⓑ 5.8
3. ⓒ 6

Try it.

\(1.638\)

Try it.

\(63.479\)

Solution

1. ⓐ 63.48
2. ⓑ 63.5
3. ⓒ 63

Try it.

\(84.281\)

Condensed — the full section is in OpenStax Prealgebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

1. Name the number \(4,926,015\) in words.
   If you missed this problem, review .

   Révèle la réponse

   Four million, nine hundred twenty-six thousand, fifteen

2. Round \(748\) to the nearest ten.
   If you missed this problem, review .

   Révèle la réponse

   \(750\)

3. Locate \(\frac{3}{10}\) on a number line.
   If you missed this problem, review .

   Révèle la réponse

4. Name each decimal: ⓐ \(\ 4.3\) ⓑ \(\ 2.45\) ⓒ \(\ 0.009\) ⓓ \(\ -15.571.\)

   Révèle la réponse

   ⓐ
   	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

5. Name each decimal:

   ⓐ \(\ 6.7\) ⓑ \(\ 19.58\) ⓒ \(\ 0.018\) ⓓ \(\ -2.053\)

   Révèle la réponse

   1. ⓐ six and seven tenths
   2. ⓑ nineteen and fifty-eight hundredths
   3. ⓒ eighteen thousandths
   4. ⓓ negative two and fifty-three thousandths

6. Name each decimal:

   ⓐ \(\ 5.8\) ⓑ \(\ 3.57\) ⓒ \(\ 0.005\) ⓓ \(\ -13.461\)

   Révèle la réponse

   1. ⓐ five and eight tenths
   2. ⓑ three and fifty-seven hundredths
   3. ⓒ five thousandths
   4. ⓓ negative thirteen and four hundred sixty-one thousandths

7. Write fourteen and thirty-seven hundredths as a decimal.

   Révèle la réponse

   	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.

8. Write as a decimal: thirteen and sixty-eight hundredths.

   Révèle la réponse

   13.68

9. Write as a decimal: five and eight hundred ninety-four thousandths.

   Révèle la réponse

   5.894

10. Write twenty-four thousandths as a decimal.

    Révèle la réponse

    	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

11. Write as a decimal: fifty-eight thousandths.

    Révèle la réponse

    0.058

12. Write as a decimal: sixty-seven thousandths.

    Révèle la réponse

    0.067

13. Write each of the following decimal numbers as a fraction or a mixed number:

    ⓐ \(\ 4.09\) ⓑ \(\ 3.7\) ⓒ \(\ -0.286\)

    Révèle la réponse

    ⓐ
    	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.

14. Write as a fraction or mixed number. Simplify the answer if possible.

    ⓐ \(\ 5.3\) ⓑ \(\ 6.07\) ⓒ \(\ -0.234\)

    Révèle la réponse

    1. ⓐ \(\ 5\frac{3}{10}\)
    2. ⓑ \(\ 6\frac{7}{100}\)
    3. ⓒ \(\ -\frac{117}{500}\)

15. Write as a fraction or mixed number. Simplify the answer if possible.

    ⓐ \(\ 8.7\) ⓑ \(\ 1.03\) ⓒ \(\ -0.024\)

    Révèle la réponse

    1. ⓐ \(\ 8\frac{7}{10}\)
    2. ⓑ \(\ 1\frac{3}{100}\)
    3. ⓒ \(\ -\frac{3}{125}\)

16. Locate \(0.4\) on a number line.

    Révèle la réponse

    The decimal \(0.4\) is equivalent to \(\frac{4}{10},\) 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.
    

17. Locate \(0.6\) on a number line.

    Révèle la réponse

    
    

18. Locate \(0.9\) on a number line.

    Révèle la réponse

    
    

19. Locate \(-0.74\) on a number line.

    Révèle la réponse

    The decimal \(-0.74\) is equivalent to \(-\frac{74}{100},\) 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 \(-0.70\).
    

20. Locate \(-0.63\) on a number line.

    Révèle la réponse

    
    

21. Locate \(-0.25\) on a number line.

    Révèle la réponse

    
    

22. Order the following decimals using \(<\ \text{or}\ \text{>:}\)

    1. ⓐ \(\ 0.64\ __0.6\)
    2. ⓑ \(\ 0.83\ __0.803\)
    Révèle la réponse

    ⓐ
    	\(\ 0.64\ __0.6\)
    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.	\(0.64>0.60\) \(0.64>0.6\)

    ⓑ
    	\(\ 0.83\ __0.803\)
    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.	\(0.830>0.803\) \(0.83>0.803\)

23. Order each of the following pairs of numbers, using \(<\ \text{or}\ \text{>:}\)

    ⓐ \(\ 0.42__0.4\) ⓑ \(\ 0.76__0.706\)

    Révèle la réponse

    1. ⓐ >
    2. ⓑ >

24. Order each of the following pairs of numbers, using \(<\ \text{or}\ \text{>:}\)

    ⓐ \(\ 0.1__0.18\) ⓑ \(\ 0.305__0.35\)

    Révèle la réponse

    1. ⓐ <
    2. ⓑ <

25. Use \(<\ \text{or}\ >\) to order. \(-0.1__-0.8.\)

    Révèle la réponse

    	\(\ -0.1\ __-0.8\)
    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.	\(-0.1>-0.8\)

26. Order each of the following pairs of numbers, using \(<\ \text{or}\ \text{>:}\)

    \(-0.3___-0.5\)

    Révèle la réponse

    >

27. Order each of the following pairs of numbers, using \(<\ \text{or}\ \text{>:}\)

    \(-0.6___-0.7\)

    Révèle la réponse

    >

28. Round \(18.379\) to the nearest hundredth.

    Révèle la réponse

    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.

29. Round to the nearest hundredth: \(1.047.\)

    Révèle la réponse

    1.05

30. Round to the nearest hundredth: \(9.173.\)

    Révèle la réponse

    9.17

31. Round \(18.379\) to the nearest ⓐ tenth ⓑ whole number.

    Révèle la réponse

    ⓐ 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.

32. Round \(6.582\) to the nearest ⓐ hundredth ⓑ tenth ⓒ whole number.

    Révèle la réponse

    1. ⓐ 6.58
    2. ⓑ 6.6
    3. ⓒ 7

33. Round \(15.2175\) to the nearest ⓐ thousandth ⓑ hundredth ⓒ tenth.

    Révèle la réponse

    1. ⓐ 15.218
    2. ⓑ 15.22
    3. ⓒ 15.2

34. Eight and three hundredths

    Révèle la réponse

    8.03

35. Nine and seven hundredths

36. Twenty-nine and eighty-one hundredths

    Révèle la réponse

    29.81

37. Sixty-one and seventy-four hundredths

38. Seven tenths

    Révèle la réponse

    0.7

39. Six tenths

40. One thousandth

    Révèle la réponse

    0.001

Symbols used here

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\approx

approximately equal
Equal to the precision shown, not exactly.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

a \bmod n

remainder
What is left after dividing a by n.

\%

per cent
Per hundred: 15% = 15/100.

a : b,\ \frac{a}{b}

ratio, fraction
a for every b; a divided by b.

How to: Decimals

1. Name decimals
2. Write decimals
3. Convert decimals to fractions or mixed numbers
4. Locate decimals on the number line
5. Order decimals
6. Round decimals
7. 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.
8. 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.

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

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Parts of this page are adapted from OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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