5.1 Decimals

Learning objectives

Learning objectives

• Name decimals

• Write decimals

• Convert decimals to fractions or mixed numbers

• Locate decimals on the number line

• Order decimals

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

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

Because $\mathrm{100 pennies}=\text{\textdollar 1},$ each penny is worth $\frac{1}{100}$ of a dollar. We write the value of one penny as $\textdollar 0.01,$ 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. Table 5.1 shows the counting numbers.

Table

5.1

How are decimals related to fractions? Table 5.2 shows the relation.

Table

5.2

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 Figure 5.2 relate to the names of the fractions from Table 5.2.

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 $\text{\textdollar 5}.03$ lunch? We read $\text{\textdollar 5.03}$ 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.

The number $15.68$ is read fifteen and sixty-eight hundredths.

Example 5.1

Solution

1. Separate the whole-number part. Read the digits left of the decimal point, then read the digits right of it as one whole number before naming their final place value.
2. Name the first decimal. For $4.3$, the whole-number part is four, the other digit is three, and the final digit is in the tenths place.
3. Name the second decimal. For $2.45$, the whole-number part is two, the other digits are forty-five, and the final digit is in the hundredths place.
4. Name the remaining decimals. The zero in $0.009$ is not read, while the final digit of $-15.571$ is in the thousandths place after five hundred seventy-one.
5. State all four names. They are four and three tenths, two and forty-five hundredths, nine thousandths, and negative fifteen and five hundred seventy-one thousandths.

Takeaway: The final decimal digit determines the place-value word in each decimal name, while a zero whole-number part is omitted.

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:

Example 5.2

Solution

1. Locate the decimal point. The word "and" in fourteen and thirty-seven hundredths marks the position of the decimal point.
2. Write the whole-number part. Translate fourteen into $14$ and place it to the left of the decimal point.
3. Count the decimal places. Hundredths requires two digits to the right of the decimal point, so reserve two places for thirty-seven.
4. Write the fractional digits. Place $3$ in the tenths place and $7$ in the hundredths place, giving $14.37$.
5. Check the place values. The $7$ is in the hundredths place, so $14.37$ is exactly fourteen and thirty-seven hundredths.

Takeaway: The place-value word fixes the number of digits after the decimal point when a number is written from its name.

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.

Example 5.3

Solution

1. Identify the decimal place. Twenty-four thousandths means the last digit sits in the thousandths place, so the written number needs three decimal places.
2. Assign each digit to a place. From the fraction $\frac{24}{1000}$, the digit $2$ is in the hundredths place and $4$ is in the thousandths place, and the tenths place is empty.
3. Fill the empty place with zero. Write $0$ in the tenths place, $2$ in the hundredths place, and $4$ in the thousandths place, giving $0.024$.
4. State the result. The first figure supplies the three labeled decimal positions and the second figure shows the completed number, so twenty-four thousandths is written as $\boxed{0.024}$.

Takeaway: Writing a fraction with a denominator of $1000$ in decimal form places its digits in the tenths, hundredths, and thousandths positions.

Before we move on to our next objective, think about money again. We know that $\text{\textdollar 1}$ is the same as $\text{\textdollar 1.00}.$ The way we write $\text{\textdollar 1}\,(\mathrm{or}\,\text{\textdollar 1.00})$ depends on the context. In the same way, integers can be written as decimals with as many zeros as needed to the right of the decimal.

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 $\text{\textdollar 5.03}$ 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 $0.03.$

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

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.

Example 5.4

Solution

1. Convert $4.09$. The two decimal places require a denominator of $100$, and the $9$ occupies the hundredths place, so $4.09=4\frac{9}{100}$. The figures labeled tenths and hundredths and the mixed-number figures confirm that the digit $9$ belongs over $100$.
2. Convert $3.7$. The single decimal place requires a denominator of $10$, and the $7$ occupies the tenths place, so $3.7=3\frac{7}{10}$. The three figures for this part show $3.7$, the fraction $3\frac{7}{10}$, and the equation connecting them.
3. Convert $-0.286$. There are three decimal places, so use a denominator of $1000$: $-0.286=-\frac{286}{1000}$. Reducing by a common factor of $2$ gives $-\frac{143}{500}$, which agrees with the final figure.
4. Check the values. The denominators contain one zero for each decimal place, and the numerators preserve the digits $9$, $7$, and $286$ in their correct positions.
5. State the result. The three conversions are $\boxed{4.09=4\frac{9}{100}}$, $\boxed{3.7=3\frac{7}{10}}$, and $\boxed{-0.286=-\frac{286}{1000}=-\frac{143}{500}}$.

Takeaway: The number of decimal places determines the denominator, while the digits after the decimal point become the numerator.

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 5.5

Solution

1. Rewrite the decimal as a fraction. The decimal $0.4$ is equivalent to $\frac{4}{10}$, so it lies between $0$ and $1$.
2. Divide the unit interval. Split the interval from $0$ to $1$ into ten equal parts and label them from $0.1$ through $0.9$, with $1.0$ at the end.
3. Count the fractional parts. The denominator $10$ means that $0.4$ occupies four of those equal intervals, counting from $0$.
4. Mark the location. Place the point at the tick labeled $0.4$, which is the fourth tenth after $0$ and before $1$.
5. Check the position. The marked point is between $0.3$ and $0.5$ and is four tenths of the way from $0$ to $1$.

Takeaway: A decimal with one place can be located by counting its numerator over ten equal intervals from zero.

Example 5.6

Solution

1. Rewrite the decimal as a fraction. The decimal $-0.74$ is equivalent to $-\frac{74}{100}$, which places it between $0$ and $-1$.
2. Mark tenths on the negative side. Label $-0.10$, $-0.20$, and the remaining multiples of $-0.10$ as they move left from $0$.
3. Identify the bounding marks. Since $74$ lies between $70$ and $80$, the value lies between $-0.70$ and $-0.80$.
4. Refine the position. The distance from $-0.70$ is $0.04$, while the distance to $-0.80$ is $0.06$, so the point is closer to $-0.70$.
5. Check the direction. Both numbers in the interval are negative, and the larger value $-0.70$ lies to the right of $-0.80$.

Takeaway: For a negative decimal, count leftward from zero and compare distances to the adjacent marked tenths.

Order Decimals

Which is larger, $0.04$ or $0.40?$

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

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

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.

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

Notice what we did in converting $0.31$ to a fraction—we started with the fraction $\frac{31}{100}$ and ended with the equivalent fraction $\frac{310}{1000}.$ Converting $\frac{310}{1000}$ back to a decimal gives $0.310.$ So $0.31$ is equivalent to $0.310.$ Writing zeros at the end of a decimal does not change its value.

If two decimals have the same value, they are said to be equivalent decimals.

We say $0.31$ and $0.310$ are equivalent decimals.

Remember, writing zeros at the end of a decimal does not change its value.

Example 5.7

Solution

1. Equalize the first decimal places. Write $0.6$ as $0.60$, so compare $0.64$ with $0.60$ in hundredths.
2. Compare the first pair. Since $64$ is greater than $60$, the correct sign is $0.64>0.6$.
3. Equalize the second decimal places. Write $0.83$ as $0.830$, so compare $0.830$ with $0.803$ in thousandths.
4. Compare the second pair. Since $830$ is greater than $803$, the correct sign is $0.83>0.803$.
5. Check both results. Each comparison uses matching place values, and adding trailing zeros did not change either original number.

Takeaway: Comparing $64$ with $60$ and $830$ with $803$ shows that both original decimal pairs have the greater value on the left.

When we order negative decimals, it is important to remember how to order negative integers. Recall that larger numbers are to the right on the number line. For example, because $-2$ lies to the right of $-3$ on the number line, we know that $-2>-3.$ Similarly, smaller numbers lie to the left on the number line. For example, because $-9$ lies to the left of $-6$ on the number line, we know that $-9<-6.$

If we zoomed in on the interval between $0$ and $-1,$ we would see in the same way that $-0.2>-0.3\,\mathrm{and}\,-0.9<-0.6.$

Example 5.8

Solution

1. Align the decimal points. Write $-0.1$ directly above $-0.8$ so that both tenths digits occupy the same place-value column.
2. Read the tenths digits. The signed tenths digits are $-1$ and $-8$, with the remaining places understood to be zero.
3. Compare the signed values. Because $-1$ is greater than $-8$, one tenth is greater than eight tenths.
4. Use the number-line direction. Negative $-0.1$ lies to the right of negative $-0.8$, confirming that it is the larger number.
5. State the ordered pair. The required inequality is $-0.1>-0.8$.

Takeaway: The number line confirms that $-0.1$ lies to the right of $-0.8$, so one negative tenth exceeds eight negative tenths.

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 $\text{\textdollar 3.279}$ per gallon. But if you were to buy exactly one gallon of gas at this price, you would pay $\text{\textdollar 3.28}$, 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 $\text{\textdollar 2.72}$ to the nearest dollar. Is it closer to $\text{\textdollar 2}$ or to $\text{\textdollar 3}?$ What if we wanted to round $\text{\textdollar 2.72}$ to the nearest ten cents; is it closer to $\text{\textdollar 2.70}$ or to $\text{\textdollar 2.80}?$ The number lines in Figure 5.4 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 5.9

Solution

1. Identify the hundredths digit. In $18.379$, the digits $3$, $7$, and $9$ occupy the tenths, hundredths, and thousandths places respectively, as the figures label the $7$ as the hundredths digit.
2. Inspect the thousandths digit. The thousandths digit is $9$. Because $9\ge 5$, round the hundredths digit upward rather than leaving it unchanged.
3. Increase the hundredths digit. Change the hundredths digit $7$ to $8$, which gives $18.38$ before any digits beyond the hundredths place are removed.
4. State the result. The final figure shows the rounded value, so $18.379$ rounded to the nearest hundredth is $\boxed{18.38}$.

Takeaway: The thousandths digit determines whether the hundredths digit stays the same or increases by one.

Example 5.10

Solution

1. Round to the nearest tenth. In $18.379$, the tenths digit is $3$ and the hundredths digit is $7$. Because $7\ge 5$, increase the tenths digit $3$ by $1$ and remove the digits beyond the tenths place, giving $18.4$.
2. Verify the tenth with aligned decimals. Aligning the decimal points in $18.379$ and $18.400$ shows that $18.400$ is greater than $18.379$ but is still closer to $18.4$ than to $18.3$. The figures marking the tenths place and the rounded value $18.4$ agree with this result.
3. Round to the nearest whole number. The tenths digit of $18.379$ is $3$. Because $3<5$, do not increase the ones digit; instead delete the decimal portion and leave $18$.
4. State the result. The nearest tenth is $\boxed{18.4}$, and the nearest whole number is $\boxed{18}$.

Takeaway: Round to a tenth using the hundredths digit, and round to a whole number using the tenths digit.

Section 5.1 Exercises

Section 5.1 Exercises

Name Decimals

In the following exercises, name each decimal.

Write Decimals

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

Convert Decimals to Fractions or Mixed Numbers

In the following exercises, convert each decimal to a fraction or mixed number.

Locate Decimals on the Number Line

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

Order Decimals

In the following exercises, order each of the following pairs of numbers, using $<\,\mathrm{or}\,\mathrm{>.}$

Round Decimals

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

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

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

Everyday Math

Writing Exercises

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.

1. Name each decimal:

ⓐ $\,6.7$ ⓑ $\,19.58$ ⓒ $\,0.018$ ⓓ $\,\,-2.053$

2. Name each decimal:

ⓐ $\,5.8$ ⓑ $\,3.57$ ⓒ $\,0.005$ ⓓ $\,-13.461$

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

ⓐ $\,5.3$ ⓑ $\,6.07$ ⓒ $\,-0.234$

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

ⓐ $\,8.7$ ⓑ $\,1.03$ ⓒ $\,-0.024$

2. Order each of the following pairs of numbers, using $<\,\mathrm{or}\,\mathrm{>:}$

ⓐ $\,0.42\mathrm{\_\_}0.4$ ⓑ $\,0.76\mathrm{\_\_}0.706$

3. Order each of the following pairs of numbers, using $<\,\mathrm{or}\,\mathrm{>:}$

ⓐ $\,0.1\mathrm{\_\_}0.18$ ⓑ $\,0.305\mathrm{\_\_}0.35$

1. Order each of the following pairs of numbers, using $<\,\mathrm{or}\,\mathrm{>:}$

$-0.3\mathrm{\_\_\_}-0.5$

2. Order each of the following pairs of numbers, using $<\,\mathrm{or}\,\mathrm{>:}$

$-0.6\mathrm{\_\_\_}-0.7$

3. Round $6.582$ to the nearest ⓐ hundredth ⓑ tenth ⓒ whole number.

1. Round $15.2175$ to the nearest ⓐ thousandth ⓑ hundredth ⓒ tenth.

2. $5.5$

3. $5.01$

1. $8.71$

2. $0.002$

3. $0.381$

1. $-17.9$

2. Negative eleven and nine ten-thousandths

3. $1.99$

1. $15.7$

2. $0.239$

3. $0.13$

1. $0.011$

2. $-0.00007$

3. $6.4$

1. $7.05$

2. $4.006$

3. $10.25$

1. $1.324$

2. $14.125$

3. $0.9\mathrm{\_\_}0.6$

1. $0.37\mathrm{\_\_}0.63$

2. $0.6\mathrm{\_\_}0.59$

3. $0.91\mathrm{\_\_}0.901$

1. $-0.5\mathrm{\_\_}-0.3$

2. $-0.62\mathrm{\_}-0.619$

3. $5.781$

1. $63.479$

2. Salary Increase Danny got a raise and now makes $\text{\textdollar 58,965.95}$ a year. Round this number to the nearest:

ⓐ dollar

ⓑ thousand dollars

ⓒ ten thousand dollars.

3. Sales Tax Hyo Jin lives in San Diego. She bought a refrigerator for $\text{\textdollar 1624.99}$ and when the clerk calculated the sales tax it came out to exactly $\text{\textdollar 142.186625}.$ Round the sales tax to the nearest ⓐ penny ⓑ dollar.

1. How does your knowledge of money help you learn about decimals?

2. Jim ran a $100-meter$ race in $\mathrm{12.32 seconds}.$ Tim ran the same race in $\mathrm{12.3 seconds}.$ Who had the faster time, Jim or Tim? How do you know?