Chapter 6. Decimals
6.1. Objectives*
After completing this chapter, you should
Reading and Writing Decimals (Section 6.2)
understand the meaning of digits occurring to the right of the ones position
be familiar with the meaning of decimal fractions
be able to read and write a decimal fraction
Converting a Decimal to a Fraction (Section 6.3)
be able to convert an ordinary decimal and a complex decimal to a fraction
Addition and Subtraction of Decimals (Section 6.5)
understand the method used for adding and subtracting decimals
be able to add and subtract decimals
be able to use the calculator to add and subtract decimals
Multiplication of Decimals (Section 6.6)
understand the method used for multiplying decimals
be able to multiply decimals
be able to simplify a multiplication of a decimal by a power of 10
understand how to use the word "of" in multiplication
Division of Decimals (Section 6.7)
understand the method used for dividing decimals
be able to divide a decimal number by a nonzero whole number and by another, nonzero, decimal number
be able to simplify a division of a decimal by a power of 10
Nonterminating Divisions (Section 6.8)
understand the meaning of a nonterminating division
be able to recognize a nonterminating number by its notation
Combinations of Operations with Decimals and Fractions (Section 6.10)
be able to combine operations with decimals
6.2. Reading and Writing Decimals*
Section Overview
Digits to the Right of the Ones Position
Decimal Fractions
Reading Decimal Fractions
Writing Decimal Fractions
Digits to the Right of the Ones Position
We began our study of arithmetic (Section 1.2) by noting that our number system is called a positional number system with base ten. We also noted that each position has a particular value. We observed that each position has ten times the value of the position to its right.
This means that each position has 
the value of the position to its left.
Thus, a digit written to the right of the units position must have a value of
of 1. Recalling that the word "of" translates to multiplication
(⋅), we can see that the value of the first position to the right of the units digit is
of 1, or
The value of the second position to the right of the units digit is
of
, or
The value of the third position to the right of the units digit is
of
, or
This pattern continues.
We can now see that if we were to write digits in positions to the right of the units positions, those positions have values that are fractions. Not only do the positions have fractional values, but the fractional values are all powers of 10 (10,102 ,103 ,…).
Decimal Fractions
If we are to write numbers with digits appearing to the right of the units digit, we must have a way of denoting where the whole number part ends and the fractional part begins. Mathematicians denote the separation point of the units digit and the tenths digit by writing a decimal point. The word decimal comes from the Latin prefix "deci" which means ten, and we use it because we use a base ten number system. Numbers written in this form are called decimal fractions, or more simply, decimals.
Notice that decimal numbers have the suffix "th."
A decimal fraction is a fraction in which the denominator is a power of 10.
The following numbers are examples of decimals.
-
42.6
The 6 is in the tenths position.
-
9.8014
The 8 is in the tenths position. The 0 is in the hundredths position. The 1 is in the thousandths position. The 4 is in the ten thousandths position.
- 0.93
The 9 is in the tenths position. The 3 is in the hundredths position.
Note
Quite often a zero is inserted in front of a decimal point (in the units position) of a decimal fraction that has a value less than one. This zero helps keep us from overlooking the decimal point.
-
0.7
The 7 is in the tenths position.
Note
We can insert zeros to the right of the right-most digit in a decimal fraction without changing the value of the number.
Reading Decimal Fractions
To read a decimal fraction,
Read the whole number part as usual. (If the whole number is less than 1, omit steps 1 and 2.)
Read the decimal point as the word "and."
Read the number to the right of the decimal point as if it were a whole number.
Say the name of the position of the last digit.
Sample Set A
Read the following numbers.
Example 6.1.
6.8
Note
Some people read this as "six point eight." This phrasing gets the message across, but technically, "six and eight tenths" is the correct phrasing.
Example 6.2.
14.116
Example 6.3.
0.0019
Example 6.4.
81
Eighty-one
In this problem, the indication is that any whole number is a decimal fraction. Whole numbers are often called decimal numbers.
81 = 81 . 0
Practice Set A
Read the following decimal fractions.
Writing Decimal Fractions
To write a decimal fraction,
Write the whole number part.
Write a decimal point for the word "and."
Write the decimal part of the number so that the right-most digit appears in the position indicated in the word name. If necessary, insert zeros to the right of the decimal point in order that the right-most digit appears in the correct position.
Sample Set B
Write each number.
Example 6.5.
Thirty-one and twelve hundredths.
The decimal position indicated is the hundredths position.
31.12
Example 6.6.
Two and three hundred-thousandths.
The decimal position indicated is the hundred thousandths. We'll need to insert enough zeros to the immediate right of the decimal point in order to locate the 3 in the correct position.
2.00003
Example 6.7.
Six thousand twenty-seven and one hundred four millionths.
The decimal position indicated is the millionths position. We'll need to insert enough zeros to the immediate right of the decimal point in order to locate the 4 in the correct position.
6,027.000104
Example 6.8.
Seventeen hundredths.
The decimal position indicated is the hundredths position.
0.17
Practice Set B
Write each decimal fraction.
Exercises
For the following three problems, give the decimal name of the position of the given number in each decimal fraction.
Exercise 6.2.8. (Go to Solution)
1. 3.941 9 is in the ____________________ position. 4 is in the ____________________ position. 1 is in the ____________________ position.
Exercise 6.2.9.
17.1085 1 is in the ____________________ position. 0 is in the ____________________ position. 8 is in the ____________________ position. 5 is in the ____________________ position.
Exercise 6.2.10. (Go to Solution)
652.3561927 9 is in the ____________________ position. 7 is in the ____________________ position.
For the following 7 problems, read each decimal fraction by writing it.
Exercise 6.2.11.
9.2
Exercise 6.2.13.
10.15
Exercise 6.2.15.
0.78
Exercise 6.2.17.
10.00011
For the following 10 problems, write each decimal fraction.
Exercise 6.2.19.
Fourteen and sixty seven-hundredths.
Exercise 6.2.21.
Sixty-one and five tenths.
Exercise 6.2.23.
Thirty-three and twelve ten-thousandths.
Exercise 6.2.25.
Two millionths.
Exercise 6.2.27.
One and ten ten-millionths.
For the following 10 problems, perform each division using a calculator. Then write the resulting decimal using words.
Exercise 6.2.29.
1÷8
Exercise 6.2.31.
2÷5
Exercise 6.2.33.
1÷50
Exercise 6.2.35.
15÷8
Exercise 6.2.37.
9÷40
Exercises for Review
Solutions to Exercises
Solution to Exercise 6.2.34. (Return to Exercise)
one thousand eight hundred seventy-five ten thousandths
6.3. Converting a Decimal to a Fraction*
Section Overview
Converting an Ordinary Decimal to a Fraction
Converting a Complex Decimal to a Fraction
Converting an Ordinary Decimal to a Fraction
We can convert a decimal fraction to a fraction, essentially, by saying it in words, then writing what we say. We may have to reduce that fraction.
Sample Set A
Convert each decimal fraction to a proper fraction or a mixed number.
Example 6.9.
Reading: six tenths→
.
Reduce:
.
Example 6.10.
Reading: nine hundred three thousands→
.
Example 6.11.
Reading: eighteen and sixty-one hundredths→
.
Example 6.12.
Reading: five hundred eight and five ten thousandths→
.
Reduce:
.
Practice Set A
Convert the following decimals to fractions or mixed numbers. Be sure to reduce.
Converting A Complex Decimal to a Fraction
Numbers such as
are called complex decimals. We can also convert complex decimals to fractions.
Sample Set B
Convert the following complex decimals to fractions.
Example 6.13.
The
appears to occur in the thousands position, but it is referring to
of a hundredth. So, we read
as "eleven and two-thirds hundredths."
Example 6.14.
Note that
Practice Set B
Convert each complex decimal to a fraction or mixed number. Be sure to reduce.
Exercises
For the following 20 problems, convert each decimal fraction to a proper fraction or a mixed number. Be sure to reduce.
Exercise 6.3.10.
0.1
Exercise 6.3.12.
0.71
Exercise 6.3.14.
0.811
Exercise 6.3.16.
2.6
Exercise 6.3.18.
25.88
Exercise 6.3.20.
1.355
Exercise 6.3.22.
0.375
Exercise 6.3.24.
21.1875
Exercise 6.3.26.
1.0055
Exercise 6.3.28.
22.110
For the following 10 problems, convert each complex decimal to a fraction.
Exercise 6.3.30.
Exercise 6.3.32.
Exercise 6.3.34.
Exercise 6.3.36.
Exercise 6.3.38.
Exercises for Review
Exercise 6.3.39. (Go to Solution)
(Section 3.6) Find the greatest common factor of 70, 182, and 154.
Exercise 6.3.43. (Go to Solution)
(Section 6.2) In the decimal number 26.10742, the digit 7 is in what position?
Solutions to Exercises
6.4. Rounding Decimals*
Section Overview
Rounding Decimal Numbers
Rounding Decimal Numbers
We first considered the concept of rounding numbers in Section 1.4 where our concern with rounding was related to whole numbers only. With a few minor changes, we can apply the same rules of rounding to decimals.
To round a decimal to a particular position:
Mark the position of the round-off digit (with an arrow or check).
Note whether the digit to the immediate right of the marked digit is
less than 5. If so, leave the round-off digit unchanged.
5 or greater. If so, add 1 to the round-off digit.
If the round-off digit is
to the right of the decimal point, eliminate all the digits to its right.
to the left of the decimal point, replace all the digits between it and the decimal point with zeros and eliminate the decimal point and all the decimal digits.
Sample Set A
Round each decimal to the specified position. (The numbers in parentheses indicate which step is being used.)
Example 6.15.
Round 32.116 to the nearest hundredth.
|
1)
|
|
2b)
The digit immediately to the right is 6, and
6 > 5, so we add 1 to the round-off digit:
1 + 1 = 2 |
|
3a)
The round-off digit is to the right of the decimal point, so we eliminate all digits to its right.
32.12 |
The number 32.116 rounded to the nearest hundredth is 32.12.
Example 6.16.
Round 633.14216 to the nearest hundred.
|
1)
|
| 2a) The digit immediately to the right is 3, and 3 < 5 so we leave the round-off digit unchanged. |
|
3b)
The round-off digit is to the left of 0, so we replace all the digits between it and the decimal point with zeros and eliminate the decimal point and all the decimal digits.
600 |
The number 633.14216 rounded to the nearest hundred is 600.
Example 6.17.
1,729.63 rounded to the nearest ten is 1,730.
Example 6.18.
1.0144 rounded to the nearest tenth is 1.0.
Example 6.19.
60.98 rounded to the nearest one is 61.
Sometimes we hear a phrase such as "round to three decimal places." This phrase means that the round-off digit is the third decimal digit (the digit in the thousandths position).
Example 6.20.
67.129 rounded to the second decimal place is 67.13.
Example 6.21.
67.129558 rounded to 3 decimal places is 67.130.
Practice Set A
Round each decimal to the specified position.
Exercises
For the first 10 problems, complete the chart by rounding each decimal to the indicated positions.
Exercise 6.4.9.
3.52612
| Tenth | Hundredth | Thousandth | Ten Thousandth |
| 3.53 |
Exercise 6.4.11.
36.109053
| Tenth | Hundredth | Thousandth | Ten Thousandth |
| 36.1 |
Exercise 6.4.13.
7.4141998
| Tenth | Hundredth | Thousandth | Ten Thousandth |
| 7.414 |
Exercise 6.4.15.
0.00008
| Tenth | Hundredth | Thousandth | Ten Thousandth |
| 0.0001 |
Exercise 6.4.17.
0.0876543
| Tenth | Hundredth | Thousandth | Ten Thousandth |
For the following 5 problems, round 18.4168095 to the indicated place.
Exercise 6.4.19.
1 decimal place.
Exercise 6.4.21.
6 decimal places.
For the following problems, perform each division using a calculator.
Exercise 6.4.23.
4 ÷ 3 and round to 2 decimal places.
Exercise 6.4.25.
1 ÷ 27 and round to 6 decimal places.
Exercise 6.4.27.
3 ÷ 16 and round to 3 decimal places.
Exercise 6.4.29.
26 ÷ 7 and round to 5 decimal places.
Exercises for Review
Solutions to Exercises
Solution to Exercise 6.4.8. (Return to Exercise)
| Tenth | Hundredth | Thousandth | Ten Thousandth |
| 20.0 | 20.01 | 20.011 | 20.0107 |
Solution to Exercise 6.4.10. (Return to Exercise)
| Tenth | Hundredth | Thousandth | Ten Thousandth |
| 531.2 | 531.22 | 531.219 | 531.2188 |
Solution to Exercise 6.4.12. (Return to Exercise)
| Tenth | Hundredth | Thousandth | Ten Thousandth |
| 2.0 | 2.00 | 2.000 | 2.0000 |
Solution to Exercise 6.4.14. (Return to Exercise)
| Tenth | Hundredth | Thousandth | Ten Thousandth |
| 0.0 | 0.00 | 0.000 | 0.0000 |
Solution to Exercise 6.4.16. (Return to Exercise)
| Tenth | Hundredth | Thousandth | Ten Thousandth |
| 9.2 | 9.19 | 9.192 | 9.1919 |
6.5. Addition and Subtraction of Decimals*
Section Overview
The Logic Behind the Method
The Method of Adding and Subtracting Decimals
Calculators
The Logic Behind the Method
Consider the sum of 4.37 and 3.22. Changing each decimal to a fraction, we have
Performing the addition, we get
Thus, 4.37 + 3.22 = 7.59.
The Method of Adding and Subtracting Decimals
When writing the previous addition, we could have written the numbers in columns.
This agrees with our previous result. From this observation, we can suggest a method for adding and subtracting decimal numbers.
To add or subtract decimals:
Align the numbers vertically so that the decimal points line up under each other and the corresponding decimal positions are in the same column.
Add or subtract the numbers as if they were whole numbers.
Place a decimal point in the resulting sum or difference directly under the other decimal points.
Sample Set A
Find the following sums and differences.
Example 6.22.
9.813 + 2.140
Example 6.23.
841.0056 + 47.016 + 19.058
To insure that the columns align properly, we can write a 0 in the position at the end of the numbers 47.016 and 19.058 without changing their values.
Example 6.24.
1.314 – 0.58
Example 6.25.
16.01 – 7.053
Example 6.26.
Find the sum of 6.88106 and 3.5219 and round it to three decimal places.
We need to round the sum to the thousandths position. Since the digit in the position immediately to the right is 9, and 9>5, we get
10.403
Example 6.27.
Wendy has $643.12 in her checking account. She writes a check for $16.92. How much is her new account balance?
To find the new account balance, we need to find the difference between 643.12 and 16.92. We will subtract 16.92 from 643.12.
After writing a check for $16.92, Wendy now has a balance of $626.20 in her checking account.
Practice Set A
Find the following sums and differences.
Exercise 6.5.6. (Go to Solution)
Find the sum of 11.6128 and 14.07353, and round it to two decimal places.
Calculators
The calculator can be useful for finding sums and differences of decimal numbers. However, calculators with an eight-digit display cannot be used when working with decimal numbers that contain more than eight digits, or when the sum results in more than eight digits. In practice, an eight-place decimal will seldom be encountered. There are some inexpensive calculators that can handle 13 decimal places.
Sample Set B
Use a calculator to find each sum or difference.
Example 6.28.
42.0638 + 126.551
| Display Reads | ||
| Type | 42.0638 | 42.0638 |
| Press | + | 42.0638 |
| Type | 126.551 | 126.551 |
| Press | = | 168.6148 |
The sum is 168.6148.
Example 6.29.
Find the difference between 305.0627 and 14.29667.
| Display Reads | ||
| Type | 305.0627 | 305.0627 |
| Press | — | 305.0627 |
| Type | 14.29667 | 14.29667 |
| Press | = | 290.76603 |
The difference is 290.76603
Example 6.30.
51.07 + 3,891.001786
Since 3,891.001786 contains more than eight digits, we will be unable to use an eight-digit display calculator to perform this addition. We can, however, find the sum by hand.
The sum is 3,942.071786.
Practice Set B
Use a calculator to perform each operation.
Exercises
For the following 15 problems, perform each addition or subtraction. Use a calculator to check each result.
Exercise 6.5.13.
15.015 − 6.527
Exercise 6.5.15.
3.16 − 2.52
Exercise 6.5.17.
0.9162 − 0.0872
Exercise 6.5.19.
761.0808 − 53.198
Exercise 6.5.21.
19.1161 + 27.8014 + 39.3161
Exercise 6.5.23.
2.181 + 6.05 + 1.167 + 8.101
Exercise 6.5.25.
27 + 42 + 9.16 − 0.1761 + 81.6
For the following 10 problems, solve as directed. A calculator may be useful.
Exercise 6.5.27.
Add 6.1121 and 4.916 and round to 2 decimal places.
Exercise 6.5.29.
Subtract 5.2121 from 9.6341 and round to 1 decimal place.
Exercise 6.5.31.
Subtract 7.01884 from the sum of 13.11848 and 2.108 and round to 4 decimal places.
Exercise 6.5.32. (Go to Solution)
A checking account has a balance of $42.51. A check is written for $19.28. What is the new balance?
Exercise 6.5.33.
A checking account has a balance of $82.97. One check is written for $6.49 and another for $39.95. What is the new balance?
Exercise 6.5.34. (Go to Solution)
A person buys $4.29 worth of hamburger and pays for it with a $10 bill. How much change does this person get?
Exercise 6.5.35.
A man buys $6.43 worth of stationary and pays for it with a $20 bill. After receiving his change, he realizes he forgot to buy a pen. If the total price of the pen is $2.12, and he buys it, how much of the $20 bill is left?
Exercise 6.5.36. (Go to Solution)
A woman starts recording a movie on her video cassette recorder with the tape counter set at 21.93. The movie runs 847.44 tape counter units. What is the final tape counter reading?
Exercises for Review
Solutions to Exercises
Solution to Exercise 6.5.11. (Return to Exercise)
Since each number contains more than eight digits, using some calculators may not be helpful. Adding these by “hand technology,” we get 4,785.00031
6.6. Multiplication of Decimals*
Section Overview
The Logic Behind the Method
The Method of Multiplying Decimals
Calculators
Multiplying Decimals By Powers of 10
Multiplication in Terms of “Of”
The Logic Behind the Method
Consider the product of 3.2 and 1.46. Changing each decimal to a fraction, we have
Thus, (3.2)(1.46) = 4.672.
Notice that the factor
Using this observation, we can suggest that the sum of the number of decimal places in the factors equals the number of decimal places in the product.
The Method of Multiplying Decimals
To multiply decimals,
Multiply the numbers as if they were whole numbers.
Find the sum of the number of decimal places in the factors.
The number of decimal places in the product is the sum found in step 2.
Sample Set A
Find the following products.
Example 6.31.
6.5 ⋅ 4.3
Thus, 6.5⋅4.3 = 27.95.
Example 6.32.
23.4⋅1.96
Thus, 23.4⋅1.96 = 45.864.
Example 6.33.
Find the product of 0.251 and 0.00113 and round to three decimal places.
Now, rounding to three decimal places, we get
Practice Set A
Find the following products.
Calculators
Calculators can be used to find products of decimal numbers. However, a calculator that has only an eight-digit display may not be able to handle numbers or products that result in more than eight digits. But there are plenty of inexpensive ($50 - $75) calculators with more than eight-digit displays.
Sample Set B
Find the following products, if possible, using a calculator.
Example 6.34.
2.58 ⋅ 8.61
| Display Reads | ||
| Type | 2.58 | 2.58 |
| Press | × | 2.58 |
| Type | 8.61 | 8.61 |
| Press | = | 22.2138 |
The product is 22.2138.
Example 6.35.
0.006 ⋅ 0.0042
| Display Reads | ||
| Type | .006 | .006 |
| Press | × | .006 |
| Type | .0042 | 0.0042 |
| Press | = | 0.0000252 |
We know that there will be seven decimal places in the product (since 3 + 4 = 7). Since the display shows 7 decimal places, we can assume the product is correct. Thus, the product is 0.0000252.
Example 6.36.
0.0026 ⋅ 0.11976
Since we expect 4 + 5 = 9 decimal places in the product, we know that an eight-digit display calculator will not be able to provide us with the exact value. To obtain the exact value, we must use "hand technology." Suppose, however, that we agree to round off this product to three decimal places. We then need only four decimal places on the display.
| Display Reads | ||
| Type | .0026 | .0026 |
| Press | × | .0026 |
| Type | .11976 | 0.11976 |
| Press | = | 0.0003114 |
Rounding 0.0003114 to three decimal places we get 0.000. Thus, 0.0026 ⋅ 0.11976 = 0.000 to three decimal places.
Practice Set B
Use a calculator to find each product. If the calculator will not provide the exact product, round the result to four decimal places.
Multiplying Decimals by Powers of 10
There is an interesting feature of multiplying decimals by powers of 10. Consider the following multiplications.
| Multiplication | Number of Zeros in the Power of 10 | Number of Positions the Decimal Point Has Been Moved to the Right |
| 10 ⋅ 8 . 315274 = 83 . 15274 | 1 | 1 |
| 100 ⋅ 8 . 315274 = 831 . 5274 | 2 | 2 |
| 1, 000 ⋅ 8 . 315274 = 8, 315 . 274 | 3 | 3 |
| 10 , 000 ⋅ 8 . 315274 = 83 , 152 . 74 | 4 | 4 |
To multiply a decimal by a power of 10, move the decimal place to the right of its current position as many places as there are zeros in the power of 10. Add zeros if necessary.
Sample Set C
Find the following products.
Example 6.37.
100⋅34.876. Since there are 2 zeros in 100, Move the decimal point in 34.876 two places to the right.
Example 6.38.
1,000⋅4.8058. Since there are 3 zeros in 1,000, move the decimal point in 4.8058 three places to the right.
Example 6.39.
10,000⋅56.82. Since there are 4 zeros in 10,000, move the decimal point in 56.82 four places to the right. We will have to add two zeros in order to obtain the four places.
Since there is no fractional part, we can drop the decimal point.
Example 6.40.
Example 6.41.
Practice Set C
Find the following products.
Multiplication in Terms of “Of”
Recalling that the word "of" translates to the arithmetic operation of multiplication, let's observe the following multiplications.
Sample Set D
Example 6.42.
Find 4.1 of 3.8.
Translating "of" to "×", we get
Thus, 4.1 of 3.8 is 15.58.
Example 6.43.
Find 0.95 of the sum of 2.6 and 0.8.
We first find the sum of 2.6 and 0.8.
Now find 0.95 of 3.4
Thus, 0.95 of (2.6 + 0.8) is 3.230.
Practice Set D
Exercises
For the following 30 problems, find each product and check each result with a calculator.
Exercise 6.6.25.
4.5⋅6.1
Exercise 6.6.27.
6.1⋅7
Exercise 6.6.29.
(1.99)(0.05)
Exercise 6.6.31.
(5.116)(1.21)
Exercise 6.6.33.
(16.527)(9.16)
Exercise 6.6.35.
1.0037⋅1.00037
Exercise 6.6.37.
(4.2)(4.2)
Exercise 6.6.39.
1.11⋅1.11
Exercise 6.6.41.
9.0168⋅1.2
Exercise 6.6.43.
(0.000001)(0.01)
Exercise 6.6.45.
(10)(36.17)
Exercise 6.6.47.
10⋅8.0107
Exercise 6.6.49.
100⋅0.779
Exercise 6.6.51.
1000⋅42.7125571
Exercise 6.6.53.
100,000⋅9.923
Exercise 6.6.55.
(8.09)(7.1)
| Actual product | Tenths | Hundreds | Thousandths |
Exercise 6.6.56. (Go to Solution)
(11.1106)(12.08)
| Actual product | Tenths | Hundreds | Thousandths |
Exercise 6.6.57.
0.0083⋅1.090901
| Actual product | Tenths | Hundreds | Thousandths |
For the following 15 problems, perform the indicated operations
Exercise 6.6.59.
Find 5.2 of 3.7.
Exercise 6.6.61.
Find 16 of 1.04
Exercise 6.6.63.
Find 0.09 of 0.003
Exercise 6.6.65.
Find 0.01 of the sum of 3.6 and 12.18
Exercise 6.6.67.
Find the difference of 6.1 of 2.7 and 2.7 of 4.03
Exercise 6.6.69.
If a person earns $8.55 an hour, how much does he earn in twenty-five hundredths of an hour?
Exercise 6.6.71.
In the problem above, how much is the total cost if 0.065 sales tax is added?
Exercise 6.6.72. (Go to Solution)
A river rafting trip is supposed to last for 10 days and each day 6 miles is to be rafted. On the third day a person falls out of the raft after only
of that day’s mileage. If this person gets discouraged and quits, what fraction of the entire trip did he complete?
Exercise 6.6.73.
A woman starts the day with $42.28. She buys one item for $8.95 and another for $6.68. She then buys another item for sixty two-hundredths of the remaining amount. How much money does she have left?
For the following 10 problems, use a calculator to determine each product. If the calculator will not provide the exact product, round the results to five decimal places.
Exercise 6.6.75.
0.261 ⋅ 1.96
Exercise 6.6.77.
(9.46)2
Exercise 6.6.79.
0.00037 ⋅ 0.0065
Exercise 6.6.81.
0.1286 ⋅ 0.7699
Exercise 6.6.83.
0.00198709 ⋅ 0.03
Exercises for Review
Exercise 6.6.87.
(Section 6.2) Write "fourteen and one hundred twenty-one ten-thousandths, using digits."
Exercise 6.6.88. (Go to Solution)
(Section 6.5) Subtract 6.882 from 8.661 and round the result to two decimal places.
Solutions to Exercises
Solution to Exercise 6.6.54. (Return to Exercise)
| Actual product | Tenths | Hundreds | Thousandths |
| 28.382 | 28.4 | 28.38 | 28.382 |
Solution to Exercise 6.6.56. (Return to Exercise)
| Actual product | Tenths | Hundreds | Thousandths |
| 134.216048 | 134.2 | 134.22 | 134.216 |
Solution to Exercise 6.6.58. (Return to Exercise)
| Actual product | Tenths | Hundreds | Thousandths |
| 185.626 | 185.6 | 185.63 | 185.626 |
6.7. Division of Decimals*
Section Overview
The Logic Behind the Method
A Method of Dividing a Decimal By a Nonzero Whole Number
A Method of Dividing a Decimal by a Nonzero Decimal
Dividing Decimals by Powers of 10
The Logic Behind the Method
As we have done with addition, subtraction, and multiplication of decimals, we will study a method of division of decimals by converting them to fractions, then we will make a general rule.
We will proceed by using this example: Divide 196.8 by 6.
We have, up to this point, divided 196.8 by 6 and have gotten a quotient of 32 with a remainder of 4. If we follow our intuition and bring down the .8, we have the division 4.8÷6.
Thus, 4.8÷6 = .8.
Now, our intuition and experience with division direct us to place the .8 immediately to the right of 32.
From these observations, we suggest the following method of division.
A Method of Dividing a Decimal by a Nonzero Whole Number
To divide a decimal by a nonzero whole number:
Write a decimal point above the division line and directly over the decimal point of the dividend.
Proceed to divide as if both numbers were whole numbers.
If, in the quotient, the first nonzero digit occurs to the right of the decimal point, but not in the tenths position, place a zero in each position between the decimal point and the first nonzero digit of the quotient.
Sample Set A
Find the decimal representations of the following quotients.
Example 6.44.
114.1÷7 = 7
Thus, 114.1÷7 = 16.3.
Check: If 114.1÷7 = 16.3, then 7⋅16.3 should equal 114.1.
Example 6.45.
0.02068÷4
Place zeros in the tenths and hundredths positions. (See Step 3.)
Thus, 0.02068÷4 = 0.00517.
Practice Set A
Find the following quotients.
A Method of Dividing a Decimal By a Nonzero Decimal
Now that we can divide decimals by nonzero whole numbers, we are in a position to divide decimals by a nonzero decimal. We will do so by converting a division by a decimal into a division by a whole number, a process with which we are already familiar. We'll illustrate the method using this example: Divide 4.32 by 1.8.
Let's look at this problem as
.
The divisor is
. We can convert
into a whole number if we multiply it by 10.
But, we know from our experience with fractions, that if we multiply the denominator of a fraction by a nonzero whole number, we must multiply the numerator by that same nonzero whole number. Thus, when converting
to a whole number by multiplying it by 10, we must also multiply the numerator
by 10.
We have converted the division 4.32÷1.8 into the division 43.2÷18, that is,
Notice what has occurred.
If we "move" the decimal point of the divisor one digit to the right, we must also "move" the decimal point of the dividend one place to the right. The word "move" actually indicates the process of multiplication by a power of 10.
To divide a decimal by a nonzero decimal,
Convert the divisor to a whole number by moving the decimal point to the position immediately to the right of the divisor's last digit.
Move the decimal point of the dividend to the right the same number of digits it was moved in the divisor.
Set the decimal point in the quotient by placing a decimal point directly above the newly located decimal point in the dividend.
Divide as usual.
Sample Set B
Find the following quotients.
Example 6.46.
32.66÷7.1
| The divisor has one decimal place. |
| Move the decimal point of both the divisor and the dividend 1 place to the right. |
| Set the decimal point. |
| Divide as usual. |
Thus, 32.66÷7.1 = 4.6.
Check: 32.66÷7.1 = 4.6 if 4.6 × 7.1 = 32.66
Example 6.47.
1.0773÷0.513
| The divisor has 3 decimal places. |
| Move the decimal point of both the divisor and the dividend 3 places to the right. |
| Set the decimal place and divide. |
Thus, 1.0773÷0.513 = 2.1.
Checking by multiplying 2.1 and 0.513 will convince us that we have obtained the correct result. (Try it.)
Example 6.48.
12÷0.00032
| The divisor has 5 decimal places. |
| Move the decimal point of both the divisor and the dividend 5 places to the right. We will need to add 5 zeros to 12. |
| Set the decimal place and divide. |
This is now the same as the division of whole numbers.
Checking assures us that 12÷0.00032 = 37,500.
Practice Set B
Find the decimal representation of each quotient.
Calculators
Calculators can be useful for finding quotients of decimal numbers. As we have seen with the other calculator operations, we can sometimes expect only approximate results. We are alerted to approximate results when the calculator display is filled with digits. We know it is possible that the operation may produce more digits than the calculator has the ability to show. For example, the multiplication
produces 5 + 4 = 9 decimal places. An eight-digit display calculator only has the ability to show eight digits, and an approximation results. The way to recognize a possible approximation is illustrated in problem 3 of the next sample set.
Sample Set C
Find each quotient using a calculator. If the result is an approximation, round to five decimal places.
Example 6.49.
12.596÷4.7
| Display Reads | ||
| Type | 12.596 | 12.596 |
| Press | ÷ | 12.596 |
| Type | 4.7 | 4.7 |
| Press | = | 2.68 |
Since the display is not filled, we expect this to be an accurate result.
Example 6.50.
0.5696376÷0.00123
| Display Reads | ||
| Type | .5696376 | 0.5696376 |
| Press | ÷ | 0.5696376 |
| Type | .00123 | 0.00123 |
| Press | = | 463.12 |
Since the display is not filled, we expect this result to be accurate.
Example 6.51.
0.8215199÷4.113
| Display Reads | ||
| Type | .8215199 | 0.8215199 |
| Press | ÷ | 0.8215199 |
| Type | 4.113 | 4.113 |
| Press | = | 0.1997373 |
There are EIGHT DIGITS — DISPLAY FILLED! BE AWARE OF POSSIBLE APPROXIMATIONS.
We can check for a possible approximation in the following way. Since the division 
can be checked by multiplying 4 and 3, we can check our division by performing the multiplication
This multiplication produces 3 + 7 = 10 decimal digits. But our suspected quotient contains only 8 decimal digits. We conclude that the answer is an approximation. Then, rounding to five decimal places, we get 0.19974.
Practice Set C
Find each quotient using a calculator. If the result is an approximation, round to four decimal places.
Dividing Decimals By Powers of 10
In problems 4 and 5 of the section called “Practice Set B”, we found the decimal representations of 8,162.41÷10 and 8,162.41÷100. Let's look at each of these again and then, from these observations, make a general statement regarding division of a decimal number by a power of 10.
Thus, 8,162.41÷10 = 816.241.
Notice that the divisor 10 is composed of one 0 and that the quotient 816.241 can be obtained from the dividend 8,162.41 by moving the decimal point one place to the left.
Thus, 8,162.41÷100 = 81.6241.
Notice that the divisor 100 is composed of two 0's and that the quotient 81.6241 can be obtained from the dividend by moving the decimal point two places to the left.
Using these observations, we can suggest the following method for dividing decimal numbers by powers of 10.
To divide a decimal fraction by a power of 10, move the decimal point of the decimal fraction to the left as many places as there are zeros in the power of 10. Add zeros if necessary.
Sample Set D
Find each quotient.
Example 6.52.
9,248.6÷100 Since there are 2 zeros in this power of 10, we move the decimal point 2 places to the left.
Example 6.53.
3.28÷10,000
Since there are 4 zeros in this power of 10, we move the decimal point 4 places to the left. To do so, we need to add three zeros.
Practice Set D
Find the decimal representation of each quotient.
Exercises
For the following 30 problems, find the decimal representation of each quotient. Use a calculator to check each result.
Exercise 6.7.22.
16.8÷8
Exercise 6.7.24.
12.33÷3
Exercise 6.7.26.
73.56÷12
Exercise 6.7.28.
12.16÷64
Exercise 6.7.30.
439.35÷435
Exercise 6.7.32.
46.41÷9.1
Exercise 6.7.34.
0.68÷1.7
Exercise 6.7.36.
2.832÷0.4
Exercise 6.7.38.
16.2409÷4.03
Exercise 6.7.40.
25.050025÷5.005
Exercise 6.7.42.
0.48÷0.08
Exercise 6.7.44.
48.87÷0.87
Exercise 6.7.46.
64,351.006÷10
Exercise 6.7.48.
64,351.006÷1,000
Exercise 6.7.50.
0.43÷100
For the following 5 problems, find each quotient. Round to the specified position. A calculator may be used.
Exercise 6.7.51. (Go to Solution)
11.2944÷6.24
| Actual Quotient | Tenths | Hundredths | Thousandths |
Exercise 6.7.52.
45.32931÷9.01
| Actual Quotient | Tenths | Hundredths | Thousandths |
Exercise 6.7.53. (Go to Solution)
3.18186÷0.66
| Actual Quotient | Tenths | Hundredths | Thousandths |
Exercise 6.7.54.
4.3636÷4
| Actual Quotient | Tenths | Hundredths | Thousandths |
Exercise 6.7.55. (Go to Solution)
0.00006318÷0.018
| Actual Quotient | Tenths | Hundredths | Thousandths |
For the following 9 problems, find each solution.
Exercise 6.7.56.
Divide the product of 7.4 and 4.1 by 2.6.
Exercise 6.7.57. (Go to Solution)
Divide the product of 11.01 and 0.003 by 2.56 and round to two decimal places.
Exercise 6.7.58.
Divide the difference of the products of 2.1 and 9.3, and 4.6 and 0.8 by 0.07 and round to one decimal place.
Exercise 6.7.59. (Go to Solution)
A ring costing $567.08 is to be paid off in equal monthly payments of $46.84. In how many months will the ring be paid off?
Exercise 6.7.60.
Six cans of cola cost $2.58. What is the price of one can?
Exercise 6.7.61. (Go to Solution)
A family traveled 538.56 miles in their car in one day on their vacation. If their car used 19.8 gallons of gas, how many miles per gallon did it get?
Exercise 6.7.62.
Three college students decide to rent an apartment together. The rent is $812.50 per month. How much must each person contribute toward the rent?
Exercise 6.7.63. (Go to Solution)
A woman notices that on slow speed her video cassette recorder runs through 296.80 tape units in 10 minutes and at fast speed through 1098.16 tape units. How many times faster is fast speed than slow speed?
Exercise 6.7.64.
A class of 34 first semester business law students pay a total of $1,354.90, disregarding sales tax, for their law textbooks. What is the cost of each book?
For the following problems, use calculator to find the quotients. If the result is approximate (see Sample Set C Example 6.51) round the result to three decimal places.
Exercise 6.7.66.
0.067444÷0.052
Exercise 6.7.68.
219,709.36÷9941.6
Exercise 6.7.70.
2.4858225÷1.11611
Exercise 6.7.72.
2.102838÷1.0305
Exercises for Review
Solutions to Exercises
Solution to Exercise 6.7.14. (Return to Exercise)
3.5197645 is an approximate result. Rounding to four decimal places, we get 3.5198
Solution to Exercise 6.7.51. (Return to Exercise)
| Actual Quotient | Tenths | Hundredths | Thousandths |
| 1.81 | 1.8 | 1.81 | 1.810 |
Solution to Exercise 6.7.53. (Return to Exercise)
| Actual Quotient | Tenths | Hundredths | Thousandths |
| 4.821 | 4.8 | 4.82 | 4.821 |
Solution to Exercise 6.7.55. (Return to Exercise)
| Actual Quotient | Tenths | Hundredths | Thousandths |
| 0.00351 | 0.0 | 0.00 | 0.004 |
6.8. Nonterminating Divisions*
Section Overview
Nonterminating Divisions
Denoting Nonterminating Quotients
Nonterminating Divisions
Let's consider two divisions:
9.8 ÷ 3.5
4÷ 3
Previously, we have considered divisions like example 1, which is an example of a terminating division. A terminating division is a division in which the quotient terminates after several divisions (the remainder is zero).
The quotient in this problem terminates in the tenths position. Terminating divisions are also called exact divisions.
The division in example 2 is an example of a nonterminating division. A non-terminating division is a division that, regardless of how far we carry it out, always has a remainder.
We can see that the pattern in the brace is repeated endlessly. Such a decimal quotient is called a repeating decimal.
Denoting Nonterminating Quotients
We use three dots at the end of a number to indicate that a pattern repeats itself endlessly.
4 ÷ 3 = 1 . 333 …
Another way, aside from using three dots, of denoting an endlessly repeating pattern is to write a bar ( ¯ ) above the repeating sequence of digits.
The bar indicates the repeated pattern of 3.
Repeating patterns in a division can be discovered in two ways:
As the division process progresses, should the remainder ever be the same as the dividend, it can be concluded that the division is nonterminating and that the pattern in the quotient repeats. This fact is illustrated in Example 6.54 of the section called “Sample Set A”.
As the division process progresses, should the "product, difference" pattern ever repeat two consecutive times, it can be concluded that the division is nonterminating and that the pattern in the quotient repeats. This fact is illustrated in Example 6.55 and 4 of the section called “Sample Set A”.
Sample Set A
Carry out each division until the repeating pattern can be determined.
Example 6.54.
100 ÷ 27
When the remainder is identical to the dividend, the division is nonterminating. This implies that the pattern in the quotient repeats.
100÷27 = 3.70370370… The repeating block is 703.
Example 6.55.
1 ÷ 9
We see that this “product, difference”pattern repeats. We can conclude that the division is nonterminating and that the quotient repeats.
1÷9 = 0.111… The repeating block is 1.
Example 6.56.
Divide 2 by 11 and round to 3 decimal places.
Since we wish to round the quotient to three decimal places, we'll carry out the division so that the quotient has four decimal places.
The number .1818 rounded to three decimal places is .182. Thus, correct to three decimal places,
2 ÷ 11 = 0 . 182
Example 6.57.
Divide 1 by 6.
We see that this “product, difference” pattern repeats. We can conclude that the division is nonterminating and that the quotient repeats at the 6.
Practice Set A
Carry out the following divisions until the repeating pattern can be determined.
Exercises
For the following 20 problems, carry out each division until the repeating pattern is determined. If a repeating pattern is not apparent, round the quotient to three decimal places.
Exercise 6.8.8.
8÷11
Exercise 6.8.10.
5÷6
Exercise 6.8.12.
3÷1.1
Exercise 6.8.14.
10÷2.7
Exercise 6.8.16.
8.08÷3.1
Exercise 6.8.18.
0.213÷0.31
Exercise 6.8.20.
6.03÷1.9
Exercise 6.8.22.
1.55÷0.27
Exercise 6.8.24.
0.444÷0.999
Exercise 6.8.26.
3.8÷0.99
For the following 10 problems, use a calculator to perform each division.
Exercise 6.8.28.
8÷11
Exercise 6.8.30.
1÷44
Exercise 6.8.34.
0.0707÷0.7070
Exercise 6.8.36.
1÷0.9999999
Exercise for Review
Exercise 6.8.37. (Go to Solution)
(Section 1.2) In the number 411,105, how many ten thousands are there?
Exercise 6.8.40.
(Section 6.5) Subtract 8.01629 from 9.00187 and round the result to three decimal places.
Solutions to Exercises
6.9. Converting a Fraction to a Decimal*
Now that we have studied and practiced dividing with decimals, we are also able to convert a fraction to a decimal. To do so we need only recall that a fraction bar can also be a division symbol. Thus,
not only means "3 objects out of 4," but can also mean "3 divided by 4."
Sample Set A
Convert the following fractions to decimals. If the division is nonterminating, round to two decimal places.
Example 6.58.
. Divide 3 by 4.
Thus,
.
Example 6.59.
Divide 1 by 5.
Thus,
Example 6.60.
. Divide 5 by 6.
We are to round to two decimal places.
Thus,
to two decimal places.
Example 6.61.
. Note that
.
Convert
to a decimal.
Thus,
.
Example 6.62.
. This is a complex decimal.
Note that the 6 is in the hundredths position.
The number
is read as "sixteen and one-fourth hundredths."
Now, convert
to a decimal.
Thus,
.
Practice Set A
Convert the following fractions and complex decimals to decimals (in which no proper fractions appear). If the divison is nonterminating, round to two decimal places.
Exercises
For the following 30 problems, convert each fraction or complex decimal number to a decimal (in which no proper fractions appear).
Exercise 6.9.8.
Exercise 6.9.10.
Exercise 6.9.12.
Exercise 6.9.14.
Exercise 6.9.16.
Exercise 6.9.18.
Exercise 6.9.20.
Exercise 6.9.22.
Exercise 6.9.24.
Exercise 6.9.26.
Exercise 6.9.28.
Exercise 6.9.30.
Exercise 6.9.32.
Exercise 6.9.34.
Exercise 6.9.36.
For the following 18 problems, convert each fraction to a decimal. Round to five decimal places.
Exercise 6.9.38.
Exercise 6.9.40.
Exercise 6.9.42.
Exercise 6.9.44.
Exercise 6.9.46.
Exercise 6.9.48.
Exercise 6.9.50.
Exercise 6.9.52.
Exercise 6.9.54.
For the following problems, use a calculator to convert each fraction to a decimal. If no repeating pattern seems to exist, round to four decimal places.
Exercise 6.9.56.
Exercise 6.9.58.
Exercise 6.9.60.
Exercises for Review
Solutions to Exercises
6.10. Combinations of Operations with Decimals and Fractions*
Having considered operations with decimals and fractions, we now consider operations that involve both decimals and fractions.
Sample Set A
Perform the following operations.
Example 6.63.
. Convert both numbers to decimals or both numbers to fractions. We’ll convert to decimals.
To convert
to a decimal, divide 1 by 4.
Now multiply 0.38 and .25.
Thus,
.
In the problems that follow, the conversions from fraction to decimal, or decimal to fraction, and some of the additions, subtraction, multiplications, and divisions will be left to you.
Example 6.64.
Convert
to a decimal.
1.85 + 0.375⋅4.1 Multiply before adding.
1.85 + 1.5375 Now add.
3.3875
Example 6.65.
Convert 0.28 to a fraction.
Example 6.66.
Practice Set A
Perform the following operations.
Exercises
Exercise 6.10.6.
Exercise 6.10.8.
Exercise 6.10.10.
Exercise 6.10.12.
Exercise 6.10.14.
Exercise 6.10.16.
Exercise 6.10.18.
Exercise 6.10.20.
Exercise 6.10.22.
Exercise 6.10.24.
Exercise 6.10.26.
Exercise 6.10.28.
Exercise 6.10.30.
Exercise 6.10.32.
Exercises for Review
Solutions to Exercises
6.11. Summary of Key Concepts*
Summary of Key Concepts
A decimal point is a point that separates the units digit from the tenths digit.
A decimal fraction is a fraction whose denominator is a power of ten.
Decimals can be converted to fractions by saying the decimal number in words, then writing what was said.
Decimals are rounded in much the same way whole numbers are rounded.
To add or subtract decimals,
Align the numbers vertically so that the decimal points line up under each other and the corresponding decimal positions are in the same column.
Add or subtract the numbers as if they were whole numbers.
Place a decimal point in the resulting sum directly under the other decimal points.
To multiply two decimals,
Multiply the numbers as if they were whole numbers.
Find the sum of the number of decimal places in the factors.
The number of decimal places in the product is the number found in step 2.
To multiply a decimal by a power of 10, move the decimal point to the right as many places as there are zeros in the power of ten. Add zeros if necessary.
To divide a decimal by a nonzero decimal,
Convert the divisor to a whole number by moving the decimal point until it appears to the right of the divisor's last digit.
Move the decimal point of the dividend to the right the same number of digits it was moved in the divisor.
Proceed to divide.
Locate the decimal in the answer by bringing it straight up from the dividend.
To divide a decimal by a power of 10, move the decimal point to the left as many places as there are zeros in the power of ten. Add zeros if necessary.
A terminating division is a division in which the quotient terminates after several divisions. Terminating divisions are also called exact divisions.
A nonterminating division is a division that, regardless of how far it is carried out, always has a remainder. Nonterminating divisions are also called nonexact divisions.
A fraction can be converted to a decimal by dividing the numerator by the denominator.
6.12. Exercise Supplement*
Exercise Supplement
Reading and Writing Decimals (Section 6.2)
Exercise 6.12.1. (Go to Solution)
The decimal digit that appears two places to the right of the decimal point is in the _______________ position.
Exercise 6.12.2.
The decimal digit that appears four places to the right of the decimal point is in the _______________ position.
For problems 3-8, read each decimal by writing it in words.
Exercise 6.12.4.
8.105
Exercise 6.12.6.
5.9271
Exercise 6.12.8.
4.01701
For problems 9-13, write each decimal using digits.
Exercise 6.12.10.
Two and one hundred seventy-seven thousandths.
Exercise 6.12.12.
Four tenths.
Converting a Decimal to a Fraction (Section 6.3)
For problem 14-20, convert each decimal to a proper fraction or a mixed number.
Exercise 6.12.14.
1.07
Exercise 6.12.16.
0.05
Exercise 6.12.18.
Exercise 6.12.20.
Rounding Decimals (Section 6.4)
For problems 21-25, round each decimal to the specified position.
Exercise 6.12.22.
4.087 to the nearest tenth.
Exercise 6.12.24.
817.42 to the nearest ten.
Addition, Subtraction, Multiplication and Division of Decimals, and Nonterminating Divisions (Section 6.5,Section 6.6,Section 6.7,Section 6.8)
For problem 26-45, perform each operation and simplify.
Exercise 6.12.26.
7.10 + 2.98
Exercise 6.12.28.
1.2⋅8.6
Exercise 6.12.30.
57.51÷2.7
Exercise 6.12.32.
32,051.3585÷23,006.9999
Exercise 6.12.34.
1,000⋅1,816.001
Exercise 6.12.36.
0.135888÷16.986
Exercise 6.12.38.
4.119÷10,000
Exercise 6.12.40.
6.9÷12
Exercise 6.12.42.
8.8÷19. Round to one decimal place.
Exercise 6.12.44.
1.1÷9.9
Converting a Fraction to a Decimal (Section 6.9)
For problems 46-55, convert each fraction to a decimal.
Exercise 6.12.46.
Exercise 6.12.48.
Exercise 6.12.50.
Exercise 6.12.52.
Exercise 6.12.54.
Combinations of Operations with Decimals and Fractions (Section 6.10)
For problems 56-62, perform each operation.
Exercise 6.12.56.
Exercise 6.12.58.
Exercise 6.12.60.
Exercise 6.12.62.
Solutions to Exercises
6.13. Proficiency Exam*
Proficiency Exam
Exercise 6.13.1. (Go to Solution)
(Section 6.2) The decimal digit that appears three places to the right of the decimal point is in the __________ position.
Exercise 6.13.3. (Go to Solution)
(Section 6.2) Write eighty-one and twelve hundredths using digits. 81.12
Exercise 6.13.4. (Go to Solution)
(Section 6.2) Write three thousand seventeen millionths using digits.
For problems 10-20, perform each operation.
