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Arithmetic Review: Decimal Fractions

Module by: Wade Ellis, Denny Burzynski. E-mail the authors

Summary: This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. This chapter contains many examples of arithmetic techniques that are used directly or indirectly in algebra. Since the chapter is intended as a review, the problem-solving techniques are presented without being developed. Therefore, no work space is provided, nor does the chapter contain all of the pedagogical features of the text. As a review, this chapter can be assigned at the discretion of the instructor and can also be a valuable reference tool for the student.

Overview

  • Decimal Fractions
  • Adding and Subtracting Decimal Fractions
  • Multiplying Decimal Fractions
  • Dividing Decimal Fractions
  • Converting Decimal Fractions to Fractions
  • Converting Fractions to Decimal Fractions

Decimal Fractions

Fractions are one way we can represent parts of whole numbers. Decimal fractions are another way of representing parts of whole numbers.

Decimal Fractions

A decimal fraction is a fraction in which the denominator is a power of 10.

A decimal fraction uses a decimal point to separate whole parts and fractional parts. Whole parts are written to the left of the decimal point and fractional parts are written to the right of the decimal point. Just as each digit in a whole number has a particular value, so do the digits in decimal positions.

The positions of the digits lying to the left and to the right of the decimal point are labeled. Moving towards left from the decimal point, the positions are labeled: the first as 'ones', the second as 'Tens', the third as 'Hundreds', the fourth as 'Thousands', the fifth as 'Ten Thousands', the sixth as 'Hundred Thousands', and the seventh as 'Millions'. Moving towards right from the decimal point, the positions are labeled: the first position as 'Tenths', the second position as 'Hundredths', the third as 'Thousandths', the fourth as 'Ten Thousandths', the fifth as 'Hundred Thousandths', and the sixth position as 'Millionths'. There is a comment written below the decimal positons as "Notice that decimal number have the suffix 'th'."

Sample Set A

The following numbers are decimal fractions.

Example 1

57.9 The 9 is in thetenthsposition.57.9=57 9 10 . 57.9 The 9 is in thetenthsposition.57.9=57 9 10 .

Example 2

6.8014 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.  6.8014=6 8014 10000 . 6.8014 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.  6.8014=6 8014 10000 .

Adding and Subtracting Decimal Fractions

Adding/Subtracting Decimal Fractions

To add or subtract decimal fractions,

  1. Align the numbers vertically so that the decimal points line up under each other and corresponding decimal positions are in the same column. Add zeros if necessary.
  2. Add or subtract the numbers as if they were whole numbers.
  3. Place a decimal point in the resulting sum or difference directly under the other decimal points.

Sample Set B

Find each sum or difference.

Example 3

9.183+2.140 The decimal points are aligned in the same column.  9.183 +  2.140 ¯ 11.323 9.183+2.140 The decimal points are aligned in the same column.  9.183 +  2.140 ¯ 11.323

Example 4

841.0056+47.016+19.058 The decimal points are aligned in the same column.  841.0056 47.016 Place a 0 into the thousandths position. + 19.058 ¯ Place a 0 into the thousandths position.  The decimal points are aligned in the same column.  841.0056 47.0160 + 19.0580 ¯ 907.0796 841.0056+47.016+19.058 The decimal points are aligned in the same column.  841.0056 47.016 Place a 0 into the thousandths position. + 19.058 ¯ Place a 0 into the thousandths position.  The decimal points are aligned in the same column.  841.0056 47.0160 + 19.0580 ¯ 907.0796

Example 5

16.017.053 The decimal points are aligned in the same column.  16.01 Place a 0 into the thousandths position.  7.053 ¯ The decimal points are aligned in the same column.  16.010 7.053 ¯ 8.957 16.017.053 The decimal points are aligned in the same column.  16.01 Place a 0 into the thousandths position.  7.053 ¯ The decimal points are aligned in the same column.  16.010 7.053 ¯ 8.957

Multiplying Decimal Fractions

Multiplying Decimal Fractions

To multiply decimals,

  1. Multiply tbe numbers as if they were whole numbers.
  2. Find the sum of the number of decimal places in the factors.
  3. The number of decimal places in the product is the sum found in step 2.

Sample Set C

Find the following products.

Example 6

6.5×4.3 6.5×4.3

The vertical multiplication of two decimals; six point five, and four point three. See the longdesc for a full description.

6.5×4.3=27.95 6.5×4.3=27.95

Example 7

23.4×1.96 23.4×1.96

The vertical multiplication of two decimals; twenty-three point four, and one point nine six. See the longdesc for a full description.

23.4×1.96=45.864 23.4×1.96=45.864

Dividing Decimal Fractions

Dividing Decimal Fractions

To divide a decimal by a nonzero decimal,

  1. 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.
  2. Move the decimal point of the dividend to the right the same number of digits it was moved in the divisor.
  3. Set the decimal point in the quotient by placing a decimal point directly above the decimal point in the dividend.
  4. Divide as usual.

Sample Set D

Find the following quotients.

Example 8

32.66÷7.1 32.66÷7.1

A long division problem showing seven point one dividing thirty-two point six six. See the longdesc for a full description.

32.66÷7.1=4.6 Check: 32.66÷7.1=4.6if4.6×7.1=32.66 4.6 7.1 ¯ 4.6 322 ¯ 32.66 True 32.66÷7.1=4.6 Check: 32.66÷7.1=4.6if4.6×7.1=32.66 4.6 7.1 ¯ 4.6 322 ¯ 32.66 True

Example 9

A long division problem showing zero point five one three dividing one point zero seven seven three. See the longdesc for a full description.

Check by multiplying 2.1 2.1 and 0.513. 0.513. This will show that we have obtained the correct result.

Example 10

12÷0.00032 12÷0.00032 A long division problem showing zero point zero zero zero three two dividing twelve. See the longdesc for a full description.

Converting Decimal Fractions to Fractions

We can convert a decimal fraction to a fraction by reading it and then writing the phrase we have just read. As we read the decimal fraction, we note the place value farthest to the right. We may have to reduce the fraction.

Sample Set E

Convert each decimal fraction to a fraction.

Example 11

0.6 0. 6 ¯ tenths position Reading: six tenths 6 10 Reduce: 0.6= 6 10 = 3 5 0.6 0. 6 ¯ tenths position Reading: six tenths 6 10 Reduce: 0.6= 6 10 = 3 5

Example 12

21.903 21.90 3 ¯ thousandths position Reading: twenty-one and nine hundred three thousandths21 903 1000 21.903 21.90 3 ¯ thousandths position Reading: twenty-one and nine hundred three thousandths21 903 1000

Converting Fractions to Decimal Fractions

Sample Set F

Convert the following fractions to decimals. If the division is nonterminating, round to 2 decimal places.

Example 13

3 4 3 4

A long division problem showing four dividing three point zero zero. See the longdesc for a full description.

3 4 =0.75 3 4 =0.75

Example 14

1 5 1 5

A long division problem showing five dividing one point zero. The quotient is point two. The result of point two, multiplied by the divisor, five, is one point zero written under the dividend one point zero. The remainder zero is written as a result of the subtraction of one point zero and one point zero. A comment 'Divide one by five' is written on the right side of the problem.

1 5 =0.2 1 5 =0.2

Example 15

5 6 5 6

A long division problem showing six dividing five point zero zero zero. See the longdesc for a full description.

5 6 = 0.833... We are to round to 2 decimal places. 5 6 = 0.83 to 2 decimal places. 5 6 = 0.833... We are to round to 2 decimal places. 5 6 = 0.83 to 2 decimal places.

Example 16

5 1 8 Note that 5 1 8 =5+ 1 8 . 5 1 8 Note that 5 1 8 =5+ 1 8 .

One point zero zero zero is being divided by eight, using long division method. See the longdesc for a full description.

1 8 =.125 Thus, 5 1 8 =5+ 1 8 =5+.125=5.125. 1 8 =.125 Thus, 5 1 8 =5+ 1 8 =5+.125=5.125.

Example 17

0.16 1 4 0.16 1 4

This is a complex decimal. The “6” is in the hundredths position. The number 0.16 1 4 0.16 1 4 is read as “sixteen and one-fourth hundredths.”

0.16 1 4 = 16 1 4 100 = 16·4+1 4 100 = 65 4 100 1 = 65 13 4 · 1 100 20 = 13×1 4×20 = 13 80 0.16 1 4 = 16 1 4 100 = 16·4+1 4 100 = 65 4 100 1 = 65 13 4 · 1 100 20 = 13×1 4×20 = 13 80

Now, convert 13 80 13 80 to a decimal.

Thirteen point zero zero zero zero is being divided by eighty, using long division method. See the longdesc for a full description.

0.16 1 4 =0.1625. 0.16 1 4 =0.1625.

Exercises

For the following problems, perform each indicated operation.

Exercise 1

1.84+7.11 1.84+7.11

Solution

8.95 8.95

Exercise 2

15.0156.527 15.0156.527

Exercise 3

4.9042.67 4.9042.67

Solution

2.234 2.234

Exercise 4

156.3324.095 156.3324.095

Exercise 5

.0012+1.53+5.1 .0012+1.53+5.1

Solution

6.6312 6.6312

Exercise 6

44.98+22.812.76 44.98+22.812.76

Exercise 7

5.00043.00004+1.6837 5.00043.00004+1.6837

Solution

3.68406 3.68406

Exercise 8

1.11+12.121213.131313 1.11+12.121213.131313

Exercise 9

4.26·3.2 4.26·3.2

Solution

13.632 13.632

Exercise 10

2.97·3.15 2.97·3.15

Exercise 11

23.05·1.1 23.05·1.1

Solution

25.355 25.355

Exercise 12

5.009·2.106 5.009·2.106

Exercise 13

0.1·3.24 0.1·3.24

Solution

0.324 0.324

Exercise 14

100·12.008 100·12.008

Exercise 15

1000·12.008 1000·12.008

Solution

12,008 12,008

Exercise 16

10,000·12.008 10,000·12.008

Exercise 17

75.642÷18.01 75.642÷18.01

Solution

4.2 4.2

Exercise 18

51.811÷1.97 51.811÷1.97

Exercise 19

0.0000448÷0.014 0.0000448÷0.014

Solution

0.0032 0.0032

Exercise 20

0.129516÷1004 0.129516÷1004

For the following problems, convert each decimal fraction to a fraction.

Exercise 21

0.06 0.06

Solution

3 50 3 50

Exercise 22

0.115 0.115

Exercise 23

3.7 3.7

Solution

3 7 10 3 7 10

Exercise 24

48.1162 48.1162

Exercise 25

712.00004 712.00004

Solution

712 1 25000 712 1 25000

For the following problems, convert each fraction to a decimal fraction. If the decimal form is nonterminating,round to 3 decimal places.

Exercise 26

5 8 5 8

Exercise 27

9 20 9 20

Solution

0.45 0.45

Exercise 28

15÷22 15÷22

Exercise 29

7 11 7 11

Solution

0.636 0.636

Exercise 30

2 9 2 9

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