1. Chapter 1 / Whole Numbers and Introduction to Algebra
Chapter Five
Decimals
5.1 Introductions to Decimals
5.2 Adding & Subtracting Decimals
5.3 Multiplying Decimals & Circumference of a Circle
5.4 Dividing Decimals
5.5 Fractions, Decimals, & Order of Operations
5.6 Equations Containing Decimals

2. Introduction to Decimals
Chapter 1 / Whole Numbers and Introduction to Algebra
Section 5.1
Introduction to Decimals

3. Chapter 1 / Whole Numbers and Introduction to Algebra
Like fractional notation, decimal notation is used to denote a part of a whole. Numbers written in decimal notation are called decimal numbers, or simply decimals. The decimal has three parts.
16.743
Whole- number part
Decimal part
Decimal point

4. The position of each digit in a number determines its place value.
hundred-thousandths
hundreds
tens
tenths
hundredths
thousandths
ten-thousandths
ones 1 10 1
1000 1
100,000
Place Value
100 10 1 1
100 1
10,000
decimal point

5. of the value of the place to its left.
Notice that the value of each place is
of the value of the place to its left. 1 10
Martin-Gay, Prealgebra, 5ed

6. Martin-Gay, Prealgebra, 5ed
16.734
The digit 3 is in the hundredths place, so
its value is 3 hundredths or 3
100
Martin-Gay, Prealgebra, 5ed

7. Writing (or Reading) a Decimal in Words
Chapter 1 / Whole Numbers and Introduction to Algebra
Writing (or Reading) a Decimal in Words
Step 1. Write the whole-number part in words.
Step 2. Write “and” for the decimal point.
Step 3. Write the decimal part in words as though it were a whole number, followed by the place value of the last digit.
Martin-Gay, Prealgebra, 5ed

8. Writing a Decimal in Words
Write the decimal in words.
whole-number part
decimal part
one hundred forty-three
and
fifty-six thousandths
Martin-Gay, Prealgebra, 5ed

9. Writing Decimals in Standard Form
A decimal written in words can be written in standard form by reversing the procedure.
Write one hundred six and five hundredths in standard form.
one hundred six and five hundredths
whole-number part
decimal
decimal part
5 must be in the
hundredths place
106 . 05

10. three and fifty-four thousandths is 3.054 Martin-Gay, Prealgebra, 5ed
Helpful Hint
When writing a decimal from words to decimal notation, make sure the last digit is in the correct place by inserting 0s after the decimal point if necessary.
For example,
three and fifty-four thousandths is 3.054
thousandths place
Martin-Gay, Prealgebra, 5ed

11. Chapter 1 / Whole Numbers and Introduction to Algebra
Writing Decimals as Fractions
Once you master writing and reading decimals correctly, then you write a decimal as a fraction using the fractions associated with the words you use when you read it.
0.9
is read “nine tenths” and written as a fraction as
Martin-Gay, Prealgebra, 5ed

12. Martin-Gay, Prealgebra, 5ed
twenty-one hundredths
and written as a fraction as
0.21 is read as 21
100
0.011 is read as
eleven thousandths
and written as a fraction as 11
1000
Martin-Gay, Prealgebra, 5ed

13. 37
100 . =
029 29
1000 . =
2 decimal places
2 zeros
3 decimal places
3 zeros
Notice that the number of decimal places in a decimal number is the same as the number of zeros in the denominator of the equivalent fraction. We can use this fact to write decimals as fractions.

14. Martin-Gay, Prealgebra, 5ed
Comparing Decimals
One way to compare decimals is to compare their graphs on a number line. Recall that for any two numbers on a number line, the number to the left is smaller and the number to the right is larger. To compare 0.3 and 0.7 look at their graphs.
0.3
0.7 1
0.3 < 0.7 or > 0.3 3 10 7 10
Martin-Gay, Prealgebra, 5ed

15. Chapter 1 / Whole Numbers and Introduction to Algebra
Comparing decimals by comparing their graphs on a number line can be time consuming, so we compare the size of decimals by comparing digits in corresponding places.
Martin-Gay, Prealgebra, 5ed

16. Comparing Two Positive Decimals
Chapter 1 / Whole Numbers and Introduction to Algebra
Comparing Two Positive Decimals
Compare digits in the same places from left to right. When two digits are not equal, the number with the larger digit is the larger decimal. If necessary, insert 0s after the last digit to the right of the decimal point to continue comparing.
Compare hundredths place digits
35.638
35.657 3
< 5
35.638
<
35.657
Martin-Gay, Prealgebra, 5ed

17. Chapter 1 / Whole Numbers and Introduction to Algebra
Helpful Hint
For any decimal, writing 0s after the last digit to the right of the decimal point does not change the value of the number.
8.5 = 8.50 = 8.500, and so on
When a whole number is written as a decimal, the decimal point is placed to the right of the ones digit.
15 = = 15.00, and so on
Martin-Gay, Prealgebra, 5ed

18. Chapter 1 / Whole Numbers and Introduction to Algebra
Rounding Decimals
We round the decimal part of a decimal number in nearly the same way as we round whole numbers. The only difference is that we drop digits to the right of the rounding place, instead of replacing these digits by 0s. For example,
rounded to the nearest hundredth is
63.78
Martin-Gay, Prealgebra, 5ed

19. Rounding Decimals To a Place Value to the Right of the Decimal Point
Chapter 1 / Whole Numbers and Introduction to Algebra
Rounding Decimals To a Place Value to the Right of the Decimal Point
Step 1. Locate the digit to the right of the given place value.
Step 2. If this digit is 5 or greater, add 1 to the digit in the given place value and drop all digits to the right. If this digit is less than 5, drop all digits to the right of the given place.
Martin-Gay, Prealgebra, 5ed

20. Rounding Decimals to a Place Value
Round to the nearest tenth.
Locate the digit to the right of the tenths place.
tenths place
digit to the right
Since the digit to the right is less than 5, drop it and all digits to its right.
rounded to the nearest tenths is 326.4
Martin-Gay, Prealgebra, 5ed

21. Adding and Subtracting Decimals
Chapter 1 / Whole Numbers and Introduction to Algebra
Section 5.2
Adding and Subtracting Decimals 21

22. Adding or Subtracting Decimals
Chapter 1 / Whole Numbers and Introduction to Algebra
Adding or Subtracting Decimals
Step 1. Write the decimals so that the decimal points line up vertically.
Step 2. Add or subtract as for whole numbers.
Step 3. Place the decimal point in the sum or difference so that it lines up vertically with the decimal points in the problem. 22
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23. Helpful Hint
Recall that 0s may be inserted to the right of the decimal point after the last digit without changing the value of the decimal. This may be used to help line up place values when adding or subtracting decimals.
becomes
two 0s inserted
71.74 23

24. Martin-Gay, Prealgebra, 5ed
Chapter 1 / Whole Numbers and Introduction to Algebra
Helpful Hint
Don’t forget that the decimal point in a whole number is after the last digit. 24
Martin-Gay, Prealgebra, 5ed 24

25. Martin-Gay, Prealgebra, 5ed
Chapter 1 / Whole Numbers and Introduction to Algebra
Estimating Operations on Decimals
Estimating sums, differences, products, and quotients of decimal numbers is an important skill whether you use a calculator or perform decimal operations by hand.
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26. Martin-Gay, Prealgebra, 5ed
Chapter 1 / Whole Numbers and Introduction to Algebra
Estimating When Adding Decimals
Add
Exact
Estimate
rounds to
rounds to
This is a reasonable answer.
Martin-Gay, Prealgebra, 5ed 26

27. Martin-Gay, Prealgebra, 5ed
Chapter 1 / Whole Numbers and Introduction to Algebra
Helpful Hint
When rounding to check a calculation, you may want to round the numbers to a place value of your choosing so that your estimates are easy to compute mentally.
Martin-Gay, Prealgebra, 5ed 27

28. Evaluating with Decimals
Chapter 1 / Whole Numbers and Introduction to Algebra
Evaluating with Decimals
Evaluate x + y for x = 5.5 and y = 2.8.
Replace x with 5.5 and y with 2.8 in x + y.
x + y = ( ) + ( )
5.5
2.8
= 8.3 28
Martin-Gay, Prealgebra, 5ed 28

29. Multiplying Decimals and Circumference of a Circle
Chapter 1 / Whole Numbers and Introduction to Algebra
Section 5.3
Multiplying Decimals and Circumference of a Circle 29

30. Martin-Gay, Prealgebra, 5ed
Multiplying Decimals
Multiplying decimals is similar to multiplying whole numbers. The difference is that we place a decimal point in the product. 7 10 3
100 x 21
1000
0.7 x = =
1 decimal place
2 decimal places =
3 decimal places 30
Martin-Gay, Prealgebra, 5ed

31. Martin-Gay, Prealgebra, 5ed
Chapter 1 / Whole Numbers and Introduction to Algebra
Multiplying Decimals
Step 1. Multiply the decimals as though they were whole numbers.
Step 2. The decimal point in the product is placed so the number of decimal places in the product is equal to the sum of the number of decimal places in the factors. 31
Martin-Gay, Prealgebra, 5ed 31

32. Martin-Gay, Prealgebra, 5ed
Chapter 1 / Whole Numbers and Introduction to Algebra
Estimating when Multiplying Decimals
Multiply 32.3 x 1.9.
Exact
Estimate
rounds to
rounds to
This is a reasonable answer.
Martin-Gay, Prealgebra, 5ed 32

33. Multiplying Decimals by Powers of 10
Chapter 1 / Whole Numbers and Introduction to Algebra
Multiplying Decimals by Powers of 10
There are some patterns that occur when we multiply a number by a power of ten, such as 10, 100, 1000, 10,000, and so on. 33
Martin-Gay, Prealgebra, 5ed 33

34. Multiplying Decimals by Powers of 10
Chapter 1 / Whole Numbers and Introduction to Algebra
Multiplying Decimals by Powers of 10
x 10 =
x 100 =
x 100,000 = 7,654,300
Decimal point moved 1 place to the right.
1 zero
Decimal point moved 2 places to the right.
2 zeros
Decimal point moved 5 places to the right.
5 zeros
The decimal point is moved the same number of places as there are zeros in the power of 10. 34
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35. Martin-Gay, Prealgebra, 5ed
Chapter 1 / Whole Numbers and Introduction to Algebra
Multiplying by Powers of 10 such as 10, 100, 1000 or 10,000, . . .
Move the decimal point to the right the same number of places as there are zeros in the power of 10.
Multiply: x 100
Since there are two zeros in 100, move the decimal place two places to the right.
x 100 = =
343.05 35
Martin-Gay, Prealgebra, 5ed 35

36. Notice that zeros had to be inserted.
Chapter 1 / Whole Numbers and Introduction to Algebra
Multiplying by Powers of 10 such as 0.1, 0.01, 0.001, , . . .
Move the decimal point to the left the same number of places as there are decimal places in the power of 10.
Multiply: 8.57 x 0.01
Since there are two decimal places in 0.01, move the decimal place two places to the left.
8.57 x 0.01 = =
0.0857
Notice that zeros had to be inserted. 36 36

37. Martin-Gay, Prealgebra, 5ed
Chapter 1 / Whole Numbers and Introduction to Algebra
Finding the Circumference of a Circle
The distance around a polygon is called its perimeter.
The distance around a circle is called the circumference.
This distance depends on the radius or the diameter of the circle. 37
Martin-Gay, Prealgebra, 5ed 37

38. Circumference of a Circle Martin-Gay, Prealgebra, 5ed
Chapter 1 / Whole Numbers and Introduction to Algebra
Circumference of a Circle r d
Circumference = 2·p ·radius or
Circumference = p ·diameter
C = 2 p r or C = p d 38
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39. Martin-Gay, Prealgebra, 5ed
The symbol p is the Greek letter pi, pronounced “pie.” It is a constant between 3 and 4. A decimal approximation for p is 3.14.
A fraction approximation for p is 22 7 39
Martin-Gay, Prealgebra, 5ed

40. Find the circumference of a circle whose radius is 4 inches.
Chapter 1 / Whole Numbers and Introduction to Algebra
4 inches
Find the circumference of a circle whose radius is 4 inches.
C = 2pr = 2p ·4 = 8p inches
8p inches is the exact circumference of this circle.
If we replace  with the approximation 3.14, C = 8  8(3.14) = inches.
25.12 inches is the approximate circumference of the circle. 40 40

41. Dividing Decimals Section 5.4
Chapter 1 / Whole Numbers and Introduction to Algebra
Section 5.4
Dividing Decimals 41

42. Division of decimal numbers is similar to division of whole numbers.
The only difference is the placement of a decimal point in the quotient.
If the divisor is a whole number, divide as for whole numbers; then place the decimal point in the quotient directly above the decimal point in the dividend. 8 4
quotient
divisor 63
52.92
dividend
2 5 2
-2 52 42

43. Martin-Gay, Prealgebra, 5ed
If the divisor is not a whole number, we need to move the decimal point to the right until the divisor is a whole number before we divide.
divisor 6 3 52 92 .
dividend 63
529 2 . 8 4 63 52
9.2
- 504 25 2
-252 43
Martin-Gay, Prealgebra, 5ed

44. Martin-Gay, Prealgebra, 5ed
Chapter 1 / Whole Numbers and Introduction to Algebra
Dividing by a Decimal
Step 1. Move the decimal point in the divisor to the right until the divisor is a whole number.
Step 2. Move the decimal point in the dividend to the right the same number of places as the decimal point was moved in Step 1.
Step 3. Divide. Place the decimal point in the quotient directly over the moved decimal point in the dividend. 44
Martin-Gay, Prealgebra, 5ed 44

45. Martin-Gay, Prealgebra, 5ed
Chapter 1 / Whole Numbers and Introduction to Algebra
Estimating When Dividing Decimals
Divide ÷ 2.8
Exact
Estimate
rounds to
This is a reasonable answer.
Martin-Gay, Prealgebra, 5ed 45

46. Martin-Gay, Prealgebra, 5ed
There are patterns that occur when dividing by powers of 10, such as 10, 100, 1000, and so on. . 45 62 10 =
456.2
The decimal point moved 1 place to the left.
1 zero 1
000
4562 , . =
456.2
The decimal point moved 3 places to the left.
3 zeros
The pattern suggests the following rule. 46
Martin-Gay, Prealgebra, 5ed

47. To divide by a number is the same as multiplying by its reciprocal.
Dividing Decimals by Powers of 10 such as 10, 100, or 1000, . . .
Move the decimal point of the dividend to the left the same number of places as there are zeros in the power of 10.
Notice that this is the same pattern as multiplying by powers of 10 such as 0.1, 0.01, or Because dividing by a power of 10 such as 100 is the same as multiplying by its reciprocal , or 0.01.
To divide by a number is the same as multiplying by its reciprocal. 47

48. Fractions, Decimals, and Order of Operations
Section 5.5
Fractions, Decimals, and Order of Operations

49. Martin-Gay, Prealgebra, 5ed
Chapter 1 / Whole Numbers and Introduction to Algebra
Writing Fractions as Decimals
To write a fraction as a decimal, divide the numerator by the denominator. 49
Martin-Gay, Prealgebra, 5ed 49

50. Martin-Gay, Prealgebra, 5ed
Chapter 1 / Whole Numbers and Introduction to Algebra
Comparing Fractions and Decimals
To compare decimals and fractions, write the fraction as an equivalent decimal.
Compare and
Therefore, < 0.25 50
Martin-Gay, Prealgebra, 5ed 50

51. Martin-Gay, Prealgebra, 5ed
Chapter 1 / Whole Numbers and Introduction to Algebra
Order of Operations
1. Do all operations within grouping symbols such as parentheses or brackets.
2. Evaluate any expressions with exponents.
3. Multiply or divide in order from left to right.
4. Add or subtract in order from left to right. 51
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52. Using the Order of Operations Martin-Gay, Prealgebra, 5ed
Simplify ( – 2.3) ( )
( – 2.3) ( )
= ( – 2.3) (5.3)
Simplify inside parentheses.
Objective D Continued
= (5.3)
Write ( – 2.3)2 as 5.29. =
Multiply.
= 27.02
Add.
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53. Martin-Gay, Prealgebra, 5ed
Finding the Area of a Triangle
height A
base • height = 1 2
base A bh = 1 2 53
Martin-Gay, Prealgebra, 5ed

54. Equations Containing Decimals
Chapter 1 / Whole Numbers and Introduction to Algebra
Section 5.6
Equations Containing Decimals 54

55. Steps for Solving an Equation in x
Chapter 1 / Whole Numbers and Introduction to Algebra
Steps for Solving an Equation in x
Step 1. If fractions are present, multiply both sides of the equation by the LCD of the fractions.
Step 2. If parentheses are present, use the distributive property.
Step 3. Combine any like terms on each side of the equation. 55
Martin-Gay, Prealgebra, 5ed 55

56. Steps for Solving an Equation . . .
Chapter 1 / Whole Numbers and Introduction to Algebra
Steps for Solving an Equation . . .
Step 4. Use the addition property of equality to rewrite the equation so that variable terms are on one side of the equation and constant terms are on the other side.
Step 5. Divide both sides by the numerical coefficient of x to solve.
Step 6. Check the answer in the original equation. 56
Martin-Gay, Prealgebra, 5ed 56

57. Solving Equations with Decimals
– 0.01(5a + 4) = 0.04 – 0.01(a + 4)
– 1(5a + 4) = 4 – 1(a + 4)
Multiply both sides by 100.
– 5a – 4 = 4 – a – 4
Apply the distributive property.
– 4a – 4 = 4 – 4
Add a to both sides.
– 4a = 4
Add 4 to both sides and simplify.
a = – 1
Divide both sides by 4.