3.3: Combining Decimals- Addition and Subtraction with Decimals

Last updated
Save as PDF

Page ID: 30988

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $

$ \newcommand{\dsum}{\displaystyle\sum\limits} $

$ \newcommand{\dint}{\displaystyle\int\limits} $

$ \newcommand{\dlim}{\displaystyle\lim\limits} $

$ \newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$

( \newcommand{\kernel}{\mathrm{null}\,}\) $ \newcommand{\range}{\mathrm{range}\,}$

$ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$

$ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$

$ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$ \newcommand{\Span}{\mathrm{span}}$

$ \newcommand{\id}{\mathrm{id}}$

$ \newcommand{\Span}{\mathrm{span}}$

$ \newcommand{\kernel}{\mathrm{null}\,}$

$ \newcommand{\range}{\mathrm{range}\,}$

$ \newcommand{\RealPart}{\mathrm{Re}}$

$ \newcommand{\ImaginaryPart}{\mathrm{Im}}$

$ \newcommand{\Argument}{\mathrm{Arg}}$

$ \newcommand{\norm}[1]{\| #1 \|}$

$ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\AA}{\unicode[.8,0]{x212B}}$

$ \newcommand{\vectorA}[1]{\vec{#1}} % arrow$

$ \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$

$ \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$ \newcommand{\vectorC}[1]{\textbf{#1}} $

$ \newcommand{\vectorD}[1]{\overrightarrow{#1}} $

$ \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} $

$ \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} $

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$\newcommand{\longvect}{\overrightarrow}$

$ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $

$\newcommand{\ket}[1]{\left| #1 \right>}$
$\newcommand{\bra}[1]{\left< #1 \right|}$
$\newcommand{\braket}[2]{\left< #1 \vphantom{#2} \right| \left. #2 \vphantom{#1} \right>}$
$\newcommand{\braopket}[3]{\left< #1 \vphantom{#2}\vphantom{#3} \right| #2 \vphantom{#1}\vphantom{#3} \left| #3 \vphantom{#1}\vphantom{#2} \right>}$
$\newcommand{\qmvec}[1]{\mathbf{\vec{#1}}}$
$\newcommand{\op}[1]{\hat{\mathbf{#1}}}$
$\newcommand{\expect}[1]{\langle #1 \rangle}$
$\newcommand{\dfn}[1]{\emph{\textbf{#1}}}$

$\newcommand{\avec}{\mathbf a}$ $\newcommand{\bvec}{\mathbf b}$ $\newcommand{\cvec}{\mathbf c}$ $\newcommand{\dvec}{\mathbf d}$ $\newcommand{\dtil}{\widetilde{\mathbf d}}$ $\newcommand{\evec}{\mathbf e}$ $\newcommand{\fvec}{\mathbf f}$ $\newcommand{\nvec}{\mathbf n}$ $\newcommand{\pvec}{\mathbf p}$ $\newcommand{\qvec}{\mathbf q}$ $\newcommand{\svec}{\mathbf s}$ $\newcommand{\tvec}{\mathbf t}$ $\newcommand{\uvec}{\mathbf u}$ $\newcommand{\vvec}{\mathbf v}$ $\newcommand{\wvec}{\mathbf w}$ $\newcommand{\xvec}{\mathbf x}$ $\newcommand{\yvec}{\mathbf y}$ $\newcommand{\zvec}{\mathbf z}$ $\newcommand{\rvec}{\mathbf r}$ $\newcommand{\mvec}{\mathbf m}$ $\newcommand{\zerovec}{\mathbf 0}$ $\newcommand{\onevec}{\mathbf 1}$ $\newcommand{\real}{\mathbb R}$ $\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$ $\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$ $\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$ $\newcommand{\laspan}[1]{\text{Span}\{#1\}}$ $\newcommand{\bcal}{\cal B}$ $\newcommand{\ccal}{\cal C}$ $\newcommand{\scal}{\cal S}$ $\newcommand{\wcal}{\cal W}$ $\newcommand{\ecal}{\cal E}$ $\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$ $\newcommand{\gray}[1]{\color{gray}{#1}}$ $\newcommand{\lgray}[1]{\color{lightgray}{#1}}$ $\newcommand{\rank}{\operatorname{rank}}$ $\newcommand{\row}{\text{Row}}$ $\newcommand{\col}{\text{Col}}$ $\renewcommand{\row}{\text{Row}}$ $\newcommand{\nul}{\text{Nul}}$ $\newcommand{\var}{\text{Var}}$ $\newcommand{\corr}{\text{corr}}$ $\newcommand{\len}[1]{\left|#1\right|}$ $\newcommand{\bbar}{\overline{\bvec}}$ $\newcommand{\bhat}{\widehat{\bvec}}$ $\newcommand{\bperp}{\bvec^\perp}$ $\newcommand{\xhat}{\widehat{\xvec}}$ $\newcommand{\vhat}{\widehat{\vvec}}$ $\newcommand{\uhat}{\widehat{\uvec}}$ $\newcommand{\what}{\widehat{\wvec}}$ $\newcommand{\Sighat}{\widehat{\Sigma}}$ $\newcommand{\lt}{<}$ $\newcommand{\gt}{>}$ $\newcommand{\amp}{&}$ $\definecolor{fillinmathshade}{gray}{0.9}$

Adding Decimals

Addition of decimal numbers is quite similar to addition of whole numbers. For example, suppose that we are asked to add 2.34 and 5.25. We could change these decimal numbers to mixed fractions and add.

\[ \begin{aligned} 2.34 + 5.25 & = 2 \frac{34}{100} + 5 \frac{25}{100} \\ & = 7 \frac{59}{100} \end{aligned}\nonumber \]

However, we can also line the decimal numbers on their decimal points and add vertically, as follows.

\[ \begin{array}{r} 2.34 \\ + 5.25 \\ \hline 7.59 \end{array}\nonumber \]

Note that this alignment procedure produces the same result, “seven and fifty nine hundredths.” This motivates the following procedure for adding decimal numbers.

Adding Decimals

To add decimal numbers, proceed as follows:

Place the numbers to be added in vertical format, aligning the decimal points.
Add the numbers as if they were whole numbers.
Place the decimal point in the answer in the same column as the decimal points above it.

Example 1

Add 3.125 and 4.814.

Solution

Place the numbers in vertical format, aligning on their decimal points. Add, then place the decimal point in the answer in the same column as the decimal points that appear above the answer.

\[ \begin{array}{r} 3.125 \\ +4.814 \\ \hline 7.939 \end{array}\nonumber \]

Thus, 3.125 + 4.814 = 7.939.

Exercise

Add: 2.864 + 3.029

Answer: 5.893

Example 2

Jane has $4.35 in her purse. Jim has $5.62 in his wallet. If they sum their money, what is the total?

Solution

Arrange the numbers in vertical format, aligning decimal points, then add.

\[ \begin{array}{r} \$ 4.35 \\ + \$ 5.62 \\ \hline \$ 9.97 \end{array}\nonumber \]

Exercise

Alice has $8.63 in her purse and Joanna has $2.29. If they combine sum their money, what is the total?

Answer: $10.91

Before looking at another example, let’s recall an important observation.

Important Observation

Adding zeros to the end of the fractional part of a decimal number does not change its value. Similarly, deleting trailing zeros from the end of a decimal number does not change its value.

For example, we could add two zeros to the end of the fractional part of 7.25 to obtain 7.2500. The numbers 7.25 and 7.2500 are identical as the following argument shows:

\[ \begin{aligned} 7.2500 & = 7 \frac{2500}{10000} \\ & = 7 \frac{25}{100} \\ & = 7.25 \end{aligned}\nonumber \]

Example 3

Add 7.5 and 12.23.

Solution

Arrange the numbers in vertical format, aligning their decimal points in a column. Note that we add a trailing zero to improve columnar alignment.

\[ \begin{array}{r} 7.50 \\ +12.23 \\ \hline 19.73 \end{array}\nonumber \]

Hence, 7.5 + 12.23 = 19.73.

Exercise

Add: 9.7 + 15.86

Answer: 25.56

Example 4

Find the sum: 12.2+8.352 + 22.44.

Solution

Arrange the numbers in vertical format, aligning their decimal points in a column. Note that we add trailing zeros to improve the columnar alignment.

\[ \begin{array}{r} 12.200 \\ 8.352 \\ + 22.440 \\ \hline 42.992 \end{array}\nonumber \]

Hence, 12.2+8.352 + 22.44 = 42.992.

Exercise

Add: 12.9+4.286 + 33.97

Answer: 51.156

Subtracting Decimals

Subtraction of decimal numbers proceeds in much the same way as addition of decimal numbers.

Subtracting Decimals

To subtract decimal numbers, proceed as follows:

Place the numbers to be subtracted in vertical format, aligning the decimal points.
Subtract the numbers as if they were whole numbers.
Place the decimal point in the answer in the same column as the decimal points above it.

Example 5

Subtract 12.23 from 33.57.

Solution

Arrange the numbers in vertical format, aligning their decimal points in a column, then subtract. Note that we subtract 12.23 from 33.57.

\[ \begin{array}{r} 33.57 \\ -12.23 \\ \hline 21.34 \end{array}\nonumber \]

Hence, 33.57 − 12.23 = 21.34.

Exercise

Subtract: 58.76 − 38.95

Answer: 19.81

As with addition, we add trailing zeros to the fractional part of the decimal numbers to help columnar alignment.

Example 6

Find the difference: 13.3 − 8.572.

Solution

Arrange the numbers in vertical format, aligning their decimal points in a column. Note that we add trailing zeros to the fractional part of 13.3 to improve columnar alignment.

\[ \begin{array}{r} 13.300 \\ -8.572 \\ \hline 4.728 \end{array}\nonumber \]

Hence, 13.3 − 8.572 = 4.728.

Exercise

Subtract: 15.2 − 8.756

Answer: 6.444

Adding and Subtracting Signed Decimal Numbers

We use the same rules for addition of signed decimal numbers as we did for the addition of integers.

Adding Two Decimals with Like Signs

To add two decimals with like signs, proceed as follows:

Add the magnitudes of the decimal numbers.
Prefix the common sign.

Example 7

Simplify: −3.2+(−18.95).

Solution

To add like signs, first add the magnitudes.

\[ \begin{array}{r} 3.20 \\ +18.95 \\ \hline 22.15 \end{array}\nonumber \]

Prefix the common sign. Hence, −3.2+(−18.95) = −22.15

Exercise

Simplify: −5.7 + (−83.85)

Answer: −89.55

We use the same rule as we did for integers when adding decimals with unlike signs.

Adding Two Decimals with Unlike Signs

To add two decimals with unlike signs, proceed as follows:

Subtract the smaller magnitude from the larger magnitude.
Prefix the sign of the decimal number with the larger magnitude.

Example 8

Simplify: −3 + 2.24.

Solution

To add unlike signs, first subtract the smaller magnitude from the larger magnitude.

\[ \begin{array}{r} 3.00 \\ -2.24 \\ \hline 0.76 \end{array}\nonumber \]

Prefix the sign of the decimal number with the larger magnitude. Hence, −3+2.24 = −0.76.

Exercise

Simplify: −8 + 5.74

Answer: −2.26

Subtraction still means add the opposite.

Example 9

Simplify: −8.567 − (−12.3).

Solution

Subtraction must first be changed to addition by adding the opposite.

\[−8.567 − (−12.3) = −8.567 + 12.3\nonumber \]

We have unlike signs. First, subtract the smaller magnitude from the larger magnitude.

\[ \begin{array}{r} 12.300 \\ − 8.567 \\ \hline 3.733 \end{array}\nonumber \]

Prefix the sign of the decimal number with the larger magnitude. Hence:

\[ \begin{aligned} −8.567 − (−12.3) & = −8.567 + 12.3 \\ & = 3.733 \end{aligned}\nonumber \]

Exercise

Simplify: −2.384 − (−15.2)

Answer: 12.816

Order of operations demands that we simplify expressions contained in parentheses first.

Example 10

Simplify: −11.2 − (−8.45 + 2.7).

Solution

We need to add inside the parentheses first. Because we have unlike signs, subtract the smaller magnitude from the larger magnitude.

\[ \begin{array}{r} 8.45 \\ − 2.70 \\ \hline 5.75 \end{array}\nonumber \]

Prefix the sign of the number with the larger magnitude. Therefore,

\[−11.2 − (−8.45 + 2.7) = −11.2 − (−5.75)\nonumber \]

Subtraction means add the opposite.

\[−11.2 − (−5.75) = −11.2+5.75\nonumber \]

Again, we have unlike signs. Subtract the smaller magnitude from the larger magnitude.

\[ \begin{array}{r} 11.20 \\ − 5.75 \\ \hline 5.45 \end{array}\nonumber \]

Prefix the sign of the number with the large magnitude.

\[ −11.2+5.75 = −5.45\nonumber \]

Exercise

Simplify: −12.8 − (−7.44 + 3.7)

Answer: −9.06

Writing Mathematics

The solution to the previous example should be written as follows:

\[ \begin{aligned} −11.2 − (−8.45 + 2.7) & = −11.2 − (−5.75) \\ & = −11.2+5.75 \\ & = −5.45 \end{aligned}\nonumber \]

Any scratch work, such as the computations in vertical format in the previous example, should be done in the margin or on a scratch pad.

Example 11

Simplify: −12.3 −|− 4.6 − (−2.84)|.

Solution

We simplify the expression inside the absolute value bars first, take the absolute value of the result, then subtract.

\[ \begin{aligned} -12.3 - |-4.6 -(-2.84)| ~ \\ = -12.3 -|-4.6 + 2.84| ~ & \textcolor{red}{ \text{ Add the opposite.}} \\ = -12.3 -|-1.76| ~ & \textcolor{red}{ \text{ Add: } -4.6 + 2.84 = -1.76.} \\ = -12.3-1.76 ~ & \textcolor{red}{ |-1.76|=1.76.} \\ =-12.3 + (-1.76) ~& \textcolor{red}{ \text{ Add the opposite.}} \\ = -14.06 ~ & \textcolor{red}{ \text{ Add: } -12.3 + (-1.76) = -14.06.} \end{aligned}\nonumber \]

Exercise

Simplify: −8.6 −|− 5.5 − (−8.32)|

Answer: −11.42

Exercises

In Exercises 1-12, add the decimals.

1. $31.9 + 84.7$

2. $9.39 + 7.7$

3. $4 + 97.18$

4. $2.645 + 2.444$

5. $4 + 87.502$

6. $23.69 + 97.8$

7. $95.57 + 7.88$

8. $18.7+7$

9. $52.671 + 5.97$

10. $9.696 + 28.2$

11. $4.76 + 2.1$

12. $1.5 + 46.4$

In Exercises 13-24, subtract the decimals.

13. $9 − 2.261$

14. $98.14 − 7.27$

15. $80.9 − 6$

16. $9.126 − 6$

17. $55.672 − 3.3$

18. $4.717 − 1.637$

19. $60.575 − 6$

20. $8.91 − 2.68$

21. $39.8 − 4.5$

22. $8.210 − 3.7$

23. $8.1 − 2.12$

24. $7.675 − 1.1$

In Exercises 25-64, add or subtract the decimals, as indicated.

25. $−19.13 − 7$

26. $−8 − 79.8$

27. $6.08 − 76.8$

28. $5.76 − 36.8$

29. $−34.7+(−56.214)$

30. $−7.5+(−7.11)$

31. $8.4+(−6.757)$

32. $−1.94 + 72.85$

33. $−50.4+7.6$

34. $1.4+(−86.9)$

35. $−43.3+2.2$

36. $0.08 + (−2.33)$

37. $0.19 − 0.7$

38. $9 − 18.01$

39. $−7 − 1.504$

40. $−4.28 − 2.6$

41. $−4.47 + (−2)$

42. $−9+(−43.67)$

43. $71.72 − (−6)$

44. $6 − (−8.4)$

45. $−9.829 − (−17.33)$

46. $−95.23 − (−71.7)$

47. $2.001 − 4.202$

48. $4 − 11.421$

49. $2.6 − 2.99$

50. $3.57 − 84.21$

51. $−4.560 − 2.335$

52. $−4.95 − 96.89$

53. $−54.3 − 3.97$

54. $−2 − 29.285$

55. $−6.32 + (−48.663)$

56. $−8.8+(−34.27)$

57. $−8 − (−3.686)$

58. $−2.263 − (−72.3)$

59. $9.365 + (−5)$

60. $−0.12 + 6.973$

61. $2.762 − (−7.3)$

62. $65.079 − (−52.6)$

63. $−96.1+(−9.65)$

64. $−1.067 + (−4.4)$

In Exercises 65-80, simplify the given expression.

65. $−12.05 − |17.83 − (−17.16)|$

66. $15.88 −|− 5.22 − (−19.94)|$

67. $−6.4 + |9.38 − (−9.39)|$

68. $−16.74 + |16.64 − 2.6|$

69. $−19.1 − (1.51 − (−17.35))$

70. $17.98 − (10.07 − (−10.1))$

71. $11.55 + (6.3 − (−1.9))$

72. $−8.14 + (16.6 − (−15.41))$

73. $−1.7 − (1.9 − (−16.25))$

74. $−4.06 − (4.4 − (−10.04))$

75. $1.2 + |8.74 − 16.5|$

76. $18.4 + |16.5 − 7.6|$

77. $−12.4 − |3.81 − 16.4|$

78. $13.65 − |11.55 − (−4.44)|$

79. $−11.15 + (11.6 − (−16.68))$

80. $8.5 + (3.9 − 6.98)$

81. Big Banks. Market capitalization of nation’s four largest banks (as of April 23, 2009)

JPMorgan Chase & Co	$124.8 billion
Wells Fargo & Co	$85.3 billion
Goldman Sachs Group Inc.	$61.8 billion
Bank of America	$56.4 billion

What is the total value of the nation’s four largest banks? Associated Press Times-Standard 4/22/09

82. Telescope Mirror. The newly launched Herschel Telescope has a mirror 11.5 feet in diameter while Hubble’s mirror is 7.9 feet in diameter. How much larger is Herschel’s mirror in diameter than Hubble’s?

83. Average Temperature. The average temperatures in Sacramento, California in July are a high daytime temperature of 93.8 degrees Fahrenheit and a low nighttime temperature of 60.9 degrees Fahrenheit. What is the change in temperature from day to night? Hint: See Section 2.3 for the formula for comparing temperatures.

84. Average Temperature. The average temperatures in Redding, California in July are a high daytime temperature of 98.2 degrees Fahrenheit and a low nighttime temperature of 64.9 degrees Fahrenheit. What is the change in temperature from day to night? Hint: See Section 2.3 for the formula for comparing temperatures.

85. Net Worth. Net worth is defined as assets minus liabilities. Assets are everything of value that can be converted to cash while liabilities are the total of debts. General Growth Properties, the owners of the Bayshore Mall, have $29.6 billion in assets and $27 billion in liabilities, and have gone bankrupt. What was General Growth Properties net worth before bankruptcy? Times-Standard 4/17/2009

86. Grape crush. The California Department of Food and Agriculture’s preliminary grape crush report shows that the state produced 3.69 million tons of wine grapes in 2009. That’s just shy of the record 2005 crush of 3.76 million tons. By how many tons short of the record was the crush of 2009? Associated Press-Times-Standard Calif. winegrapes harvest jumped 23% in ’09.

87. Turnover. The Labor Department’s Job Openings and Labor Turnover Survey claims that employers hired about 4.08 million people in January 2010 while 4.12 million people were fired or otherwise left their jobs. How many more people lost jobs than were hired? Convert your answer to a whole number. Associated Press-Times-Standard 03/10/10 Job openings up sharply in January to 2.7M.

Answers

1. 116.6

3. 101.18

5. 91.502

7. 103.45

9. 58.641

11. 6.86

13. 6.739

15. 74.9

17. 52.372

19. 54.575

21. 35.3

23. 5.98

25. −26.13

27. −70.72

29. −90.914

31. 1.643

33. −42.8

35. −41.1

37. −0.51

39. −8.504

41. −6.47

43. 77.72

45. 7.501

47. −2.201

49. −0.39

51. −6.895

53. −58.27

55. −54.983

57. −4.314

59. 4.365

61. 10.062

63. −105.75

65. −47.04

67. 12.37

69. −37.96

71. 19.75

73. −19.85

75. 8.96

77. −24.99

79. 17.13

81. $328.3 billion

83. −32.9 degrees Fahrenheit

85. $2.6 billion

87. 40, 000