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# 3.3: Combining Decimals- Addition and Subtraction with Decimals


## 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:

1. Place the numbers to be added in vertical format, aligning the decimal points.
2. Add the numbers as if they were whole numbers.
3. 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: 1. Place the numbers to be subtracted in vertical format, aligning the decimal points. 2. Subtract the numbers as if they were whole numbers. 3. 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: 1. Add the magnitudes of the decimal numbers. 2. 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: 1. Subtract the smaller magnitude from the larger magnitude. 2. 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 & Co124.8 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

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