2A Rounding, Comparing, Adding and Subtracting Decimals

Look at the following formulas. The first formula (P = A + B + C) is one we use to calculate perimeter of a triangular region where A, B and C are the lengths of the sides of the triangle. Suppose we want to find the perimeter of a triangle with sides 3.1 cm, 2.67 cm and 4.129 cm. Since many of the numbers we want to plug into formulas involve decimals, we need to be able to add and subtract decimals.

As with whole numbers, the key to adding and subtracting decimals is place value. So let us review the place values of decimals. For example, look at the number 3.12589

3 . 1 2 5 8 9

Ones Tenths Hundredths Thousandths Ten-Thousandths Hundred-Thousandths

For example tenths means out of 10. So 0.1 means 1 tenth or 1/10. 0.02 means 2 hundredths or 2/100, and so on.

We cannot add or subtract the tenths place with the thousandths place. The place values have to be the same.

Another key to decimals is that you can add zeros to the end of a decimal and it does not change its value. For example 2.7 is the same as 2.70 which is the same as 2.700 which is the same as 2.7000 and so on. This can be helpful when lining up decimal places.

Comparing Decimals

Decimals are often difficult to compare. For example, in statistics we often have to compare a P-value decimal of 0.0035 to 0.05 and see which is larger. The key is to add zeros until both numbers end in the same place value.

In the above example, to compare 0.0035 to 0.05 we can add two zeros to the end of 0.05. Remember this does not change its value.

0.0500

0.0035

Notice it is now easy to compare them. Which is larger, 500 or 35? Obviously 500 is larger than 35 so 0.05 is larger than 0.0035. (In statistics this tells us that the P-value is significantly low.)

Try another one. Compare 24.3 to 24.2997. Again if we add zeros to the 24.3 so that it ends in the same place value, we get the following.

24.3000

24.2997

Which is larger, 243000 or 242997? Since 243000 is larger than 242997, we see that 24.3 is larger than 24.2997.

Try the following examples with your instructor.

Rounding Decimals

To correctly round a decimal, we need to decide to which place value we want to round to. We then look to the right of that place value. If the number to the right is 5 or above, we will round up by adding one to the place value. If the number to the right is 4 or below, we will round down by leaving the place value alone. The key to remember is that after rounding, we want to stop after the place value so that the number is simpler to use.

For example, round 0.03792 to the hundredths place. The hundredths place is the 3. Looking to the right of it we see a 7. Since 7 is large (5 or above) we round up by adding 1 to the place value. So the 3 becomes 4 and we cut off the rest of the numbers.

$$0.03792\approx 0.04$$

Let’s try another example. Round 1.83597 to the tenths place. The tenths place is the 8. Looking to the right of it we see a 3. Since 3 is small (4 or below) we round down by leaving the place value alone. So the 8 stays as it is and we cut off the rest of the numbers.

$$1.83597\approx 1.8$$

Try the following examples with your instructor. Pay close attention to the place values.

Addition and Subtraction

Let’s look at an example. Let’s add 3.258 + 7.9; It is important to line up the place values before adding or subtracting. Let us rewrite 7.9 as 7.900 so that both numbers end in the thousandths place. Now we add exactly as if the numbers were whole numbers.

$$\begin{array}{l}3.258\\ \underset{\_}{+7.900}\\ 11.158\end{array}$$

Similarly to subtract, we can add zeros to make sure the numbers end in the same place value. Now we can line up the place values and subtract. Let’s subtract 9 – 3.2751. First let’s rewrite the 9 as 9.0000 so that the place values line up. Now subtract. Remember an easy way to borrow is to borrow 1 from 9000, which leaves 8999. Then add 10 to the last place value.

$$\begin{array}{rccccc}& 8.& 9& 9& 9& 10\\ & 9.& 0& 0& 0& 0\\ -& 3.& 2& 7& 5& 1\\ & 5.& 7& 2& 4& 9\end{array}$$

Try the next few examples with your instructor.

Practice Problems Section 2A

1. Which is larger, 0.082 or 0.0399?

   Check answer

   0.082

2. Which is smaller, 1.13 or 1.028?

3. Which is smaller, 0.034 or 0.0099?

   Check answer

   0.0099

4. Which is larger, 34.5 or 34.499?

5. Which is larger, 0.05 or 0.00016?

   Check answer

   0.05

6. Which is smaller, 8.719 or 8.7?

7. Which is smaller, 0.378 or 0.4?

   Check answer

   0.378

8. Which is larger, 2.41 or 2.4086?

9. Which is larger, 0.0058 or 0.005?

   Check answer

   0.0058

10. Which is smaller, 12.6 or 12.499?

11. Which is smaller, 0.0175 or 0.00175?

    Check answer

    0.00175

12. Which is larger, 68.327 or 68.7?

13. Round 0.7829 to the hundredths place.

    Check answer

    0.78

14. Round 3.529 to the ones place.

15. Round 9.647 to the tenths place.

    Check answer

    9.6

16. Round 0.7956 to the thousandths place.

17. Round 0.58699 to the thousandths place.

    Check answer

    0.587

18. Round 78.462 to the ones place.

19. Round 1.375 to the tenths place.

    Check answer

    1.4

20. Round 0.7956 to the hundredths place.

21. Round 72.973 to the ones place.

    Check answer

    73

22. Round 0.053289 to the thousandths place.

23. Round 0.053289 to the tenths place.

    Check answer

    0.1

24. Round 8.00374 to the hundredths place.

25. Round 3.745117 to the ten-thousandths place.

    Check answer

    3.7451

26. Round $48.1277 to the nearest cent (hundredths place).

27. Round $359.0244 to the nearest cent. (hundredths place).

    Check answer

    $359.02

28. Round $1372.631 to the nearest cent (hundredths place).

29. Round $74.538 to the nearest dollar. (ones place).

    Check answer

    $75

30. Round $671.284 to the nearest dollar (ones place).

31. Round $481.723 to the nearest dollar (ones place).

    Check answer

    $482

32. Round $10,457.43 to the nearest dollar (ones place).

For #33-47, add the following decimals. Don’t forget to line up the place values.

33. 0.2 + 0.135

    Check answer

    0.335

34. 0.89 + 1.35

35. 2.83 + 1.17

    Check answer

    4

36. 12 + 17.431

37. 0.064 + 1.23

    Check answer

    1.294

38. 13.9145 + 7.528

39. 3.73 + 13.898

    Check answer

    17.628

40. 0.00827 + 1.443

41. 6.834 + 7.0029

    Check answer

    13.8369

42. 46.1+ 0.439

43. 3.408 + 13.93

    Check answer

    17.338

44. 0.845 + 6.0851

45. 51.92 + 64.083

    Check answer

    116.003

46. 0.0072 + 0.02665

47. 9.31 + 7.0917

    Check answer

    16.4017

For #48-62, subtract the following decimals. Don’t forget to line up the place values.

48. 8.4 – 6.9

49. 1.003 – 0.475

    Check answer

    0.528

50. 9.31 – 6.4227

51. 14.2 – 4.75

    Check answer

    9.45

52. 2.03 – 0.187

53. 7 – 2.581

    Check answer

    4.419

54. 17 – 3.26

55. 4.931 – 2.1

    Check answer

    2.831

56. 8.4 – 0.0035

57. 3.6 – 0.047

    Check answer

    3.553

58. 0.835 – 0.3724

59. 0.9 – 0.638

    Check answer

    0.262

60. 19.2 – 3.055

61. 18.528 – 17.9

    Check answer

    0.628

62. 1.108 – 0.0266

For #63-68, add or subtract the following decimals in order from left to right.
Don’t forget to line up the place values.

63. 13.4 + 1.58 + 0.472

    Check answer

    15.452

64. 8 + 2.14 + 6.9

65. 8.235 + 7.49 + 14

    Check answer

    29.725

66. 4.75 – 1.327 + 0.8

67. 7.3 + 0.94 – 0.037

    Check answer

    8.203

68. 127 – 33.9 + 4.82

A formula we have seen before is the profit formula P = R – C where P is profit, R is revenue and C is cost.

69. Find the profit if R = $178.14 and C = $95.77

    Check answer

    $82.37

70. Find the profit if R = $13000 and C = $2879.41

71. Find the profit if R = $5,264 and C = $3095.77

    Check answer

    $2168.23

72. Find the profit if R = $7,851.26 and C = $5248.99

Another formula we have seen is the perimeter of a rectangular region in geometry.
P = L +W + L + W. (Use this formula for #73-76)

73. Find the perimeter of a rectangular field that is 1.4 miles by 0.85 miles.

    Check answer

    4.5 mi

74. Find the distance around the outside of a rectangular room that is 13.5 feet by 10.8 feet.

75. Find the perimeter of a rectangular window that is 17.5 inches by 13.85 inches.

    Check answer

    62.7 in

76. Find the distance around the outside of a rectangular garden that is 8.375 meters by 6.284 meters.

77. In statistics, we often need to add up a bunch of numbers in a data set. Let’s practice. Add up the following numbers and find the sum.

    2.84, 3.17, 1.74, 2.52, 3.25, 2.7, 1.98, 3

    Check answer

    21.2

78. Let’s practice adding up a bunch of numbers. Add the following and find the sum.

    39.4, 36.7, 37.2, 40.6, 41.3, 42, 37.5, 39

79. Greg went to the store to buy some groceries. The prices of the items he purchased (including tax) are listed below. What was the total amount spent at the store?

    Apples: $4.89
    Milk: $2.57
    Bread: $1.94
    Eggs: $3.08

    Check answer

    $12.48

80. The temperature before the storm was 22.487 degrees Celsius. After the storm the temperature had dropped to 13.529 degrees Celsius. How much did the temperature drop?

81. Lucia had $1849.22 in her checking account. She deposited a check from her work for $427.38 and then paid her mortgage of $1027.56 and her electricity bill of $72.83. How much does Lucia now have in her checking account?

    Check answer

    $1176.21

Adapted from Preparing for Algebra and Statistics, Third Edition, by Matt Teachout. License: CC BY 4.0. Copyright and sources.