2.4 Arithmetic Operations with Decimal Numbers

Addition of Decimal Numbers

Addition of decimal numbers refers to combining decimal numbers to find the total or sum. It is similar to adding whole numbers.

Follow these steps to add decimal numbers:

1. Write the numbers one under the other by aligning the decimal points of the numbers.

2. Add zeros to the end of any decimal number that has fewer decimal places, if necessary, to ensure that each number has the same number of decimal places. Draw a horizontal line underneath.

3. Starting from the right-most place value, add all the numbers in that column.

   • If the total is less than 10, write the total under the horizontal line in the same column.
   • If the total is 10 or more, write the ‘ones’ digit of the total under the horizontal line in the same column, and write the ‘tens’ digit of the total above the column to the left.

4. Follow this procedure for each column going from right to left. Write the decimal point in the answer, aligned with the other decimal points in the sum.

Example 2.4-a: Adding Decimal Numbers

Perform the following additions:

1. 25.125 + 7.14

2. 741.87 + 135.456

3. 127 + 68.8 + 669.95

Subtraction of Decimal Numbers

Subtraction of decimal numbers refers to finding the difference between decimal numbers. It is similar to subtracting whole numbers.

Follow these steps to subtract a decimal number from another decimal number:

1. Write the numbers one under the other by aligning the decimal points of the numbers. Ensure that the number from which subtraction is indicated (the minuend) is in the top row and that the number that is being subtracted (the subtrahend) is below.

2. Add zeros to the end of any decimal number that has fewer decimal places, if necessary, to ensure that each number has the same number of decimal places. Draw a horizontal line underneath.

3. Starting from the right-most place value, subtract the bottom number from the top number.

   • If the top digit is greater than (or equal to) the bottom digit, write the difference under the horizontal line in the same column.
   • If the top digit is less than the bottom digit, borrow ‘one’ from the digit to the left in the top number, and add ‘ten’ to the digit in the current place value of the top number. Then, find the difference and write it under the horizontal line.

4. Follow this procedure for each column going from right to left. Write the decimal point in the answer, aligned with the other decimal points in the difference.

Example 2.4-b: Subtracting Decimal Numbers

Perform the following subtractions:

1. Subtract 29.02 from 135.145

2. Subtract 38.7 from 457

Multiplication of Decimal Numbers

Multiplication of decimal numbers refers to finding the product of two decimal numbers.

Follow these steps to multiply one decimal number by another decimal number:

1. Line up the numbers on the right without aligning the decimal points.

2. Multiply the number assuming that there are no decimal points; i.e., multiply each digit in the top number by each digit in the bottom number and add the products, just like the process for multiplying whole numbers.

3. Count the total number of decimal places in the numbers that are being multiplied (the factors).

4. The number obtained in Step 3 is equal to the number of decimal places in the answer. Starting at the right of the answer, move towards the left by the total number of decimal places counted, and place the decimal point there.

Example 2.4-c: Multiplying Decimal Numbers

Multiply 12.56 and 1.8.

Solution

Therefore, multiplying 12.56 and 1.8 results in 22.608.

Division of Decimal Numbers

Division of decimal numbers is the process of determining how many times one decimal number is contained in another decimal number.

Follow these steps to divide a decimal number:

1. If the divisor is not a whole number, convert it to a whole number by moving the decimal point to the right. Move the decimal point in the dividend by the same number of places.

2. Divide by following a similar process to the process of dividing whole numbers. Add zeros to the right of the last digit of the dividend and keep dividing until there is no remainder or a repeating pattern shows up in the quotient.

Example 2.4-d: Dividing Decimal Numbers

Perform the following divisions:

1. Divide 8.25 by 0.6

2. Divide: 0.166 by 0.03

Powers and Square Roots of Decimal Numbers

Similar to fractions, powers of decimal numbers are usually written within brackets.

For example, [latex] (0.12)^3 [/latex] is read as “twelve hundredths raised to the power of three”.

• This means that 0.12 is used as a factor three times.

• i.e., [latex] (0.12)^3 = (0.12)(0.12)(0.12) = 0.001728 [/latex]

Example 2.4-e: Evaluating Powers of Decimal Numbers

Evaluate the power: [latex] (1.25)^3 [/latex]

Determining square roots of decimal numbers is simple if the decimal number can first be converted to a decimal fraction with an even power of ten as the denominator (i.e., [latex] 10^2 = 100, 10^4 = 10,000[/latex], etc.). Then, follow the procedure for evaluating the square root of a fraction.

For example, [latex] \displaystyle{\sqrt{0.25} = \sqrt{\frac{25}{100}} = \frac{\sqrt{25}}{\sqrt{100}} = \frac{5}{10} = 0.5} [/latex]

Example 2.4-f: Evaluating Square Roots of Decimal Numbers

Evaluate the square root: [latex] \sqrt{0.49} [/latex]

2.4 Exercises

For Problems 1 to 8, perform the additions.

1. 619.985 + 52.82 + 3.187

2. 927.896 + 659.50 + 128.649

3. 17 + 3.48 + 0.278 + 78.24

4. 74 + 129.258 + 0.32 + 666.015

5. 396.716 + 191.68 + 90.6

6. 292.454 + 121.69 + 65.3

7. 625.365 + 27.97 + 0.613

8. 948.684 + 15.17 + 0.717

9. Calculate the sum of the following numbers:
   Six and thirty-nine thousandths; Eighty and fourteen hundredths; Sixteen and eight tenths.

10. Calculate the sum of the following numbers:
    Twenty and ninety-five hundredths; Two hundred and seventy-two thousandths; Nineteen and nine tenths.

For Problems 11 to 18, perform the subtractions.

11. 9.555 – 7.18

12. 423.92 − 185.728

13. 15.7 − 7.92

14. 29.28 – 13.4

15. 848.62 – 495.476

16. 539.64 – 258.357

17. 475.3 – 281.375

18. 409.5 – 179.832

19. Subtract eight hundred twenty and four hundredths from one thousand, one hundred one and six tenths.

20. Subtract three hundred five and thirty-nine hundredths from seven hundred twenty and four tenths.

For Problems 21 to 28, perform the multiplications.

21. 189.945 × 6.3

22. 137.89 × 5.4

23. 92.74 × 3.25

24. 62.095 × 4.18

25. 25. 0.59 × 0.9

26. 0.43 × 0.8

27. 145.75 × 3.74

28. 109.78 × 2.91

For Problems 29 to 36, perform the divisions.

29. 261.31 ÷ 7

30. 67.78 ÷ 9

31. 413.9 ÷ 6

32. 732.6 ÷ 8

33. 9.155 ÷ 0.7

34. 14.6 ÷ 0.6

35. 2.7 ÷ 0.15

36. 3.1 ÷ 0.25

For Problems 37 to 40, evaluate the powers of the decimal numbers.

37. a. [latex] (1.1)^3 [/latex]   b. [latex] (1.2)^3 [/latex]

38. a. [latex] (0.1)^3 [/latex]   b. [latex] (0.3)^2 [/latex]

39. a. [latex] (0.9)^2 [/latex]   b. [latex] (0.05)^3 [/latex]

40. a. [latex] (0.4)^2 [/latex]   b. [latex] (0.02)^3 [/latex]

For Problems 41 to 46, evaluate the square roots of the decimal numbers.

41. a. [latex] \sqrt{0.36} [/latex]   b. [latex] \sqrt{0.64} [/latex]

42. a. [latex] \sqrt{0.25} [/latex]   b. [latex] \sqrt{0.49} [/latex]

43. a. [latex] \sqrt{2.56} [/latex]   b. [latex] \sqrt{1.44} [/latex]

44. a. [latex] \sqrt{1.21} [/latex]   b. [latex] \sqrt{1.69} [/latex]

45. a. [latex] \sqrt{0.09} [/latex]   b. [latex] \sqrt{0.0004} [/latex]

46. a. [latex] \sqrt{0.01} [/latex]   b. [latex] \sqrt{0.0049} [/latex]

For Problems 47 to 54, formulate arithmetic expressions and evaluate.

47. Find the amount that is $45.27 less than $90.75.

48. Find the amount that is $248.76 less than $627.40.

49. Find the difference in the amounts $235.62 and $115.75.

50. Find the difference in the amounts $30.75 and $15.89.

51. Find the sum of $252.34 and $297.90.

52. Find the sum of $52.43 and $23.95.

53. Find the amount that is $412.78 more than $634.25.

54. Find the amount that is $38.89 more than $25.67.

55. The cost of an item is $125.69. If Arun gave $150.00 to the cashier, how much change would Arun receive?

56. The cost of an item is $88.46. If you gave $90.00 to the cashier, how much change would you receive?

57. Last week Carol spent $96.75 more on food than on transportation. She spent $223.15 on transport. How much did Carol spend on both food and transportation last week?

58. Bill saved $578.50 this week. He saved $124.85 more last week than this week. How much did Bill save during the two-week period?

59. A car driver filled gas when the odometer reading was 35,894.9 km. The odometer reading now is 39,894.4 km. How many kilometres did the driver travel, rounded to the nearest kilometre?

60. The normal selling price of an item is $237.75. When this item was on sale Dave paid $49.89 less for it. How much did Dave pay for that item?

61. After paying $515.09 for a car lease and $379.92 for property tax, Elisa’s bank balance was $675.45. How much money did Elisa have initially?

62. After spending $38.96 on toys and $1.75 on wrapping paper, Ann still had $45.75. How much money did Ann have initially?

63. Andy bought a TV that was on sale for $2,249.95. He agreed to pay $130.45 every month for 18 months. How much more money than the sale price did Andy pay for the TV?

64. Simon bought a camera that was on sale for $799.99. He agreed to pay $70.35 every month for 12 months. How much more money than the sale price did Simon pay for the camera?

65. Danny leased a car on a four-year term at $694.38 per month. At the end of the lease period, she paid an additional $18,458.74 to purchase the car. Calculate the total amount Danny paid for the car.

66. A salesperson earns a salary of $725.35 every week. During the past three weeks, he also received commissions of $375.68, $578.79, and $338.57. Calculate his total income for the past three weeks.

67. Taylor bought 3 kg of walnuts at $8.69 per kg and 4 kg of almonds at $7.72 per kg. He gave the cashier a $100 bill. How much change should Taylor receive from the cashier?

68. John bought two shirts at $20.95 each and three pairs of pants at $34.55 each. He gave $200 to the cashier. Calculate the balance he should receive from the cashier.

69. A cake that weighed 0.82 kg was cut into slices that weighed 0.1025 kg each. How many slices were there?

70. A string that measured 0.875 m was cut into pieces of 0.0625 m each. How many pieces were there?

71. Gilbert bought 2 kg of grapes at $3.29 per kg and 1.5 kg of strawberries at $5.99 per kg. He gave a $20 bill to the cashier. How much should he expect to receive in change from the cashier?

72. Marion bought three dresses at $22.49 per dress and two pairs of shoes at $14.99 per pair. She gave a $100 bill to the cashier. What change should she expect to receive from the cashier?