1.2 Arithmetic Operations with Whole Numbers

Addition of Whole Numbers

Addition of whole numbers refers to combining two or more whole numbers to find the total.

The numbers that are added are referred to as the addends, and the result or answer is called the total or sum. The symbol ‘+’ denotes addition.

For example, 9 + 5 refers to adding 9 and 5. This is read as ‘nine plus five’.

Follow these steps to add whole numbers:

1. Write the numbers one under the other by aligning the place values (ones, tens, hundreds, etc.) of the numbers and draw a horizontal line underneath.
2. Starting from the ones place value, add all the numbers in the 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. This is called ‘carrying’.
3. Add the numbers in the tens column, followed by the hundreds column, etc., by following the same procedure for each column.

Example 1.2-a: Adding Whole Numbers

Perform the following additions:

1. 3,514 + 245

2. 8,578 + 3,982 + 564 + 92

Subtraction of Whole Numbers

Subtraction of whole numbers refers to finding the difference between numbers. This is the reverse process of addition.

The number from which another number is subtracted is referred to as the minuend and the number that is being subtracted is referred to as the subtrahend. The result or answer is called the difference. The symbol ‘–’ denotes subtraction.

For example, 8 – 5 refers to subtracting 5 from 8. This is read as ‘eight minus five’.

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

1. Write the numbers one under the other by aligning the place values (ones, tens, hundreds, etc.) of the numbers and draw a horizontal line underneath. Ensure that the number that is being subtracted from (the minuend) is in the top row and that the number that is being subtracted (the subtrahend) is below.
2. Starting from the ones 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. This is called ‘borrowing’.
3. Subtract the numbers in the tens column, followed by the hundreds column, etc., by following the same procedure for each column.

Note: To check your answer, add the difference to the number subtracted (subtrahend); the result should be the number from which it was subtracted (minuend). e.g., 8 – 5 = 3; therefore 3 + 5 = 8

Example 1.2-b: Subtracting Whole Numbers

Perform the following subtractions:

1. Subtract 1,314 from 3,628

2. Subtract 789 from 8,357

Note: If a larger whole number is subtracted from a smaller whole number, the result will be a negative number. To do this, reverse the question to deduct the smaller number from the larger number following the above steps, and set the answer to be negative.

Multiplication of Whole Numbers

Multiplication is the process of finding the product of two numbers. Multiplication of whole numbers can be thought of as repeated additions. The symbol ‘×’ denotes multiplication.

For example, 5 × 4 refers to repeatedly adding 5, four times. This is read as ‘five times four’, and can also be written as 5 ∙ 4 or 5(4).

5 × 4 can be represented pictorially as:

Here, the size of the set is 5 and it is repeated 4 times:
5 + 5 + 5 + 5 = 20

This can also be viewed as 4 × 5:

Here, the size of the set is 4 and it is repeated 5 times:
4 + 4 + 4 + 4 + 4 = 20

The numbers that are being multiplied are referred to as factors and the result is referred to as the product.

In this example, 5 and 4 are factors of 20.

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

1. Write the numbers one under the other by aligning the place values and draw a horizontal line underneath, as done for addition and subtraction.
2. Starting from the ones digit of the bottom factor, multiply the digit by each digit of the top factor, starting with the right-most digit.
   • If the product is less than 10, write the product under the horizontal line.
   • If the product is 10 or more, write the ‘ones’ digit of the product under the horizontal line, carry the ‘tens’ digit, and add it to the subsequent product.
3. Multiply the next digit of the bottom factor by each digit of the top factor by following the same procedure, and write the product below the first product, but one place value to the left. Continue for each digit of the bottom factor.
4. Sum all the products to obtain the answer.

The above steps for multiplying numbers are provided in detail in Example 1.2-c below.

Example 1.2-c: Multiplying Whole Numbers

Perform the following multiplications:

1. Multiply 38 by 6

2. Multiply 36 by 24

3. Multiply 263 by 425

Division of Whole Numbers

Division is the process of determining how many times one number is contained in another. This is the inverse process of multiplication. When a larger number is divided by a smaller number, this division can be thought of as repeated subtractions. The symbol ‘÷’ denotes division.

For example, 20 ÷ 5 refers to repeatedly subtracting 5 from 20. This is read as ‘twenty divided by five’, and can also be written as 5 |20.

In a set of 20 items, we can create four groups of 5:

Therefore, 20 ÷ 5 = 4.

The number that is being divided is referred to as the dividend, and the number by which the dividend is divided is referred to as the divisor. The result or answer is called the quotient. If the dividend cannot be divided evenly by the divisor, the number left over is referred to as the remainder.

For example, consider 25 ÷ 7:

The following relationship exists between the four components of a division problem:

Dividend = Divisor × Quotient + Remainder

25 = 7 × 3 + 4

The steps to be followed in dividing whole numbers are provided in detail in Example 1.2-d below.

Example 1.2-d: Dividing Whole Numbers

Perform the following divisions and state the quotient and remainder:

1. Divide 76 by 3

2. Divide 637 by 25

3. Divide 6,543 by 12

Arithmetic Operations with Zero and One

Table 1.2-a: Arithmetic Operations with Zero

Table 1.2-b: Arithmetic Operations with One

Powers of Whole Numbers

We learned previously that multiplication is a shorter way to write repeated additions of a number. Similarly, when a number is multiplied by itself repeatedly, we can represent this repeated multiplication using exponential notation.

When 2 is multiplied 5 times, in repeated multiplication, it is represented by:

2 × 2 × 2 × 2 × 2

However, it can be tedious to represent repeated multiplication using this notation. Instead, exponential notation can be used.

When 2 is multiplied 5 times, in exponential notation, it is represented by:

In this example, 2 is known as the base, 5 is known as the exponent, and the whole representation 2⁵ is known as the power. The exponent is written in superscript to the right of the base, and represents the number of times that the base is multiplied by itself. The whole representation is read as “2 raised to the power of 5” or “2 to the 5^th power”.

Example 1.2-e: Converting Exponential Notation to Repeated Multiplication and Evaluating

Expand 8² to show the repeated multiplication and evaluate.

Solution

\(8^2 + 8 × 8 = 64\)

Example 1.2-f: Converting Repeated Multiplication to Exponential Notation

Express the repeated multiplication 9 × 9 × 9 × 9 × 9 × 9 in exponential notation.

Example 1.2-g: Evaluating Expressions by Writing in Standard Form

Express the following in standard form and then evaluate:

1. \(5 × 3^4\)

2. \(2^3 × 3^2\)

3. \(5^4 × 2^2\)

4. \(5^2 × 4^2\)

Example 1.2-h: Evaluating Powers using a Different Power of the Same Base

Use the equation \(6^5 = 7,776\) to evaluate the following expressions:

1. \(6^6\)

2. \(6^4\)

Table 1.2-c: Examples of Powers with Exponents of 2, 3, 4, and 5

Perfect Squares and Square Roots

Any whole number base with an exponent of 2 is known as a perfect square. For example, 9 is the perfect square produced by 3 raised to an exponent 2. The first twenty perfect squares are shown in Table 1.2-c under Powers of 2.

Finding the square root of a perfect square is the inverse of raising a whole number to the power of 2.

For example,

• 2 is the square root of 4 because \(2^2 = 4\) or \(2 × 2 = 4\)

• 5 is the square root of 25 because \(5^2 = 25\) or \(5 × 5 = 25\)

That is, a whole number multiplied by itself results in a perfect square. The whole number multiplied by itself to get that perfect square is called the square root of that perfect square.

For example,

• 16 is a perfect square because it is the product of two identical factors of 4; i.e., 16 = 4 × 4.

• Therefore, 4 is the square root of 16.

The radical sign \(\sqrt{ }\) indicates the root of a number (or expression). The square root of 16, using the radical sign, is represented by \(\sqrt[2]{16}\) , where 2 is the index. The index indicates which root is to be taken, and is written as a small superscript number to the left of the radical symbol. For square roots, the index 2 does not need to be written as it is understood to be there; i.e. \(\sqrt[2]{16}\) is written as \(\sqrt{16}\).

Note: In future chapters, we will discuss roots with indexes other than 2.

Example 1.2-i: Determining the Square Root of Perfect Squares

Determine the square root of the following:

1. \(\sqrt{36}\)

2. \(\sqrt{81}\)

3. \(\sqrt{144}\)

1.2 Exercises

For the addition Problems 1 to 8, (i) estimate the answer by rounding each number to the nearest ten, and (ii) calculate the exact answer.

1. a. 48 + 29
   b. 38 + 95
2. a. 16 + 79
   b. 69 + 47
3. a. 875 + 48
   b. 574 + 79
4. a. 459 + 27
   b. 356 + 65
5. a. 286 + 109 + 15
   b. 839 + 645 + 27
6. a. 989 + 215 + 25
   b. 798 + 237 +12
7. a. 195 + 459 + 8
   b. 996 + 816 + 6
8. a. 896 + 642 + 9
   b. 995 + 724 + 8

For the addition Problems 9 to 14, (i) estimate the answer by rounding each number to the nearest hundred, and (ii) calculate the exact answer.

9. a. 745 + 668
   b. 427 + 225
10. a. 357 + 245
    b. 451 + 645
11. a. 1,883 + 5,466
    b. 2,157 + 3,459
12. a. 6,950 + 2,367
    b. 3,765 + 1,992
13. a. 2,635 + 372 + 1,524
    b. 653 + 2,188 + 891
14. a. 1,650 + 1,647 + 875 + 167
    b. 3,869 + 1,967 + 550 + 745

For the subtraction Problems 15 to 22, (i) estimate the answer by rounding each number to the nearest ten, and (ii) calculate the exact answer.

15. a. 62 − 25
    b. 33 – 18
16. a. 43 − 27
    b. 71 − 59
17. a. 208 − 79
    b. 315 − 47
18. a. 327 − 28
    b. 500 − 73
19. a. 767 − 159
    b. 804 − 308
20. a. 904 − 629
    b. 584 − 167
21. a. 8,302 − 7,244
    b. 2,927 − 888
22. a. 9,185 − 6,728
    b. 5,765 − 777

For the subtraction Problems 23 to 28, (i) estimate the answer by rounding each number to the nearest hundred, and (ii) calculate the exact answer.

23. a. 946 – 452
    b. 855 – 251
24. a. 868 – 74
    b. 495 – 357
25. a. 2,950 – 2,275
    b. 3,961 – 1,833
26. a. 2,955 – 1,350
    b. 1,967 – 352
27. a. 3,513 – 2,846
    b. 3,981 – 1,657
28. a. 7,676 – 3,969
    b. 5,789 – 5,626

For the multiplication Problems 29 to 34, (i) estimate the answer by rounding each number to the nearest ten, and (ii) calculate the exact answer.

29. a. 58 × 75
    b. 63 × 59
30. a. 35 × 97
    b. 95 × 71
31. a. 764 × 53
    b. 799 × 68
32. a. 482 × 95
    b. 755 × 55
33. a. 1,995 × 37
    b. 2,150 × 59
34. a. 2,996 × 32
    b. 2,995 × 38

For the division Problems 35 to 44, (i) estimate the answer by rounding each number to the nearest ten, and (ii) perform the exact division and state the quotient and the remainder.

35. a. 78 ÷ 5
    b. 36 ÷ 8
36. a. 69 ÷ 7
    b. 85 ÷ 9
37. a. 86 ÷ 27
    b. 78 ÷ 19
38. a. 59 ÷ 12
    b. 95 ÷ 18
39. a. 654 ÷ 14
    b. 396 ÷ 24
40. a. 777 ÷ 16
    b. 255 ÷ 19
41. a. 2,562 ÷ 18
    b. 1,225 ÷ 28
42. a. 2,097 ÷ 55
    b. 1,795 ÷ 64
43. a. 3,004 ÷ 204
    b. 6,501 ÷ 498
44. a. 3,895 ÷ 264
    b. 5,195 ÷ 255

For Problems 45 to 48, express the repeated multiplication in exponential notation.

45. a. 6 × 6 × 6 × 6
    b. 12 × 12
46. a. 5 × 5 × 5
    b. 7 × 7 × 7 × 7 × 7
47. a. 3 × 3 × 3 × 3 × 3 × 3
    b. 9 × 9 × 9 × 9
48. a. 8 × 8 × 8 × 8 × 8
    b. 4 × 4 × 4 × 4 × 4

For Problems 49 to 52, write the base and exponent for the powers.

49. a. \(2^9\)
    b. \(5^7\)
50. a. \(6^2\)
    b. \(10^9\)
51. a. \(1^{20}\)
    b. \(8^3\)
52. a. \(7^5\)
    b. \(12^1\)

For Problems 53 and 54, express the powers in standard notation and then evaluate.

53. a. \(10^6\)
    b. \(3^5\)
54. a. \(2^8\)
    b. \(5^4\)
55. If \(3^{10} = 59,049\), evaluate \(3^9\) and \(3^{11}\)
56. If \(5^8 = 390,625\), evaluate \(5^7\) and \(5^9\)

For Problems 53 and 54, express the powers in standard notation and then evaluate.

57. a. \(\sqrt{64}\)
    b. \(\sqrt{121}\)
58. a. \(\sqrt{25}\)
    b. \(\sqrt{225}\)
59. a. \(\sqrt{1}\)
    b. \(\sqrt{324}\)
60. a. \(\sqrt{100}\)
    b. \(\sqrt{400}\)
61. a. \(\sqrt{8^2}\)
    b. \(\sqrt{20^2}\)
62. a. \(\sqrt{12^2}\)
    b. \(\sqrt{45^2}\)
63. a. What amount is $65 less than $784?
    b. What amount is $35 more than $98?
64. a. What amount is $79 less than $487?
    b. What amount is $97 more than $52?
65. a. What amount is $515 increased by $847?
    b. What amount is $745 decreased by $125?
66. a. What amount is $745 increased by $1,274?
    b. What amount is $526 decreased by $346?
67. a. If there are 24 pens in a box, how many pens are there in 15 boxes?
    b. A wire of length 420 centimetres is to be cut into 35-centimetre pieces. How many 35-centimetre pieces can be cut?
68. a. If you save $125 every month, how much will you save in one year?
    b. If 8 people can be seated at a dinner table, how many tables are required to seat 280 people?
69. a. If you save $15 a week, how many weeks will it take you to save $675?
    b. If there are 16 chairs in a row, how many chairs are there in 22 rows?
70. a. If you work 25 hours per week, how many weeks will it take you to complete a job that requires 450 hours?
    b. If you earn $18 per hour, how much will you earn in a week in which you have worked 32 hours?
71. Mythili went to a bookstore and bought a dictionary. She gave the cashier a $100 note and received $47 as change. How much did the dictionary cost?
72. There were 744 students in a primary school. In May, after some students left to join another school, 576 students remained. How many students left for the other school?
73. Andy has $1,238 and Bill has $346 less than Andy. How much money do both of them have together?
74. A television costs $649 more than a Blu-ray player. If the Blu-ray player costs $235, how much will it cost to buy the TV and the Blu-ray player?
75. Peter earns $18 per hour. If he worked 25 hours last week, calculate his earning for that week.
76. Sam’s overtime rate is $37 per hour. If he worked 29 hours overtime last week, calculate his overtime pay for that week.
77. Earl wants to save $3,150. If he saves $75 per week, how many weeks will it take for him to achieve his goal?
78. How long will it take to travel 1,190 km at 85 km per hour?
79. A large group of students visited an exhibition on Friday. The first 275 students were given one free balloon each. 487 students were disappointed that they did not receive any balloons. How many students visited the exhibition?
80. There were 2,415 boys and 1,875 girls that attended a fair. The first 2,650 visitors to the fair received gifts. How many visitors did not receive gifts?
81. There were 450 passengers on a train. Half of them were men, 125 were children, and the rest were women. How many more men than women were there on the train?
82. Out of the 3,678 visitors at an art exhibition, 1,469 were men, 1,234 were women, and the rest were children. How many more adults than children went to the exhibition?
83. Aran had some money saved up. On his birthday, Aran’s grandparents gave him $125. After spending $98 on toys and $75 on clothes, he had $115 remaining. What amount did Aran have in his savings before his birthday?
84. Girija had some biscuits. She ate 19 and gave 27 to her brother and 5 to her parents. She had 12 biscuits left. How many biscuits did Girija have originally?
85. There are 19 girls in a class of 43 students. 15 of the girls and 11 of the boys are wearing eyeglasses.
    a. How many boys are there in the class?
    b. How many students do not wear eyeglasses?
86. There are 22 players on a soccer field. 10 players are wearing blue and 2 are wearing green. The rest of the players are wearing red. How many players are wearing red?

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