Mixed Operations of Integers: Addition, Subtraction and Order of Operations

Mixed Operations of Integers: Integers are the set of whole numbers along with negative numbers. The set of integers is represented as \(I\) or \(Z = \left\{{ – 3, – 2, – 1,0,1,2,3…} \right\}.\) All the basic mathematical operations done on whole numbers or natural numbers can be performed on integers. In this article, we will study the addition and subtraction of integers using a number line.
We will also study the mixed operations on integers and the rules to be followed while operating them. The mixed operation with integers is solved by using the PEDMAS or BODMAS rule. Mixed operations are multiple operations involved like addition, subtraction, multiplication and division. Let us study these topics in detail.

Number Line Meaning

A number line represents zero, positive and negative integers. Every number on it is located with respect to zero. The positive integers are on the right side of the zero, and the negative integers are on the left side. Thus the value of integers is determined by the position of integers.

Addition and Subtraction of Integers on Number Line

Addition and subtraction of integers can be performed on a number line to understand the operations better. First, draw the number line with zero at the middle of the number line, then write down the positive integers on the right side of zero and negative integers on the left side. After locating the integers on the number line, we begin to perform the operations.
The operation always starts with the first number, whether the addition or subtraction of two integers. Locate the first number on the number line based on the sign of the second number; we decide the direction of the movement. For positive sign, we have to move to the right while to the left for negative sign. How much is to be moved is decided by the integer to be added or subtracted.

Addition of Integers

Positive integers: On adding two positive integers, the result will always be a positive integer. To add a positive integer, we always move towards the right side direction. For example, the addition of \(2\) and \(5\left({2 + 5 = 7}\right)\)
Here the first number is \(2,\) and the second number is \(5;\) both are positive. First, locate \(2\) on a number line. Then move \(5\) places to the right will give \(7.\)

Negative integers: On adding two negative integers, the result will always be a negative integer. To add a negative integer, we always move towards the left side direction. For example, the addition of \( – 1\) and \( – 3.\)

Here the first number is \( – 1,\) and the second number is \( – 3;\) both are negative. Mark \( – 1\) on a number line. Then moving \(3\) places to the left will give \( – 4.\)

Subtraction of Integers

Positive integers: On subtracting two positive numbers, move to the left as far as the value of the second number.
For example, subtract \(5\) from \(4.\)
Here the first number is \(4,\) and the second number is \(5;\) both are positive. But the sign between them is negative. First, mark \(4\) on a number line. Then move \(5\) places to the left to get \( – 1.\)

Negative numbers: On subtracting two negative numbers, move to the right as far as the value of the second number.
For example, subtract \( – 4\) from \( – 3.\)
We can write this as \(\left({ – 3} \right) – \left({ – 4} \right) = \left({ – 3} \right) + 4\)

Multiplication and Division of Integers

The following are the rules in performing multiplication on integers:
Rule 1: The product of a positive integer and a negative integer is negative.
Rule 2: The product of two positive integers is positive.
Rule 3: The product of two negative integers is positive.
For example, \(2 \times 4 = 8\)
\( – 2 \times – 4 = 8\)
\( – 2 \times – 4 = – 8\)
\(2 \times – 4 = – 8\)
The following are the rules in performing the division on integers:
Rule 1: The quotient of a positive integer and a negative integer is negative.
Rule 2: The quotient of two positive integers is positive.
Rule 3: The quotient of two negative integers is positive.

What is the Order of Operations?

In a mathematical expression, there may be several operations to be done. If certain rules are not followed, you may end up with different answers that may or may not be correct. For an expression, we will have only one correct answer. To obtain that correct answer, we need to follow certain rules. These rules depend on basic operations such as addition, subtraction, multiplication and division.
Order of operation is the sequence in which we evaluate any expression. We start from parenthesis (brackets), the exponents (orders), division or multiplication (from left to right, whichever comes first) and addition and subtraction (from left to right, whichever comes first). The figure below gives an idea of how the order of operations looks.

Order of Operations Definition

Order of operation is the sequence in which the operations are carried out using basic rules, especially while solving any expression with multiple operations. Expressions that appear between two operators, we operate in the order given below. The order of operation of integers are given below:

1. Brackets \((),\{ \} ,[]\)
2. Exponents
3. Division \(\left( \div \right)\) and Multiplication \(\left( \times \right)\)
4. Addition \(\left( + \right)\) and Subtraction \(\left( – \right)\)

Mixed Operations Rules

We follow these basic rules in a given sequence while performing any operations that are mentioned below.

Operation rule 1: First thing to do while operating the expression is to solve the numbers inside the parenthesis or bracket. We solve the parenthesis inside to out, grouping the operations. Observe the pattern of brackets present in the expression. There is a order to solve brackets, that is \(\left[{\left\{{\left({} \right)} \right\}} \right].\) First we solve the round bracket \(()\) then curly bracket \(\{ \} \) then box(square) bracket \([].\) The order of operations to be followed inside the brackets.

Operation rule 2: After operating on parenthesis, we look for exponents. If present, solve them.

Operation rule 3: Now, we operate on the four basic operations. In this step, we look for numbers with multiplication and division operations. If present, solve them from left to right.

Operation rule 4: Last operations to be carried out are addition or subtraction and solving them from left to right.

These rules in shorts can be remembered by their acronym PEDMAS or BODMAS. Let’s look into PEDMAS and BODMAS

How to Use Mixed Operations of Integers?

When multiple operations are involved in an expression, we call them mixed operations, and we follow certain rules for solving them. Let us understand the importance of the rules used in order of operations with the help of examples.

1) Solving parentheses according to the order of operations:
Expression: \(4 \times \left({5 + 2} \right)\)
Solution: \(4 \times \left( 7 \right) = 28\) (Correct (✔). This is a correct way to solve the parentheses)
Let us look at another approach for the same expression.
\(4 \times \left({5 + 2} \right) = 20 + 2 = 22\) (Incorrect (✘). This is an incorrect way to solve the parentheses)

2) Solving exponents according to the order of operations
Expression: \(4 \times {\left({ – 5} \right)^2}\)
Solution: \(4 \times \left({25} \right) = 100\) (Correct (✔). This is a correct way to solve the exponents)
Another approach for the same expression may be as follows
\(4 \times {\left({ – 5} \right)^2} = – {20^2} = – 400\) ((Incorrect (✘). This is an incorrect way to solve the exponents)

3) For solving multiplication or division and addition or subtraction
Expression: \(3 + 5 \times 2\)
Solution: \(3 + 5 \times 2 = 3 + 10 = 13\) (Correct (✔). Correct order.)
Another approach for the same expression may be as follows
\(3 + 5 \times 2 = 8 \times 2 = 16\) (Incorrect (✘). Incorrect order.)
Expression: \(3 – 6 \div 2\)
Solution: \(3 – 6 \div 2 = 3 – 3 = 0\) (Correct (✔). Correct order.)
Another approach for the same expression may be as follows
\(3 – 6 \div 2 = \left({ – 3} \right) \div 2 = – \frac{3}{2}\) (Incorrect (✘). Correct order not followed.)

While following the rules of order of operations, always remember to do multiplication or division before addition or subtraction.

Real-Life Applications of Mixed Operations of Integers

Many real-life examples require some order of operations to perform them correctly. Let us take an example. Suppose you went to purchase \(4\) pizzas that cost \(₹200\) each, and you want to split the total cost amount \(4\) people evenly. To find out each person share let us use the order of operations here.
Total number of people \( = 4\)
Total number of pizzas \( = 4\)
Cost of one pizza \(=₹200\)
Let us frame an expression using PEMDAS:
Expression: \(\left({200 + 200 + 200 + 200} \right) \div 4\) or \(\left({4 \times 200} \right) \div 4\)
Solution: According to BODMAS or PEDMAS we are going first to solve the parentheses.
\(\left({800} \right) \div 4 = 200\)
According to the order of operations, each person needs to pay \(₹200.\)
We can find many such day-to-day activities that involve the mixed operation of integers.

Solved Examples – Mixed Operations of Integers

Q.1. Simplify the given expression using the rules of the order of operations.
\(\left[{130 + \left\{{ – 18 – \left({45 – 55} \right) \times 22} \right\}} \right] \div 2\)
Ans: Given, \(\left[{130 + \left\{{ – 18 – \left({45 – 55} \right) \times 22} \right\}} \right] \div 2\)
Using the BODMAS rule
\(\left[{130 + \left\{{ – 18 – \left({45 – 55} \right) \times 22} \right\}} \right] \div 2 = \left[{130 + \left\{{ – 18 – \left({ – 10} \right) \times 2} \right\}} \right] \div 2\)
\(\left[{130 + \left\{{ – 18 + 10 \times 2} \right\}} \right] \div 2\)
\(\left[{130 + 2} \right] \div 2\)
\(\frac{{132}}{2}\)
\( = 66\)

Q.2. Simplify: \(42 \div – 6 + 5\)
Ans: \(42 \div – 6 + 5 = \frac{{42}}{{ – 6}} + 5\) (Using the BODMAS rule)
\( = – 7 + 5\)
\( = – 2\)

Q.3. Simplify \(4\left({ – 12 + 6} \right) \div 3\)
Ans: Given, \(4\left({ – 12 + 6} \right) \div 3 = \frac{{4\left({ – 12 + 6} \right)}}{3}\) (Using the BODMAS rule)
\( = \frac{{4\left({ – 6} \right)}}{3}\)
\( = 4 \times – 2\)
\( = – 8\)

Q.4. Simplify \(7\left({5 + 3} \right) \div 4\left({9 – 2} \right)\)
Ans: \(7\left({5 + 3} \right) \div 4\left({9 – 2} \right) = 7\left( 8 \right) \div 4\left( 7 \right)\) (Using the BODMAS rule)
\( = \frac{{7 \times 8}}{{4 \times 7}}\)
\( = 84\)
\( = 2\)

Q.5. Write an expression that shows the meaning of these words using the integers. Then evaluate the expression.
Half of the sum of five and two is then divided by seven.
Ans: Sum of five and two \( = 5 + 2\)
Half of the sum of five and two \( = \frac{1}{2}\left({5 + 2} \right)\)
It is further divided by seven \( = \frac{1}{2}\left({5 + 2} \right) \div 7\) is the required expression
\( = \frac{1}{2}\left({5 + 2} \right) \div 7 = \frac{7}{2} \div 7\)
\( = \frac{7}{2} \times \frac{1}{7}\)
\( = \frac{1}{2}\)

Summary

In this article, we studied the addition and subtraction of integers using the number line. We defined the number line with a diagram. Then we discussed the addition and subtraction of positive and negative integers using the number line. We moved to the operation of integers. We learnt how the operations are carried out using certain rules, namely BODMAS.
Without the rules, we may end up with different answers every time we solve them. We have solved examples with mixed operations in an expression to make students understand exactly how the operations are carried out and in which order. Also, real-life examples where we come across multiple operations on integers are explained.

Learn All Concepts on Integers

Frequently Asked Questions (FAQs) – Mixed Operations of Integers

Q.1. How do you solve mixed operations with integers?
Ans: The mixed operation with integers are solved by using the PEDMAS or BODMAS rule.

Q.2. What are mixed operations in math?
Ans: Mixed operations are multiple operations involved in an expression.

Q.3. What are the 4 operations of integers?
Ans: The basic arithmetic operations for real numbers are addition, subtraction, multiplication, and division.

Q.4. What are the rules in performing multiplication on integers?
Ans: The following are the rules in performing multiplication on integers:
Rule 1: The product of a positive integer and a negative integer is negative.
Rule 2: The product of two positive integers is positive.
Rule 3: The product of two negative integers is positive.

Q.5. Do you multiply first if no brackets?
Ans: In an expression, if parentheses and exponents are not present, we start with the multiplication and then division, working from left to right.

Now you are provided with all the necessary information on the mixed operations of integers and we hope this detailed article is helpful to you. If you have any queries regarding this article, please ping us through the comment section below and we will get back to you as soon as possible.