What is the order of operations?
The rules that state the order in which operations in an expression should be solved are known as the order of operations in mathematics.
We have to follow some basic rules to solve the order of operations. They are mentioned below.
1. First, we perform operations in grouping symbols.
2. Then, from right to left, we multiply and then divide.
3. Finally, we add and then subtract from right to left.
The order of operations can be expressed by another rule, the PEMDAS. PEMDAS is an easier way of remembering the order of operations, with each letter representing a mathematical operation.
To obtain accurate results, we should follow the order of operations.
Writing numerical expressions
A numerical expression is a mathematical sentence made up entirely of numbers and one or more operation symbols.
The symbols for addition, subtraction, multiplication, and division are examples of operation symbols.
Only numbers and operation symbols are allowed in a numerical expression. Here are some examples of numerical expressions:
1) 3 + 7
2) 213 – 11
3) \(43\times2 -33\)
4) \(\frac{980}{20}\times2 + 21 – 11\)
What are the grouping symbols? Why do we use them?
The grouping symbols are used to indicate that a set of numbers and meaningful operations should be grouped together and treated as if they were a single number. Parentheses, brackets, and braces are the three basic types of grouping symbols.
In a math expression, the grouping symbols are used to show what should be done first. The most common grouping symbol is parentheses. When parentheses have already been utilized, brackets and braces can be used for further group sections of a math expression.
Order of the grouping symbols
The order of the grouping symbols is solved from the inside. The symbol that is in the innermost part of the given numerical expression is solved first. We reach out by solving the expression sequentially.
Solved Examples
Example 1: Simplify 3{2 + [6 (6 + 1) + 5]}.
Solution:
We have to solve the equation from the inside out.
= 3{2 + [6(6 + 1) + 5]} Here, ( 6 + 1) is the innermost bracket.
= 3{2 + [6(7) + 5]}
= 3{2 + [42 + 5]} Now, [42 + 5] becomes the innermost bracket
= 3{2 + 47} Finally, {2 + 47} is the innermost bracket
= \(3\times49\)
= 147
Example 2: Jessica spends $100 on materials to make a dress.
She makes 12 dresses and sells ,eight of these for ten dollars and the remaining four for fifteen dollars.
Create a numerical expression to represent this situation, and then calculate Jessica’s profit.
Solution:
\(8\times 10+4\times 15-100\)
Going by the order of operations, we multiply first, then add, and finally subtract.
= 80 + 60 – 100
= 140 – 100
= 40
Therefore, Jessica will have a profit of $40.
Example 3: A pair of shoes costs $50. The store manager offers a 10-dollar discount. A man and his brother purchased four pairs of shoes and split the cost between them. Create a numerical expression to represent this situation, and then calculate the cost each brother bears for the shoes
Solution:
(Original price of shoes – discount) × number of shoes purchased ÷ number of brothers
\((50 – 10)\times 4 ÷ 2\)
Going by the order of operations, we first solve the operation in the parentheses, then multiply, and finally divide.
\( = (40)\times 4\) ÷ 2
= 160 ÷ 2
= 80
Example 4: Jonas withdraws $2,000 from his bank account. He gets his bike repaired with $500. Then he dividesd the money into five equal parts, giving four to charity and keeping one for himself. He finally takes his wife to a restaurant for dinner and spends $100 on food. Create a numerical expression to represent this situation, and calculate Jonas’ current financial situation.
Solution:
(Total money – the cost of repairing the bike) ÷ Money distribution – cost of the dinner)
Going by the order of operations, we solve the operation in the parentheses first, then we divide, and finally subtract.
(2000 – 500) ÷ 5 – 100
= (1500) ÷ 5 – 100
($1500 split into 5 parts implies $300 in each part. 4 parts , i.e., $1200 is given to charity, and he keeps the remaining part, which is $300)
= 300 – 100
= 200
Jonas has $200 left.
MDAS stands for Multiplication, Division, Addition, and Subtraction. It’s part of the order of operations, which is a set of rules that establish a method for simplifying mathematical operations. It’s used when there are multiple operations in an expression or equation.
When an expression includes entities such as parentheses, exponents, multiplication and division, as well as addition and subtraction, the PEMDAS rule will prevent you from getting the wrong answer.