What are the types of lines?

In geometry there are four types of lines: Horizontal Line Vertical lines Parallel lines Perpendicular lines

A ray is a part of a line with one endpoint (i.e., starting point) and continues forever in the other direction.

What is perpendicular lines?

When two straight lines meet or intersect at an angle of 90 degrees, they are perpendicular to each other.

What is parallel lines?

Parallel lines are two straight lines that are always the same distance apart. They never meet or intersect at any point, even at infinity.

Order of Operations in Math

In this article, you will be learning about the order of operations, GEMDAS or PEMDAS rules, and how this rule applies to the math calculation. So let’s start.

Have you ever heard this phrase? You must be wondering what this phrase is and why we are discussing it here.

This saying acts as a magic wand to memorize an important concept of Math; the order of operations. But before decoding the phrase mentioned above, let’s understand what an operation is?

What is an operation?

In math, an operation means addition, subtraction, multiplication, division, exponentiation, and grouping.

Let’s understand with an example:

If you have been asked to choose the correct way to solve a question “15 – 2 x 3,”.

Way 1. 15 – 2 x 3 = 13 x 3 = 39 ( operation starts from left to right)
Way 2. 15 – 2 x 3 = 15 – 6 = 9 ( operation starts from right to left)

Which way is correct?
Maybe you can be confused about choosing the correct answer. To avoid this confusion, let’s understand the concept of “order of operations.”

What is an order of operations?

Order of operations is a rule that must be followed in order to solve the math problems such as addition, subtraction, multiplication, division, groupings, etc. If you don’t follow the order of operations rule, you can get a wrong answer.

In simple words, the order of operations tells us how to solve a math problem.

“Please excuse my dear Aunt Sally” (PEMDAS) is a math mnemonic that tells us in which order we should solve a mathematical problem.

PEMDAS stands for

Please stands for Parenthesis
Excuse stands for Exponents (Powers and Square Roots, etc.)
My stands for Multiplication (Multiplication/Division: left-to-right)
Dear stands for Division
Aunt stands for Addition (Addition/Subtraction: left -to-right)
Sally stands for Subtraction

PEMDAS Rules: Follow these orders to do math calculation

1st. Any calculation that comes inside the parenthesis or groups or brackets should be done first. This includes: ( ), { }, and [ ]. If you have all of these three parentheses, perform in the order of ( ), { }, [ ].
If there are no parentheses, skip this step.

2nd. Next, after solving operations inside of parenthesis (if any), exponents, roots, and absolute value are to be calculated from left to right.
If there are no exponents, roots, and absolute value, skip this step.

3rd. Next, perform multiplication/division – whichever operation comes first when you calculate from left to right.

NOTE: As in the PEMDAS rule, Multiplication comes before Division, but it doesn’t mean that Multiplication operation will always be performed before Division. Divide and Multiply rank equally. Therefore start the calculation from left to right and perform either division or multiplication depending upon whatever comes first.

For example: 24 ÷ 3 × 4
24 ÷ 3 × 4 (calculating from right to left)
= 24 ÷ 12
= 2, is wrong answer
24 ÷ 3 × 4 (calculating from left to right)
= 8 × 4
= 32 is the correct answer, because, going from left to right, the operation of division comes first.

4th. Next, perform addition/subtraction – whichever operation comes first when you calculate from left to right.
IMP: Add and Subtract rank equally. You can subtract before addition as long as you are calculating from left to right.

For example: 24 – 3 + 4
24 – 3 + 4
= 24 – 7
= 17, is wrong answer
24 – 3 + 4
= 21 + 4
= 25 is the correct answer, because, going from left to right, the operation of subtraction comes first.

GEMDAS Rules

GEMDAS stands for

Good stands for Groupings (first performs)
Evening stands for Exponents (Powers and Square Roots, etc.)
My stands for Multiplication (Multiplication/Division: left-to-right)
Dear stands for Division
Aunt stands for Addition (Addition/Subtraction: left -to-right)
Sally stands for Subtraction

Few more examples of order of operations

Example 1. 5 x (2 + 2)

First, solve whatever is in the groupings (parentheses)
5 x (2 + 2)
= 5 x 4
= 20
Answer is 20.

Example 2. 36 ÷ (12 – 10)^2

First, start calculation whatever is in the groupings (parentheses)
= 36 ÷ (12 – 10)^2
= 36 ÷ 2^2 ( second step to solve exponents i.e 2^2 )
= 36 ÷ 4 ( last, divide)
= 9
Answer is 9.

Example 3. 4 + 3 x 2

First, multiply the 3 and 2.
= 4 + 3 x 2
= 4 + 6 (last, add)
= 10
Answer is 10.

Example 4. 15 – 5 x 8 ÷ 5

= 15 – 5 x 8 ÷ 5
= 15 – 40 ÷ 5
= 15 – 8
= 7
Answer is 7.

Example 5. 4(5 − 3)² − 10 ÷ 5 + 8

In this problem there are parentheses, exponents, division, multiplication, addition, and subtraction. So as per PEMDAS rule, we’ll start by calculating the expression inside the parentheses.

4 (5 − 3)² − 10 ÷ 5 + 8

Now, calculate the exponent.
= 4 (2)² − 10 ÷ 5 + 8

Now, calculate multiplication because it comes first from left to right.
= 4 x 4 − 10 ÷ 5 + 8

Now perform division.
= 16 − 10 ÷ 5 + 8

Subtract now because it comes first from left to right.
= 16 − 2 + 8
= 14 + 8
= 22
Answer is 22.