Order of Operation, How to simplify and solve a complex Addition, Subtraction, Multiplication Problem

Problem: 3 + 4 x 2 = ?

Let's look at the question shown above, two students solve the problem with different result:

Student 1                                Student 2

3 + 4 x 2 =                            3 + 4 x 2 =

7 x 2 = 14                             3 + 8 = 11

It seems that each student interpreted the problem differently, resulting in two different answers. 

When performing arithmetic operations there can be only one correction answer. We need a set of rules in order to avoid this kid of confusion. Mathematicians have devised a standard order of operations for calculations involving more than one arithmetic operation:

Rule 1:  First, perform any calculations inside parentheses. (parentheses : (), [], {}, <>)

Rule 2:  Next perform all multiplications and divisions, working from left to right.

Rule 3:  Lastly, perform all additions and subtractions, working from left to right.

10105 x 97 +99 x 105 = ?

Have you ever tried to solve the question shown above without going through a complex multiplication? Here is the way how:

10105 x 97 + 99 x 105 = (10000 + 100 + 5) x 97 + (100 - 1）x 105

                                 = 970000 + 9700 + 97 x 5 + 10500 - 105

                                 = (970000 + 9700) + (100 - 3) x 5 + (10500-105)

                                 = 979700 + 500 - 15 + 10395

                                 = (979700 + 500) + (10395 - 15)

                                 = 980200 + 10380

                                 = 990580

To simplify and solve the problem shown above, We need  to remember the following equations and concepts show as below: 

Please be note that, when we are solving a problem with +, -, x and /, without any bracket in the question, 

A x B = AB      (For instance, 4 x 5 = 4(5) = 20)

For most of the problems, the multiplication of two numbers are expressed in the way shown above. 

A(B+C）= AB + AC   (For instance, 4(5) = 4(2 + 3) = 4 x 2 + 4 x 3 = 8 + 12 = 20)

A(B - C) = AB - AC    (For instance, 4(9) = 4(10-1) = 4 x 10 - 4 x 1 = 40 - 4 = 36)

-A (B - C) = -AB + AC   (For instance -5(-2) = -5(1 - 3) = -5 x 1 + -5(-3) = -5 + 15 = 10 )

A (B+C-D) = AB + AC -AD

PLEASE BE REMINDED THAT 

BOTH POSITIVE (+) MULTIPLIES TOGETHER WILL GET A POSITIVE (+) 

BOTH NEGATIVE (-) MULTIPLIES TOGETHER WILL GET A POSITIVE (+)

A NEGATIVE (-) MULTIPLIES WITH A POSITIVE (+) WILL GET A NEGATIVE (-)

Let's go for a more complicated question：

101 x 99 - 102 x 98 + 103 x 97 - 104 x 96

= (100 + 1) x 99 - (100 + 2) x 98 + (100 + 3) x 97 - (100 + 4) x 96

= [99(100) + 99] - [100(98) + 2(98)] + [100(97) + 3(97)] - [(100)(96) + 4(96)]

= (9900 + 99) - [9800 + 2(100-2)] + [9700 + 3(100 - 3)] - [9600 + 4(100-4)]

= (9900 - 9800 + 9700 - 9600) + 99 - [2(100) - 2(2)] + [3(100) - 3(3)] - [4(100) - 4(4)]

= 200 + 99 -(200-4) + (300-9) - (400 - 16)

= 200 + 99 -200 + 4 +300 - 9 - 400 + 16

= (200 - 200 + 300 - 400) + (99 + 4 - 9 + 16)

= -100 + 110

= 10