# Simplification and Approximation

Ankur Kulhari

###### Simplification

The most important thing to learn under simplification topic is the great BODMAS rule.
The BODMAS Rule
It is important to follow the correct sequence of execution of operators in an expression. This means that to simplify an expression, the following order must be followed –

1. B = Bracket, {the order – (), {} and [] – should be strictly followed}
2. O = Of (orders i.e. Powers and Square Roots, Cube Roots, etc.)
3. D = Division {evaluated left to right}
4. M = Multiplication {evaluated left to right}
5. A = Addition {evaluated left to right}
6. S = Subtraction {evaluated left to right}

Note:

1. First Brackets are needed to be evaluated strictly in the order, first (), then {} and finally [].
2. Next, exponents (for instance powers, roots etc.) or Of operators are evaluated.
3. Next, division and multiplication are to be performed, and if multiple division or multiplication operators are there in an expression, they will be evaluated from left to right with equal priority.
4. Finally, addition and subtraction are performed, and if multiple addition or subtraction operators are there in an expression, they will be evaluated from left to right with equal priority.

Example 1. Solve 12 + 22 ÷ 11 × (18 ÷ 3)^2 – 10

Solution:
= 12 + 22 ÷ 11 × 6^2 – 10 (Brackets first)
= 12 + 22 ÷ 11 × 36 – 10 (Exponents)
= 12 + 2 × 36 – 10 = 12 + 72 – 10 (Division and multiplication, left to right)
= 84 – 10 = 74 (Addition and Subtraction, left to right)

Example 2. Solve 4 + 10 – 3 × 6 / 3 + 4

Solution:
= 4 + 10 – 18/3 + 4 = 4 + 10 – 6 + 4 (Division and multiplication, left to right)
= 14 – 6 + 4 = 8 + 4 = 12 (Addition and Subtraction, left to right)

###### Modulus of a Real Number

The Modulus (absolute value) of x is considering the magnitude value of x, and is denoted by |x| and is defined as

|x|= x {if x ≥ 0} and −x {if x < 0}

Example 3. Solve |8|.

Solution:
|8| = |-8| = 8

###### Learn square roots up till 20.

To save time and do calculation quickly it is required to mug-up things as much as you can. Square roots and cube roots can be a part of the simplification/approximation question. Remembering these values can help you save some time and reach the approximate value quickly.

Number Square root Cube root
1 1 1
2 1.414 1.26
3 1.73 1.44
4 2 1.59
5 2.23 1.71
6 2.45 1.82
7 2.65 1.91
8 2.83 2
9 3 2.08
10 3.16 2.15

###### Approximation

1. Conversion of decimal numbers to nearest number.
To solve such questions, first convert the decimal to nearest value. Then simplify the given equation using the new values that you have obtained.

Example 4. Solve 4433.764 – 2211.993 – 1133.667 + 3377.442

Solution:
We can assume,
4433.764 = 4434
2211.993 = 2212
1133.667 = 1134
3377.442 = 3378
Now simplify, 4434 – 2212 – 1134 + 3378 = 4466

Example 5. Solve 530 x 20.3% + 225 x 16.8%

Solution:
Approximating, 20.3% = 20% and 16.8% = 17%
Now, simplify 530 x 20% + 225 x 17% {53*2 = 106, 225*17/100 = 2.25*17 = (2+0.25)*17 = 34+(1/4)*17 = 34+4.25 = 38.25}
= 106 + 38.25 = 144.25

Note:

• We have increased all the decimal values.
• Keep in mind the increase or decrease you considered and accordingly approximate your calculated answer to get the final value.

Example 6. Solve 37.74+121.8+321.66+512.20

Solution:
Approximating as,
37.74 = 38
121.8 = 122
321.66 = 322
512.20 = 513
=> 38+122+322+513 = 995

While approximating, we rounded up all the values so, the actual value will be less than our calculated value.

2. Approximation of Square Roots
3. Following tricks can be applied to simplify a square root:

1. Factor the number inside the square root sign.
2. Bring out the perfect square factors from the square root.
3. Now simplify the remaining values inside the square root.

Example 7. Simplify 121/2.

Solution:
√12 = √2*2*3 = 2√3 = 2*1.73 = 3.46

Example 8. Simplify 5001/2.

Solution:
√500 = √5*10*10 = 10√5 = 10 * 2.23 = 22.3

Here are some Unsolved Questions, try solving them and share your answer in the comments section below:

Exercise 1. 99 + 25 / 51 * (22 / 2) 2 – 10
Exercise 2. 43.99 Of 1202 – 13.44% of 1599 =?

1. 