(**Part V**) consists of shortcut methods, tips and tricks to solve question from Simple and Compound Interest, Partnership and Calendar. Now (**Part VI**) includes another three very important section from quantitative aptitudes including *“Numbers and Decimal Fraction”, “Logarithm” and “Height and Distance”*.

Shortcut methods and tricks to solve questions from ‘**Numbers and Decimal Fraction’** are given below:

__Decimal Fraction__

It’s a type of fractions in which denominators are powers of 10.

i.e. (1/10) = 1 tenth = 0.1;

(1/100) = 1 hundredths = 0.01, etc;

__Fraction__

If a particular unit is divided into any number of equal parts, then one or more of this part is called **fraction** of that particular unit.

Suppose 1 is divided by two equal parts 2 and 3;

Then, ½ means 2 equal parts of 1,

⅓ means 3 equal parts of 1,

⅜ means 8 equal parts of 3

Finally the equation stands on: Fraction = (Numerator / Denominator) = (N/D)

__Types of Fraction__

There are two types of fractions: **Proper Fraction** and **Improper Fraction.**

**Proper Fraction: **In case of Numerator is less than Denominator, then fraction is called Proper Fraction. e.g. ⅓, ⅔, ⅞ etc.

**Improper Fraction: **In case of Numerator is greater than Denominator, then fraction is called Proper Fraction. e.g. 5/3, 7/3, 9/6 etc.

e.g. 3+ (2 / [3+ {3 / (7 + 3/5)}]

**Compound Fraction:** Fraction of a fraction is called Compound Fraction. e.g. ⅓ of ⅔, ⅔ of ⅞ etc.

**Complex Fractions:** In such fractions, numerator or denominator or both are fractions. e.g. [(1/7 + 2/5) / (3/7 + 4/9)]

**Inverse Fractions:** Inverse if a given fraction. e.g if the given fraction is 2/3, then the inverse fraction is 3/2

**Continued Fraction:** Contains an additional fraction in the numerator or in the denominator.

**Decimal Fraction to Vulgar Fraction conversion:** We first need to write down the given number without the decimal point in the numerator and in the denominator, writing down 1 followed by as many ‘0’s as many are there figures after the decimal point.

**E.g.**

1.) 0.75

2.) 0.0075

3.) 2.012

**Solution:**

1.) 0.75 = (75/100)

2.) 0.0075 = (75/10000)

3.) 2.012 = 2(012/1000)

**Addition & Subtraction of Decimals:**

Write down the numbers under one another, placing the decimal points in one column. The numbers can now be added or subtracted in the usual way.

e.g. Add together 6.033, 0.7, 150.03 and 50

Then, 6.033

0.7

150.03

50.00

_________________

206.763

e.g. Subtracting 6.033 from 0.7

Then, 0.700

6.033

_________________

-5.333

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Shortcut methods and tricks to solve questions from ‘**Logarithm’** are given below:

If p is positive real number , other than 1 and p^{n} = x, then

n = log_{p}x, and we can say that the value of log x to the base p is n.

**Try to remember the logarithm of these below 7 numbers** for your own shake, then you can calculate many more complex equations without the help of calculators or computers:

**Log 2 = 0.30, log 3 = 0.48, log 7 = 0.85, log 11= 1.04, log 13= 1.11, log 17 = 1.23 and log 19=1.28.**

The logarithm of a composite number is equal to the sum of the logarithms of its prime factors.

__Some important equations of logarithm given below:__

- log
_{a}(xy) = log_{a}x + log_{a}y. - log
_{y}y = 1. - log
_{a}1 = 0. - log
_{a }(x^{p}) = p(log_{a}x) - log(5p) = log(5) +log(p)
- log (5 + p) = log(5) +log(p)
- log (p
^{5) }= (log(p))^{5} - log (p
^{5) }= 5log(p) - log (5/p) = [log(5)] / [log(p)]
- log (5/p) = [log(5)] – [log(p)]

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Shortcut methods and tricks to solve questions from **‘Height and Distance’** are given below:

*Let’s consider the following image:*

Then,

1.) sin ɵ = (Perpendicular / Hypotenuse) = AB / OB,

2.) cos ɵ = (Base / Hypotenuse) = OA / OB,

3.) tan ɵ = (Perpendicular / Base) = AB / OA,

4.) cosec ɵ = (1 / sin ɵ) = OB / AB,

5.) sec ɵ = (1 / cos ɵ) = OB / OA,

6.) cot ɵ = (1 / tan ɵ) = OA / AB,

Trigonometrical identities and values:

sin2 ɵ + cos2 ɵ = 1,

1 + tan2 ɵ =sec2 ɵ,

1 + cot2 ɵ = cosec2 ɵ

The below chart is very important to remember the value for your exam:

__Angel of Elevation:__

See the below diagram:

Let AB be a tower standing at any point C on the level ground is viewing at A. The angle ,which the line AC makes with the horizontal line BC is called angle of elevation .so angle ACB is angle of elevation.

Angel of Depression:

See the below diagram:

If observer is at Q and is viewing an object R on the ground ,then angle between PQ and QR is the angle of depression . So angle PQR is angle of depression.

Another important calculation is shown in the following diagram:

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Awesome post.