Aptitude, Reasoning and Quantitative Aptitude Formulas & Notes - APSC CSAT, ADRE & other Govt exams

Aptitude, Reasoning and Quantitative Aptitude Formulas for APSC CSAT, ADRE & other Govt exams

Download – Important Formulas for Quantitative Aptitude – APSC Prelim CSAT Paper

Number system

Natural Numbers: 1, 2, 3, 4…..
Whole Numbers: 0, 1, 2, 3, 4…..
Integers: ….-2, -1, 0, 1, 2 …..
Rational Numbers: Any number which can be expressed as a ratio of two integers for example a p/q format where ‘p’ and ‘q’ are integers. Proper fraction will have (p<q) and improper fraction will have (p>q)
Factors: A positive integer ‘f’ is said to be a factor of a given positive integer ‘n’ if f divides n without leaving a remainder. e.g. 1, 2, 3, 4, 6 and 12 are the factors of 12.
Prime Numbers: A prime number is a positive number which has no factors besides itself and unity.
Composite Numbers: A composite number is a number which has other factors besides itself and unity.
Factorial: For a natural number ‘n’, its factorial is defined as: n! = 1 x 2 x 3 x 4 x …. x n (Note: 0! = 1)
Absolute value: Absolute value of x (written as |x|) is the distance of ‘x’ from 0 on the number line. |x| is always positive. |x| = x for x > 0 OR -x for x <0

Sum of n numbers

Sum of first n natural numbers = n(n+1)/2
Sum of the squares of first n natural numbers = n(n+1)(2n+1)/6
Sum of the cubes of first n natural numbers = [n(n+1)/2]2
Sum of first n natural odd numbers = n²
Average = (Sum of Observation / Number of Observations )
If a car cover a certain Distance at X kmph and an equal distance at Y kmph . Then , the average speed during the whole journey is [ 2XY / (X+Y) ]

BODMAS Rule

This Rule depicts the correct sequence in which the operations are to be executed, so as to find out the value of a given expression.

Calculation should be done the following order:

B – Bracket

O – Of

D – Division

M – Multiplications

A – Addition

S – Subtractions

Arithmetic Progression (A.P.)

An A.P. is of the form a, a+d, a+2d, a+3d, … where a is called the ‘first term’ and d is called the ‘common difference’

nth term of an A.P. tn = a + (n-1)d
Sum of the first n terms of an A.P. Sn = n/2[2a+(n-1)d] or

Sn = n/2(first term + last term)


Geometrical Progression (G.P.)

A G.P. is of the form a, ar, ar², ar³, … where a is called the ‘first term’ and r is called the ‘common ratio’.
nth term of a G.P. tn = ar^n-1
Sum of the first n terms in a G.P. Sn =S_n= a(rⁿ – 1) / (r – 1)
[OR] Sn = a(1 – rⁿ) / (1 – r), if r ≠ 1.
The sum of GP (of n terms) is: S_n= na, when r = 1.
The sum of GP (of infinite terms) is: S_∞= a/(1-r), when |r| < 1.


Divisibility Rules

A number is divisible by:

2, 4 & 8 when the number formed by the last, last two, last three digits are divisible by 2, 4 & 8 respectively.
3 & 9 when the sum of the digits of the number is divisible by 3 & 9respectively.
6, 12 & 15 when it is divisible by 2 and 3, 3 and 4 & 3and 5 respectively.
7, if the number of tens added to five times the number of units is divisible by 7.
A number is divisible by 10 if the units digit is 0.
11 when the difference between the sum of the digits in the odd places and of those in even places is 0 or a multiple of 11.
13, if the number of tens added to four times the number of units is divisible by 13.
19, if the number of tens added to twice the number of units is divisible by 19.

H.C.F and L.C.M :

C.F stands for Highest Common Factor. The H.C.F. of two or more numbers is the greatest number that divides each one of them exactly.
The least number which is exactly divisible by each one of the given numbers is called their L.C.M.
Two numbers are said to be co-prime if their HCF is 1.
HCF of fractions = (HCF of numerators)/(LCM of denominators)
LCM of fractions = (LCM of numerators)/(HCF of denominators )
Product of two numbers = Product of their HCF and LCM

PERCENTAGES

To express x% as a fraction: We have, x% = x/100
To express a/b as a percentage: We have, a/b %= (a/b x 100)
If A is R% more than B, then B is less than A by R / (100+R) * 100
If A is R% less than B, then B is more than A by R / (100-R) * 100
If the price of a commodity increases by R%, then reduction in consumption, not to increase the expenditure is : R/(100+R)*100
If the price of a commodity decreases by R%, then the increase in consumption, not to decrease the expenditure is : R/(100-R)*100

PROFIT & LOSS :

Gain = Selling Price(S.P.) – Cost Price(C.P)
Loss = C.P. – S.P.
Gain % = Gain * 100 / C.P.
Loss % = Loss * 100 / C.P.
S.P. = (100+Gain%)/100*C.P.
S.P. = (100-Loss%)/100*C.P.

CP. = [100/ (100 + Gain%) ] x S.P
CP. = [100/ (100 – Loss%) ] x S.P
When a shopkeeper sell two similar items , one at a gain of say x% , and other at a loss of x% then the seller always incure a loss given by – Loss % = ( Common loss & gain % / 10 )²
If a trader sell his goods at cost price, but uses false weight , then Gain% = [ Error / (True value – Error ) ] x 100 %

SIMPLE & COMPOUND INTERESTS

Let P be the principal, R be the interest rate percent per annum, and N be the time period.

Simple Interest = (P*N*R)/100
Compound Interest = P(1 + R/100)N – P
Amount = Principal + Interest
When interest is compound Annually, Amount = P(1 + R/100)ⁿ
When interest is compounded Half-yearly: Amount = P[1 + (R/2)/100]²ⁿ
Population after n years : P(1 + R /100)ⁿ, R is the population growth rate
Population before n years : P(1 – R /100)ⁿ, R is the population growth rate

RATIO & PROPORTIONS:

The ratio a:b represents a fraction a/b. a is called antecedent and b is called consequent.
The equality of two different ratios is called proportion.
If a : b = c : d then a, b, c, d are in proportion. This is represented by a : b :: c : d.
In a : b = c : d, then we have a* d = b * c.
If a/b = c/d then ( a + b ) / ( a – b ) = ( d + c ) / ( d – c ).


TIME & DISTANCE

Distance = Speed * Time

1 km/hr = 5/18 m/sec

1 m/sec = 18/5 km/hr

Suppose a man covers a certain distance at x kmph and an equal distance at y kmph. Then, the average speed during the whole journey is 2xy/(x+y) kmph


Upstream & Downstream

In water, the direction along the stream is called downstream. And, the direction against the stream is called upstream.

If the speed of a boat in still water is u km/hr and the speed of the stream is v km/hr:

Speed downstream = (u + v) km/hr

Speed upstream= (u – v) km/hr

If the speed downstream is a km/hr and the speed upstream is b km/hr:

Speed in strill water = 1/2 (a + b) km/hr

Rate of stream = 1/2 (a – b) km/hr


TIME & WORK

If A can do a piece of work in n days, then A’s 1 day’s work = 1/n
If A and B work together for n days, then (A+B)’s 1 days’s work = 1/n
If A is twice as good workman as B, then ratio of work done by A and B = 2:1


Area & Volume

Sum of the angle of a triangle is = 180 degree
The sum of any two side of a triangle is greater than the third side .
Pythagorous Theorem = Hypotenuse² =( Base ) ² + (Height )²
Area of a rectangle = (Length x Breadth)
Perimeter of a rectangle = 2 ( Length + Breadth )
Area of a square = (side)² = 1/2 (diagonal)²
Area of an equilateral triangle = √3/4 (Side) ²
Area of 4 walls of a room = 2 (Length + Breadth) x Height
Area of a triangle =1/2 x Base x Height
Area of a circle = πR² , where R is the radius
Circumference of a circle = 2πR, thus, Circumference of a semi-circle = πR.

CUBE

Let each edge of a cube be of length a. Then, Volume = a3 cubic units.

CUBOID

Let length = l, breadth = b and height = h units. Then Volume = (l x b x h) cubic units

Types of Angle

Acute angle = 0° – 90°
Right Angle = 90°
Obtuse angle = 90° – 180°
Straight Angle = 180°
Reflex Angle = 180° – 360°
Complete angle = 360°
Complementary Angle = sum of two angles = 90°
Supplementary angle = sum of two angles = 180°

Triangle Properties

Based on sides

Equilateral Triangle : All three sides equal
Isosceles Triangle : Two sides equal
Scalene Triangle : all three sides different

Based on Angles

Right Angle Triangle : One angle 90°
Obtuse Angle Triangle : One angle more than 90°
Acute Angle Triangle : All angles less than 90°
When AC2 < AB2 + BC2 (Acute angle triangle )
When AC2 > AB2 + BC2 (Obtuse angle triangle )
When AC2 = AB2 +BC2 (Right angle triangle )

Some Basic Formulae

(a+ b)(a – b) = (a² – b²)
(a+ b)² = (a² + b² + 2ab)
(a– b)² = (a² + b² – 2ab)
(a+ b + c)² = a² + b² + c² + 2(ab + bc + ca)
(a³+ b³) = (a + b)(a² – ab + b²)
(a³– b³) = (a – b)(a² + ab + b²)
(a³+ b³ + c³ – 3abc) = (a + b + c)(a² + b² + c² – ab – bc – ac)
When a+ b + c = 0, then a³ + b³ + c³ = 3abc.

AssamExam

APSC Online Coaching – Tests, Study Materials & Assam Current Affairs