Quantitative Aptitude Important Formulas

Ratio

Ratio : The comparison between two quantities in terms of magnitude is called ratio i.e, it tells that the one quantity is how many times the other quantity.

So ratio of any two quantities is expressed as a :b or a/b. The two quantities that are being compared are called terms.In a:b , the first term a is called antecedent and second term b is called consequent .

Properties of Ratio :
a : b = m a : m b, where m is a constant

a : b : c = A : B : C is equivalent to ,a/A = b/B = c/C

Componendo
If a/b = c/d then, (a+b)/b = (c+d)/d ,called Componendo

Dividendo
If a/b = c/d then, (a-b)/b = (c-d)/d ,called Dividendo

Componendo and Dividendo
If a/b = c/d then, (a+b)/(a-b) = (c+d)/(c-d) ,called Componendo and Dividendo

Types of Ratios :
Duplicate Ratio: a : b => a² : b²

Sub Duplicate Ratio: a : b => √a : √b

Triplicate Ratios : a : b => a³ : b³

Sub Triplicate Ratio : a : b => ³√a : ³√b

Inverse Ratio or Reciprocal Ratio:If a :b be the given ratio , then 1/a :1/b or b:a is its inverse ratio

Compound Ratio: If a:b , c:d and e:f are three given ratios , then a x c x e : b x d x f is the compound ratio of the given ratios.

Proportion

Proportion:The equality of two ratios is called a proportion and we say that the four numbers are in proportion.

If a/b = c/d , then a,b,c and d are said to be in proportion and we write a:b:: c:d . This is read as a is to b as c is to d.
the all terms a, b, c and d are called proportional’s . a,b,c and d respectively called first , second (mean), third and fourth proportional.

Types of Proportion :
Continued Proportion : If a,b and c are three numbers such that a:b = b:c , then these numbers a,b and c are said to be in continued proportion

Fourth Proportional: If a : b = c: d, then d is called the fourth proportional to a, b, c.

Third Proportional: If a: b = b: c, then c is called the third proportional to a and b.

Mean Proportional: Mean proportional between a and b is square root of ab.

Direct Proportion: x is directly proportional to y, if x = ky for some constant k and we write, x ∞ y.

Inverse Proportion:x is inversely proportional to y, if xy = k for some constant k and we write, x∞(1/y)

The common number system is decimal number system.
The Decimal number system used base (or radix) as 10 . This means that the system has ten symbols or numerals to represent any quantity. These symbols are called Digits and they are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

Types of Numbers

Natural Numbers:The numbers that are used for counting called Natural Numbers.These are infinite and start from the number 1 .
Ex : 1, 2, 3, 4, 5, 6 ........

Whole numbers:The whole numbers are just all the natural numbers plus zero.
Ex : 0, 1, 2, 3, 4, 5 .............

Integers: Integers incorporate all positive and negative numbers with zero.
Ex : .... –3, –2, –1, 0, 1, 2, 3 .....

Even Numbers: An even number is one that can be divided evenly by two leaving no remainder, such as 2, 4, 6, and 8.

Odd Numbers: An odd number is one that does not divide evenly by two, such as 1, 3, 5, and 7.

Rational Numbers: All numbers of the form p/q where p and q are integers (q ≠ 0) called Rational numbers.
Ex : 4, 3/4, 0, ….

Irrational Numbers: Irrational numbers are the opposite of rational numbers. An irrational number cannot be written as a fraction as p/q where a and b are integers.
the decimal values for irrational numbers never end and do not have a repeating pattern in them.please note ‘pi’ has never ending decimal places, is irrational.
Ex : pi, 2^1/2 , 3^1/2, 5^1/2 , 7^1/2 ..........

Real numbers: Real numbers include counting numbers, whole numbers, integers, rational numbers and irrational numbers.
Ex : 8, 6, 2, (3)^1/2 , 3/5 etc.

Prime Number: A prime number is a number which can be divided only by 1 and itself. The prime number has only two factors, 1 and itself.
Ex : 2, 3, 7, 11, 13, 17, …. are prime numbers.

Composite Number: A Composite Number is a number which can be divided evenly. Any composite number has additional factors than 1 and itself.
Ex : 4, 6, 8, 9, 10 …..

Co-primes or Relatively prime numbers: A pair of numbers not having any common factors other than 1 or –1. (Or alternatively their greatest common factor is 1 or –1)
Ex : 15 and 28 are co-prime, because the factors of 15 (1,3,5,15) , and the factors of 28 (1,2,4,7,14,28) are not in common (except for 1) .

Twin Primes: A pair of prime numbers is said to be twin primes if they differ by 2.
Ex : (3,5) , (5,7) , (11,13) , …

Algebra

(a+b)² = a² + 2ab + b²
(a-b)² = a² - 2ab + b²

(a+b)² - (a-b)² = 4ab
(a+b)² + (a-b)² = 2 (a² + b²)

a²-b² = (a-b)(a+b)

(a+b)³ = a³+b³+3ab(a+b) = a³+b³+3a²b+3ab²
(a-b)³ = a³-b³-3ab(a-b) = a³-b³-3a²b+3ab²

a³+b³ =(a+b)³-3ab(a+b) = (a+b)(a²-ab+b²)
a³-b³ =(a-b)³+3ab(a-b) = (a-b)(a²+ab+b²)

Test of Divisibility

Divisibility By 2 : A number is divisible by 2, if its unit's digit is zero or even (2, 4, 6, 8..).

Divisibility By 3 : A number is divisible by 3, if the sum of its digits is divisible by 3.

Divisibility By 4 : A number is divisible by 4, if the number formed by the last two digits is divisible by 4.

Divisibility By 5 : A number is divisible by 5, if its unit's digit is either 0 or 5.

Divisibility By 6 : A number is divisible by 6, if it is divisible by both 2 and 3.

Divisibility By 8 : A number is divisible by 8, if the number formed by the last three digits of the given number is divisible by 8.

Divisibility By 9 : A number is divisible by 9, if the sum of its digits is divisible
by 9.

Divisibility By 10 : A number is divisible by 10, if it ends with 0.

Divisibility By 11 : A number is divisible by 11, if the difference of the sum of its digits at odd places and the sum of its digits at even places, is either 0 or a number divisible by 11.

Divisibility By 12: A number is divisible By 12 if the number is divisible By both 4 and 3.

Divisibility By 13: A number is divisible By 13 if its unit’s place digit is multiplied By 4 and added to the remaining digits and the number obtained is divisible By 13.

Divisibility By 14: A number is divisible By 14 if the number is divisible By both 2 and 7.

Divisibility By 15: A number is divisible By 15 if the number is divisible By both 3 and 5.

Divisibility By 16: A number is divisible By 16 if its last 4 digits is divisible By 16 or if the last four digits are zeros.

Divisibility By 17: A number is divisible By 17 if its unit’s place digit is multiplied By 5 and subtracted from the remaining digits and the number obtained is divisible By 17.

Divisibility By 18: A number is divisible By 18 if the number is divisible By both 2 and 9.

Divisibility By 19: A number is divisible By 19 if its unit’s place digit is multiplied By 2 and added to the remaining digits and the number obtained is divisible By 19.

Divisibility By 20: A number is divisible by 20 if it is divisible by 10 and the tens digit is even.