Quantitative Aptitude Important Formulas

X percent means X hundredths, written as X% => X/100

If X % of Y = X * Y/100

If the price of the commodity decreases by R%,then the increase in consumption
so that the expenditure remains same is [(R/(100-R)*100]%.

If the price of a commodity increases by R%, then the reduction in consumption
so that the expenditure remains same is [R/(100+R))*100]%.

If the present population of the town is P now and it increases at the rate of R% per annum, then:
Population after n yeras = P [1+(R/100)]ⁿ.
Population n years ago = P /[1+(R/100)]ⁿ.

If the present population of the town is P now and it decreases at the rate of R% per annum, then:
Population after n yeras = P [1-(R/100)]ⁿ.
Population n years ago = P /[1-(R/100)]ⁿ.

If the present value of a machine is P now and it depreciates at the rate R% per annum, then:
Value of the machine after n years = P[1-(R/100)]ⁿ.
Value of the machine n years ago = P/[1-(R/100)]ⁿ.

If a number A is increased successively by x% followed by y% and then by z%,
then the final value of A will be =A[1+(x/100)][1+(y/100)][1+(z/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 value of a number is first increase by x% and later decrease by x% , then the net effect is always a decrease (x/10)² %.

If the value of a number is first changed ( increased +ve or decreased -ve) by x% and then changed (increased or decreased ) by y% ,then net change % = { ± x ± y + [(±x±y)/100]} %

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.