Quantitative Aptitude Important Formulas

Fractions - Important Formulas,Tricks and Examples

Fractions: A fraction is an expression that indicates the quotient of two quantities.

Type of Fractions
Common Fraction:A common fraction also called Vulgar fraction is a fraction in which both numerator and denominator are integers.

Decimal Fraction:A decimal fraction is a fraction in which denominator is an integer power of ten. The term decimals are commonly used to refer decimal fractions.

Conversion of a Decimal into Common Fraction
Put 1 in the denominator under the decimal point and annex with it as many zeros as is the number of digits after the decimal point. Now, remove the decimal point and reduce the fraction to its lowest terms.

Basic Arithmetic operations - Decimal Fractions
Addition and Subtraction of Decimal Fractions
To add/subtract decimals,
(i) Write down the numbers, one under the other, with the decimal points lie in one column.
(ii) Now the numbers can be added normally (remember to put the decimal point in the answer).

Multiplication of a Decimal Fraction by a Power of 10
Shift the decimal point to the right by as many places as the power of 10.

Multiplication of Decimal Fractions
To multiply decimals,
(i) Multiply the given numbers considering them without decimal point.
(ii) In the product obtained, place the decimal point by starting at the right and moving a number of places equal to the sum of the decimal places in the given numbers.


Division of Decimal Fraction by a Counting Number
To divide a decimal by a counting number,
(i) Divide the given numbers considering them without decimal point.
(ii) In the quotient obtained, place the decimal point by starting at the right and moving a number of places equal to the sum of the decimal places in the given numbers.

Division of Decimal Fraction/Counting Number by a Decimal Fraction
To divide a decimal/counting number by a decimal,
(i) Multiply both the dividend and the divisor by a suitable power of 10 to make divisor a whole number.
(ii) Now divide the numbers normally.

Recurring Decimal
If a figure or a set of figures is repeated continuously in a decimal fraction, it is called a recurring decimal (also known as repeating decimal and circulating decimal).

A repeating decimal is denoted by placing a horizontal line above the repeated numerals. Another way to denote a recurring decimal is placing dots over the first and last digits.

Type of Recurring Decimals
Recurring Decimals can be classified into two categories - Pure Recurring Decimals and Mixed Recurring Decimals

Pure Recurring Decimals: Pure Recurring Decimal is a decimal fraction in which all the figures after the decimal point are repeated.

Mixed Recurring Decimals: Mixed Recurring Decimal is a decimal fraction in which some figures are not repeated and some figures are repeated.


Conversion of Recurring Decimals into Vulgar Fractions

To convert a pure recurring decimal into a vulgar fraction,
(1) Remove the number left to the decimal point, if any.
(2) Write the repeated figures only once in the numerator without the decimal point.
(3) Write as many nines in the denominator as the number of repeating figures.
(4) Add the number removed in step 1(if any) with the fraction obtained in the above steps.

To convert a mixed recurring decimal into a vulgar fraction,
(1) Remove the number left to the decimal point, if any.
(2) Numerator is the difference between the number formed by all the digits (taking repeated digits only once) and that formed by the digits which are not repeated.
(3) Denominator is the number formed by taking as many nines as the number of repeating figures followed by as many zeros as the number of non-repeating digits.
(4) Add the number removed in step 1(if any) with the fraction obtained in the above steps.


Comparison of Fractions

Fractions with same denominators.
if fractions have same denominator. So just compare the numerators. Bigger the numerator, bigger the number.

Fractions with same numerators.
if fractions have same numerator. So just compare the denominators. Bigger the denominator, smaller the number.

Fractions with different numerators and denominators.
To compare such fractions, find out LCM of the denominators.then convert each of the given fractions into an equivalent fraction with LCM as the denominator.And compare the numerators. Bigger the numerator, bigger the number.

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Numbers - Important Formulas,Tricks and Examples

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.

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Pipes and Cistern

Two pipes A and B can fill a tank in 6 hours and 4 hours respectively. If they are opened on alternate hours and if pipe A s opened first, in how many hours, the tank shall be full?

A large tanker can be filled by two pipes A and B in 60 min and 40 min respectively. How many minutes will it take to fill the tanker from empty state if B is used for first half of the time, and A and B fill it together for the other half?

A tank is filled by 3 pipes with uniform flow. The first two pipes operating simultaneously fill the tank in the same time during which the tank is filled by the third pipe alone. The 2nd pipe fills the tank 5 hours faster than first pipe and 4 hours slower than third pipe. The time required by first pipe is?

Two pipes A and B together can fill a cistern in 4 hours. Had they been opened separately, then B would have taken 6 hours more than A to fill the cistern. How much time will be taken by A to fill the cistern separately?

Two pipes A and B can fill a tank in 24 min and 32 min respectively. If both the pipes are opened simultaneously, after how much time B should be closed so that the tank is full in 18 min?

Two pipes A and B can fill a tank in 36 min and 45 min respectively. A water pipe C can empty the tank in 30 min. First A and B are opened. after 7 min, C is also opened. In how much time, the tank is full?

Two pipes can fill a cistern in 14 hours and 16 hours respectively. The pipes are opened simultaneously and it is found that due to leakage in the bottom it took 32 min more to fill the cistern. When the cistern is full, in what time will the leak empty it?

Bucket P has thrice the capacity as Bucket Q. It takes 60 turns for Bucket P to fill the empty drum. How many turns it will take for both the buckets P and Q, having each turn together to fill the empty drum?

Probability

In a class , 30 % of the students offered English, 20 % offered Hindi and 10 % offered Both.If a student is offered at random, what is the probability that he has offered English or Hindi?

In a lottery ,there are 10 prizes and 25 blanks.A lottery is drawn at random. what is the probability of getting a prize ?

Two dice are thrown simultaneously .what is the probability of getting two numbers whose product is even?

Two diced are tossed the probability that the total score is a prime number?

Two cards are drawn at random from a pack of 52 cards What is the probability that either both are black or both are queens?

Two dice are thrown together .What is the probability that the sum of the number on the two faces is divisible by 4 or 6?

A bag contains 6 white and 4 black balls .Two balls are drawn at random .Find the probability that they are of the same colour?

In a simultaneous throw of a pair of dice,find the probability of getting a total more than 7?

An unbiased die is tossed.Find the probability of getting a multiple of 3?

Problems on Ages

My brother is 3 years elder to me. My father was 28 years of age when my sister was born while my mother was 26 years of age when i was born. If my sister was 4 years of age when my brother was born,then what was the age my father and mother respectively when my brother was born?

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