Average or Arithmetic mean

Average, (Arithmetic Mean)   Definition:  The sum of all numbers in a group divided by the count of numbers in that group.   Example: Mark took 3 exams during the semester getting the scores 76, 84, and 92. The final grade is based on the average of his test scores. What is Mark's final grade? average = 84 Mark's final grade would be an 84.

Inequality/equality signs

The following are inequality/equality signs you should be familiar with: < less than x < y x is less than y > greater than x > y x is greater than y ≤ less than or equal to x ≤y x is less than or equal to y ≥ greater than or equal to x ≥y x is greater than or equal to y

Permutations and Combination quesitons

Question 1: In how many ways can the letters of the word ABACUS be rearranged such that the vowels always appear together?

Question 2: How many different four letter words can be formed (the words need not be meaningful) using the letters of the word MEDITERRANEAN such that the first letter is E and the last letter is R?

Question 3:What is the probability that the position in which the consonants appear remain unchanged when the letters of the word "Math" are re-arranged?

Question 4: There are 6 boxes numbered 1, 2, ... 6. Each box is to be filled up either with a red or a green ball in such a way that at least 1 box contains a green ball and the boxes containing green balls are consecutively numbered. The total number of ways in which this can be done is:

Question 5: A man can hit a target once in 4 shots. If he fires 4 shots in succession, what is the probability that he will hit his target?

Question 6: In how many ways can 5 letters be posted in 3 post boxes, if any number of letters can be posted in all of the three post boxes?

Question 7: Ten coins are tossed simultaneously. In how many of the outcomes will the third coin turn up a head?

Question 8: In how many ways can the letters of the word "PROBLEM" be rearranged to make seven letter words such that none of the letters repeat?

Combinations

Combinations

Combinations are of two types:

1 In which repetition is allowed
2. In which repetion is not allowed

Combinations with repetition

(n+r-1)!/r!(n-1)!

Where n is the number of things to choose from and you choose r of them

Combinations without repetition

A simple example of this would be lottery numbers.

If we assume that the order does matter, i.e., permutations, alter it so the order does not matter.

Formula: n!/r!(n-r)!

Where n is the number of things to choose from, and you choose r of them

It is also known as "n choose r" or binomial coefficient.

Permutations

Permutations

Permutation is the technique of permuting or rearranging objects or values in a particular order.

In permutations the order matters, however, in combination the order does not matter.

Permutations are of two types:
1. Where repetition is allowed
2. Where no repetition is allowed

How to calculate permutation with repetition?

If you have n things and n choices each time. And you  have to choose r of them:

The Permutations are as follows: n x n x ........(r times) or n^r

How to calculate permutation without repetition?

In this you have to reduce the number of available choices each time.
Formula: n!/(n-r)!

Where n is the number of things to choose from, and you choose r of them.

Arithmetic and Geometric Progressions

Arithmetic Progressions

These are series of numbers e.g. 2, 6, 10, 14, 18 ... or 100, 97.5, 95, 92.5, 90, . . . . where a constant number is added to or subtracted from each term to form the next.
Assuming that the first term is a and that the number added/subtracted (the "common difference") is d, then the series is: a, a + d, a + 2d, a + 3d .... The nth term of the series is a + (n - 1)d.
The sum of the first n terms of the series is: 0.5n(2a + [n - 1]d).

Geometric Progressions

These are series of numbers, e.g. 2, 6, 18, 54, 162, ... or 100, -50, 25, -12.5, 6.25 ... where each term is multiplied/divided by a constant number (the "common ratio") to form the next. The number can be positive (in which case all the terms are the same sign) or negative (in which case alternate terms have alternate signs). The first term is usually called a. The common ratio is usually called r.
If r is outside the range -1 to 1, the terms of the series get bigger and bigger (even if they change sign), and the series diverges. If r is within the range -1 to 1, the terms get smaller and smaller (closer to 0) and the series converges.

The sum of the first n terms of a geometric progression is  a(1 - r n)/1 - r 

If r lies within the range -1 < r < 1, then the series has a sum to infinity (i.e. if you added up an infinite number of terms of the series, you would still get a finite number). An example is the series 1, 1/2, 1/4, 1/8, 1/16 ... etc. where a = 1 and r = 1/2. In this case, if you add the terms of the series, you get closer and closer to 2 without ever reaching it. If you could add up an infinite number of terms, then the sum would be 2.
To find the sum to infinity of the series, simply put n = ¥ in the equation above. Providing -1 < r < 1, the term r¥ will be 0, and the whole formula reduces to this:
a/1 - r

Number Series

How to solve number series questions?

There are different ways and relations to solve the number series questions, some of them are as follows:
1. Difference
2. Product
3. Squares/Cubes
4. Prime/Composite
5. Combination
6. Counting
7. Miscellaneous

What is Mode?

Mode is the number that appears most often in a list.

For example: In a series: 2, 4, 5, 6, 8, 5, 6, 7, 9, 2, 1, 5

The mode is 5 as it appears three times.

GMAT Practice Questions

1-     The smallest number which, when divided by 4,6,or 7 leaves a remainder of 2, is:

 A: 44

 B: 62

 C: 80

 D: 86

2-    If ROPE is coded as 6821 & CHAIR is coded as 73456 then what will be the code for CRAPE?

 A: 73456

 B: 76421

 C: 77246

 D: 77123

3-    What is the smallest number which when increased by 5 is completely 

divisible by 8,11, and 24?

 A: 264

 B: 259

 C: 269

 D: None of these         

GMAT Practice Questions

1. Find the wrong number in the following sequence: 22, 33, 66, 99, 121, 279, 594

 (a) 279
 (b) 594
 (c) 121
 (d) 33
 (e) None of these.

2. A father is now three times as old as his son. Five years back he was four times as old as his son. What is the age of the son now?

 (a) 12
 (b) 15
 (c) 18
 (d) 20

Composite number

A composite number is a positive integer which has a positive divisor other than one or itself. In other words, if n > 0 is an integer and there are integers 1 < a, b < n such that n = a × b, then n is composite. By definition, every integer greater than one is either a prime number or a composite number. The number one is a unit – it is neither prime nor composite. For example, the integer 14 is a composite number because it can be factored as 2 × 7. Likewise, the integers 2 and 3 are not composite numbers because each of them can only be divided by one and itself.


Every composite number can be written as the product of two or more (not necessarily distinct) primes; furthermore, this representation is unique up to the order of the factors. 

Prime numbers

Prime numbers are positive, non-zero numbers that have exactly two factors.

A prime number (or a prime) is a natural number that has exactly two distinct natural number divisors: 1 and itself. The smallest twenty-five prime numbers (all the prime numbers under 100) are:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

Fractions

To understand fractions you should be familiar with the terms part and whole. The whole is everything and the part is just a piece of it. Fractions are used to figure out just how much of the whole the part represents.

Part = Fraction x Whole

Ex: A whole pizza is ordered for a family of 4 and cut into 8 slices. If everyone agrees to eat the same amount, how much of the pizza will each person eat?

Part = Fraction x Whole

Part = 1/4 x 8

Part = 2 slices

Ratio:

When you discuss the ratio between different things you are relating the quantity of one of those things to the other. Ratios are usually represented as fractions or with colons.

What is the ratio of books to magazines if there are 4 times as many books as there are magazines?

As a fraction:

Using a colon:

4 : 1 or 4x : 1x

The ratio is 4 to 1, so for every 4 books there is 1 magazine. If there are 100 books, (or 25 x 4 books), then there are 25 magazines, (25 x 1 magazines).

There will always be 4 times as many books as there are magazines.