## Number Series, Logic, & Analogy

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## Number Series Introduction

Let us start with these simple series of numbers to get your intellectual juices flowing. Generally, you would not see a direction question in CAT, XLRI, Jamnalal Bajaj or any of the other competitive exams from this section. However, the knowledge and ‘mastery’ of this section is needed to help you crack the actual questions. If you have gone through the quantitative section, this may come across as repetitive and too simple. Jump to next section if that is the case.

Number series questions measure your ability to reason without words. To answer these questions, you must determine the pattern of the numbers in each series before you will be able to choose which number comes next. In each series, look for the degree and direction of change between the numbers. Sometimes the pattern may contain two alternating series. Let us look at some examples

Q1) Look at the series –  17 17 34 20 20 31 23…….what numbers should come next ?

1. 26  23
2. 34  20
3. 23  33
4. 27  28
5. 23  28

A1) Correct answer is (5), 23 28

This is an alternating subtraction series with repetition. There are two different patterns here. In the first, a number repeats itself (17 17), then 3 is added that number to arrive at the next number that also repeats itself (20 20), so the next numbers in this series are 23, 23. The second series starts from 34 and 3 is subtracted from each number to arrive at the next : 34, 31, 28……so the numbers that come next are 23 (from 1st series) and 28 (from 2nd series)

Q2) Look at this series 8  11 21 15 18 21 22……..

What are the next two numbers

1. 25 18
2. 25 21
3. 25 29
4. 24 21
5. 22 26

A2) Correct answer is (2)  25 21

There are 3 series here.

First starts with 8, then 7 is added to get the next number, so the series is 8, 15, 22……

Second starts with 11, then 7 is added to get the next number, so the series is 11, 18, 25….

The third starts with 21 which remains constant and doesn’t change

So the next set of numbers are 25 (from series 2) and 21 (from series 3)

Q3) 4 7 26 10 13 20 16………

What are the next two numbers ?

1. 14 4
2. 14 17
3. 18 14
4. 19 13
5. 19 14

A3) Correct answer is (5) 19 14

Two patterns alternate here, with every 3rd number following the alternate pattern. In the main series, beginning with 4, 3 is added to each number to arrive at the next. In the alternating series, beginning with 26, 6 is subtracted from each number to arrive at the next.

Q4) Look at the series 8, 43, 11, 41, ____, 39, 17, ….

Which number shall fill the blank ?

1. 8
2. 14
3. 43
4. 44

A4) The correct answer is (2), 14. This is a simple alternating addition and subtraction series. The 1st series begins with 8 and adds 3; the second begins with 43 and subtracts 2

Q5 ) U32, V29, ___, X23, Y20….

1. W26
2. W23
3. Z17
4. Z26

A5) The correct answer is (2), W23.

In this series, the letters progress by one, the numbers decrease by 3

You will see more questions like this in the Practice Tests. Try to get those solved correct and fast since these are some of the easiest questions you will run across

## How to Solve Number Series?

See some general solving tips for number series test.
1. Examine the difference between adjacent numbers.

→ In a simple series, the difference between two consecutive numbers is constant.

Example: 27, 24, 21, 18, __

Rule: There is a difference of (-3) between each item. The missing number in this case is 15.

→ In a more complex series the differences between numbers may be dynamic rather than fixed, but there still is a clear logical rule.

Example: 3, 4, 6, 9, 13, 18, __

Rule: Add 1 to the difference between two adjacent items. After the first
number add 1, after the second number add 2 and after the third number
add 3, etc. In this case, the missing number is 24.

2.See whether there is a multiplication or division pattern between two adjacent numbers.

Example: 64, 32, 16, 8, __

Rule: Divide each number by 2 to get the next number in the series.
The missing number is 4.

3.Check whether adjacent numbers in the series change based on a logical pattern.

Example: 2, 4, 12, 48, __

Rule: Multiply the first number by 2, the second number by 3 and the third number by 4, etc. The missing item is 240.

4.See if you can find a rule that involves using two or more basic arithmetic functions (+, -, ÷, x). In the below series, the functions alternate in an orderly fashion.

Example: 5, 7, 14, 16, 32, 34, __

Rule: Add 2, multiply by 2, add 2, multiply by 2, etc. The missing item is 68.

Tip: Series’ in this category are easy to identify. Just look at the numbers that do not appear to have a set pattern.

Important:
In a series that involves two or more basic arithmetic functions, the differences between adjacent items effectively create their own series. We recommend that you try to identify each pattern separately.

Example: 4, 6, 2, 8, 3, __

Rule: In this series, the differences themselves create a series: +2, ÷3, x4, -5
The numbers advance by intervals of 1 and the arithmetic functions change in an orderly sequence. The next arithmetic function in the series should be +6, and so the next item in the series is 9 (3+6 = 9).

## Numerical Logic

In numerical reasoning questions information is provided that requires you to interpret it and then apply the appropriate logic to answer the questions. In other words, you need to work out how to get the answer rather than what calculations to apply.

Sometimes the questions are designed to approximate the type of reasoning required in the workplace. The questions will often use very specific illustrations, for example the question may present financial data or use information technology jargon. However, an understanding of these areas is not required to answer the question.

Q1) If May 19, 2007 falls on Saturday the last day of that month would fall on:

1. Monday
2. Thursday
3. Friday
4. Saturday

A1) Correct answer is (2).

May has 31 days so last day of the month is 12 days after May 19th, a Saturday. May 26th is a Saturday and adding 5 days after that takes you to may 31st being a Thursday.

2) A shopkeeper had 3 weights totaling 13 kgs. He wants to be able to weigh all things that weight between 1 kg to 13 kg. Which three weights should he keep with him ?

1. 2, 5, 6
2. 1, 3, 9
3. 1, 5, 7
4. 1, 4, 8

A2) Correct answer is (2)

The three weight should be such that in order to measure all items between 1 – 13 kg (a) a single weight should suffice  (b) sum of two or three weights should suffice (c) the difference between the two weights should suffice, or (d) difference between sum of two weights and the third should suffice. The only one that meets all these conditions are weights 1, 3 and 9.

1 kg = 1

2 kg = 3 -1

3 kg = 3

4 kg = 1 + 3

5 kg = 9 – 1 – 3

6 kg = 9 – 3

7 kg = 9 +1 – 3

8 kg= 9 – 1

9 kg = 9

10 kg = 9 + 1

11 kg = 9 + 3 – 1

12 kg = 9 + 3

13 kg = 9 + 3 + 1

For questions 3 – 5, read the following information carefully and answer the questions that follow.

In a game between two people A and B, a heap of pebbles is kept on the floor, A person can pick 1, 2 or 3 pebbles. The one who picks the last pebble loses.

Q3).  If B is to begin the game, and A to win, what is the minimum number of pebbles greater than 5 that must have been lying on the floor?

1. 7
2. 9
3. 13
4. 15

Q4)  If A were to begin the game and win it, what is the number of pebbles on the floor that is impossible for the game to have begun with assuming that both players are playing intelligently?

1. 13
2. 14
3. 15
4. 16

Q5) If the number of pebbles is 17 in how many steps does the game end?

1.  6
2. 7
3. 8
4. 9

A3) Correct answer is (2).

The minimum number of pebbles cannot be 7 as B will start with 2 pebbles and then even if A picks one pebble, B will be in a position to leave just one pebble and pick the remainder which will be a maximum of 3. Thus the minimum number of pebbles has to be at least 9 so that A will have a chance of manipulating the remaining number of pebbles. If B picks 1 pebble, A will pick 3 pebbles which will mean that there are 5 pebbles left. Even if B picks 1 pebble. A will be in a position to ensure that B picks that last pebble. Thus, the minimum number of pebbles is 9.

A4) Correct answer is (1).

The number of pebbles cannot be 13. Basically, if both play intelligently, B has to ensure that he will pick the 12th pebble. For this he will also ensure that he picks the 4th and 8th pebble since he will be in a position to. manipulate later on. If A picks 3, B will pick 1 and so on. After this, if A picks 1, 2 or 3 B will pick 3, 2 and 1 respectively so that he will pick the 8th pebble. The above step will be repeated and A will be forced to pick the last pebble. Thus he cannot win with 13 pebbles.

A5) Correct answer is 3.

If the number of pebbles is 17 the one playing second will have to ensure that he picks the 4th, 8th. 12th and the 16th pebbles. For each cycle there will be 2 steps and hence a total of 8 steps before the last pebble is picked.

6) A, B, C, D & E are five different integers. When written in the ascending order of values, the difference between any two adjacent integers is 3. D is the greatest and A is the least. B is greater than E but less than C. The sum of the five integers is equal to E- 1. A + B is:

1. 15
2. -8
3. -9
4. 16

A6) Correct answer is (b)

We know that D>C>B>E>A and D+C+B+E+A=E-1. We also know that the numbers are E-3, E, E+3, E+6, E+9 or E = -4.

B = -1, A = -7, A+B = -8

## Numerical Analogy

Analogy means ‘correspondence’. In question based on analogy a particular relationship is given and another  similar relationship has to be identified from the alternatives provided. Analogy test are , therefore , meant  to test a candidates overall knowledge , power of reasoning and ability to think concisely and accurately.

### Solved Examples

Q1-6 : 18 : : 4 : ?

1. 2
2. 6
3. 8
4. 16

Justification:

The relationship is x : x^2/2.

Answer:      C . 8

Q2-21 : 3 : : 574 : ?

1. 23
2. 82
3. 97
4. 113

Justification:       The relationship is 7x : x.

Answer:      B . 82

Q3-1 : 1 : : 25 : ?

1. 26
2. 125
3. 240
4. 625

Justification:       The relationship is x : x^2.

Answer:      D . 625

Q4-121 : 12 : : 25 : ?

1. 1
2. 2
3. 6
4. 7

Justification:       The relationship is x^2 : (x + 1)

Answer:      C . 6

Q5-. 25: 37: : 49: ?

1.   41
2.   56
3.   60
4.   65

Answer : 4.   65

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