Analogies and OMO

It is important to clarify one’s concept on Series at the following link before heading into these topics: 

 

https://analyticalandlogicalreasoningconcept.blogspot.com/2017/02/series.html

 

Keeping the approaches discussed in the above link in mind, we shall proceed with these topics without repeating the points already mentioned in Series (above link).

Odd Man Out

It is important to understand that identifying odd one out is to spot the aberration in an ongoing pattern rather than finding an exceptional quality in a particular term.

#1. 12, 16, 20, 24, 26 – Here the odd term shall not be 16 as it is the only square, but it will be 26 as it is not following the pattern of difference of 4 from the previous term.

Property > Order

This is one of the most important principles to remember while attempting questions from Series and related topics. A pattern pertaining to the property of numbers or letters will be given more priority than mere application of operators like +/-/×/÷. (Eg #2 and #3)

#2. 2, 3, 5, 9, 17, 19 – Here the difference between the terms are doubling each time from the previous one, based on which 19 can be termed to be the odd one. Also apart from 9 all other numbers are prime, which is based on property of the numbers. However the strongest case is made with the most intrinsic properties which in this case will be of odd numbers; and 2 being the only non-odd number is the answer.

(It is important to note that while explaining this example, “non-odd” term was used instead of “even”. This is because 2 is not the answer as it is the only even number but because it is the only one which is not odd among all that are odd.)

#3. A, E, I, M, O – Here the letters are at an interval of 4 – A is 1^st, E is 5^th, I is 9^th, M is 13^th and O is 15^th. Going by this pattern O seems to be the odd one out; but looking at the property of letters – all given letters are vowels except M, which makes it the odd one out.

Analogies

While the basic principles remain same as with Series, in case of verbal analogies it is advisable to frame a sentence in mind with the two words given.

#4. Tennis : Sport is similar to  - a) Piano : Music b) Game : Chess c) Dog : Animal d) Fish : Water  ➙ Here although (b) seems similar to the given analogy, it is not correct as it’s completely reverse of the given logic. We can form a sentence like 'Tennis is a Sport' and then try to fill the first word in the first blank and second one in the second blank from the options, which will make 'Dog is an Animal' the correct answer. Hence ans is (c).

Series

Series is one of the most traditional topics of Reasoning and also usually the most familiar one even for a first timer at competitive and aptitude tests. While some of the popular management entrance tests have curtailed the number of questions from this chapter in recent times, it will not be advisable to ignore this topic, for it could mean missing out on some crucial score which could affect one’s overall percentile and prove costly especially if it’s anywhere near to the cut off.

There is not much to offer in this topic conceptually; however it is important to have an approach in mind while taking on its questions. Questions from Series are mostly based on one or more of the following patterns:

i) Addition/Subtraction – Here numbers are added or subtracted in a pattern. Usually terms are relatively close to each other in such patterns; however there are exceptions (eg #3)

#1. 3, 4, 7, 12, ? – Here the series begins with 3 and increasing order of odd numbers are being added. After adding 1, 3, 5, it is now turn for adding 7 which will give the answer 19.

#2. 31, 38, 33, 40, 35, ? – Here the numbers 7 and 5 are added and subtracted alternately. After subtracting 5 from 40 and making it 35, it is now turn to add 7 and the answer shall be 42.

#3. 1, 5, 32, 288, ? – Beginning with 1_­­­­¹ numbers are added in the form of nⁿ by adding 4 i.e. 2², 27 i.e. 3³, 256 i.e. 4⁴. Next number to add should be 3125 i.e. 5⁵ which will make the next term (288 + 3125) = 3413.

ii) Multiplication/Division - Where certain numbers are multiplied or divided in a pattern. Mostly this pattern has terms far apart from each other to allow the scope of multiplication/division; however there are exceptions (eg #5)

#4. 362880, 45360, 7560, 1890, ? – Here the numbers are being divided by decreasing consecutive even numbers starting with 8 followed by 6 and 4. Next divisor should be 2 which will result in 945.

#5. 8, 4, 4, 6, 12,? – Here the first term has been multiplied with 0.5 and then subsequent terms have been multiplied with numbers at an interval of 0.5 i.e. 1, 1.5 and 2. Next term should hence be multiplied with 2.5 giving a figure of 30.

iii) Prime Numbers – These are patterns pertaining to prime numbers and may also be combined with above mentioned patterns (like addition in #7)

#6. 97, 79, 67, 53, ? – Here prime numbers have been listed in decreasing order skipping two prime numbers in between. So after 97, 89 & 83 have been skipped and similarly 73, 71, 61, 59 have been skipped till 53. Similarly after 53, 47 & 43 should be skipped to result in 41 as the next term.

#7. 1, 3, 6, 11, 18, ? – Here 2, 3, 5 and 7 have been added on the 4 terms respectively. As these are consecutive prime numbers, the next number to add should be 11 which is the next prime number after 7 and shall result in 29 as the next term

iv) Square/Cube Numbers – These are any patterns related with square and cube numbers.

#8. 1, 2, 6, 15, 31, ? – Here consecutive square numbers are being added to each term. So after adding 4² i.e. 16 next term should be added with 5² i.e. 25 which will give us 56.

#9. 9, 28, 65, 126, 217, ? – Starting with 2³+1, consecutive cube numbers have been listed after adding 1 to each. Next term should be 7³+1 = 344.

v) Sum of Digits/Difference of Digits – When next term is derived by adding the digits of previous term or finding the difference between the digits of previous term.

#10. 24, 30, 33, 39, 51, ? – Here the numbers added on the terms are a sum of the digits of the previous term: 24 + (2+4), 30 + (3+0), 33 + (3+3) and so on. Hence next term should be 51 + (5+1) = 57.

#11. 29, 36, 39, 45, 46, ? – Here the numbers added on the terms are a difference of the digits of the previous term: 29 + (9-2), 36 + (6-3), 39 + (9-3), 45 + (5-4). Hence the next term should be 46 + (6-4) = 48.

#12. 3, 7, 10, 17, 27, 44, ? – Here third term onwards the series is the sum of the previous two terms; so the next term should be (27 + 44) = 71.

vi) Mix Series – When along with addition/subtraction, division/multiplication is mixed together it leads to another pattern.

#13. 2, 7, 17, 37, 77, ? – Here the terms are being multiplied with 2 and then 3 is added to them. (2 x 2) + 3 = 7, (7 x 2) + 3 = 17, (17 x 2) + 3, and so on. Hence the next term should be (77 x 2) + 3 = 157.

vii) Alternate Series – Alternate series is when more than one series is mixed together.

#14. 1, 2, 3, 3, 5, 5, 7, 7, 9, 11, 11, 13, ? – Here the odd and even positions are two different series running alternately. The odd positions are a series of consecutive odd numbers, while the even positions are a series of prime numbers. Hence the next term will be the next prime number after 11 i.e. 13.

#15. 3, 10, 16, 9, 20, 8, 27, 30, 4, 81, 40, ? – Here 3 different series have been mixed together such that 1^st, 4^th, 7^th… terms are being multiplied with 3 (3, 9, 27, 81); the 2^nd, 5^th, 8^th …terms are being added with 10 (10, 20, 30, 40); and every third term is being divided by 2 (16, 8, 4). Hence the next term i.e. 12^th term should be 4 divided by 2 i.e. 2.

Letter Series

The following points need to be kept in mind while solving letter series:

i) Vowels/Consonants: Unlike numbers, alphabet letters do not have too many properties. The most intrinsic property is the very nature of the letter i.e. vowel or consonant. It can be identified immediately if the series is based on this pattern.

#16. B, F, J, P, ? – Here the consonants following the vowels have been arranged in ascending order of positions. Hence after P, the next vowel is U which will be followed by V and that will be our answer.

ii) Convert in numbers: When a series is given in letter form and there is no apparent pattern with vowel/consonants, then it is advisable to convert the letters into their position number in the alphabet and treat it as number series and apply all that has been mentioned above.

iii) Opposite letters: Any pair of letters whose position can be interchanged when counted from A-Z and reverse Z-A are called Opposite letters.

Eg. A is first letter from beginning and 26^th from the end; so its opposite shall be Z which is 26^th from the beginning and first from the end. Similarly J is 10^th from the beginning and 17^th from the end; so its opposite shall be Q which is 17^th if counted from A-Z and 10^th when considered from Z-A.

An easy way to identify such opposite pairs is to find the sum of their positions from A onwards and the total shall always be 27.

#17. EV, JQ, OL, TG, ?? – Here all are pairs of opposite letters where the first letter in each pair is at an interval of 5 (based on position), 10, 15, 20. As the interval between first letter of consecutive pairs is 5, first letter of the next pair shall be (20 + 5) = 25 i.e. Y and its opposite letter shall be (27 – 25) = 2 i.e. B

iv) Corresponding letters: Any pair of letters wherein the interval is consistent between them, both when counted from front or backwards are called Corresponding letters.

Eg. When counting from A to N, there is an interval of 13 and if we continue forward and come back to A after Z, then again the interval will be of 13. Similarly we can take a pair like F-S which when counted from F to S or from S to F gives us the same result.

A simple way to identify such pairs will be to verify an interval of 13 between the two letters.

#18. AN, DQ, IV, PC, ?? – Each pair here are at an interval of 13 and hence are corresponding letters. Also the first letters in every pair are at positions which are consecutive perfect squares A (1 i.e 1²), D (4 i.e. 2²), I (9 i.e. 3²), P (16 i.e. 4²). Hence, the first letter of the next pair shall be position 25 i.e. 5² which will be Y and it will be paired with (25 – 13) = 12 i.e. L which means the next pair shall be YL.

The chapter of Series can be mastered by practice alone, as one gets familiar with more and more patterns. However it is important to have an approach in mind so that one doesn’t forget to try any of the patterns.

The chapter extends to influence some other popular topics like Odd Man Out and Analogies which have been discussed on another post of this blog:

https://analyticalandlogicalreasoningconcept.blogspot.com/2017/02/analogies-and-omo.html

 

Coding & Decoding and Input-Output also have certain questions based on the concepts of Series. Following is an example –
 

#19. In a certain code language if,

‘Put on your thinking cap’ is written as #N2, ?P3, %G8, @T3, *R4

‘Life is not easy’ is written as {Y4, !E4, ]S2, ^T3

‘It cannot be helped’ is written as &E2, +D6, ?T6, ]T2

‘If you miss the obvious’ is written as ]F2, %E3, #S7, =S4, *U3

What will be the code for ‘Triumph’?

a) ^T6   b) ]M8 c) %H7 d) None of these

- Here the codes of words have been jumbled and can be identified as per following pattern:

i) The first letter has been assigned a unique symbol (eg. ‘I’ as ‘]’ can be checked in second, third and fourth statements)

ii) The letter in the code represents the last letter of the word (eg. the last letters of the words in first statement are T, N, R, G, P which are also in the codes)

iii) The number in the code represents the number of letters in the word (eg. the number of letters of words in second statement are 4, 2, 3, 4 which are also in the respective codes)

Hence, from the first and fourth sentence it can be understood that the first letter ‘T’ is denoted by the symbol ‘%’ and as Triumph is a seven letter word ending with H, its code will be %H7 i.e. option c)

Calendar

It is pertinent to define the term “odd days” before commencing the discussion on this chapter. “Odd days” as a concept is understood by all of us and we use it very frequently without trying to find a name for it.

The number of days in excess of sets of weeks in any number of days is called odd days.

Alternately, the remainder on dividing any number of days with 7 is said to be the number of odd days in that many days.

Eg: 10 days ➙ 1 week, 3 days or 10/7 gives a remainder of 3 ➙ no of odd days = 3

4 days ➙ 0 week, 4 days or 4/7 gives a remainder of 4 ➙ no of odd days = 4

30 days ➙ 4 weeks, 2 days or 30/7 gives a remainder of 2 ➙ no of odd days = 2

Basically odd days allows us to calculate the day on a particular date with reference to another date. If for instance, the present day is Tuesday, 100 days later will be Tuesday + 2 = Thursday (100 days is equivalent to 14 weeks and 2 days. Any number of exact weeks will give us the same day; hence we add the remainder after dividing with 7).

Also we should understand that odd days can range from 0-6 only (as our divisor is 7). Hence if it is said that an event will take place after 5 weeks and 42 days from the present day, which is a Monday; it implies that the event will occur on a Monday + remainder of 42/7 (i.e. 0) = Monday. (If number of odd days is in excess of 6, we divide it further by 7 to find the final remainder)

Leap Year

A year consisting of 366 days with 29 of them featuring in the month of February is known as a leap year. A normal year on the other hand has 365 days with 28 days in the month of February.

A gap of one year need not necessarily be 1^st Jan – 31^st Dec and can begin from any date of the year. If we compare two one year durations – 

i) 23 Feb’2015 – 22 Feb’2016

ii) 4 Mar’2015 – 3 Mar’2016

We begin by comparing the part of the year which defines a year to be normal/leap year which is Feb end –

In the first case we pass through Feb end of 2015 – as 2015 is not a leap year, this gap is of 365 days and equivalent to a normal year gap. In the second case we pass through Feb end of 2016 – as 2016 is a leap year, this one year gap is of 366 days and is equivalent to a leap year gap.

The distribution of odd days month wise and annually is as follows:

Month	Normal Year		Leap Year
	no of days	equivalent odd days	no of days	equivalent odd days
January	31	3	31	3
February	28	0	29	1
March	31	3	31	3
April	30	2	30	2
May	31	3	31	3
June	30	2	30	2
July	31	3	31	3
August	31	3	31	3
September	30	2	30	2
October	31	3	31	3
November	30	2	30	2
December	31	3	31	3
TOTAL	365	29 or 1	366	30 or 2

From the above we can conclude that any one year gap is a gap of 1 odd day or 2 odd days which can be verified by the number of days in the month of February of the year in which we are passing through February-end during that 1 year duration.

Pattern of Years

As we know that a day is one complete rotation of Earth while revolving around the Sun and as the number of rotations during one full revolution around the Sun is not a whole number, the need arises to accommodate for the approximations by having leap years.

The general perception of every 4^th year being a leap year is not entirely correct. While every 4^th year (i.e. every year which is a multiple of 4) is a leap year; the rule changes when referred to century years like 1800/1900/2000. Although multiples of 100 are also multiples of 4, in case of hundredth years, only multiples of 400 are leap years.

So while 1600 and 2000 are leap years, 1700, 1800 and 1900 are not leap years.

The leap year pattern is as follows –

Century Years (100,200,300,400….)	every multiple of 400 years is a leap year	eg. 400, 800,1200, 1600, 2000
Other Years (1,2,3,4,….)	every multiple of 4 years (except century years) is a leap year	eg. 4, 8, 12, 16….96, 104,108…..

Calculation of Day for any given date

While calculating the day of the week for any given date we find the number of odd days between the first day of 1AD till the given date. For this we need to know that the first day of the week is Monday and hence 1^st January 1AD was a Monday.

Putting together what has been mentioned above, we can find the day of the week for any given date. Illustrating the same by finding the day of the week for 15^th August 1947 –

Description	Period	Odd Days
Till 1947, 1600 years have gone by with 0 odd days. (As there are no odd days in 400 years, there will be no odd days in multiples of 400)	1600 years	0
After 1600, another 300 years have gone by. Every 100 years has 24 leap years and 76 normal years (as 100th year is not a leap year). Number of odd days in next 100 years: [(24 x 2) + (76 x 1)] = 124 which when further divided by 7 leaves a remainder of 5.	1601 - 1700 years	5
Same as above	1701 - 1800	5
Same as above	1801 - 1900	5
After 1900, 46 complete years have gone by with 11 leap years (every 4th year) and 35 (46 - 11) normal years with following number of odd days: [(11 x 2) + (35 x 1)] = 57 which when further divided by 7 leaves a remainder of 1.	1901 - 1946	1
Jan to Jul, no off odd days will be as discussed above	Jan 1947	3
	Feb 1947	0
	Mar 1947	3
	Apr 1947	2
	May 1947	3
	Jun 1947	2
	Jul 1947	3
Till 15th there is 1 odd day in August (15/7 leaves a remainder of 1)	Aug 1947	1
Total odd days	till 15 Aug 1947	33

 

33 days when further divided by 7 leaves a remainder of 5. Hence 15^th Aug 1947 was 5^th day of the week i.e. Friday (counting from 1 as Monday).