Tenses and Verb Tense Agreement What are tenses? Tenses are tools used to indicate when a situation or action takes place i.e. to show the position of the situation with respect to time. They are also used as tools to harass hapless CAT aspirants with extremely difficult questions. This COTD hopes to guide students through the concept of tenses and verb tense agreement. The situation being described in the sentence could have occurred “now” - present tense, “before now” - past tense or will happen “after now” - future tense. So far it's simple. But it gets a bit complicated with the introduction of aspects - simple, continuous (also called progressive), perfect and perfect continuous. The perfect aspect describes events occurring in the past but linked to a later point in time (either the present or the future). They are generally formed with has/have (present), had (past) or will have (future). The continuous aspect describes situations that take place during a given time period - in the past, present of future. They can be easily identified most of the time as they are of the form be verb + ing. They generally indicate actions that were/are/will be ongoing. The perfect continuous combines both aspects. It is of the form [ has/have (present), had (past) or will have (future) ] + been + verb + ing. Past Present Simple I went to the market yesterday I go to the market Continuous I was doing my homework I am doing my h Perfect I had washed all the plates I have washed all Perfect Continuous I had been waiting for you for over an hour by then I have been waiting for hour Tense Agreement More than identification of tense, CAT questions test a student on tense consistency. The tenses in the main clause and the subordinate clause need not be the same but they must be consistent with the relationship that is implied by the sentence. The rules governing tenses are a bit complex and
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Tenses and Verb Tense Agreement
What are tenses?
Tenses are tools used to indicate when a situation or action takes place i.e. to show the position of
the situation with respect to time. They are also used as tools to harass hapless CAT aspirants with
extremely difficult questions. This COTD hopes to guide students through the concept of tenses and
verb tense agreement.
The situation being described in the sentence could have occurred “now” - present tense, “before
now” - past tense or will happen “after now” - future tense. So far it's simple. But it gets a bit
complicated with the introduction of aspects - simple, continuous (also called progressive), perfect
and perfect continuous.
The perfect aspect describes events occurring in the past but linked to a later point in time (either the
present or the future). They are generally formed with has/have (present), had (past) or will have
(future).
The continuous aspect describes situations that take place during a given time period - in the past,
present of future. They can be easily identified most of the time as they are of the form be verb + ing.
They generally indicate actions that were/are/will be ongoing.
The perfect continuous combines both aspects. It is of the form [ has/have (present), had (past) or will
have (future) ] + been + verb + ing.
Past Present Future
Simple I went to the market yesterday I go to the market every day I will go to the market tomorrow
Continuous I was doing my homework I am doing my homework I will be doing my homework tonight
Perfect I had washed all the plates I have washed all the plates I will have washed all the plates by the
time you reach
Perfect Continuous I had been waiting for you for over an
hour by then
I have been waiting for you for over an
hour
I will have been waiting for you for
more than an hour by the time your
plane arrives
Tense Agreement
More than identification of tense, CAT questions test a student on tense consistency. The tenses in
the main clause and the subordinate clause need not be the same but they must be consistent with
the relationship that is implied by the sentence. The rules governing tenses are a bit complex and
solving these questions requires the student to rely more on what “sounds” right than on the actual
rules.
In complex sentences, the tense of the verb in the subordinate clause will be determined by the tense
of the verb in the main clause depending on the time relation between the clauses that is implied by
the sentence.
Suppose, the tense in the main clause is simple present tense.
E.g. She feels that the current system is biased against women.
Suppose the subordinate clause is as follows:
As Sheila failed to land a job in any of the big consulting companies, she feels that the current system is
biased against women.
The tense of the verb in the subordinate clause is not consistent with the relationship that is implied
by the sentence. As the sentence implies that Sheila is unable to find a job from some point in the
past to now, the correct tense for the subordinate clause will be present perfect tense.
Hence the correct statement is as follows:
As Sheila has failed to land a job in any of the big consulting companies, she feels that the current
system is biased against women.
Sample Questions:
o 1. Choose the option that would be grammatically correct for the underlined part:
Since the team’s performance figures dropped last month, the players have been determined to put in their
hundred percent in every match, and they are.
A) have been determined to put in their hundred percent in every match, and they are
B) are determined to put in their hundred percent in every match, and have been
C) were determined to put in their hundred percent in every match, and they were
D) have been determined to put in their hundred percent in every match, and they do
Answer: D
Solution: From the sentence, we understand the subordinate clause (Since the team’s performance figures
dropped last month) is an event that occurred at a single point of time in the past, the main clause (the
players have been determined to put in their hundred percent in every match) has been occurring since that point
in the past to today and subordinate clause 2 (and they do) has occurred concurrently with the main clause.
Hence, the subordinate clause should be simple past tense, the main clause should be present perfect and the
subordinate clause 2 should be simple present. Thus, the correct sentence construction is “Since the team’s
performance figures dropped last month, the players have been determined to put in their hundred percent in
every match, and they do”.
o 2. Choose the option that would be grammatically correct for the underlined part:
As people become more aware of the corrupt practices of the government, the demand for a change is increasing
at a rapid rate, the new leadership of technocrats providing more transparency
A) is increasing at a rapid rate, the new leadership of technocrats providing more transparency
B) has been increasing at a rapid rate, because of the new leadership of technocrats providing more transparency
C) has increased at a rapid rate, since the transparency of a new leadership of technocrats provides more
transparency
D) will increase at a rapid rate because the transparency of a new leadership of technocrats will provide more
transparency
Answer: D
Solution: The sentence can be split into the following clauses: subordinate clause(“As people become more
aware of the corrupt practices of the government”), main clause (“the demand for a change is increasing at a
rapid rate”), and subordinate clause 2(“the new leadership of technocrats providing more transparency”). As the
sentence is foretelling an increasing demand for leadership of technocrats with greater awareness among the
masses, the main clause should have future tense. Hence, the subordinate clause 1 will be in simple present
tense. As the subordinate clause 2 occurs after the main clause, it should also be in simple future tense. Thus, the
correct option is option D.
Number of positive and non-negative integral solutions for linear equations
Questions related to number of integral solutions are common in CAT. A few questions which
appeared in CAT in the past are:
1) What is the number of non-negative integral solutions of the equation A+B+C = 20?
2) Three different dice are rolled, what is the probability that the sum of the numbers shown is >= 13?
These can be solved using Permutations and Combinations concepts. A basic idea of Permutations
and Combinations is required to understand the concepts explained below.
The categories into which the questions can be categorized are as follows:
A) Positive integral solutions
B) Non-negative integral solutions
C) Integral solutions with conditions
Case 1: Positive integral solutions
Consider the following equation:
A + B + C = 10, A, B, C are positive integers.
To solve this, imagine that there are 10 identical objects placed next to each other with gaps
separating them. Now, I can select 2 gaps from among the 9 (10 objects have 9 gaps between them)
in ^9 C _2 ways. These selected gaps will hold the plus signs. Now, the number of objects to the left of
the first plus sign, the number of objects between the two plus signs and the number of objects to the
right of the second plus sign will be the values of A, B and C respectively.
Case 2: Non-negative integral solutions
Consider the same equation A+B+C = 10, but now A, B and C can take whole number values. The
number of solutions to this equation is obviously different from the number of solutions in the first
case.
To solve this equation, make a substitution of the variables in such a way that the case gets reduced
to solving the first case. Since A >= 0, (A+1) >= 1. Let A+1 = P. Similarly, B+1 = Q and C+1 = R.
Therefore, the equation gets reduced to (A+1) + (B+1) +(C+1) = 13 => P+Q+R = 13, with P, Q, R >=
1. The number of solutions for this equation is ^{12} C _2 using the concept of case 1.
Another way of solving this is, imagine that there are 10 0s and 2 1s in line. These have to be
arranged in any order. The number of ways of doing this is 12!/(10!*2!) = ^{12} C _2. One such
arrangement is as follows: 010000010000. The number of 0s to the left of the first 1 is the value of A,
the number of 0s between the two 1s is the value of B and the number of 0s to the right of the second
1 is the value of C. Therefore, in this case, the value of A = 1, the value of B = 5 and the value of C =
4.
Consider another rearrangement: 100000000001. In this case, the value of A is 0, that of B is 10 and
that of C is 0. It can be easily seen that every rearrangement can be mapped to a solution of the
equation A+B+C =10 with A,B and C taking non-negative integers.
Case 3: Integral solutions with conditions on the variables
Consider the following question from one of the previous CAT papers:
Three different dice are rolled such that the sum of the numbers shown is 13. In how many ways can
this be done?
Obviously, the dice can take only values from 1 to 6. So, the problem can be represented by the
equation: A+B+C = 13, with 1<= A, B , C <= 6.
The number of solutions to the equation A+B+C = 13 when there is no upper limit on the variables
is ^{12} C _2 = 66. ( using the concept of case 1 ). From this we have to subtract the number of
solutions which violate the condition.
When A=7, the equation reduces to B+C = 6 => No. of solutions = ^5 C _1 = 5
When A=8, B+C = 5 => No. of solutions = ^4 C _1 = 4
…so on till A=11, B+C = 2 => No. of solutions = 1. Therefore, total number of cases that have to be
subtracted are 1+2+…+5 = 15. As same cases can exist for B and C>6, total number of cases to be
subtracted = 15*3 = 45. So, the number of solutions of the equation A+B+C = 13 with 1< = A,B,C <= 6
is 66-45 = 21.
Another way of solving the same question is by replacing A, B, C with P, Q, R where P=6-A and so
on. Here, (6-P)+(6-Q)+(6-R)=13 which implies P+Q+R=5. As A goes from 1 to 6, P goes from 0 to 5.
Hence, the problem reduces to finding the non-negative solutions of P+Q+R=5. The number of non-
negative solutions is ^7 C _2 = 21. However, be extremely cautious while transforming the equation
and ensure you know the range of the new variables before attempting to solve it.
One more method of finding the number of solutions is using generating functions. This problem can
be expressed as finding the coefficient of x^{13} in the expression (1+x+x^2 + …+x^6)^3. This is
because, to find the coefficient of x^{13} from the expression (1+x+x^2+…+x^6)(1+x+x^2+…)..3 times
is to select some powers of x from all the 3 units, such that the sum of the powers adds to 13.
In general if the linear equation is A_1+A_2+ . . +A_p= m and 0<=A_1, .. A_p <=n then the problem can
be reduced to finding the exponent of A^m in the expression (1+x+x^2+. . +x^n) ^p.
Escalator Problems
Problems on moving escalators are now becoming quite common in CAT. They are generally
straightforward once the equations have been properly deduced. The tricky part is correctly inferring
the mathematical relation implied by the question.
Fortunately, there is no need to remember any extra formula to solve escalator problems. It boils
down to properly formulating equations involving speed, time and distance.
One point to be kept in mind while solving escalator problems is that the number of steps covered by
the escalator plus the number of steps covered by the person is equal to the total number of steps
visible on a stationary escalator.
So, in an escalator with N steps, if a person covers n1 steps in time t and reaches the end of the
escalator, it means that the escalator has covered the remaining (N−n1) steps in the same time t.
Hence, the ratio of speed of the person to the speed of escalator is n1:(N−n1)
Solved Examples:
Problems in escalators can be broadly classified into the following categories:
• One person moving in one direction
• One person moving in direction opposite to the escalator
• Two people moving in the same direction
• Two people moving in opposite directions
Let us understand each category with examples.
Consider the following examples:
1) A person is walking up an ascending escalator. He covers 50 steps and reaches the top in 25 seconds. If
the total number of steps on the escalator is 200, find the ratio of the speed of the person to the speed of
the escalator.
Solution: For the person to reach the top of the escalator, the total number of steps to be covered =
200. Since he covers 50 steps, the remaining 150 steps are covered by the escalator. Therefore, the
speed of the escalator is 150 steps/25 seconds = 6 steps/second. The speed of the person =
50steps/25 seconds = 2 steps/sec. Therefore, the required ratio is 2:6 = 1:3.
The take-away from this problem is the concept that if a total of N steps have to be covered and the
person covers a part of them, the remaining part is covered by the escalator.
2) A man takes 20 seconds to reach the top of an ascending escalator. On one particular day, the
ascending escalator was under renovation and so he had to climb up a descending escalator. He took 80
seconds to reach the top. How long would he have taken to climb to the top of a stationary escalator? The
speed of the man is 4 steps/sec.
Solution: This is a good example to show that the escalator problems are nothing but upstream-
downstream problems in a new packaging.
In this example, when the man is moving up the escalator, the speed of the escalator is added to his
own speed. Therefore, he covers the distance in lesser time. In the case in which he is climbing a
moving-down escalator, his speed is hampered by the speed on the escalator.
Let the total length of the escalator be L steps and its speed be S steps/sec.
The speed of the man is 4 steps/sec.
Therefore, L/(4+S) =20 and L/(4-S) = 80. Solving the two equations gives, S=2.4 steps/sec and L =
128 steps.
So, to climb a stationary escalator, the man takes 128/4 = 32 seconds.
3) A walks up a moving escalator and reaches the top in 25 steps. B walks up the same elevator and
reaches the top in 20 steps. The ratio of speeds of A and B is 3:2. How many steps can be seen on the
elevator?
Solution: Let the speed of A be Sa steps per unit time and that of B be Sb steps per unit time. Sa:Sb =
3:2. Let the speed of the escalator be Se steps per unit time. Let the total number of steps on the
elevator be N. Let A take Ta time to reach the top of the elevator and B take Tb time to reach the top
of the elevator. Therefore, from the relation between speed, time and distance,
N/(Sa+Se)=Ta and N/(Sb+Se)=Tb.
Number of steps climbed by A = time taken by A* speed of A = [N/(Sa+Se)]∗Sa = 25.
Number of steps climbed by B = time taken by B* speed of B = [N/(Sb+Se)]∗Sb = 20.
Sb=(2/3)∗Sa
Solving the 3 equations gives Sa=Se, and N = 50 steps.
4) Two people A and B walk on an ascending escalator but in different directions. If the speed of A is half
of the speed of B, and the speed of the escalator is the arithmetic mean of the speeds of A and B, on which
step of the escalator from the top, would the two people meet, if the total number of steps on the escalator
is 300 and both eventually reach their destination.
Solution: Let the speed of A be 4A. So, the speed of B is 8A. Since the speed of the escalator is the
arithmetic mean of the speeds of A and B, the speed of the escalator = 6A.
Since the speed of the escalator is more than the speed of A, he is moving up on the escalator
whereas B is moving down the escalator.
Time taken by A to reach the top = 300/(4A + 6A) = 30/A.
Time taken by B to reach the bottom = 300/(8A-6A) = 150/A.
B takes 5 times the time taken by A. So, the relative speed of A on the escalator is 5 times the speed
of B on the escalator. So, while covering a distance of 300 steps, A covers 5/6th of the distance and B
covers the remaining 1/6th. Therefore, A covers 250 steps and B covers 50 steps and hence meet at
the end of the 50th step from the top of the escalator.
Para Jumbles:
Para Jumbles, like Para Completion and Reading Comprehension, test your comprehension skills. As
you improve in any one of these three topics, you will get better at the other two. Like Para
Completion, the key to solving a para-jumble is in identifying the chain of thought. Once a student is
able to identify the central point and chain of thought, solving para-jumbles becomes fairly trivial.
When students ask me how to get better at PJs or any reading comprehension related topics, I
usually ask them to read good quality publications - articles, essays or well-written books. With less
than 70 days to CAT, you do not have time to inculcate a reading habit. The most effective way for a
student to improve in RC and related topics from now to Nov 16th is to solve as many RCs/PCs/PJs
as they can.
Toolkit:
The essential tools you must have in your toolkit to tackle Para Jumble questions are as follows:
• Understand the chain of thought: This is the only sure-fire way of solving Para-Jumbles. If you
understand the paragraph, its purpose and the central idea you will know how to order the sentences.
If you identify the central idea then there is only one logical way to order the sentences- the order that
would convey that idea in the easiest way. The passage would have an implicit chain of thought -
identify that chain of thought and you have the correct order. Most sentences can be classified as
introduction to the idea, arguments for, arguments against, author’s opinion or personal stand on the
subject, information regarding a subject etc. Identify the purpose of each statement and it would
determine its position in the paragraph.
• Identifying opening sentence: The opening sentence is most likely to be the one which introduces the
subject of the passage. If the passage is about the life Edgar Allen Poe, the opening statement would
mostly be the one in which he is named. All subsequent sentences would use the pronoun ‘he’ to
refer to him.
• Identifying the concluding sentence: The sentence that either sums up the thoughts expressed in the
paragraph or brings the chain of thought to its logical conclusion is most likely to be the last line of the
paragraph. Sentences that are inferences are more likely to be the last line as compared to facts and
judgements.
• Identifying links: Links help you identify the internal structure of the paragraph. Try to make blocks of
sentences by finding two or more sentences that are logically related. You can identify sentences that
are linked by using the following clues:
Transition words: Transition words are an absolutely essential tool especially if you can make
neither head nor tail of a paragraph. Transition words like Indeed, Moreover, However, Also, In fact,
For example, Similarly etc provide hints about what was stated in the preceding statement. For
example, “however”, “but” etc indicate that this statement in a way contradicts the earlier one. Other
transitional words like “Indeed”, “Moreover” indicate that both the statements are consistent in their
implication. Using transitional words you can figure out links between different statements. Once you
have a block of statements it becomes easier to identify the central idea or chain of thought.
Enumeration: Look out for connectors like “On the one hand”, “Initially”, “Firstly,”, “One reason was “,
“The first time he . . ” etc. These sentences begin enumeration of reasons, arguments etc. They will
be closely followed by sentences that begin with “On the other hand”, “Secondly,”, “Afterwards”,
“Another was”.
Cause and Effect: Two sentences listing the cause and effect of some action would be closely linked
together with the cause preceding the effect. For example: “She was over the maximum age limit for
some competitive examinations like UPSC. Hence, she was concentrating more on other competitive
examinations like CAT”. Whenever you see a transitional word like “Hence”, “As a result”, “Therefore”
etc the sentence states the effect of something. Try to find the sentence that would state the cause for
producing this effect.
Chronology: If the information given in the paragraph describes an event, there would be a
chronological order to it. For example, a passage on the cricket world cup is likely to start with
information about the group stages and end with information about the finals.
Grammar: Pronouns within statements can be used to figure out the preceding statements. Before a
pronoun is used it must be defined in some statement. For a given pronoun, figure out which
statement contains a suitable subject. This statement will precede the given statement. Also, all
intervening statements will have the same subject.
• Coherence check: After you have a tentative order, check it for coherence. This is extremely
important. Some links make sense in when statements are taken two at a time but do not work
together in a paragraph. At the end of unjumbling you should get a paragraph that is coherent.
• First and last lines are given: If the first and last lines are given, the task becomes much easier. The
job is now to find the most suitable sentence that would immediately succeed or precede these lines.
Sample Questions
Lets put these tools to use:
1. The following sentences when ordered form a coherent paragraph. Find the correct ordering of
sentences.
A) The food and clothing of one man is not the food and clothing of another; if the supply is
insufficient, what one man has is obtained at the expense of some other man
B) There are goods in regard to which individual possession is possible, and there are goods in
which all can share alike
C) On the other hand, mental and spiritual goods do not belong to one man to the exclusion of
another
D) We may distinguish two sorts of goods, and two corresponding sorts of impulses
E) This applies to material goods generally, and therefore to the greater part of the present
economic life of the world
F) If one man knows a science, that does not prevent others from knowing it; on the contrary, it
helps them to acquire the knowledge
a) DBAEFC
b) BAECFD
c) DAEBCF
d) DBAECF
Answer: d)
Solution: From all the sentences we can infer that the author is discussing two kinds of goods -
material and mental/spiritual. The sentence that introduces this topic - sentence D - must be the first
sentence. Sentence B should immediately follow this as it states what these two goods are. We have
a classic case of enumeration here where one case is stated first and the second case is stated in the
sentence beginning with “On the other hand”. The second case is of spiritual/mental goods as stated
in C and explained in F. Hence CF should be the end of the paragraph. A and E which explain the first
case of material goods should be in the middle. Hence the order is DBAECF.
2. The following sentences when ordered form a coherent paragraph. Find the correct ordering of
sentences.
A) Even the enormous, impregnable stupidity of our High Command on all matters of psychology
was penetrated by a vague notion that a few "writing fellows" might be sent out with permission to
follow the armies in the field, under the strictest censorship, in order to silence the popular clamor for
more news
B) They could not understand, nor did they ever understand that a nation which was sending all its
sons to the field of honor desired with a deep and poignant craving to know how those boys of theirs
were living and how they were dying, and what suffering was theirs, and what chances they had
against their enemy, and how it was going with the war which was absorbing all the energy and
wealth of the people at home
C) Lord Kitchener, prejudiced against them, was being broken down a little by the pressure of public
opinion, which demanded more news of their men in the field than was given by bald communiqués
D) Dimly and nervously they apprehended that in order to stimulate the recruiting of the New Army
now being called to the colors by vulgar appeals to sentiment and passion, it might be well to "write
up" the glorious side of war as it could be seen at the base and in the organization of transport,
without, of course, any allusion to dead or dying men, to the ghastly failures of distinguished generals,
or to the filth and horror of the battlefields
E) In 1915 the War Office at last moved in the matter of war correspondents
a) ABECD
b) ADEBC
c) ECADB
d) EADBC
Answer: c)
Solution: We can deduce that the subjects of the paragraph are war correspondents. The essential
point of the paragraph is that the war office and the powers that be were reluctant to allow war
correspondents on the front. They were forced to reconsider because of the growing clamour for more
news. They grudgingly allowed some war correspondents thinking that they could help in recruitment
by writing up about the war.
As the paragraph is about war correspondents sentence E that introduces them should be the first
line of the paragraph. Sentence D and B talk of a dim subject that could not understand the need for
war correspondents. Sentence A and C provide suitable subjects for D and B. Hence A and C should
precede D and B. Between A and C, as A starts with “Even” it should follow some sentence that is
similar to it. Hence C should precede A. Hence, ECADB is the correct order.
Para Completion
Para Completion questions are deceptively easy - every good PC question gets us four different
answers and some reasoning backs each answer. That is the challenging part of PC - you have to
find not just a possible solution but also the most probable solution. A challenging PC will have 3
answers that could potentially be the missing sentence of the paragraph. But it will have one option
that is head and shoulders above the rest in terms of suitability. Your task is to put yourself into the
shoes of the paragraph author. The author is trying to tell you a thought - he is trying to communicate
his thinking, his reasons for arriving at some conclusion, some information he thinks is relevant to the
point he wants to make. If you understand what the author is trying to communicate, you will
understand what the missing sentence should be. When you see the paragraph - see it in its entirety.
Try to find the main argument or point of the paragraph - it may or may not be explicitly stated. But
everything in the paragraph would circle around this point. Every paragraph is written with the
intention of acting on the reader - to inform, persuade, amuse or entertain the reader. You can figure
out what the author is trying to do without knowing the missing sentence. The missing sentence would
thus be the one that best serves the purpose of the author.
When I was preparing for CAT I would often wonder why para completion questions are part of the
CAT examination. There is no real world situation where you as a manager will find some paragraph
with one sentence missing. Well they are there for the same reason that Para Jumbles and RCs are
there - they test your reading comprehension skills. This skill is extremely important in management
education - you will be expected to read hundreds of pages in case studies every week and
comprehend information about subjects you are not familiar with. In one week you will have to read
and understand about Apple’s HR problems, Reliance’s accounting issues and Thermax’s growth
strategy. Lack of familiarity with the subject will not be an excuse then and should not be an excuse
now.
When students ask me how to get better at PC or any reading comprehension related topics, I usually
ask them to read good quality publications - articles, essays or well-written books. With less than 70
days to CAT, you do not have time to inculcate a reading habit. The most effective way for a student
to improve in RC and related topics from now to Nov 16th is to solve as many RCs/PCs/PJs as they
can.
Toolkit
The essential tools you must have in your toolkit to tackle Para Completion questions are as follows:
• Understand the chain of thought: This in the only sure-fire way of solving a PC. All other tips and tricks
will work about 60% of the time - they are useful in eliminating options but are generally a hacky way
of going about solving PC questions. There is a reason why students who are regular readers are
good at RC-type questions. It is because they are good at this very essential skill. From today till your
CAT examination, read every paragraph with this lens on - try to find the central idea or main point of
the paragraph. Every paragraph in PC represents on chain of thought. The author is trying to inform
the user of facts, opinions etc so that the reader can go from point A to point B in his own mind. The
missing sentence will be a link in this chain of thought. Figure out what the author is driving at and you
will know what the missing sentence should be.
• Context of the paragraph: The author usually bases his arguments or information in some context -
say India’s youth, English media channels, US presidential debates etc. This context defines the
scope of the paragraph. The missing sentence, especially if it is the last sentence, will generally be
based in the same context. Thus, options that talk about things that are different from this context are
less probable options for the last sentence.
• New Ideas: If the missing sentence is the last sentence of the paragraph, it is unlikely that the author
will introduce a new idea or concept in the last sentence. When I say new idea I mean something
other than the main point or context of the paragraph, illustration of the point or generalization of the
point. If the paragraph informs the reader about Ebola in Africa, the last sentence will not be about
Malaria in India.
• Contradictory sentences: The last line of a paragraph will almost NEVER contradict the main point of
the paragraph. So if the paragraph is about how a book is complete hogwash, the last line will not be
about the valid conclusions of the book.
• Concluding sentences: If in the paragraph, the author dances around the main point without explicitly
stating it, the last line of the paragraph is likely to be an explicit statement of the point.
• Paragraph Construction: Two common paragraph constructs are A) analogy/illustration followed by
stating of the general principle B) Stating of general idea or principle followed by
illustrations/arguments. In case A, the last line of the paragraph will most likely be about the principle
in general. In case B, the last line will most likely be about how the illustration/analogy ties back to the
general principle. The hackiest tool of the toolkit and should be used sparingly.
• Grammar: Look for clues from grammar : If an option uses a pronoun then the subject should be
defined unambiguously in the preceding sentences. If it is not defined then you can eliminate the
option on that basis. Similarly, if the missing sentence is not the final sentence, and the succeeding
sentence uses a pronoun find the option that introduces a suitable subject to receive the pronoun.
• Writing Style: Find the option that is consistent with the passage in writing style. If the paragraph is
sarcastic in tone, difficult to read or written in passive voice find the option that is written in the similar
style.
Solved Examples
Lets put these tools to use:
Q1) Find the most appropriate option to complete the paragraph:
A critic cannot be fair in the ordinary sense of the word. It is only about things that do not interest one
that one can give a really unbiassed opinion, which is no doubt the reason why an unbiased opinion is
always absolutely valueless. The man who sees both sides of a question, is a man who sees
absolutely nothing at all. Art is a passion, and, in matters of art, Thought is inevitably coloured by
emotion, and so is fluid rather than fixed, and, depending upon fine moods and exquisite moments,
cannot be narrowed into the rigidity of a scientific formula or a theological dogma. It is to the soul that
Art speaks, and the soul may be made the prisoner of the mind as well as of the body. One should, of
course, have no prejudices; but, as a great Frenchman remarked a hundred years ago, it is one's
business in such matters to have preferences, and when one has preferences one ceases to be
fair. It is only an auctioneer who can equally and impartially admire all schools of Art. No; fairness is
not one of the qualities of the true critic. It is not even a condition of criticism. Each form of Art with
which we come in contact dominates us for the moment to the exclusion of every other form. We must
surrender ourselves absolutely to the work in question, whatever it may be, if we wish to gain its
secret. ______________.
A) For unless we surrender, our opinion will not be objective
B) For Art is a cruel master that demands absolute submission
C) For in complete surrender, we lose objectivity but gain perspective
D) For the time, we must think of nothing else, can think of nothing else, indeed
Solving it the right way :
The chain of thought till the penultimate sentence is as follows:
An art critic cannot be fair because to be fair means to be completely uninterested in Art. The critic
should not be prejudiced but should have preferences as Art requires an emotional response and not
scientific objective response. To become an Art critic one needs to immerse oneself in that particular
art that one is studying at the moment to have any chance of understanding it.
Option A is unlikely as it is directly contradictory to the main point of the paragraph. The author does
not want the art critic to be objective. Instead he wants them to be passionate and emotional.
Option B does not fit in this chain of thought - the author clearly is appreciative of Art and is unlikely to
call it cruel. Moreover, it does not extend the chain formed till then nor does it tie it back to the main
point of the passage.
Option C is unlikely as the author does not value objectivity in art critics - he considers it worthless.
Also, it does not fit in the chain of thought, as the paragraph makes no mention of gaining a different
perspective on Art.
Option D links perfectly with the penultimate sentence and extends it by adding that not only must one
immerse oneself in the Art under consideration, one must study it exclusively.
Solving it the hacky way:
New Ideas: Option C introduces the new idea of gaining a new perspective on Art. As this has not
been mentioned by the author anywhere else in the paragraph we can eliminate option C.
Contradictory Sentences: Option A clearly contradicts the initial part of the passage where the author
states that objectivity is not desirable in an art critic. Hence we can eliminate option A.
Writing Style: Between option B and D, option D clearly has the same writing style as the paragraph -
it has several subordinate clauses and is rather heavy on using punctuation.
Hence, the correct answer is option D.
Q2) Find the most appropriate option to complete the paragraph:
No one would have believed in the last years of the nineteenth century that this world was being
watched keenly and closely by intelligences greater than man's and yet as mortal as his own; that as
men busied themselves about their various concerns they were scrutinised and studied, perhaps
almost as narrowly as a man with a microscope might scrutinise the transient creatures that swarm
and multiply in a drop of water. With infinite complacency men went to and fro over this globe about
their little affairs, serene in their assurance of their empire over matter. It is possible that the infusoria
under the microscope do the same. No one gave a thought to the older worlds of space as sources of
human danger, or thought of them only to dismiss the idea of life upon them as impossible or
improbable. It is curious to recall some of the mental habits of those departed days. At most
terrestrial men fancied there might be other men upon Mars, perhaps inferior to themselves and ready
to welcome a missionary enterprise. Yet across the gulf of space, minds that are to our minds as ours
are to those of the beasts that perish, intellects vast and cool and unsympathetic, regarded this earth
with envious eyes, and slowly and surely drew their plans against us. ___________________.
A) Fortunately for us, their best-laid plans unraveled due to unforeseen circumstances
B) And early in the twentieth century came the great disillusionment
C) Now it is too late to do anything about it
D) No one could have foreseen the destruction they brought with them
Solving it the right way: The chain of thoughts is as follows:
Human beings were unaware that they were being watched and studied by intelligent alien life forms
much like they would study bacteria. They went about their life without considering the danger posed
by superior alien beings. Now the situation has changed where human beings are now aware of this
and our way of thinking has changed.
Option A can be eliminated as the paragraph suggests that our way of thinking has changed implying
that the aliens were not unsuccessful in their plans. Moreover this is outcome is unlikely given the
regretful tone of the author in the paragraph.
Option B is likely as it logical concludes the chain of thought constructed in the paragraph. The
passage speaks about how we were unaware, ignorant and unconcerned. Disillusionment is clearly
the next logical step indicating the end of this blissful ignorance.
Option C is unlikely as it is inconsistent with the timeline implied by the paragraph. From the
paragraph we can infer that the calamity has already struck and the human race has already been
caught unawares. We can infer that not only were humans too late in realizing but that even that
knowledge has been known for quite some time.
Option D is inconsistent with the chain of thought of the paragraph as the author rues that we did not
foresee this. Hence, it implies that it was something that could have been and should have been
foreseen.
Solving it the hacky way:
Writing style: The author has a slightly exaggerated writing style and option A, B and D are more
consistent with the style. Hence we can eliminate option C.
Concluding sentences: The author dances around the main point that aliens have attacked us and
caught us unawares in the entire paragraph without explicitly stating that. Hence it is likely that he
concludes the paragraph with the statement of the event. Hence option A and B are more likely
options. Thus, we can eliminate option D.
Contradictory sentences: Option A is clearly contradictory to the statements in the passage where the
author implies that the aliens have attacked and human way of thinking has changed as a
consequence. Hence, we can eliminate option A.
Circular Tracks and Clocks
Circular Tracks
Questions on Circular Tracks are quite common in CAT as they help test multiple concepts at once.
Students need to have clear understanding of time, distance, speed and LCM/HCF.
Two general ways of answering questions on circular tracks are:
• Absolute speed approach: This approach is useful if the runners are moving in the same direction. In
this approach, we find the time taken by each individual runner to complete one round of the track.
The LCM of these times gives us the time interval required by the runners to meet each other.
• Relative speed approach: This is a generalized approach that works well irrespective of the direction of
the runners. Generally when we calculate the speed of a person, we calculate it with respect to the
ground. Hence, the frame of reference is the ground. Another way of defining speed would be in
relation with some other frame of reference - for e.g. another runner. If A and B are moving towards
each other at a m/s and b m/s, we consider B as stationary. From B’s frame of reference, A is moving
towards it at (a+b)m/s. Suppose A and B are moving in the same direction at 5m/s and 4m/s from the
same starting point. In this case, A is moving away from B at (5-4) = 1m/s.
The relative speed method is an incredibly powerful method that easily reduces the complexity level
of many problems. Consider for example the following problem:
4 people A, B, C and D are standing at the 4 corners of a square of side 50 meters. They start moving at a
speed of 10 m/s in such a way that B always moves towards A, C always moves towards B, D always
moves towards C and A always moves towards D. Find the time in which all of them meet.
When this question is solved from the frame of reference of the ground, the problem is nightmarish in
its complexity. It becomes incredibly easy when we use one of the runners as the frame of reference.
Consider the frame of reference of runner A. We assume that A is stationary. Since B is always
moving towards A, B has to cover a distance equal to one side of the square (50 meters) to reach A.
Since the speed of B is 10m/s, he takes 5 seconds to reach A. Since all the people meet at the same
time, we can conclude that all 4 people meet after 5 seconds.
Consider the following question which can be solved using either approach:
Amit and Ravi are running on a circular track of 800m. Amit runs at 4m/s and Ravi runs at 5m/s. If the
race is of 10km, how many times do Amit and Ravi meet?
Absolute speed approach: The time taken by Amit to complete a round is 200s and Ravi is 160s. Hence
the LCM of the two times = LCM(160,200) = 800s. The total duration of the race for Ravi =
10km/(5m/s) = 2000s. Hence, Amit and Ravi meet quotient (2000/800) = 2 times.
Relative speed based approach: As the two are moving in the same direction, the relative speed of Ravi
with respect to Amit = (5m/s-4m/s) = 1m/s. Hence, if we assume that Amit is stationary at the starting
pointing and Ravi is running around the circular track at 1m/s. Hence, he takes 800m/1m/s = 800s to
complete one round and meet Amit. Hence, number of times they meet = quotient(2000s/800s) = 2
times.
This approach gives us the number of times when A and B meet. However, this method does not give
the actual points on the track where A and B meet. To find the actual number of points on the track,
we have to consider that both A and B are moving. The following solved examples make this point
clearer.
Solved Examples
3 persons A, B and C start moving on a circular track from a point. The ratio of their speeds is 1:2:3.
Case 1: When all three are moving in clockwise direction.
a) After how long do A and B meet for the first time?
b) After how long do all three meet for the first time?
c) After how long do all three meet for the first time at the starting point?
d) At how many distinct points on the circular track do all three meet?
Case 2: When A and B are moving in clockwise direction and C is moving in anti-clockwise direction.
a) After how long do they meet for the first time?
b) After how long do they meet at the starting point for the first time?
c) At how many distinct points on the circular track do they meet and what is the relation between the
points?
Solution:
Case 1: When all three of them are moving in the same direction:
Formula: When the speeds of A and B are in the ratio of a:b, they meet at |a-b| distinct points when A and B are
moving in the same direction.
Let the length of the track be 300 meters and the speeds of A, B and C be 1 m/s, 2 m/s and 3m/s to
simplify calculations.
The difference between the speeds of A and B is 1 m/s. The difference between the speeds of B and
C is 1 m/s. The difference between the speeds of A and C is 2 m/s. Therefore, A and B meet at 1
distinct point. Similarly, B and C also meet at one distinct point. A and C meet at 2 distinct points. The
point to be noted here is that the number of distinct point at which any 2 persons meet when they are
traveling on a circular track is the difference in their speeds when the speeds are in their reduced
ratios.
Formula: Time taken for three persons to meet for the first time = LCM(d/|a-b|,d/|c-b|) where a,b and c are the
speeds and d is the total length of the track.
Time taken for the nth meeting is n*LCM(d/|a-b|,d/|c-b|). Time taken for the first meeting at the starting point =
LCM(d/a,d/b,d/c).
The number of distinct points at which all three meet is the HCF of the differences in speeds of each
pair of runners, when all the speeds are in the reduced ratio. In this case, it is the HCF of 1,1 and 2,
which is 1. Therefore, all three runners meet at 1 distinct point on the circle.
To find the time interval between the meetings of A and B, consider the relative speed between them.
Since A is running at 1 m/s and B at 2m/s in the same direction, the relative speed between them is
1m/s. Imagine A is at rest and B is running at a speed of 1m/s. Therefore, to meet A who is at the
starting point, B has to cover a distance of 300 m at 1m/s. Therefore, he meets A for the first time
after 300 sec. B meets C for the first time after 300m/(3m/s-2m/s) = 300s. Since A and B meet only at
the starting point, all three of them meet at the starting point. Time after which they meet = 300s.
Case 2:
Formula: Time taken for two persons to meet for the first time when they are moving in opposite directions =
d/(a+b) where a and b are the speeds and d is the total length of the track. Time taken for the nth meeting is n*
(d/a+b). Time taken for the first meeting at the starting point = LCM(d/a,d/b).
Since A and B are running in the same direction, the equations between them from the first case will
hold true.
A and B meet at 2m/s - 1m/s = 1m/s => 1 distinct point.
Since B and C are running in the opposite direction, the number of distinct points at which they meet
is the sum of the speeds. B and C meet at 3m/s+2m/s = 5m/s => 5 distinct points. They meet at points
which are at 60m , 120m, 180m, 240m and 300m from the point at which they start. ( Gaps of d/5
where d is the circumference of the circle ).
Similarly, A and C meet at 3+1 = 4 distinct points. They meet at points which are 75m, 150m, 225m
and 300m from the starting point. ( Gaps of d/4 where d is the circumference of the circle ).
We can see that all three of them meet only at one distinct point, which is the starting point. The times
at which A and B meet at the starting point are 300s, 600s, so on. The times taken for B and C to
meet at the starting point are 1*300/(2m/s+3m/s), 2*300(2m/s+3m/s) ,… = 60s, 120s,…. Therefore,
we can conclude that all three of them meet for the first time after LCM(300s, 60s) = 300s.
Clocks
Problems of clocks can be dealt with in the same way as problems of circular tracks. The hands of the
clock are nothing but runners running on a circular track.The points to be kept in mind are:
• Since there are 60 minutes on the clock and 360 degree, the minute hand covers 6 degrees in one
minute.
• The angle between each hour is 360/12 = 30 degrees.
• So, the hour hand covers 30 degrees in one hour, or half degree in one minute.
• Therefore, the relative speed between the hour hand and the minute hand is 5 and a half degrees.
Consider the following example:
John left for his friend’s house sometime between 5 PM and 6 PM. When he returned between 8 and 9
PM, he noticed that the hour hand and minute hand had interchanged their positions. Find the number of
minutes he was out of his house.
Solution: Let the initial angle between the hands be theta. By the time the hands interchanged, the
hour hand had travelled theta, and the minute hand covered 360*2 + 360-theta( two hours plus the
angle 360 – theta).
Speed of hour hand = 1/2 degree/minute.
Distance travelled by hour hand = theta.
Therefore, time needed = theta/(1/2) = 2*theta minutes.
Speed of hour hand = 6 degrees per minute.
Distance travelled by minute hand = 360*2 + 360 – theta = 1080 – theta.
Therefore, time needed = (1080 – theta) /6.
Since both the times are equal, 2*theta = (1080 – theta)/6 => theta = 1080/13.
That means, the minute hand is ahead of the hour hand by 1080/13 degrees when John left the
house. We know that at 5 PM, the angle between the two hands is 5*30 = 150 degrees and the hour
hand is ahead. Therefore, the minute hand covers 150 + 1080/13 degrees relative to the hour hand.
Since the relative speed is 5 and a half minutes, the total time taken is (150+1080/13) /(11/2) = 42
(54/143) minutes. Therefore, John left the house at 5:42:22.
Critical Reasoning
Critical reasoning questions are comprehension questions based on a small paragraph. Your task is
to identify different elements either stated in or implied by the information in the paragraph. Like
RCs/PJs/PCs, these questions test your comprehension skills but tend to be less ambiguous.
For example consider the following paragraph:
The defense attorneys for the accused in the Aurora Theater shooting case have argued against the televising of
the trial on the grounds that it would violate the defendant’s right to a fair trial. They believe witnesses for the
defense would be worried about a backlash from the public if they gave a testimony supporting the accused.
The important elements of Critical Reasoning are as follows:
• Premise: These are the facts stated in the passage and can be assumed to be true. All conclusions
are derived on the basis of these facts and implicit assumptions hidden within the stated inferences.
Above para:
The defense attorneys have asked that the trial should not be televised. They believe this would deter
witnesses and would not allow the defendant to have a fair trial.
• Assumption: These are facts that are not explicitly stated in the paragraph but are necessary to
arrive at the conclusions stated in the paragraph. An assumption, if untrue, would invalidate the
conclusion of the paragraph even if the stated facts remain valid.
Above para:
There are two assumptions in the above paragraph:
Firstly, you cannot have a fair trial where witnesses are afraid to testify for the accused.
Secondly, televising the trial would expose the witnesses to the public and would make them
vulnerable to attacks from the public.
• Stated Inference: These are conclusions stated explicitly in the paragraph.
Above para:
The accused cannot have a fair trial if witnesses refuse to testify for him fearing a backlash.
• Unstated Inference: These are conclusions that can be derived with reasonable certainty (note: not
absolute) from the information given in the paragraph.
Above para:
The public would have negative reaction to anyone seen supporting the accused in the Aurora
shooting case.
• Supporting Argument: These arguments strengthen the conclusion stated in the paragraph.
Examples of supporting arguments are stating the implicit assumption, examples where the stated
conclusion is true etc. Any evidence that corroborates the conclusion, even if it does not remove all
doubt regarding it, can be considered as a supporting argument.
Above para:
Many witnesses have refused to testify for the defense fearing that they would be socially ostracized
in their community if they testified.
• Weakening Argument: These arguments weaken the conclusion stated in the paragraph. They may
not necessarily invalidate the conclusion, but introduce more uncertainty regarding it.
Above para:
Televising of a court trial can be done while concealing the identity of the witnesses.
Generally, the premise would be given along with stated inference (optional) and the student would be
asked to identify one of assumption, unstated inference, supporting argument or weakening
argument.
Sample Questions:
Which of the following, if true, does most to support India’s stand?
In a surprise move, India today reversed its position on the Trade Facilitation agreement and decided to veto its
ratification. India sought negotiations on the public stocking issue before it signs the agreement. India wants to
see a permanent solution to the public stocking issue that threatens its food security and could possibly hamper
its ability to feed its poor citizens during famine conditions. The Trade Facilitation agreement seeks to
streamline custom rules worldwide and has nothing to do with the public stocking issue.
a) India has a lot of support on the public stocking issue from other countries with large poor
populations including Cuba, Venezuela and Bolivia
b) If the WTO does not take a lenient stand on the public stocking issue, India would walk out of the
WTO thereby reducing the WTO’s influence in the world
c) Other WTO nations are desperate to ratify the Trade Facilitation Agreement and would be more
likely to make concessions on public stocking to pass the agreement
d) Trade Facilitation would impact the lives of far fewer people as compared to the Public Stocking
issue
Answer: c)
Solution: Premise: India is refusing to sign the Trade Facilitation agreement unless it gets a solution to
the public stocking issue. The two issues are completely unrelated.
Inference: India is using the agreement as a tool to get the public stocking issue resolved to its
satisfaction.
If C is true, then India has a much better chance of getting the public stocking issue resolved if it uses
signing the agreement as leverage. Hence, C, more than any other statement would support India’s
stand.
Which of the following accurately describes the flaw in the main argument of the following paragraph?
Despite a disappointing monsoon, India has already saved adequate amount of water for this year. This is due
to better rain-water harvesting systems that have been put in place. As a result of these systems, the amount of
water in India’s reservoirs has been rising consistently. Currently, 87% of the available reservoir capacity has
been filled by these systems. Last year, only 75% of available capacity was drawn down for use.
a) It assumes that the water stored in reservoirs was the only source of water for people of India last
year
b) It takes for granted that the rain-water harvesting systems have been working as intended and are
reducing the wastage of water
c) It presumes that India is now less dependent on the monsoon season than it was in the past
d) It fails to consider the scenario that the amount of water used in India may have been declining in
the last few years
Answer: a)
Solution: Premise: India used 75% of reservoir capacity last year and now it has stored 87% of the
capacity. The steady increase in the stored water levels has been due to better rain-water harvesting
systems in spite of having less rainfall.
Conclusion: As the stored amount exceeds amount required, we have enough water for this year.
Assumption: The reservoirs provided a 100% of the water required last year. If they did not, then the
other water sources may have less than necessary amount of water and we cannot be reasonably
sure of meeting the entire year’s water requirement.
Highest power of a number in a factorial
Questions on the highest power of a number in a factorial are some of the easiest questions one can
hope to get from Number Systems. Even though not many questions are asked on this topic, the few
that are asked can be solved easily by understanding the concept. Missing out on these questions is
unpardonable as far as CAT is concerned.
Consider the following example:
What is the highest power of 2 in 80! ?
To solve this, we have to first understand how 80! Is constructed.
80! = 1*2*3*4*….80. From this, we can see that there are 80/2 = 40 multiples of 2 in 80!. Each of
these multiples contains one power of 2. Similarly, there are 80/4 = 20 multiples of 4 in 80!.
Since 4=22, each of these 20 numbers contains an additional power of 2. Similarly, there are 80/8 =
10 multiples of 8 in 80!. Since 8=23, each of these 10 multiples contains another additional power of
2. Similarly, there are 80/16 = 5 multiples of 16 in 80!. Since 16=24, each of these 5 numbers
contains an additional power of 2. Similarly, there are [80/32] = 2 multiples of 32 in 80!. Since 32=25,
each of these 2 numbers contains an additional power of 2. Similarly, there is [80/64] = 1 multiple of
64 in 80!. Since 64=26, this number contains an additional power of 2.
Counting all the numbers, we can see that we have a total of 40+20+10+5+2+1 = 78 powers of 2 in
80!.
In general,
Formula:
The highest power of a prime number p in x! is given by the expression [x/p]+[x/p^2]+[x/p^3]+[x/p^4]+…
where [x] is the greatest integer less than or equal to x.
Now, consider the following example:
What is the highest power of 8 in 80! ?
If we solve the problem using the method used for solving case 1 ( highest power of 2 in 80! ), we get
the highest power of 8 = [80/8] + [80/64] = 10+1 = 11.
This is wrong because we are missing out on some powers of 2 and 4, which when combined give a
power of 8.
To solve this, we have to realize that 8=23. Now, if we solve for the highest power of 2 in 80! and club
every 3 powers, we get the number of powers of 8.
Since the highest power of 2 in 80! = 78, we get the highest power of 8 in 80! = [78/3] = 26.
Point to be noted here is that the method used in the first case works if the number whose highest
power is being asked is a prime number. For higher powers of the number, it has to be prime
factorized first. To make the point clearer:
In general,
Formula:
The highest power of a prime number p raised to n in x! is given by the expression
[{[x/p]+[x/p^2]+[x/p^3]+[x/p^4]+….}/n].
Consider another example:
What is the highest power of 10 in 70! ?
Since 10 is not prime, first prime factorize 10.
10 = 2*5.
Highest power of 2 in 70! = [70/2] + [70/4] + [70/8] + [70/16] + [70/32] + [70/64] = 35+17+8+4+2+1 =
67.
Highest power of 5 in 70! = [70/5] + [70/25] = 14+2 = 16.
Now, combining one power of 2 and one power of 5 gives one power of 10. Therefore, combining 16
powers of 5 and 16 powers of 2 gives 16 powers of 10. We still have some powers of 2, but we do not
have any more powers of 5. Therefore, we cannot obtain any more powers of 10. The highest power
of 10 in 70! is therefore, 16.
In general, there are always less number of powers of the highest prime number. Therefore, a short-
cut can be to calculate only the highest power of the largest prime factor.
The takeaway from this example is:
Formula: To find the highest power of a composite number in a factorial, prime factorize the number and find
the highest powers of each of the prime factors. The highest power of the composite number is the minimum
value among the powers of each of the prime factors.