Some of the competitive examination give questions on a pattern generally known as "input and output". In these questions, input is converted into output through a few systematic steps. In the directions for these questions, an example (to show how a given input is transformed into a designed output) will be given. The input is a string of elements (these elements can be alphabets or numbers or words or any combination of these), which is processed through a few methodical steps and transformed into designed (pre-defined order of elements) output.
From the given example, we have to understand the following: (i) the method followed in the transformation; and
(ii) the Desired arrangement of the elements in the final output.
Same method has to be employed to answer the subsequent questions.
Now, let us analyse the two important phrases used in the above paragraph, viz., desired order of output and methodical transformation.
I. Desired order of output :
The form in which the final output is required, is pre-defined. The following are the most commonly followed arrangements of elements in the output.
(a) if the elements are numbers: The numbers can be arranged in Ascending order or Descending order.
(b) If the elements are letters or words:
(i)Alphabetical order: the elements appear in the same order in which they apear in a dictionary, starting from A to Z.
(ii)Reverse Alphabetical order: the elements appear in the reverse order in which they apear in a dictionary, starting from Z to A.
(iii)if the elements are a combination of letters, numbers and words, several kinds of arrangement are possible. On a case-to-case basis, we have to find out the arrangement in the output.
II. Methodical transformation:
To achieve the pre-defined output, the input string of elements is processed through a few steps. These steps follow a specific pattern while rearranging the given elements. These are most commonly followed methods. Let us name them as single element movement, and interchange of two elements. Students should note that there can be methods of transformation. Now, let us discuss these two methods mentioned above with the help of a simple input " B D C A F E". Let the Desired output be "A B C D E F".
1. Single element movement: In this method, only one element is arranged in each step. The elements, which are to the left of the place vacated by the rearranged, shift to their right to fill vacant space. The position of the other elements remain unchanged.
Illustration of this method:
Input: B D C A F E
Step I: A B D C F E
Step II: A B C D F E
Step III: A B C D E F
Step III is the final output.
In step one, letter 'A' is removed from its position and arranged in the first position. The three letters to the left of the vacant place i.e., B, D and C shift to their right by one place. Similar method followed by subsequent steps.
2. Interchange of elements: In each step, the element to ba arraanged interchanges its position with the element in its designated position. In this case, A interchanges its own position with B and the positions of other elements remain unchanged. This is the first step. Each of the subsequent steps follow the same methods till the final output is obtained.
Method of Answering a Question:
Step I: compare input and output in the example given in the question and observe the final arrangement.
Step II: Observe how each element is being rearranged and also the pattern followed by remaining elements.
Step III: Whenever an element comes into its designated position whithout consuming any step, then leave such element untouched.
Let us understand the concepts discussed above more clearly by using the following examples.
I. Arranging the words given in the input in alphabetical order:
Example: A word arrangement machine, when given an input consisting of words, rearranges them following a particular pattern in each step. The following is an illustration of input and the steps involved in the arrangement.
input : belt an area the state are tea
Step I : an belt area the state are tea
Step II : an are belt area the state tea
Step IV : an are area belt state the tea
Step V : an are area belt state tea the
Step V is the final output (last step) for the above input.
Now, let us solve one question based on the above model.
Example 1 : which of the following is the last step for the following input ?
input : from food has made case wage
(A). has made from food case wage
(B). case food from has wage made
(C). case food from has made wage
(D). case food from made has wage
Solution: On comparing the input with the output, it is clear that the given words are arranged in alphabetical order. Hence, the output is 'case food from has made wage'. Choice - 'C'
II. Arranging the numbers in the given input in increasing order:
Example: A number arrangement machine, when given an input line of numbers, rearranges them following a particular pattern in each step. The following is an example of input and the steps involved in the rearrangement.
input : 78 92 56 38 144 87
Step I : 38 92 56 78 144 87
Step II : 38 56 92 78 144 87
Step III : 38 56 78 92 144 87
Step IV : 38 56 78 87 144 92
Step V : 38 56 78 87 92 144
As all the numbers in the given input are arranged in the increasing order, Step V is the final step or output.
Explanation: The numbers are arranged in the ascending order in the output. During rearrangement, only by two numbers I.e., the number to be rearranged and the number in its designated place, interchange positions and the positions of the remaining numbers remain unchanged.
Monday 1 December 2014
Input and Output
Odd Man Out
Finding the odd man out from the given alternatives is a very common type of questions that one comes across in different competitive examinations. In the questions on odd man out, all the items - except one - follow a certain pattern (in their formation) or belong to a group. The item that does not follow the pattern or does not belong to the group has to be marked as the answer choice.
The problems of this variety often fall under the category of Classification. When a given set of elements is classified under a single head, one of the items will not have the common property, which the others will have. Hence it becomes the odd man out.
Questions on classification can be asked in any form. Some of the commonly asked ones are given below.
(1) Alphabet classification: in this type, a group of jumbled letters typically consisting of three letters, ( but can be four or two or a single word ) are put together. The pattern or order in which they are grouped is to be studied and we need to find out which groups have the same pattern or relationship between the letters. There will be one choice, which will have a different pattern from the rest and that is our answer.
Worked out examples:
1. Find the odd one among the following.
(A) ZW (B) TQ (C) SP (D) NL
Solution: Choice - 'D' is following a different pattern.
2. . Find the odd one among the following.
(A) CFD (B) GJH (C) KNM (D) JMK
Solution: Among above options 'C' is different with other options. A, B and D following a similar pattern.
(2) Word Classification: Here, different items are classified based on common properties like names, places, professions, parts of speech, etc., A few examples are illustrated below.
Worked out examples:
3. Find out the odd one among the following.
( A) Mercury. (B) Moon.
(C) Jupiter. (D) Saturn.
Solution: All others except Moon are planets where as Moon is a satellite.
(3) Number Classification :
In this case, we need to choose the odd number from the given alternatives. The numbers may belong to a particular set, I.e., they may be odd, even, prime, rational, squares or cubes only one of the choices will not follow the rule which others do and that is our answer. A few illustrations are given below.
Worked out Examples:
4. Find the odd one among the following.
(A) 17 (B) 27 (C) 37 (D) 47
Solution: All the given numbers except 27 are prime numbers whereas 27 is a composite number. Choice - B
The problems of this variety often fall under the category of Classification. When a given set of elements is classified under a single head, one of the items will not have the common property, which the others will have. Hence it becomes the odd man out.
Questions on classification can be asked in any form. Some of the commonly asked ones are given below.
(1) Alphabet classification: in this type, a group of jumbled letters typically consisting of three letters, ( but can be four or two or a single word ) are put together. The pattern or order in which they are grouped is to be studied and we need to find out which groups have the same pattern or relationship between the letters. There will be one choice, which will have a different pattern from the rest and that is our answer.
Worked out examples:
1. Find the odd one among the following.
(A) ZW (B) TQ (C) SP (D) NL
Solution: Choice - 'D' is following a different pattern.
2. . Find the odd one among the following.
(A) CFD (B) GJH (C) KNM (D) JMK
Solution: Among above options 'C' is different with other options. A, B and D following a similar pattern.
(2) Word Classification: Here, different items are classified based on common properties like names, places, professions, parts of speech, etc., A few examples are illustrated below.
Worked out examples:
3. Find out the odd one among the following.
( A) Mercury. (B) Moon.
(C) Jupiter. (D) Saturn.
Solution: All others except Moon are planets where as Moon is a satellite.
(3) Number Classification :
In this case, we need to choose the odd number from the given alternatives. The numbers may belong to a particular set, I.e., they may be odd, even, prime, rational, squares or cubes only one of the choices will not follow the rule which others do and that is our answer. A few illustrations are given below.
Worked out Examples:
4. Find the odd one among the following.
(A) 17 (B) 27 (C) 37 (D) 47
Solution: All the given numbers except 27 are prime numbers whereas 27 is a composite number. Choice - B
Exercise:
1. A. Iran: Asia b. Canberra: Australia c. Norway: Europe d. Algeria:
Africa
Ans: (b)
Explanation : In all other pairs, second is
continent to which the Country denoted by the first belongs.
2. A. Scapel : surgeon b. chisel:
soldier C. awl: cobbler d. Knife:
chef
Ans : (b)
Explanation: In all other pairs, first is
tool used by the second.
3. a. Mulder: Proteins
b. curie: radium
c. Becquerel: radioactivity d.
Einstein: television
Ans: (d)
Explanation : In all other pairs, first
is name of o scientist who discovered the second.
4 .a. Sheep: bleat b. Horse: neigh c. Ass:
grunt d. Owl: hoot
Ans: (c)
Explanation : In all other pairs second one is
the sound made by the first.
Saturday 29 November 2014
Statement and Assumptions
This is a very important topic. In recent exams 5 questions of this type are asked. So mastering this chapter is essential for success in competitive exams.
FORMAT OF THE PROBLEM
Directions : In each question below a statement (or a passage) is followed by two assumptions numbered I and II. An assumption is something supposed or taken in for granted. You have to consider the statement and the following assumptions and then decide which of the assumptions is implicit in the statement.
Give answer:
(a) if only assumption I is implicit.
(b) if only assumption II is implicit.
(c) if either assumption I or assumption II is implicit.
(d) if neither of the assumptions is implicit.
(e) if both the assumptions are implicit.
Example:
Statement: Read the study material prepared by XYZ to get a high score in the exam.
Assumptions :
I. Study material prepared by XYZ is of good quality.
II. Getting a high score in the exam is desirable.
ASSUMPTION: An assumption is something which is assumed, supposed and taken for granted.
When someone says something he may not put every aspect of his idea into words. That which is left unsaid or taken for granted is called an assumption.
IMPLICATION: Implication is something which is implied. It is the hidden meaning of the statement. Sometimes implications are also taken as assumptions.
DIFFERENCE BETWEEN ASSUMPTION AND IMPLICATION
An assumption is something on which the statement is based, while implication is something which is derived from and therefore based upon the statement.While doing problems on statement assumption
one should read each statement very carefully and whenever find anything presupposed or taken for granted, immediately notice it. Misreading of some facts leads to incorrect answer. So while reading one should be conscious about words because sometimes a single word's presence or absence changes
the entire meaning of a sentence. Points to be remembered while evaluating assumptions :
The use of definite words may lend a different tone to a statement. So one should be careful about them. For example some key words like 'only', 'best', 'strongest', 'all', 'definitely', 'certainly' etc. impact a kind of exclusiveness to the sentence and thus reduce its range.
Example:
Statement : You should use computers to increase the efficiency of your office.
Assumptions: I. Only the use of computers can increase the efficiency of office.
II. The use of computers can increase the efficiency of the office.
Assumption II is a valid one but I is not valid. In the statement it is suggested that computers may be used to increase the efficiency but it is not said that only computers can increase the efficiency. It is only one of the methods by which the efficiency can be increased.
1) When two suggestions / events / facts, A and B are connected by the conjuction,
a) A because / as a result of B. Assumption - B leads to A
b) A therefore / hence B. Assumption - A leads to B
c) A even after / inspite of / despite B. Assumption - Usually A does not occur, when B occurs.
Ex: The house was robbed inspite of the tight security in that area.
2) Whenever you come across a connotative or connotive phrases note that,
a) "It is true that" can be expressed as
i) It would be correct to say that....
ii) Even the most sceptic men would agree that......
iii) It can be claimed with reasonable degree of truth that.....
b) It is false can be expressed as
i) It is highly misleading to say that.....
ii) Nothing could be farther from the truth that....
iii) It is baseless to say that.....
3) If the statement tells about the existence of something the assumption should be that it does exists. If it is said that something is absent, the assumption should be that it not exists.
Statement :The increase in the improved number of educated women.
Assumption : There are educated women in the society.
4) If an adjective is attached (unconditionally) to any subject, it must be assumed that the subject, does have the quality.
Ex. Statement : The talkative nature of Sheela attracted everybody's attention.
Valid assumption: Sheela is talkative.
5) Suppose a fact / report / observation / study / data 'A' followed by a suggested course of action B is given. If some negative aspect of A is mentioned,The valid assumptions can be
i) A needs improvement.
ii) The negative aspects of A are undesirable
iii) B will improve A
iv) The advantages of adopting B far outweigh the disadvantages of net adapting it.
6) Advertisements / notices / appeals Statements in the form of an advertisement or an official notice or a notice issued in the public interest or an appeal are often asked in the recent exams.The following assumptions are valid in these cases.
i) An advertisement / appeal / notice does have some effect.
ii) In the case of an advertisement, that which is being highlighted is looked for and expected by the people.
iii) In the case of a public - interest notice, it is the duty of those who issue it, to issue such notices.
iv) In case of a public interest notice, what is being advised must be beneficial for people and its non-practice harmful in some way.
v) In case of an appeal, the reason for issuing it exists.
vi) In case of an official notice, the effect of its implementation will be beneficial for the organisation.
TIPS FOR QUICK ANSWER
From the explanations given above you can find that there may be more than one valid assumption for a statement. In almost all cases they are not mutually exclusive. So if you find that both the assumptions are valid mark the answer 'both are implicit'. The answer choice "Either I or II is implicit" is to be marked only, when one assumption excludes the other. This case occurs only very rarely. So be careful before marking this choice.
In questions with more than two assumptions its combinations like I & II are implicit, II & III are implicit, I & III are implicit etc. are given as answer choices. If you are sure that assumption III is not implicit you can eliminate answer choices which contains III thus reducing the number of answer choices. Then select answer from the remaining choices. You can keep in mind some typical
examples which will help you while answering questions in the examination. If the given assumption is just contrary to the given statement or if it is not connected with the given statement then the assumption can be immediately rejected. In some cases the assumption is a restatement of the given
statement. The candidate should be careful in evaluating this type of assumptions.
Eg Statement : Small things are beautiful.
Invalid assumptions:
I. Big things are not beautiful.
II. Big things are ugly.
III. Small things are not beautiful.
Assumption III is just the contrary of what is given in the statement. More saying of small things are beautiful does not mean that big things are not beautiful or ugly. So assumptions I & II are not implicit.
Directions : In each question below is given a statement followed by two assumptions - numbered I and II. An assumption is something supposed or taken for granted. You have to consider the statement and the following assumptions and decide which of the assumptions is implicit in the statement.
Give answer (a) if only assumption I is implicit;
(b) If only assumption II is implicit;
(c) If either I or II is implicit;
(d) if neither I nor II is implicit and
(e) if both I and II are implicit.
Solved Examples:
1. Statement : Read the study material prepared by XYZ to get a high score in the exam.
Assumptions :
I Study material prepared by XYZ is of good quality.
II Getting a high score in the exams is desirable.
Ans (e). Both the assumptions are implicit in the statement. It is said in the statement that the study material prepared by XYZ helps to get a high score, which means that the study material by XYZ is of good quality thus assumption I is implicit. Reading the study material is recommended to get a high score in the exam, that is getting a high score, in the exams is desirable. Hence II is also implicit.
2. Statement : This book is invested to guide the layman to study tailoring in the absence of a teacher.
Assumptions :
I. A teacher of tailoring may not be available to everyone.
II. Tailoring can be learn with the help of a book.
Ans(e). Both the assumptions are implicit. The book is intended to teach in the absence of a teacher. From this we can enter that the absence of teacher is a possibility, thus assumption I is valid. Since the book is intended to teach tailoring, assumption II is also implicit.
3. Statement : Buy pure and natural pearls of company `X' - an advertisement.
Assumption :
I. Artificial pearls can be prepared.
II. People do not mind paying more for pure and natural pearls.
Ans (a). The advertisement tells that the pearls of company X is natural, this means that artificial pearls may be available in the market. So assumption I is implicit. But the advertisement tells nothing about the price of the pearls. So II cannot be implicit
FORMAT OF THE PROBLEM
Directions : In each question below a statement (or a passage) is followed by two assumptions numbered I and II. An assumption is something supposed or taken in for granted. You have to consider the statement and the following assumptions and then decide which of the assumptions is implicit in the statement.
Give answer:
(a) if only assumption I is implicit.
(b) if only assumption II is implicit.
(c) if either assumption I or assumption II is implicit.
(d) if neither of the assumptions is implicit.
(e) if both the assumptions are implicit.
Example:
Statement: Read the study material prepared by XYZ to get a high score in the exam.
Assumptions :
I. Study material prepared by XYZ is of good quality.
II. Getting a high score in the exam is desirable.
ASSUMPTION: An assumption is something which is assumed, supposed and taken for granted.
When someone says something he may not put every aspect of his idea into words. That which is left unsaid or taken for granted is called an assumption.
IMPLICATION: Implication is something which is implied. It is the hidden meaning of the statement. Sometimes implications are also taken as assumptions.
DIFFERENCE BETWEEN ASSUMPTION AND IMPLICATION
An assumption is something on which the statement is based, while implication is something which is derived from and therefore based upon the statement.While doing problems on statement assumption
one should read each statement very carefully and whenever find anything presupposed or taken for granted, immediately notice it. Misreading of some facts leads to incorrect answer. So while reading one should be conscious about words because sometimes a single word's presence or absence changes
the entire meaning of a sentence. Points to be remembered while evaluating assumptions :
The use of definite words may lend a different tone to a statement. So one should be careful about them. For example some key words like 'only', 'best', 'strongest', 'all', 'definitely', 'certainly' etc. impact a kind of exclusiveness to the sentence and thus reduce its range.
Example:
Statement : You should use computers to increase the efficiency of your office.
Assumptions: I. Only the use of computers can increase the efficiency of office.
II. The use of computers can increase the efficiency of the office.
Assumption II is a valid one but I is not valid. In the statement it is suggested that computers may be used to increase the efficiency but it is not said that only computers can increase the efficiency. It is only one of the methods by which the efficiency can be increased.
1) When two suggestions / events / facts, A and B are connected by the conjuction,
a) A because / as a result of B. Assumption - B leads to A
b) A therefore / hence B. Assumption - A leads to B
c) A even after / inspite of / despite B. Assumption - Usually A does not occur, when B occurs.
Ex: The house was robbed inspite of the tight security in that area.
2) Whenever you come across a connotative or connotive phrases note that,
a) "It is true that" can be expressed as
i) It would be correct to say that....
ii) Even the most sceptic men would agree that......
iii) It can be claimed with reasonable degree of truth that.....
b) It is false can be expressed as
i) It is highly misleading to say that.....
ii) Nothing could be farther from the truth that....
iii) It is baseless to say that.....
3) If the statement tells about the existence of something the assumption should be that it does exists. If it is said that something is absent, the assumption should be that it not exists.
Statement :The increase in the improved number of educated women.
Assumption : There are educated women in the society.
4) If an adjective is attached (unconditionally) to any subject, it must be assumed that the subject, does have the quality.
Ex. Statement : The talkative nature of Sheela attracted everybody's attention.
Valid assumption: Sheela is talkative.
5) Suppose a fact / report / observation / study / data 'A' followed by a suggested course of action B is given. If some negative aspect of A is mentioned,The valid assumptions can be
i) A needs improvement.
ii) The negative aspects of A are undesirable
iii) B will improve A
iv) The advantages of adopting B far outweigh the disadvantages of net adapting it.
6) Advertisements / notices / appeals Statements in the form of an advertisement or an official notice or a notice issued in the public interest or an appeal are often asked in the recent exams.The following assumptions are valid in these cases.
i) An advertisement / appeal / notice does have some effect.
ii) In the case of an advertisement, that which is being highlighted is looked for and expected by the people.
iii) In the case of a public - interest notice, it is the duty of those who issue it, to issue such notices.
iv) In case of a public interest notice, what is being advised must be beneficial for people and its non-practice harmful in some way.
v) In case of an appeal, the reason for issuing it exists.
vi) In case of an official notice, the effect of its implementation will be beneficial for the organisation.
TIPS FOR QUICK ANSWER
From the explanations given above you can find that there may be more than one valid assumption for a statement. In almost all cases they are not mutually exclusive. So if you find that both the assumptions are valid mark the answer 'both are implicit'. The answer choice "Either I or II is implicit" is to be marked only, when one assumption excludes the other. This case occurs only very rarely. So be careful before marking this choice.
In questions with more than two assumptions its combinations like I & II are implicit, II & III are implicit, I & III are implicit etc. are given as answer choices. If you are sure that assumption III is not implicit you can eliminate answer choices which contains III thus reducing the number of answer choices. Then select answer from the remaining choices. You can keep in mind some typical
examples which will help you while answering questions in the examination. If the given assumption is just contrary to the given statement or if it is not connected with the given statement then the assumption can be immediately rejected. In some cases the assumption is a restatement of the given
statement. The candidate should be careful in evaluating this type of assumptions.
Eg Statement : Small things are beautiful.
Invalid assumptions:
I. Big things are not beautiful.
II. Big things are ugly.
III. Small things are not beautiful.
Assumption III is just the contrary of what is given in the statement. More saying of small things are beautiful does not mean that big things are not beautiful or ugly. So assumptions I & II are not implicit.
Directions : In each question below is given a statement followed by two assumptions - numbered I and II. An assumption is something supposed or taken for granted. You have to consider the statement and the following assumptions and decide which of the assumptions is implicit in the statement.
Give answer (a) if only assumption I is implicit;
(b) If only assumption II is implicit;
(c) If either I or II is implicit;
(d) if neither I nor II is implicit and
(e) if both I and II are implicit.
Solved Examples:
1. Statement : Read the study material prepared by XYZ to get a high score in the exam.
Assumptions :
I Study material prepared by XYZ is of good quality.
II Getting a high score in the exams is desirable.
Ans (e). Both the assumptions are implicit in the statement. It is said in the statement that the study material prepared by XYZ helps to get a high score, which means that the study material by XYZ is of good quality thus assumption I is implicit. Reading the study material is recommended to get a high score in the exam, that is getting a high score, in the exams is desirable. Hence II is also implicit.
2. Statement : This book is invested to guide the layman to study tailoring in the absence of a teacher.
Assumptions :
I. A teacher of tailoring may not be available to everyone.
II. Tailoring can be learn with the help of a book.
Ans(e). Both the assumptions are implicit. The book is intended to teach in the absence of a teacher. From this we can enter that the absence of teacher is a possibility, thus assumption I is valid. Since the book is intended to teach tailoring, assumption II is also implicit.
3. Statement : Buy pure and natural pearls of company `X' - an advertisement.
Assumption :
I. Artificial pearls can be prepared.
II. People do not mind paying more for pure and natural pearls.
Ans (a). The advertisement tells that the pearls of company X is natural, this means that artificial pearls may be available in the market. So assumption I is implicit. But the advertisement tells nothing about the price of the pearls. So II cannot be implicit
Friday 28 November 2014
Data Sufficiency
To begin understanding Data Sufficiency, it’s helpful to look at the official directions and answer choices for these problems, courtesy of the Graduate Management Admissions Council.
Directions
This Data Sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements plus your knowledge of
mathematics and everyday facts (such as the number of days in July or the meaning of counterclockwise), you must indicate whether:
(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked;
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked;
(C) Both statements (1) and (2) TOGETHER are sufficient to answer the question asked; but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient to answer the question asked;
(E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
You should see from the answer choices that the name of the game is “is the data sufficient?”, so let’s spend some time discussing what constitutes sufficient information.
More important than those solutions, however, are these takeaways:
• A statement is sufficient when it guarantees exactly one (and only one) answer to the question.
• This means that in a Yes/No question, you have sufficient information if the answer is “Definitely Yes” or if the answer is “Definitely No”. You do not have sufficient information when the answer is “Sometimes Yes but Sometimes No” (or “Maybe”).
• This means that in a “What is the Value?” question, you have sufficient information when you can pin down exactly one value for the question, but you do not have sufficient information when more than one value is possible.
• Data Sufficiency questions require attention to detail – the drills on the previous page came in pairs, and to the untrained eye each pair might have seemed the same. But subtle differences in what was given or asked – variable squared vs. variable cubed; cube vs. rectangular box; 2A + 3P vs. 3A + 2P – can make all the difference.
As tricky as Data Sufficiency can look at first, the way in which they are constructed ensures that you can attack them systematically. As you know, the answer choices are always the same. And you should also know that there are only two types of question stems that can ask:
1. Yes or No Questions
• A statement gives multiple solutions, but they all give the same answer.
• A statement provides a no answer instead of a yes answer.
2. What Is the Value? Questions
• A statement appears to be giving one value because you have assumed properties of the number that were not actually given (positive, integers, etc.).
• Restrictions were placed in the problem that you did not properly leverage (for instance, the problem is asking for the number of children, which must be an integer and cannot be negative).
Avoid assumptions. Every time you approach a Data Sufficiency problem, you must actively consider any assumptions that you may have been baited into making. Avoiding assumptions is perhaps the most important skill in all with Data Sufficiency.
1. Yes or No Question
Statement:Is x > 3?
Arguments:(1) The sum of x and the square of x is 12.
(2) square of x > 9
(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
(C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
(D) EACH statement ALONE is sufficient to answer the question asked
(E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Solution. A
Question Type: Yes/No. This question asks: “Is x > 3?”
Given information in question stem or diagram: No important information is given in the question stem.
Statement 1: The first step in this statement is to translate the wording into the following equation: x + x2 = 12. Since this is a quadratic equation, you should set everything equal to zero so that x2 + x – 12 = 0. Factoring this, you see that (x + 4)(x - 3) = 0 and x would be -4 or 3. The difficulty in this statement is that many people assume that this information is not sufficient because there are two values, one negative and one positive. However, remember that to prove sufficiency in a yes or no question requires only a definitive answer, not one value. Since each of these values (-4 and 3) gives a “no” answer to the question, this statement is sufficient. The answer is either A or D.
Statement 2: x2 > 9. If x2 > 9 then either x > 3, which gives you a “yes” answer, or x < 3, which gives you a “no” answer. For example x could be -5 (which when squared is > 9) or 5 (which when squared is also > 9). This statement is thus not sufficient, and the correct answer is A.
Note: This question is created to prey on two common mistakes, one relating to Data Sufficiency itself and one relating to algebra:
1.) People (even those who have done lots of data sufficiency) tend to forget to look for the “no” answer inYes/No questions and they often make mistakes about what is really requiredfor sufficiency on Yes/No questions.
2.) People forget about the negative possibilities when dealing with squared variables in inequalities.
2. What Is the Value? Question
Statement: If a, b, and c are distinct positive integers where a < b < c and √abc = c, what is
the value of a?
Arguments:(1) c = 8
(2) The average of a, b, and c is 14/3
(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
(C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
(D) EACH statement ALONE is sufficient to answer the question asked
(E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Solution:D
Question Type: What Is the Value? This question asks for the specific value of
positive integer a.
Given information in the question stem or diagram: There is a lot of information SOLUTIONS to leverage from this question stem. a, b, and c are distinct positive integers; a< b < c; and the square root of abc = c. You should first manipulate the last one algebraically by squaring both sides to see that abc = c2. Divide both sides by c (you can do this because you know that c cannot be 0 from the question stem) and the equation becomes ab = c. So before you even go to the statements you know that ab = c and all of the variables are different positive integers.Statement 1: c = 8. Combined with what you learned from the question stem, this means that ab = 8. Since a and b are distinct positive integers and a < b, the only possibility is a = 2 and b= 4. You might consider a = 1 and b = 8 but since
the integers must be distinct, you cannot have b = 8 since c = 8. This is sufficient but you will only see that if you properly leverage every piece of information given in the question stem. Remember: When you are given even a small piece of information in the question stem it is usually very important. The correct answer is A or D.
Statement 2: The average of a, b, and c is 14/3
. This means that the total of a+ b + c = 14. This statement is even trickier than the last but requires a similar leveraging of all available information. It may seem at first glance that there are many possibilities for the values of a, b, and c. However, the only way that ab = c and a + b + c = 14 is for a = 2, b= 4, and c = 8. There is no other way to have three distinct numbers add up to 14 and have ab = c. This statement is also sufficient and the correct answer is D. This question provides an excellent example of a phenomenon you will see often in Data Sufficiency: When a lot of information
is given in the question stem, statements are usually sufficient with much less information than you might first think.
Directions
This Data Sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements plus your knowledge of
mathematics and everyday facts (such as the number of days in July or the meaning of counterclockwise), you must indicate whether:
(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked;
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked;
(C) Both statements (1) and (2) TOGETHER are sufficient to answer the question asked; but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient to answer the question asked;
(E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
You should see from the answer choices that the name of the game is “is the data sufficient?”, so let’s spend some time discussing what constitutes sufficient information.
More important than those solutions, however, are these takeaways:
• A statement is sufficient when it guarantees exactly one (and only one) answer to the question.
• This means that in a Yes/No question, you have sufficient information if the answer is “Definitely Yes” or if the answer is “Definitely No”. You do not have sufficient information when the answer is “Sometimes Yes but Sometimes No” (or “Maybe”).
• This means that in a “What is the Value?” question, you have sufficient information when you can pin down exactly one value for the question, but you do not have sufficient information when more than one value is possible.
• Data Sufficiency questions require attention to detail – the drills on the previous page came in pairs, and to the untrained eye each pair might have seemed the same. But subtle differences in what was given or asked – variable squared vs. variable cubed; cube vs. rectangular box; 2A + 3P vs. 3A + 2P – can make all the difference.
As tricky as Data Sufficiency can look at first, the way in which they are constructed ensures that you can attack them systematically. As you know, the answer choices are always the same. And you should also know that there are only two types of question stems that can ask:
1. Yes or No Questions
• A statement gives multiple solutions, but they all give the same answer.
• A statement provides a no answer instead of a yes answer.
2. What Is the Value? Questions
• A statement appears to be giving one value because you have assumed properties of the number that were not actually given (positive, integers, etc.).
• Restrictions were placed in the problem that you did not properly leverage (for instance, the problem is asking for the number of children, which must be an integer and cannot be negative).
Avoid assumptions. Every time you approach a Data Sufficiency problem, you must actively consider any assumptions that you may have been baited into making. Avoiding assumptions is perhaps the most important skill in all with Data Sufficiency.
1. Yes or No Question
Statement:Is x > 3?
Arguments:(1) The sum of x and the square of x is 12.
(2) square of x > 9
(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
(C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
(D) EACH statement ALONE is sufficient to answer the question asked
(E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Solution. A
Question Type: Yes/No. This question asks: “Is x > 3?”
Given information in question stem or diagram: No important information is given in the question stem.
Statement 1: The first step in this statement is to translate the wording into the following equation: x + x2 = 12. Since this is a quadratic equation, you should set everything equal to zero so that x2 + x – 12 = 0. Factoring this, you see that (x + 4)(x - 3) = 0 and x would be -4 or 3. The difficulty in this statement is that many people assume that this information is not sufficient because there are two values, one negative and one positive. However, remember that to prove sufficiency in a yes or no question requires only a definitive answer, not one value. Since each of these values (-4 and 3) gives a “no” answer to the question, this statement is sufficient. The answer is either A or D.
Statement 2: x2 > 9. If x2 > 9 then either x > 3, which gives you a “yes” answer, or x < 3, which gives you a “no” answer. For example x could be -5 (which when squared is > 9) or 5 (which when squared is also > 9). This statement is thus not sufficient, and the correct answer is A.
Note: This question is created to prey on two common mistakes, one relating to Data Sufficiency itself and one relating to algebra:
1.) People (even those who have done lots of data sufficiency) tend to forget to look for the “no” answer inYes/No questions and they often make mistakes about what is really requiredfor sufficiency on Yes/No questions.
2.) People forget about the negative possibilities when dealing with squared variables in inequalities.
2. What Is the Value? Question
Statement: If a, b, and c are distinct positive integers where a < b < c and √abc = c, what is
the value of a?
Arguments:(1) c = 8
(2) The average of a, b, and c is 14/3
(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
(C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
(D) EACH statement ALONE is sufficient to answer the question asked
(E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Solution:D
Question Type: What Is the Value? This question asks for the specific value of
positive integer a.
Given information in the question stem or diagram: There is a lot of information SOLUTIONS to leverage from this question stem. a, b, and c are distinct positive integers; a< b < c; and the square root of abc = c. You should first manipulate the last one algebraically by squaring both sides to see that abc = c2. Divide both sides by c (you can do this because you know that c cannot be 0 from the question stem) and the equation becomes ab = c. So before you even go to the statements you know that ab = c and all of the variables are different positive integers.Statement 1: c = 8. Combined with what you learned from the question stem, this means that ab = 8. Since a and b are distinct positive integers and a < b, the only possibility is a = 2 and b= 4. You might consider a = 1 and b = 8 but since
the integers must be distinct, you cannot have b = 8 since c = 8. This is sufficient but you will only see that if you properly leverage every piece of information given in the question stem. Remember: When you are given even a small piece of information in the question stem it is usually very important. The correct answer is A or D.
Statement 2: The average of a, b, and c is 14/3
. This means that the total of a+ b + c = 14. This statement is even trickier than the last but requires a similar leveraging of all available information. It may seem at first glance that there are many possibilities for the values of a, b, and c. However, the only way that ab = c and a + b + c = 14 is for a = 2, b= 4, and c = 8. There is no other way to have three distinct numbers add up to 14 and have ab = c. This statement is also sufficient and the correct answer is D. This question provides an excellent example of a phenomenon you will see often in Data Sufficiency: When a lot of information
is given in the question stem, statements are usually sufficient with much less information than you might first think.
Wednesday 26 November 2014
Syllogism
The word "syllogism" is given by Greeks which means 'inference' or 'deduction'. It was introduced by Aristotle.An example of a question of syllogism is given below.
Directions : In the following questions, two statements are given followed by two conclusions. You have to study the two statements and then decide which of the conclusions follow from the statements. Mark the right answer from (1), (2), (3), (4) and (5)
Statements : All plants are trees.
No trees are green.
Conclusions : I. Some plants are green.
II. No plants are green.
1) Only I follows
2) Only II follows
3) Both I and II follow
4) Either I or II follows
5) Neither I nor II follows
This is a typical question of syllogism. Here the choice (2) is true. Later on we can discuss
the method to reach at the answer choice. Now let us see some definitions related to syllogism.
PROPOSITION
A proposition is a sentence that makes a statement and gives a relation between two or more terms.
In logic, any statement is termed a proposition.
Eg : i) All windows are rods
ii) No cloth is a bay
iii) Some students are members
iv) Some green are not white
The parts of proposition are given below.
i) Subject : A subject is the part of the proposition about which something is being said.
ii) Predicate : Predicate is the part of the proposition denoting that which is affirmed
or denied about the subject.
eg : In the proposition All novels are songs, something is being said about novels. So novels is the subject. Songs is the predicate here because it affirmed about the subject.
CLASSIFICATION OF PROPOSITIONS
i) Universal positive proposition: A proposition of the form All S are P is called a universal positive proposition. A universal positive proposition is denoted by A.
eg : All girls are disciplined.
All bulbs are lions.
i) Universal negative proposition :A proposition of the form No S is P is called a universal negative proposition. It is usually denoted by E.
eg : No professors is lazy.
No boxes are baskets.
ii) Particular positive Proposition : A proposition of the form Some S are P is called a particular positive proposition. It is usually denoted by I.
eg : Some boys are smarts.
Some boys are cats.
iv) Particular negative proposition : A proposition of the form Some S are not P is called particular negative proposition. It is denoted by the letter O.
eg : Some flowers are not grapes.
Some fans are not black.
In syllogism, there are two types of inferences.
1) Mediate inference : Here conclusion is drawn from two propositions. For example, if you are given All cats are dogs and All dogs are animals, then a conclusion of the form All cats are animals could
be drawn from it.
2) Immediate inference :
Here conclusion is drawn from only one given proposition. For example if a given statement is All gates are blue, then based on this a conclusion could be drawn that Some blue are gates. This is a case of immediate inference.
Two important cases of immediate inference is given below.
a) Implications : If a given proposition is A - type, then it also implies that the I - type conclusion must be true. Let us verify it by considering the proposition, All elephants are big. This statement
naturally implies that the conclusion Some elephants are big must be true. Similarly we
can prove that an E - type proposition also implies an O - type conclusion.
b) Conversion
Two steps are to be followed in conversion. The first step is to change the subject as the
predicate and the predicate as the subject. The second step is to change the type of the given
proposition to the pattern given in the following table.
Type of the given proposition Type of the proposition after conversion
A I
E E
I I
O Cannot be converted
Let us consider the statement Some posters are good looking. This can be converted by using the above table as Some good looking are posters. In the same way, No books are pencils can be converted as No pencils are books.
HIDDEN PROPOSITION
You may find it difficult to categorise some propositions of the form Rahim is brilliant, Every man talks English, Not a single student passed the exam, No student except Prem was present,etc. We shall know, how to find the hidden propositions in such sentences.
A - type hidden propositions :
• All positive propositions beginning with 'each', 'every' and 'any'.
• A positive sentence with a particular person as its subject.
• A positive sentence with a very definite exception.
eg : Each of them plays football.
He should be awarded.
All members except Kavitha have a share of profit.
E - type hidden proposition
• All negative sentences beginning with 'no one', 'none' and 'not a single'
• A sentence with a particular person as its subject but a negative sense.
• A negative sentence with a very definite exception.
• An interrogative sentence which is used to make an assertion.
eg : None can escape from death.
Swathi is not an IAS officer.
No student except Salim has attend the party. Is there any person who can cheat himself?
I - type hidden propositions :
• Positive propositions beginning with words such as 'most', 'a few', 'mostly', 'generally',
'almost', `frequently', and negative propositions beginning with words such
as 'few', 'seldom', `hardly', 'scarcely','rarely' and 'little'.
• A positive sentence with an exception which is not definite.
eg : Very few writers research before they write.
Seldom are people not jealous.
All students except five have failed.
O - type hidden propositions :
• All negative propositions beginning with words such as 'all', 'every', 'any' and 'each'.
• Negative propositions with words as 'most', 'a few', 'mostly', 'generally', 'almost', and `frequently'.
• Positive words beginning with 'few', 'seldom', 'hardly', scarcely', 'rarely' and little.
• A negative sentence with an exception which is not definite..
e.g. : All men are not honest
Most of the books have not been read.
Girls are usually not brave.
Rarely is a rich man worried.
No students except a few are absent.
EXCLUSIVE PROPOSITIONS
A statement beginning with'only', 'alone','none but' or 'none else but' is called exclusive proposition. Such propositions can be reduced to A or E or I type.
Only brave men are pilots.
This sentence means that "No coward man is a pilot" and "All pilots are brave men".
SOLUTION OF SYLLOGISM BY ANALYTICAL METHOD
There are two steps to be followed for solving syllogism by analytical method. A problem of syllogism consists of two propositions which have one common term. This common term will be the predicate of the first proposition and the subject of the second. If this condition is not satisfied in the given propositions, they should be aligned accordingly.
eg : Statement : All birds are trees.
Some trees are cows.
Here the common term is 'trees'. Also it satisfies the above said condition. Hence the statements are properly aligned.
Let us consider another example.
eg : Statement : All pencils are bottles
All bricks are pencils.
Here the common term is 'pencil'. But it does not satisfy the given condition. So we have to align this pair. This can be aligned easily by changing the order of the statements. The aligned pair will be
All bricks are pencils.
All pencils are bottles.
eg : Statements : No watch is hat
All pins are hats.
In this pair, the common term is 'hat' and it is the predicate of both the sentences. So we have to align the sentences by converting any of the sentences and changing the order if needed. After alignment, the above example will become
All pins are hats
No hat is watch.
While aligning a given pair of statements, the priority should be given while converting, to I- type statements to E-type statements and then to A - type statement, in that order. That is, the rule of IEA should be followed.
After aligning the given pair of statements, the conclusion can be easily drawn by using the following table.
Statement - I Statement - II Conclusion
A + A = A
A + E = E
A + E = O*
E + I = O*
I + A = I
I + E = O
No definite conclusion can be drawn for other combinations like A+I, O+A etc, which are not mentioned in the above table.
For the above given combinations which are aligned properly, the conclusion is a proposition whose subject is the subject of the first statement and whose predicate is the predicate of the second statements. The common terms disappears.
In the above table, O* implies that the conclusion is of type - O, whose subject is the predicate of the second statement and the predicate of the conclusion is the subject of the first statement.
SOLVED EXAMPLES.
1. Statements :All bags are toys.
All toys are keys.
The sentences are already aligned. From the above given Table, A+A=A. Hence the conclusion is of type - A whose subject is the subject of the first proposition and the predicate is the predicate of the second proposition. So the conclusion is All bags are keys.
2. Statements : All teachers are readers.
All teachers are writers.
This pair is not properly aligned because the subject of both the sentences is 'teachers'. Since both the sentences are of type - A, we may convert any of them. So the aligned pair is Some readers are teachers.
All teachers are writers.
Here the conclusion will be of type - I because I+A=I. The conclusion is Some readers are writers.
3. Statements : Some chocolates are toffees.
All chocolates are pastries.
The subject of both the sentences is the same.By the rule of IEA, we convert the I - type statement.
So the aligned pair is, Some toffees are chocolates.All chocolates are pastries I+A=I. So the conclusion is Some toffees are pastries.
4. Statements :All lights are balls
No bats are lights
By changing the order of the statements itself we can align the sentences. The aligned pair is
No bats are lights. All lights are balls.E+A=O*. So the conclusion is,
Some balls are not bats.
5. Statements :Some caps are red.
No clip is red.
Here the common term is 'red' which is the predicate of both the sentences. By the rule of IEA, we convert the I - type statement. After conversion, the given pair becomes,
Some red are caps.
No clip is red.
Now by changing the order of the statements, we can align the sentences. So the aligned pair is,
No clip is red.
Some red are caps.
The conclusion is of type O* since E+I=O*. Hence the conclusion is Some caps are not clips.
6. Statements : Some powders are not soaps.
All soaps are detergents.
The given pair is properly aligned. But no definite conclusion can be drawn from this type
because it is a O+A - type combination.
IMMEDIATE INFERENCE
Now let us consider an example which has two statements as well as two conclusions.
eg. Statements: All novels are stories.
All stories are songs.
Conclusion : (i) All novels are songs.
(ii) Some songs are novels.
First of all let us consider only the statements . The sentences are already aligned.Since A+A = A, the conclusion will be All novels are songs If we convert this conclusion, we get Some songs are novels.Hence both the conclutions given in the question are true.
eg: Statements :Some roses are leaves.
Some leaves are throns.
Conclusions : (i) Some roses are thorns.
(II) Some leaves are roses.
We know that for a combination of I+I - type no conclusion could be drawn. But if we convert the first statement, we get Some leaves are roses. Which is conclusion (ii) Also on converting the second statement, we get some thorns are leaves. This proposition is not given in the conclusion part. So in this example, conclusion (ii) alone is true.
So while solving the problems on syllogism, we should also take the immediate inferences of the given statements as well as the immediate inference of the conclusion drawn from the table.
COMPLEMENTARY PAIR
Consider the following. Conclusions :i) Some buses are trucks.
ii) Some buses are not trucks.
We know that either some buses will be trucks or some buses will not be trucks. Hence either (i) or (ii) is true. Such pair of statements are called complementary pairs. So in a complementary pair, at least one of the two statements is always true. We can call a pair as a complementary pair if
i) The subject and predicate of both the sentences are the same.
ii) They are an I + O - type pair or an A + O type pair or an I + E - type pair.
Some complementary pairs are given below.
i) All birds are swans .
Some birds are not swans.
ii) Some tables are watches.
Some tables are not watches.
iii) Some girls are cute.
No girls are cute.
Note :The steps to be followed to do a syllogism problem by analytical method are mentioned below.
i) Align the sentences properly
ii) Draw conclusion using the table
iii) Check for immediate inferences.
iv) Check for complementary pair if steps ii and iii fail.
SOLVED EXAMPLES
1. Statement : No rooms are stones
Some houses are rooms.
Conclusions : i) Some houses are stones
ii) Some houses are not stones.
We can easily align the statements by changing the order of the sentences. The aligned pair is :
Some houses are rooms.
No rooms are stones.
I + E = O. So the conclusion is Some houses are not stones, Hence we obtain a definite conclusion that conclusion (ii) is correct. Hence step IV becomes unnecessary.
2. Statements :Some cows are horses
All cows are tigers.
Conclusions : i) Some tigers are horses.
ii) Some tigers are cows.
To align the sentences, it is sufficient to convert the first statement. So the aligned pair is
Some horses are cows.
All cows are tigers.
I + A = I. Hence the conclusion will be Some horses are tigers. If we convert this conclusion, we get Some tigers are horses which is conclusion (i). Also if we convert the second statement, conclusion (ii) is obtained. Hence both the conclusions given above should be taken as true. There is no need to
check for complementary pair because definite conclusion has already been obtained.
3. Statements :Some poets are teachers.
Some teachers are saints
Conclusions : i) Some poets are saints.
ii) Some poets are not saints.
This pair is already aligned. But there is no definite conclusion for I + I type combinations. Also none of the given conclusions is the immediate inference of any of the statements. So let us check for the complementary pair. The conclusions given are in the form of 'some' and 'some not'. Hence either conclusion (i) or (ii) follows.
THREE - STATEMENT SYLLOGISM
This type of syllogism problems consist of 3 statements which are followed by 4 or more conclusions. A typical three - statement syllogism problem is given below.
Directions : Below are given three statements followed by several conclusions based on them.
Examine the conclusions and decide whether they logically follow from the given statements.
You have to take the given statements as true even if they appear to be at variance with
commonly known facts.
Statements : A) All bags are hats.
B) Some pins are bags.
C) No hats are needles.
Conclusions : I) Some pins are hats.
II) No needles are bags.
III) Some pins are needles.
IV) Some pins are not needles.
(a) Only I and II follow
(b) Only I and IV follow
(c) I, II and IV follow
(d) Either III or IV, and I follow
(e) Either III or IV and I and II follow.
Before solving this example, let us see the steps in solving a three-statement syllogism problems.
Step I
i) Consider a given conclusion.
ii) Note the subject and predicate of this given conclusion.
iii) Now find which of the two given statements has this subject and predicate.
iv) a) If there is a common term between the two statements chosen in the previous part, then consider only these two statements.
b) If there is no common term between the two statements chosen in the previous part, then we should consider all the three statements.
Step II
i) If two statements are relevant for a given conclusion, align them.
ii) If three statements are relevant, write them as a chain. That is, align them in such a way that the predicate of the first sentence and subject of the second are the same, and the predicate of the second sentence and the subject of the third sentence are the same.
iii) Now arrive at the conclusion using the table.
iv) Now compare the given conclusion with the conclusion drawn using the tables. If they match, the given conclusion is true. If they do not match, it is false.
Step III
i) If a given statement has already been marked as a valid conclusion after step II, then leave it. Otherwise check if it is an immediate inference of any of the three given statements of the conclusion
derived.
ii) Search for complementary pair :
a) Check if any two given conclusions have the same subject and the same predicate.
b) If (a) is satisfied, then check whether any of them has been marked as a valid conclusion after step II or as an immediate inference.
c) If none of them has been marked as a valid conclusion, then they will form a complementary pair if they are an A - O or I - O or I - E pair.
d) If they do make a complementary pair, then mark the choice "either of the two follows".
If a conclusion is marked as a valid conclusion after step II, then it is not necessary to perform step III (i). Again if a given conclusion has already been accepted in step III (i), then it is not necessary to perform step III
(ii). The learner should understand these steps clearly. Now follow the solution to the example which is already given. Here we have to check the validity of each and every conclusions one by one.
Conclusion I : Here the subject is pin and the predicate is hat. So let us consider (A) and (B) as our relevant statements because they have a common term 'bags'. The second step is to align the sentences.
The aligned pair is,
Some pins are bags.
All bags are hats.
I + A = I. So we arrive at the conclusion, 'Some pins are hats'. So conclusion I is valid.
Conclusion II : Here the subject is 'needles' and the predicate is 'bags'. Statement C contains the subject 'Needles'. But 'bags' appears in both A and B. We should select A because there is a common term between A and C. This is an aligned pair and so we arrive at the conclusion No bags are needles which implies No needles are bags. Hence conclusion II is valid.
Conclusion III : Here the subject is 'pins' and the 'predicate' is needles. These words appear in statements (B) and (C) respectively which have no term in common. So all the three statements
should be taken as relevant. Now align the statements as Step II (ii) So we get,
Some pins are bags
All bags are hats.
No hats are needles.
I + A + E = (I + A) + E= I + E = O.
So the conclusion is 'Some pins are not needles', which is conclusion IV. So conclusion IV is valid.
Since conclusion III is not valid in step II, let us perform step III (i). The conclusion, Some pins are not needles is not an immediate inference of any of the three given statements. So the next step is to check the existence of a complementary pair in the given conclusions.We see that conclusion III and conclusion IV form a complementary pair of I - O type. So the choice "either III or IV follows" could be selected. But we find that conclusion IV is valid from the previous step. So conclusion III is not valid. Hence for this given example, the third choice which is 'I, II and IV follow' is true.
Directions : In the following questions, two statements are given followed by two conclusions. You have to study the two statements and then decide which of the conclusions follow from the statements. Mark the right answer from (1), (2), (3), (4) and (5)
Statements : All plants are trees.
No trees are green.
Conclusions : I. Some plants are green.
II. No plants are green.
1) Only I follows
2) Only II follows
3) Both I and II follow
4) Either I or II follows
5) Neither I nor II follows
This is a typical question of syllogism. Here the choice (2) is true. Later on we can discuss
the method to reach at the answer choice. Now let us see some definitions related to syllogism.
PROPOSITION
A proposition is a sentence that makes a statement and gives a relation between two or more terms.
In logic, any statement is termed a proposition.
Eg : i) All windows are rods
ii) No cloth is a bay
iii) Some students are members
iv) Some green are not white
The parts of proposition are given below.
i) Subject : A subject is the part of the proposition about which something is being said.
ii) Predicate : Predicate is the part of the proposition denoting that which is affirmed
or denied about the subject.
eg : In the proposition All novels are songs, something is being said about novels. So novels is the subject. Songs is the predicate here because it affirmed about the subject.
CLASSIFICATION OF PROPOSITIONS
i) Universal positive proposition: A proposition of the form All S are P is called a universal positive proposition. A universal positive proposition is denoted by A.
eg : All girls are disciplined.
All bulbs are lions.
i) Universal negative proposition :A proposition of the form No S is P is called a universal negative proposition. It is usually denoted by E.
eg : No professors is lazy.
No boxes are baskets.
ii) Particular positive Proposition : A proposition of the form Some S are P is called a particular positive proposition. It is usually denoted by I.
eg : Some boys are smarts.
Some boys are cats.
iv) Particular negative proposition : A proposition of the form Some S are not P is called particular negative proposition. It is denoted by the letter O.
eg : Some flowers are not grapes.
Some fans are not black.
In syllogism, there are two types of inferences.
1) Mediate inference : Here conclusion is drawn from two propositions. For example, if you are given All cats are dogs and All dogs are animals, then a conclusion of the form All cats are animals could
be drawn from it.
2) Immediate inference :
Here conclusion is drawn from only one given proposition. For example if a given statement is All gates are blue, then based on this a conclusion could be drawn that Some blue are gates. This is a case of immediate inference.
Two important cases of immediate inference is given below.
a) Implications : If a given proposition is A - type, then it also implies that the I - type conclusion must be true. Let us verify it by considering the proposition, All elephants are big. This statement
naturally implies that the conclusion Some elephants are big must be true. Similarly we
can prove that an E - type proposition also implies an O - type conclusion.
b) Conversion
Two steps are to be followed in conversion. The first step is to change the subject as the
predicate and the predicate as the subject. The second step is to change the type of the given
proposition to the pattern given in the following table.
Type of the given proposition Type of the proposition after conversion
A I
E E
I I
O Cannot be converted
Let us consider the statement Some posters are good looking. This can be converted by using the above table as Some good looking are posters. In the same way, No books are pencils can be converted as No pencils are books.
HIDDEN PROPOSITION
You may find it difficult to categorise some propositions of the form Rahim is brilliant, Every man talks English, Not a single student passed the exam, No student except Prem was present,etc. We shall know, how to find the hidden propositions in such sentences.
A - type hidden propositions :
• All positive propositions beginning with 'each', 'every' and 'any'.
• A positive sentence with a particular person as its subject.
• A positive sentence with a very definite exception.
eg : Each of them plays football.
He should be awarded.
All members except Kavitha have a share of profit.
E - type hidden proposition
• All negative sentences beginning with 'no one', 'none' and 'not a single'
• A sentence with a particular person as its subject but a negative sense.
• A negative sentence with a very definite exception.
• An interrogative sentence which is used to make an assertion.
eg : None can escape from death.
Swathi is not an IAS officer.
No student except Salim has attend the party. Is there any person who can cheat himself?
I - type hidden propositions :
• Positive propositions beginning with words such as 'most', 'a few', 'mostly', 'generally',
'almost', `frequently', and negative propositions beginning with words such
as 'few', 'seldom', `hardly', 'scarcely','rarely' and 'little'.
• A positive sentence with an exception which is not definite.
eg : Very few writers research before they write.
Seldom are people not jealous.
All students except five have failed.
O - type hidden propositions :
• All negative propositions beginning with words such as 'all', 'every', 'any' and 'each'.
• Negative propositions with words as 'most', 'a few', 'mostly', 'generally', 'almost', and `frequently'.
• Positive words beginning with 'few', 'seldom', 'hardly', scarcely', 'rarely' and little.
• A negative sentence with an exception which is not definite..
e.g. : All men are not honest
Most of the books have not been read.
Girls are usually not brave.
Rarely is a rich man worried.
No students except a few are absent.
EXCLUSIVE PROPOSITIONS
A statement beginning with'only', 'alone','none but' or 'none else but' is called exclusive proposition. Such propositions can be reduced to A or E or I type.
Only brave men are pilots.
This sentence means that "No coward man is a pilot" and "All pilots are brave men".
SOLUTION OF SYLLOGISM BY ANALYTICAL METHOD
There are two steps to be followed for solving syllogism by analytical method. A problem of syllogism consists of two propositions which have one common term. This common term will be the predicate of the first proposition and the subject of the second. If this condition is not satisfied in the given propositions, they should be aligned accordingly.
eg : Statement : All birds are trees.
Some trees are cows.
Here the common term is 'trees'. Also it satisfies the above said condition. Hence the statements are properly aligned.
Let us consider another example.
eg : Statement : All pencils are bottles
All bricks are pencils.
Here the common term is 'pencil'. But it does not satisfy the given condition. So we have to align this pair. This can be aligned easily by changing the order of the statements. The aligned pair will be
All bricks are pencils.
All pencils are bottles.
eg : Statements : No watch is hat
All pins are hats.
In this pair, the common term is 'hat' and it is the predicate of both the sentences. So we have to align the sentences by converting any of the sentences and changing the order if needed. After alignment, the above example will become
All pins are hats
No hat is watch.
While aligning a given pair of statements, the priority should be given while converting, to I- type statements to E-type statements and then to A - type statement, in that order. That is, the rule of IEA should be followed.
After aligning the given pair of statements, the conclusion can be easily drawn by using the following table.
Statement - I Statement - II Conclusion
A + A = A
A + E = E
A + E = O*
E + I = O*
I + A = I
I + E = O
No definite conclusion can be drawn for other combinations like A+I, O+A etc, which are not mentioned in the above table.
For the above given combinations which are aligned properly, the conclusion is a proposition whose subject is the subject of the first statement and whose predicate is the predicate of the second statements. The common terms disappears.
In the above table, O* implies that the conclusion is of type - O, whose subject is the predicate of the second statement and the predicate of the conclusion is the subject of the first statement.
SOLVED EXAMPLES.
1. Statements :All bags are toys.
All toys are keys.
The sentences are already aligned. From the above given Table, A+A=A. Hence the conclusion is of type - A whose subject is the subject of the first proposition and the predicate is the predicate of the second proposition. So the conclusion is All bags are keys.
2. Statements : All teachers are readers.
All teachers are writers.
This pair is not properly aligned because the subject of both the sentences is 'teachers'. Since both the sentences are of type - A, we may convert any of them. So the aligned pair is Some readers are teachers.
All teachers are writers.
Here the conclusion will be of type - I because I+A=I. The conclusion is Some readers are writers.
3. Statements : Some chocolates are toffees.
All chocolates are pastries.
The subject of both the sentences is the same.By the rule of IEA, we convert the I - type statement.
So the aligned pair is, Some toffees are chocolates.All chocolates are pastries I+A=I. So the conclusion is Some toffees are pastries.
4. Statements :All lights are balls
No bats are lights
By changing the order of the statements itself we can align the sentences. The aligned pair is
No bats are lights. All lights are balls.E+A=O*. So the conclusion is,
Some balls are not bats.
5. Statements :Some caps are red.
No clip is red.
Here the common term is 'red' which is the predicate of both the sentences. By the rule of IEA, we convert the I - type statement. After conversion, the given pair becomes,
Some red are caps.
No clip is red.
Now by changing the order of the statements, we can align the sentences. So the aligned pair is,
No clip is red.
Some red are caps.
The conclusion is of type O* since E+I=O*. Hence the conclusion is Some caps are not clips.
6. Statements : Some powders are not soaps.
All soaps are detergents.
The given pair is properly aligned. But no definite conclusion can be drawn from this type
because it is a O+A - type combination.
IMMEDIATE INFERENCE
Now let us consider an example which has two statements as well as two conclusions.
eg. Statements: All novels are stories.
All stories are songs.
Conclusion : (i) All novels are songs.
(ii) Some songs are novels.
First of all let us consider only the statements . The sentences are already aligned.Since A+A = A, the conclusion will be All novels are songs If we convert this conclusion, we get Some songs are novels.Hence both the conclutions given in the question are true.
eg: Statements :Some roses are leaves.
Some leaves are throns.
Conclusions : (i) Some roses are thorns.
(II) Some leaves are roses.
We know that for a combination of I+I - type no conclusion could be drawn. But if we convert the first statement, we get Some leaves are roses. Which is conclusion (ii) Also on converting the second statement, we get some thorns are leaves. This proposition is not given in the conclusion part. So in this example, conclusion (ii) alone is true.
So while solving the problems on syllogism, we should also take the immediate inferences of the given statements as well as the immediate inference of the conclusion drawn from the table.
COMPLEMENTARY PAIR
Consider the following. Conclusions :i) Some buses are trucks.
ii) Some buses are not trucks.
We know that either some buses will be trucks or some buses will not be trucks. Hence either (i) or (ii) is true. Such pair of statements are called complementary pairs. So in a complementary pair, at least one of the two statements is always true. We can call a pair as a complementary pair if
i) The subject and predicate of both the sentences are the same.
ii) They are an I + O - type pair or an A + O type pair or an I + E - type pair.
Some complementary pairs are given below.
i) All birds are swans .
Some birds are not swans.
ii) Some tables are watches.
Some tables are not watches.
iii) Some girls are cute.
No girls are cute.
Note :The steps to be followed to do a syllogism problem by analytical method are mentioned below.
i) Align the sentences properly
ii) Draw conclusion using the table
iii) Check for immediate inferences.
iv) Check for complementary pair if steps ii and iii fail.
SOLVED EXAMPLES
1. Statement : No rooms are stones
Some houses are rooms.
Conclusions : i) Some houses are stones
ii) Some houses are not stones.
We can easily align the statements by changing the order of the sentences. The aligned pair is :
Some houses are rooms.
No rooms are stones.
I + E = O. So the conclusion is Some houses are not stones, Hence we obtain a definite conclusion that conclusion (ii) is correct. Hence step IV becomes unnecessary.
2. Statements :Some cows are horses
All cows are tigers.
Conclusions : i) Some tigers are horses.
ii) Some tigers are cows.
To align the sentences, it is sufficient to convert the first statement. So the aligned pair is
Some horses are cows.
All cows are tigers.
I + A = I. Hence the conclusion will be Some horses are tigers. If we convert this conclusion, we get Some tigers are horses which is conclusion (i). Also if we convert the second statement, conclusion (ii) is obtained. Hence both the conclusions given above should be taken as true. There is no need to
check for complementary pair because definite conclusion has already been obtained.
3. Statements :Some poets are teachers.
Some teachers are saints
Conclusions : i) Some poets are saints.
ii) Some poets are not saints.
This pair is already aligned. But there is no definite conclusion for I + I type combinations. Also none of the given conclusions is the immediate inference of any of the statements. So let us check for the complementary pair. The conclusions given are in the form of 'some' and 'some not'. Hence either conclusion (i) or (ii) follows.
THREE - STATEMENT SYLLOGISM
This type of syllogism problems consist of 3 statements which are followed by 4 or more conclusions. A typical three - statement syllogism problem is given below.
Directions : Below are given three statements followed by several conclusions based on them.
Examine the conclusions and decide whether they logically follow from the given statements.
You have to take the given statements as true even if they appear to be at variance with
commonly known facts.
Statements : A) All bags are hats.
B) Some pins are bags.
C) No hats are needles.
Conclusions : I) Some pins are hats.
II) No needles are bags.
III) Some pins are needles.
IV) Some pins are not needles.
(a) Only I and II follow
(b) Only I and IV follow
(c) I, II and IV follow
(d) Either III or IV, and I follow
(e) Either III or IV and I and II follow.
Before solving this example, let us see the steps in solving a three-statement syllogism problems.
Step I
i) Consider a given conclusion.
ii) Note the subject and predicate of this given conclusion.
iii) Now find which of the two given statements has this subject and predicate.
iv) a) If there is a common term between the two statements chosen in the previous part, then consider only these two statements.
b) If there is no common term between the two statements chosen in the previous part, then we should consider all the three statements.
Step II
i) If two statements are relevant for a given conclusion, align them.
ii) If three statements are relevant, write them as a chain. That is, align them in such a way that the predicate of the first sentence and subject of the second are the same, and the predicate of the second sentence and the subject of the third sentence are the same.
iii) Now arrive at the conclusion using the table.
iv) Now compare the given conclusion with the conclusion drawn using the tables. If they match, the given conclusion is true. If they do not match, it is false.
Step III
i) If a given statement has already been marked as a valid conclusion after step II, then leave it. Otherwise check if it is an immediate inference of any of the three given statements of the conclusion
derived.
ii) Search for complementary pair :
a) Check if any two given conclusions have the same subject and the same predicate.
b) If (a) is satisfied, then check whether any of them has been marked as a valid conclusion after step II or as an immediate inference.
c) If none of them has been marked as a valid conclusion, then they will form a complementary pair if they are an A - O or I - O or I - E pair.
d) If they do make a complementary pair, then mark the choice "either of the two follows".
If a conclusion is marked as a valid conclusion after step II, then it is not necessary to perform step III (i). Again if a given conclusion has already been accepted in step III (i), then it is not necessary to perform step III
(ii). The learner should understand these steps clearly. Now follow the solution to the example which is already given. Here we have to check the validity of each and every conclusions one by one.
Conclusion I : Here the subject is pin and the predicate is hat. So let us consider (A) and (B) as our relevant statements because they have a common term 'bags'. The second step is to align the sentences.
The aligned pair is,
Some pins are bags.
All bags are hats.
I + A = I. So we arrive at the conclusion, 'Some pins are hats'. So conclusion I is valid.
Conclusion II : Here the subject is 'needles' and the predicate is 'bags'. Statement C contains the subject 'Needles'. But 'bags' appears in both A and B. We should select A because there is a common term between A and C. This is an aligned pair and so we arrive at the conclusion No bags are needles which implies No needles are bags. Hence conclusion II is valid.
Conclusion III : Here the subject is 'pins' and the 'predicate' is needles. These words appear in statements (B) and (C) respectively which have no term in common. So all the three statements
should be taken as relevant. Now align the statements as Step II (ii) So we get,
Some pins are bags
All bags are hats.
No hats are needles.
I + A + E = (I + A) + E= I + E = O.
So the conclusion is 'Some pins are not needles', which is conclusion IV. So conclusion IV is valid.
Since conclusion III is not valid in step II, let us perform step III (i). The conclusion, Some pins are not needles is not an immediate inference of any of the three given statements. So the next step is to check the existence of a complementary pair in the given conclusions.We see that conclusion III and conclusion IV form a complementary pair of I - O type. So the choice "either III or IV follows" could be selected. But we find that conclusion IV is valid from the previous step. So conclusion III is not valid. Hence for this given example, the third choice which is 'I, II and IV follow' is true.
Wednesday 29 October 2014
Syllogism
Tip #1
Solve the questions through a Venn Diagram. Always make sure common areas are shaded do give you a correct answer.
Tip #2
Shortcut rules (if Venn Diagrams are confusing you) between Statement 1 and Statement 2 in that order
Tip #3
You can cancel out common terms in two statements given, then on the remaining terms apply the syllogisms rules and solve.
E.g. Some dogs are goats, All goats are cows. Cancel out "goats" which leaves us with Some dogs are...all are cows. Important words remaining are ALL and SOME in that order.
SOME + ALL = SOME, hence conclusion is SOME dogs are cows.
Tip #4
Interchange between reading the question as well as the conclusion before arriving at the answers. Always evaluate each and every conclusion to find out how many conclusions are possible.
Tip #5
Avoid using common knowledge as Syllogisms questions usually state unnatural statements
Tip #6
Remember some implications All <=> Some, e.g. All A are B also implies Some A are B (being a subset) and Some B are A Some <=> Some, e.g. Some A are B also implies Some B are A No<=> No, e.g. No A are B also implies No B are A.
In each question below, there are two or three statements followed by four conclusions numbered I, II, III and IV. You have to take the given statements to be true even if they seem to be at variance with commonly known facts and then decide which of the given conclusions logically follow(s) from
the given statements.
1. Statements: Some boys are girls.
All girls are students.
Conclusions:
I. Some boys are students.
II. Some students are boys.
III. Some students are girls.
IV. All students are girls.
(a) I, II and III follow
(b) II, III and IV follow
(c) I, III and IV follow
(d) I, II and IV follow
(e) All follow
2. Statements: All books are watches.
Some watches are clips.
Conclusions:
I. Some watches are books.
II. No watches are books.
III. Some books are clips.
IV. No books are clips.
(a) I,and III follow
(b) Only I follow
(c) Either I or II follows
(d) Either III or IV and I follow
(e) Either I or II and III follow.
3. Statements:
A. All thieves are men.
B. All men are graduates.
C. No graduates are employed.
Conclusions :
I. Some graduates are thieves.
II. No employed are thieves.
III. Some men are thieves.
IV. Some employed are men.
(a) I, II and III follow
(b) II, III and IV follow
(c) Only I and II follow
(d) Only II and II follow
(e) Only II and IV follow.
4. Statements:
A. Some books are pens
B. All tables are chairs.
C. No pens are tables.
Conclusions:
I. Some books are not tables.
II. Some pens are not chairs.
III. Some books are not chairs.
IV. Some chairs are not pens
(a) I, and IV follow
(b) II, and IV follow
(c) I and III follow
(d) II and III follow
(e) III, and IV follow.
5. Statements: All pigs are elephants.
No pigs are bakers.
Conclusions :
I. Some bakers are not pigs.
II. Some pigs are not bakers.
III. Some elephant are not bakers.
IV. Some bakers are not elephants
(a) I, II and III follow
(b) I, II and IV follow
(c) I, III and IV follow
(d) II, III and IV follow
(e) All follow
6. Statements:
A. All books are notes.
B. Some notes are pencils.
C. No pencils are papers.
Conclusions:
I. Some notes are books.
II. Some pencils are books.
III. Some books are papers.
IV. No books are papers.
(a) Only I follows
(b) Only I and either III or IV follows
(c) Either III or IV follows
(d) Only I and III follow
(e) None of these.
Solutions:
1. a. I + A = I. The conclusion is Some boys are students which is I. This can be converted to Some students are boys, which is II. Some students are girls follows from All girls are students.
2. d. A+ I pair has no definite conclusion. But conclusion I follows directly from All books are watches. III and IV are a complementary pair.
3. a. Conclusion drawn from statements A and B is All thieves are graduates. Conclusion I is obvious from this sentence. III is obvious from statement A. Also A + A + E = A + E = E. Conclusion drawn from all the three is No thieves are employed. II is obvious from this sentence.
4. a. I follows from statements A and C. IV follows from statements C and B. No other conclusion is possible
5. a. The aligned pair is No backers are pigs. All pigs are elephants. E + A = O*. Hence the conclusion is Some elephants are not bakers. Thus III follows. No pigs are bakers implies that No bakers are pigs. I is obvious from this sentence. II follows directly from No pigs are bakers. Hence I, II and III follow.
6. b. Conversion of A gives Some notes are books. Therefore I follows. For II A and B are the relevant statements. But A + I - type pair has no conclusion. So II is not valid. For III all the statements are relevant. No definite conclusion can be obtained from this combination. But conclusions III and IV form a complementary pair.Hence either conclusion III or conclusion IV follows.
Solve the questions through a Venn Diagram. Always make sure common areas are shaded do give you a correct answer.
Tip #2
Shortcut rules (if Venn Diagrams are confusing you) between Statement 1 and Statement 2 in that order
- All + All = All
- All + No = No
- All + Some = No Conclusion
- Some + All = Some
- Some + Some = No Conclusion
- Some + No = Some Not
- No + No = No Conclusion
- No + All = Some not reversed
- No + Some = Some not reversed
Tip #3
You can cancel out common terms in two statements given, then on the remaining terms apply the syllogisms rules and solve.
E.g. Some dogs are goats, All goats are cows. Cancel out "goats" which leaves us with Some dogs are...all are cows. Important words remaining are ALL and SOME in that order.
SOME + ALL = SOME, hence conclusion is SOME dogs are cows.
Tip #4
Interchange between reading the question as well as the conclusion before arriving at the answers. Always evaluate each and every conclusion to find out how many conclusions are possible.
Tip #5
Avoid using common knowledge as Syllogisms questions usually state unnatural statements
Tip #6
Remember some implications All <=> Some, e.g. All A are B also implies Some A are B (being a subset) and Some B are A Some <=> Some, e.g. Some A are B also implies Some B are A No<=> No, e.g. No A are B also implies No B are A.
In each question below, there are two or three statements followed by four conclusions numbered I, II, III and IV. You have to take the given statements to be true even if they seem to be at variance with commonly known facts and then decide which of the given conclusions logically follow(s) from
the given statements.
1. Statements: Some boys are girls.
All girls are students.
Conclusions:
I. Some boys are students.
II. Some students are boys.
III. Some students are girls.
IV. All students are girls.
(a) I, II and III follow
(b) II, III and IV follow
(c) I, III and IV follow
(d) I, II and IV follow
(e) All follow
2. Statements: All books are watches.
Some watches are clips.
Conclusions:
I. Some watches are books.
II. No watches are books.
III. Some books are clips.
IV. No books are clips.
(a) I,and III follow
(b) Only I follow
(c) Either I or II follows
(d) Either III or IV and I follow
(e) Either I or II and III follow.
3. Statements:
A. All thieves are men.
B. All men are graduates.
C. No graduates are employed.
Conclusions :
I. Some graduates are thieves.
II. No employed are thieves.
III. Some men are thieves.
IV. Some employed are men.
(a) I, II and III follow
(b) II, III and IV follow
(c) Only I and II follow
(d) Only II and II follow
(e) Only II and IV follow.
4. Statements:
A. Some books are pens
B. All tables are chairs.
C. No pens are tables.
Conclusions:
I. Some books are not tables.
II. Some pens are not chairs.
III. Some books are not chairs.
IV. Some chairs are not pens
(a) I, and IV follow
(b) II, and IV follow
(c) I and III follow
(d) II and III follow
(e) III, and IV follow.
5. Statements: All pigs are elephants.
No pigs are bakers.
Conclusions :
I. Some bakers are not pigs.
II. Some pigs are not bakers.
III. Some elephant are not bakers.
IV. Some bakers are not elephants
(a) I, II and III follow
(b) I, II and IV follow
(c) I, III and IV follow
(d) II, III and IV follow
(e) All follow
6. Statements:
A. All books are notes.
B. Some notes are pencils.
C. No pencils are papers.
Conclusions:
I. Some notes are books.
II. Some pencils are books.
III. Some books are papers.
IV. No books are papers.
(a) Only I follows
(b) Only I and either III or IV follows
(c) Either III or IV follows
(d) Only I and III follow
(e) None of these.
Solutions:
1. a. I + A = I. The conclusion is Some boys are students which is I. This can be converted to Some students are boys, which is II. Some students are girls follows from All girls are students.
2. d. A+ I pair has no definite conclusion. But conclusion I follows directly from All books are watches. III and IV are a complementary pair.
3. a. Conclusion drawn from statements A and B is All thieves are graduates. Conclusion I is obvious from this sentence. III is obvious from statement A. Also A + A + E = A + E = E. Conclusion drawn from all the three is No thieves are employed. II is obvious from this sentence.
4. a. I follows from statements A and C. IV follows from statements C and B. No other conclusion is possible
5. a. The aligned pair is No backers are pigs. All pigs are elephants. E + A = O*. Hence the conclusion is Some elephants are not bakers. Thus III follows. No pigs are bakers implies that No bakers are pigs. I is obvious from this sentence. II follows directly from No pigs are bakers. Hence I, II and III follow.
6. b. Conversion of A gives Some notes are books. Therefore I follows. For II A and B are the relevant statements. But A + I - type pair has no conclusion. So II is not valid. For III all the statements are relevant. No definite conclusion can be obtained from this combination. But conclusions III and IV form a complementary pair.Hence either conclusion III or conclusion IV follows.
Data interpretation
Data Interpretation is one of the easy sections of one day competitive Examinations. It is an extension of Mathematical skill and accuracy. Data interpretation is nothing but drawing conclusions and inferences from a comprehensive data presented numerically in tabular form by means of an illustration, viz. Graphs, Pie Chart etc. Thus the act of organising and interpreting data to get meaningful information is Data Interpretation. A good grasp of basic geometric as well as arithmetic formulae is must to score high in this section. Familiarity with graphical representation of data like Venn diagrams, graphs,pie charts, histogram, polygon etc. should be thought. Once the data are grasped well, questions based on tables and graphs take little time.
In some competitive examinations data are presented in more than one table or graphs. The aim is to test not only quantitative skill but also relative, comparative and analytical ability. The crux of the matter is to find a relationship between the two tables or graphs before attempting the questions.
Some Useful tips:
1 . Data Interpretation questions are based on information given in tables and graphs.These questions test your ability to interpret the information presented and to select the appropriate data for answering
a question.
2 . Get a general picture of the information before reading the question. Read the given titles carefully and try to understand its nature.
3 . Avoid lengthy calculations generally, data interpretation questions do not require to do extensive calculations and computations. Most questions simply require reading the data correctly and carefully and putting them to use directly with common sense.
4 . Breakdown lengthy questions into smaller parts and eliminate impossible choices.
5 . Use only the information given and your knowledge of everyday facts, such as the number of hours in a day, to answer the questions based on tables and graphs.
6 . Answer the questions asked and not what you think the questions should be.
7 . Be careful while dealing with units.
8 . To make reading easier and to avoid errors
observe graphs keeping them straight.
9 . Be prepared to apply basic mathematical rules, principles and formulae.
10. Since one of the major benefits of graphs and tables is that they present data in a form that enables you to readily make comparisons, use this visual attribute of graphs and tables to help you answer the questions. Where possible, use your eyes instead of your computational skills.
Tables
Tables are often used in reports, magazines and newspaper to present a set of numerical facts. They enable the reader to make comparisons and to draw quick conclusions. It is one of the easiest and
most accurate ways of presenting data. They require much closer reading than graphs of charts and hence are difficult and time consuming to interpret. One of the main purposes of tables is to make complicated information easier to understand. The advantage of presenting data in a table is that one can see the information at a glance. While answering questions based on tables, carefully read the table title and the column headings. The title of the table gives you a general idea of the type and
often the purpose of the information presented. The column headings tell you the specific kind of information given in that column. Both the table title and the column headings are usually very straight forward.
Graphs
There may be four types of graphs.
1) Circle Graphs: Circle graphs are used to show how various sectors are in the whole. Circle graphs are sometimes called Pie Charts. Circle graphs usually give the percent that each sector receives In such representation the total quantity in question is distributed over a total angle of 360°. While using circle graphs to find ratios of various sectors, don't find the amounts each sector received and then
the ratio of the amounts. Find the ratio of the percents, which is much quicker.
2) Line Graphs: Line graphs are used to show how a quantity changes continuously. If the line goes up, the quantity is increasing; if the line goes down, the quantity is decreasing; if the line is horizontal, the quantity is not changing.
3) Bar Graphs: Given quantities can be compared by the height or length of a bar graph. A bar graph can have either vertical or horizontal bars. You can compare different quantities or the same quantity
at different times. In bar graph the data is discrete. Presentation of data in this form makes evaluation of parameters comparatively very easy.
4) Cumulative Graphs : You can compare several catagories by a graph of the cumulative type. These are usually bar or line graphs where the height of the bar or line is divided up proportionally among different quantities.
I.Bar Graph:
A. If the expenditure of Company A in 2004 is Rs 36 lakhs, then what is its income in that year?
(1) 42 lakhs (2) 48 lakhs (3) 54 lakhs (4) 60 lakhs (5) 75 lakhs
B. If the income of Company A in 2002 and expenditure of Company B in the same year is equal to Rs 60 lakhs then what is the difference between their net profit in 2002?
(1) 6 lakhs (2) 8 lakhs (3) 10 lakhs (4) 12 lakhs (5) None of these
C. If the income of Company A in 2001 and income of Company B in 2005 is Rs 50 lakhs and Rs 80 lakhs respectively then the profit gained by Company A in 2001 is how much percent more than that of the profit gained by Company B in 2005?
(1) 62.5% (2) 67.5% (3) 82.5% (4) 87.5% (5) 75%
D. Ratio of expenditure to income of Company B in 2004 is how much percent more than that of ratio of expenditure to income of Company A in 2005?
(1) 10% (2) 20% (3) 30% (4) 40% (5) 50%
E. If income of Company A in 2006 is Rs 75 lakhs then what is the expenditure of Company B in the same year?
(1) 60 lakhs (2) 75 lakhs (3) 90 lakhs (4) 87.5 lakhs (5) None of these
Line chart: Following line graph shows the expenditure and percentage profit of a company of 2005 to 2010.
F. What is the ratio of income in 2008 and 2010?
(1) 5 : 6 (2) 6 : 7 (3) 7 : 8 (4) 8 : 9 (5) 7 : 9
G. In which year amount of profit is minimum?
(1) 2005 (2) 2006 (3) 2007 (4) 2008 (5) 2009
H. What is the percentage rise in amount of profit gained by Company from 2007 to 2008?
(1) 27% (2) 36% (3) 44% (4) 62.5% (5) 87.5%
I. Expenditure of Company in year 2008 is how much percent more than that of expenditure in 2006?
(1) 15% (2) 20% (3) 25% (4) 27.5% (5) 30%
J. Income of Company in 2005 in what percentage of income of Company in 2010?
(1) 200/3% (2) 279/5% (3) 105/2% (4) 403/9% (5) 320/5%
Table: The table given below gives the number of days worked by employees of five grades A, B, C, D and E in different departments.
K.
The number of days worked in HR department was highest for which grade ?
1) A 2) B 3) C 4) D 5) E
L. The grade which worked least in all departments is ______?
1) Grade E in IT 2) Grade B in HR 3) Grade B in Software
4) Grade D in IT 5) Grade B in Accounts
M. What is the average of working days in IT department?
1) 305 2) 300 3) 296 4) 292 5) None of these \
N. If average working hours in a day are 8 then the amount of work put by Grade C is
1) 13240 2) 12570 3) 32600 4) 12480 5) 9912
M. If working hours in a day are 8 then average work done (in hours) by Grade A in four departments together is
1) 1267 2) 2534 3) 2436 4) 1672 5) None of these
In some competitive examinations data are presented in more than one table or graphs. The aim is to test not only quantitative skill but also relative, comparative and analytical ability. The crux of the matter is to find a relationship between the two tables or graphs before attempting the questions.
Some Useful tips:
1 . Data Interpretation questions are based on information given in tables and graphs.These questions test your ability to interpret the information presented and to select the appropriate data for answering
a question.
2 . Get a general picture of the information before reading the question. Read the given titles carefully and try to understand its nature.
3 . Avoid lengthy calculations generally, data interpretation questions do not require to do extensive calculations and computations. Most questions simply require reading the data correctly and carefully and putting them to use directly with common sense.
4 . Breakdown lengthy questions into smaller parts and eliminate impossible choices.
5 . Use only the information given and your knowledge of everyday facts, such as the number of hours in a day, to answer the questions based on tables and graphs.
6 . Answer the questions asked and not what you think the questions should be.
7 . Be careful while dealing with units.
8 . To make reading easier and to avoid errors
observe graphs keeping them straight.
9 . Be prepared to apply basic mathematical rules, principles and formulae.
10. Since one of the major benefits of graphs and tables is that they present data in a form that enables you to readily make comparisons, use this visual attribute of graphs and tables to help you answer the questions. Where possible, use your eyes instead of your computational skills.
Tables
Tables are often used in reports, magazines and newspaper to present a set of numerical facts. They enable the reader to make comparisons and to draw quick conclusions. It is one of the easiest and
most accurate ways of presenting data. They require much closer reading than graphs of charts and hence are difficult and time consuming to interpret. One of the main purposes of tables is to make complicated information easier to understand. The advantage of presenting data in a table is that one can see the information at a glance. While answering questions based on tables, carefully read the table title and the column headings. The title of the table gives you a general idea of the type and
often the purpose of the information presented. The column headings tell you the specific kind of information given in that column. Both the table title and the column headings are usually very straight forward.
Graphs
There may be four types of graphs.
1) Circle Graphs: Circle graphs are used to show how various sectors are in the whole. Circle graphs are sometimes called Pie Charts. Circle graphs usually give the percent that each sector receives In such representation the total quantity in question is distributed over a total angle of 360°. While using circle graphs to find ratios of various sectors, don't find the amounts each sector received and then
the ratio of the amounts. Find the ratio of the percents, which is much quicker.
2) Line Graphs: Line graphs are used to show how a quantity changes continuously. If the line goes up, the quantity is increasing; if the line goes down, the quantity is decreasing; if the line is horizontal, the quantity is not changing.
3) Bar Graphs: Given quantities can be compared by the height or length of a bar graph. A bar graph can have either vertical or horizontal bars. You can compare different quantities or the same quantity
at different times. In bar graph the data is discrete. Presentation of data in this form makes evaluation of parameters comparatively very easy.
4) Cumulative Graphs : You can compare several catagories by a graph of the cumulative type. These are usually bar or line graphs where the height of the bar or line is divided up proportionally among different quantities.
I.Bar Graph:
(1) 42 lakhs (2) 48 lakhs (3) 54 lakhs (4) 60 lakhs (5) 75 lakhs
B. If the income of Company A in 2002 and expenditure of Company B in the same year is equal to Rs 60 lakhs then what is the difference between their net profit in 2002?
(1) 6 lakhs (2) 8 lakhs (3) 10 lakhs (4) 12 lakhs (5) None of these
C. If the income of Company A in 2001 and income of Company B in 2005 is Rs 50 lakhs and Rs 80 lakhs respectively then the profit gained by Company A in 2001 is how much percent more than that of the profit gained by Company B in 2005?
(1) 62.5% (2) 67.5% (3) 82.5% (4) 87.5% (5) 75%
D. Ratio of expenditure to income of Company B in 2004 is how much percent more than that of ratio of expenditure to income of Company A in 2005?
(1) 10% (2) 20% (3) 30% (4) 40% (5) 50%
E. If income of Company A in 2006 is Rs 75 lakhs then what is the expenditure of Company B in the same year?
(1) 60 lakhs (2) 75 lakhs (3) 90 lakhs (4) 87.5 lakhs (5) None of these
Line chart: Following line graph shows the expenditure and percentage profit of a company of 2005 to 2010.
(1) 5 : 6 (2) 6 : 7 (3) 7 : 8 (4) 8 : 9 (5) 7 : 9
G. In which year amount of profit is minimum?
(1) 2005 (2) 2006 (3) 2007 (4) 2008 (5) 2009
H. What is the percentage rise in amount of profit gained by Company from 2007 to 2008?
(1) 27% (2) 36% (3) 44% (4) 62.5% (5) 87.5%
I. Expenditure of Company in year 2008 is how much percent more than that of expenditure in 2006?
(1) 15% (2) 20% (3) 25% (4) 27.5% (5) 30%
J. Income of Company in 2005 in what percentage of income of Company in 2010?
(1) 200/3% (2) 279/5% (3) 105/2% (4) 403/9% (5) 320/5%
Table: The table given below gives the number of days worked by employees of five grades A, B, C, D and E in different departments.
1) A 2) B 3) C 4) D 5) E
L. The grade which worked least in all departments is ______?
1) Grade E in IT 2) Grade B in HR 3) Grade B in Software
4) Grade D in IT 5) Grade B in Accounts
M. What is the average of working days in IT department?
1) 305 2) 300 3) 296 4) 292 5) None of these \
N. If average working hours in a day are 8 then the amount of work put by Grade C is
1) 13240 2) 12570 3) 32600 4) 12480 5) 9912
M. If working hours in a day are 8 then average work done (in hours) by Grade A in four departments together is
1) 1267 2) 2534 3) 2436 4) 1672 5) None of these
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