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Logical Reasoning · Logical Connectives, Syllogisms and Venn Diagrams

Venn Diagram - Formula, Key Point and Examples

🔵

\text{VENN DIAGRAM}

🟢 1. BASIC CONCEPT A Venn diagram is a graphical representation of sets using circles or closed curves.

\text{Venn Diagram} = \text{Graphical representation of sets}

The important concepts are:

\text{Union},\quad \text{Intersection},\quad \text{Difference},\quad \text{Complement}

🟢 2. SET A set is a collection of well-defined objects. For example:

A=\{1,2,3,4,5\}

The number of elements in set \(A\) is:

n(A)=5

🟢 3. UNIVERSAL SET The universal set contains all the elements under consideration. It is represented by:

U

🟢 4. UNION OF TWO SETS The union of sets \(A\) and \(B\) contains all elements belonging to \(A\), \(B\), or both. It is represented by:

A\cup B

The formula is:

n(A\cup B) = n(A)+n(B)-n(A\cap B)

🟢 5. INTERSECTION OF TWO SETS The intersection contains the elements common to both sets. It is represented by:

A\cap B

🟢 6. DIFFERENCE OF SETS The difference \(A-B\) contains elements that belong to \(A\) but not to \(B\).

A-B

Number of elements:

n(A-B) = n(A)-n(A\cap B)

Similarly:

n(B-A) = n(B)-n(A\cap B)

🟢 7. COMPLEMENT OF A SET The complement of \(A\) contains all elements of the universal set that are not in \(A\). It is represented by:

A'

or:

A^c

Formula:

n(A') = n(U)-n(A)

🟢 8. TWO-SET VENN DIAGRAM For two sets \(A\) and \(B\), the important regions are:

A\text{ only}

B\text{ only}

A\cap B

\text{Neither }A\text{ nor }B

🟢 9. ONLY A The elements belonging to \(A\) but not to \(B\) are:

A-B

Therefore:

n(A\text{ only}) = n(A)-n(A\cap B)

🟢 10. ONLY B The elements belonging to \(B\) but not to \(A\) are:

B-A

Therefore:

n(B\text{ only}) = n(B)-n(A\cap B)

🟢 11. NEITHER A NOR B Elements belonging to neither \(A\) nor \(B\) are outside the union.

n(\text{Neither}) = n(U)-n(A\cup B)

Therefore:

n(\text{Neither}) = n(U)-n(A)-n(B)+n(A\cap B)

🟢 12. UNION FORMULA For two sets:

n(A\cup B) = n(A)+n(B)-n(A\cap B)

🔴 IMPORTANT The intersection is subtracted because common elements are counted twice.

\text{Union} = \text{First Set} + \text{Second Set} - \text{Common Elements}

🟢 13. FINDING THE INTERSECTION From the union formula:

n(A\cap B) = n(A)+n(B)-n(A\cup B)

🟢 14. DISJOINT SETS Two sets are disjoint if they have no common elements. Therefore:

A\cap B=\varnothing

and:

n(A\cap B)=0

Hence:

n(A\cup B) = n(A)+n(B)

🟢 15. THREE-SET VENN DIAGRAM For three sets \(A\), \(B\), and \(C\), the important regions are:

A\text{ only}

B\text{ only}

C\text{ only}

A\cap B\text{ only}

A\cap C\text{ only}

B\cap C\text{ only}

A\cap B\cap C

and:

\text{None}

🟢 16. THREE-SET UNION FORMULA For three sets:

n(A\cup B\cup C) = n(A)+n(B)+n(C)

-n(A\cap B) -n(A\cap C) -n(B\cap C)

+n(A\cap B\cap C)

🔴 IMPORTANT For three sets, the common intersection of all three sets is added once. 🟢 17. ONLY A IN THREE SETS The number belonging only to \(A\) is:

n(A\text{ only}) = n(A) -n(A\cap B) -n(A\cap C) +n(A\cap B\cap C)

🟢 18. ONLY B IN THREE SETS

n(B\text{ only}) = n(B) -n(A\cap B) -n(B\cap C) +n(A\cap B\cap C)

🟢 19. ONLY C IN THREE SETS

n(C\text{ only}) = n(C) -n(A\cap C) -n(B\cap C) +n(A\cap B\cap C)

🟢 20. A AND B ONLY Elements belonging to \(A\) and \(B\), but not \(C\):

n(A\cap B\text{ only}) = n(A\cap B) -n(A\cap B\cap C)

🟢 21. A AND C ONLY

n(A\cap C\text{ only}) = n(A\cap C) -n(A\cap B\cap C)

🟢 22. B AND C ONLY

n(B\cap C\text{ only}) = n(B\cap C) -n(A\cap B\cap C)

🟢 23. ALL THREE SETS Elements belonging to all three sets are represented by:

A\cap B\cap C

Therefore:

n(\text{All Three}) = n(A\cap B\cap C)

🟢 24. NONE OF THE THREE SETS The number of elements belonging to none of the three sets is:

n(\text{None}) = n(U)-n(A\cup B\cup C)

🟢 25. AT LEAST ONE "At least one" means belonging to one or more sets. Therefore:

\text{At Least One} = A\cup B\cup C

Hence:

n(\text{At Least One}) = n(A\cup B\cup C)

🟢 26. AT LEAST TWO "At least two" means belonging to two or three sets.

n(\text{At Least Two}) = n(A\cap B) +n(A\cap C) +n(B\cap C)

Since the elements belonging to all three sets are counted three times:

n(\text{At Least Two}) = n(A\cap B) +n(A\cap C) +n(B\cap C) -2n(A\cap B\cap C)

🟢 27. EXACTLY TWO "Exactly two" means belonging to exactly two sets but not all three.

n(\text{Exactly Two}) = n(A\cap B) +n(A\cap C) +n(B\cap C) -3n(A\cap B\cap C)

🟢 28. EXACTLY ONE "Exactly one" means belonging to only one of the three sets.

n(\text{Exactly One}) = n(A\text{ only}) +n(B\text{ only}) +n(C\text{ only})

🟢 29. AT LEAST ONE AND NONE For the universal set:

n(\text{At Least One}) + n(\text{None}) = n(U)

Therefore:

n(\text{None}) = n(U)-n(\text{At Least One})

🟢 30. IMPORTANT VENN DIAGRAM SYMBOLS

\cup = \text{Union}

\cap = \text{Intersection}

A-B = \text{Difference}

A' = \text{Complement of A}

\varnothing = \text{Empty Set}

U = \text{Universal Set}

🔴 IMPORTANT KEY POINTS

A\cup B = \text{Elements in A or B or both}

A\cap B = \text{Elements common to A and B}

A-B = \text{Elements in A but not in B}

A' = \text{Elements not in A}

n(A\cup B) = n(A)+n(B)-n(A\cap B)

n(A\text{ only}) = n(A)-n(A\cap B)

n(B\text{ only}) = n(B)-n(A\cap B)

n(\text{Neither}) = n(U)-n(A\cup B)

n(A\cup B\cup C) = n(A)+n(B)+n(C)

-n(A\cap B) -n(A\cap C) -n(B\cap C)

+n(A\cap B\cap C)

\text{At Least One} = A\cup B\cup C

\text{None} = U-(A\cup B\cup C)

\text{Exactly Two} = \text{Pairwise Intersections} - 3\times\text{Triple Intersection}

\text{Always subtract overlapping regions when calculating a union}

\text{For three sets, add the triple intersection once}

Logical Reasoning · Data Interpretation and Data Sufficiency

Data Sufficiency - Formulas, Key Points and Examples

🔵

\text{DATA SUFFICIENCY}

🟢 1. BASIC CONCEPT Data Sufficiency questions test whether the given information is sufficient to answer a question. The objective is not always to find the answer. The objective is to determine whether the given statements provide enough information to find a unique answer.

\text{Data Sufficiency} = \text{Checking whether the given data is sufficient to answer the question}

🟡 KEY POINT

\text{Sufficient Data}\neq\text{Necessarily Calculating the Final Answer}

🟢 2. STATEMENT-BASED DATA SUFFICIENCY A typical question contains:

\text{Question}+\text{Statement I}+\text{Statement II}

You must determine whether:

\text{Statement I alone is sufficient}

or:

\text{Statement II alone is sufficient}

or:

\text{Both statements together are sufficient}

or:

\text{Even both statements together are insufficient}

🟢 3. MAIN RULE Always test the statements separately first.

\text{Step 1: Test Statement I alone}

\text{Step 2: Test Statement II alone}

\text{Step 3: If neither is sufficient, test Statements I and II together}

🟡 KEY POINT

\text{Do not combine the statements immediately}

🟢 4. STATEMENT I ALONE If Statement I gives enough information to determine a unique answer:

\text{Statement I alone is sufficient}

There is no need to use Statement II. 🟢 5. STATEMENT II ALONE If Statement II gives enough information to determine a unique answer:

\text{Statement II alone is sufficient}

There is no need to use Statement I. 🟢 6. BOTH STATEMENTS TOGETHER Sometimes neither statement is sufficient individually, but both together provide enough information.

\text{Statement I alone}=\text{Insufficient}

\text{Statement II alone}=\text{Insufficient}

But:

\text{Statement I}+\text{Statement II}=\text{Sufficient}

🟢 7. BOTH STATEMENTS TOGETHER ARE INSUFFICIENT If even after combining both statements, more than one possible answer remains:

\text{Statement I}+\text{Statement II}=\text{Insufficient}

🟡 KEY POINT

\text{Multiple Possible Answers}\Rightarrow\text{Insufficient Data}

🟢 8. UNIQUE ANSWER Data is sufficient only when the information leads to one definite answer. If:

x=10

then the value of \(x\) is uniquely determined. Therefore:

\text{Sufficient}

But if:

x>5

then many values are possible. Therefore:

\text{Insufficient}

🟢 9. YES OR NO QUESTIONS For questions asking whether a statement is true or false, the data is sufficient if the statement can be definitely answered with:

\text{YES}

or:

\text{NO}

If both possibilities remain:

\text{YES or NO}

then:

\text{Insufficient}

🟢 10. VALUE-BASED QUESTIONS Suppose the question asks:

\text{What is the value of }x?

The data is sufficient only if \(x\) can be uniquely determined. For example:

x+5=12

Therefore:

x=7

Hence:

\text{Sufficient}

🟢 11. AGE PROBLEMS For age questions, identify the number of unknown ages and the relationships between them. For example:

A+B=40

This alone does not determine \(A\) and \(B\) individually. Therefore:

\text{Insufficient}

If another independent equation is given:

A-B=10

then:

A+B=40

A-B=10

Adding both equations:

2A=50

A=25

Therefore:

B=15

Hence:

\text{Sufficient}

🟢 12. NUMBER PROBLEMS If the question asks for the value of a number and a statement gives:

x=25

then:

\text{Statement is Sufficient}

But if the statement only gives:

x>10

then:

\text{Statement is Insufficient}

🟢 13. EQUATION-BASED SUFFICIENCY One equation with two unknowns is generally insufficient. For example:

x+y=20

There are many possible values of \(x\) and \(y\). Therefore:

\text{Insufficient}

Two independent equations may be sufficient:

x+y=20

x-y=4

Adding the equations:

2x=24

x=12

Therefore:

y=8

Hence:

\text{Sufficient}

🟢 14. INEQUALITY An inequality may or may not provide a unique answer. For example:

x>10

does not determine a unique value. Therefore:

\text{Insufficient}

But if the question asks whether \(x\) is positive:

x>10

then:

x>0

Therefore:

\text{Sufficient}

🟢 15. RATIO PROBLEMS A ratio alone may be insufficient if the actual values are required. For example:

A:B=2:3

The actual values could be:

A=2,\quad B=3

or:

A=4,\quad B=6

or:

A=20,\quad B=30

Therefore, the actual values cannot be uniquely determined.

\text{Insufficient}

🟢 16. RATIO WITH TOTAL If:

A:B=2:3

and:

A+B=50

then:

2x+3x=50

5x=50

x=10

Therefore:

A=20

B=30

Hence:

\text{Sufficient}

🟢 17. PERCENTAGE PROBLEMS If the question asks for a percentage and the required part and whole are known:

\text{Percentage} = \frac{\text{Part}}{\text{Whole}}\times100

If both the part and whole can be determined:

\text{Sufficient}

If either the part or whole cannot be determined:

\text{Insufficient}

🟢 18. AVERAGE PROBLEMS If the number of observations and total are known:

\text{Average} = \frac{\text{Total}}{\text{Number of Observations}}

Therefore:

\text{Total}+\text{Number of Observations} \Rightarrow\text{Sufficient}

If only the average is known and the total is required, the data is generally insufficient unless the number of observations is also known. 🟢 19. GEOMETRY PROBLEMS For geometry questions, determine whether the given information uniquely fixes the required quantity. For a rectangle:

\text{Area}=l\times b

If both length and breadth are known:

l,\quad b\Rightarrow\text{Area}

Hence:

\text{Sufficient}

🟢 20. TRIANGLE PROBLEMS The sum of the angles of a triangle is:

A+B+C=180^\circ

If two angles are known, the third angle can be determined. For example:

A=60^\circ

B=70^\circ

Therefore:

C=180^\circ-60^\circ-70^\circ

C=50^\circ

Hence:

\text{Sufficient}

🟢 21. SPEED, TIME AND DISTANCE The basic relationship is:

\text{Distance} = \text{Speed}\times\text{Time}

Therefore:

d=st

If speed and time are known:

s,\quad t\Rightarrow d

Hence:

\text{Sufficient}

🟢 22. TIME AND WORK The basic relationship is:

\text{Work} = \text{Rate}\times\text{Time}

If the rate and time are known:

W=RT

Therefore:

R,\quad T\Rightarrow W

Hence:

\text{Sufficient}

🟢 23. PROFIT AND LOSS The basic relationships are:

\text{Profit}=SP-CP

and:

\text{Loss}=CP-SP

To determine profit or loss, the required values must be known or uniquely derivable. 🟡 KEY POINT

\text{Do Not Assume a Missing Value}

🟢 24. DATA SUFFICIENCY VS DATA INTERPRETATION Data Interpretation asks you to calculate or interpret information.

\text{Data Interpretation}\rightarrow\text{Find the Answer}

Data Sufficiency asks whether the information is enough.

\text{Data Sufficiency}\rightarrow\text{Check Whether the Answer Can Be Determined}

🟢 25. DO NOT USE UNNECESSARY INFORMATION A statement may contain information that is not required. The important question is:

\text{Can the Required Answer Be Uniquely Determined?}

If yes:

\text{Sufficient}

If no:

\text{Insufficient}

🟢 26. INDEPENDENT INFORMATION Two statements are useful together when they provide independent information. For example:

x+y=20

and:

x-y=4

These are independent equations and determine \(x\) and \(y\). Therefore:

\text{Together They Are Sufficient}

🟢 27. REDUNDANT INFORMATION Sometimes both statements provide essentially the same information. For example:

x=10

and:

2x=20

Statement I alone is sufficient. Statement II alone is also sufficient. Therefore:

\text{Either Statement Alone Is Sufficient}

🟢 28. COMMON ANSWER PATTERN Many aptitude tests use answer choices such as:

\text{A. Statement I alone is sufficient}

\text{B. Statement II alone is sufficient}

\text{C. Both statements together are sufficient}

\text{D. Either statement alone is sufficient}

\text{E. Even both statements together are insufficient}

🟡 KEY POINT

\text{Follow the Exact Answer-Choice Convention Given in the Examination}

🔴 IMPORTANT KEY POINTS

\text{Test Statement I Alone First}

\text{Test Statement II Alone Next}

\text{Combine Them Only When Necessary}

\text{Sufficient Means a Unique Answer Can Be Determined}

\text{More Than One Possible Answer Means Insufficient}

\text{Do Not Assume Information That Is Not Given}

\text{Do Not Solve More Than Necessary}

\text{A Statement Can Be Sufficient Even If the Actual Answer Is Not Calculated}

\text{For YES/NO Questions, One Definite Answer Is Sufficient}

\text{Check Each Statement Independently Before Combining Them}

Logical Reasoning · Data Interpretation and Data Sufficiency

Data Interpretation - Formulas, Key Points and Examples

🔵

\text{DATA INTERPRETATION}

🟢 1. BASIC CONCEPT Data Interpretation means analyzing given data and using it to calculate required values. Data may be presented in the form of:

\text{Tables}

\text{Bar Graphs}

\text{Line Graphs}

\text{Pie Charts}

\text{Mixed Graphs}

🟡 KEY POINT

\text{Read the data carefully before performing any calculation.}

🟢 2. DATA TABLE A table presents information in rows and columns. For example:

\text{Total} = \text{Sum of all relevant values}

If values are:

a,\ b,\ c,\ d

then:

\text{Total}=a+b+c+d

🟢 3. TOTAL VALUE To find the total of different categories:

\text{Total} = \text{Value}_1+\text{Value}_2+\text{Value}_3+\cdots

🟡 KEY POINT

\text{Always include all categories specified in the question.}

🟢 4. DIFFERENCE The difference between two values is:

\text{Difference} = \text{Larger Value}-\text{Smaller Value}

For two quantities A and B:

\text{Difference}=|A-B|

🟢 5. RATIO To find the ratio of A to B:

\text{Ratio} = A:B

or:

\text{Ratio} = \frac{A}{B}

The ratio should be simplified whenever possible. 🟢 6. PERCENTAGE To find what percentage A is of B:

\text{Percentage} = \frac{A}{B}\times100

🟡 KEY POINT

\text{The denominator must be the reference value mentioned in the question.}

🟢 7. PERCENTAGE INCREASE If a value changes from an original value to a new value:

\text{Increase} = \text{New Value}-\text{Original Value}

\text{Percentage Increase} = \frac{\text{Increase}}{\text{Original Value}}\times100

🟢 8. PERCENTAGE DECREASE

\text{Decrease} = \text{Original Value}-\text{New Value}

\text{Percentage Decrease} = \frac{\text{Decrease}}{\text{Original Value}}\times100

🟢 9. AVERAGE The average of a set of values is:

\text{Average} = \frac{\text{Sum of Values}}{\text{Number of Values}}

For n values:

\text{Average} = \frac{x_1+x_2+x_3+\cdots+x_n}{n}

🟢 10. WEIGHTED AVERAGE When different values have different frequencies or weights:

\text{Weighted Average} = \frac{\sum wx}{\sum w}

where:

w=\text{Weight}

and:

x=\text{Value}

🟢 11. BAR GRAPH A bar graph represents data using rectangular bars. The length or height of each bar represents the corresponding value.

\text{Bar Length}\propto\text{Data Value}

🟡 KEY POINT

\text{Check the scale before reading the value from a bar graph.}

🟢 12. BAR GRAPH SCALE If one division represents k units:

\text{Value} = \text{Number of Divisions}\times k

For example, if one division represents 20:

5\text{ divisions}=5\times20

=100

🟢 13. DOUBLE BAR GRAPH A double bar graph compares two sets of data. For categories A, B and C:

\text{Difference} = \text{First Data Set}-\text{Second Data Set}

The values should be compared category by category. 🟢 14. LINE GRAPH A line graph shows changes in data over time or across different categories. The change between two consecutive values is:

\text{Change} = \text{New Value}-\text{Previous Value}

🟡 KEY POINT

\text{Look at the direction and scale of the line carefully.}

🟢 15. TOTAL FROM A LINE GRAPH If the values shown are:

x_1,x_2,x_3,\ldots,x_n

then:

\text{Total} = x_1+x_2+x_3+\cdots+x_n

🟢 16. PIE CHART A pie chart represents a whole as a circle of:

360^\circ

The complete data represents:

100\%

Therefore:

360^\circ=100\%

🟢 17. PIE CHART ANGLE TO VALUE If the total quantity is T and the sector angle is θ:

\text{Value} = \frac{\theta}{360^\circ}\times T

🟢 18. PIE CHART VALUE TO ANGLE If a category has value V and the total is T:

\text{Angle} = \frac{V}{T}\times360^\circ

🟢 19. PIE CHART ANGLE TO PERCENTAGE

\text{Percentage} = \frac{\theta}{360^\circ}\times100

🟢 20. PIE CHART PERCENTAGE TO ANGLE

\text{Angle} = \frac{\text{Percentage}}{100}\times360^\circ

Therefore:

\text{Angle} = \text{Percentage}\times3.6^\circ

🟢 21. IMPORTANT PIE CHART VALUES For 50%:

50\%\rightarrow180^\circ

For 25%:

25\%\rightarrow90^\circ

For 20%:

20\%\rightarrow72^\circ

For 10%:

10\%\rightarrow36^\circ

For 5%:

5\%\rightarrow18^\circ

🟢 22. FINDING TOTAL FROM A PIE CHART If a sector represents V and has angle θ:

V = \frac{\theta}{360^\circ}\times T

Therefore:

T = \frac{V\times360^\circ}{\theta}

🟢 23. COMPARISON OF TWO VALUES To find how many times A is B:

\text{Times} = \frac{A}{B}

To find how much greater A is than B:

\text{Difference} = A-B

To find the percentage by which A is greater than B:

\text{Percentage} = \frac{A-B}{B}\times100

🟢 24. RATIO OF TWO CATEGORIES If two categories have values A and B:

\text{Ratio}=A:B

\text{If the ratio must be simplified, divide both terms by their HCF.}

🟢 25. FINDING UNKNOWN VALUE If the total and known values are given:

\text{Unknown} = \text{Total}-\text{Sum of Known Values}

🟢 26. TOTAL FROM PERCENTAGES If a total T is divided into percentages:

p_1\%,p_2\%,p_3\%,\ldots

then the corresponding values are:

\text{Value} = \frac{p}{100}\times T

🟢 27. FINDING PERCENTAGE FROM A TABLE If a category has value A and the total is T:

\text{Percentage} = \frac{A}{T}\times100

🟢 28. FINDING AVERAGE FROM A TABLE If the values are:

x_1,x_2,x_3,\ldots,x_n

then:

\text{Average} = \frac{x_1+x_2+x_3+\cdots+x_n}{n}

🟢 29. AVERAGE WHEN TOTAL IS GIVEN If the total of n observations is T:

\text{Average} = \frac{T}{n}

Therefore:

T=\text{Average}\times n

🟢 30. CHANGE IN AVERAGE If the total changes by ΔT while the number of observations remains n:

\text{Change in Average} = \frac{\Delta T}{n}

🟢 31. MISSING VALUE USING AVERAGE If n values have an average A:

\text{Total} = nA

If the sum of known values is S:

\text{Missing Value} = nA-S

🟢 32. COMBINED DATA If two groups have totals T₁ and T₂ and numbers of observations n₁ and n₂:

\text{Combined Average} = \frac{T_1+T_2}{n_1+n_2}

Using individual averages A₁ and A₂:

\text{Combined Average} = \frac{n_1A_1+n_2A_2}{n_1+n_2}

🟢 33. PERCENTAGE OF TOTAL If a category has value V and total value is T:

\text{Percentage Share} = \frac{V}{T}\times100

🟢 34. TOTAL PRODUCTION OR SALES If production or sales are given for different years:

\text{Total} = \text{Year 1}+\text{Year 2}+\text{Year 3}+\cdots

🟢 35. YEAR-TO-YEAR CHANGE For two consecutive years:

\text{Change} = \text{Current Year Value}-\text{Previous Year Value}

Percentage change:

\text{Percentage Change} = \frac{\text{Change}}{\text{Previous Year Value}}\times100

🟢 36. DATA SUFFICIENCY IN DI Some questions provide multiple statements or pieces of data. The objective is to determine whether the given information is sufficient to answer the question. 🟡 KEY POINT

\text{Do not calculate unnecessary values.}

Use only the information required to answer the question. 🟢 37. APPROXIMATION Approximation can be used when the question asks for an approximate value. For example:

49.8\approx50

99.7\approx100

🟡 KEY POINT

\text{Use approximation only when the question permits an approximate answer.}

🟢 38. UNIT CONVERSION Always make sure that quantities use the same units before comparing or calculating. For example:

1\text{ km}=1000\text{ m}

1\text{ hour}=60\text{ minutes}

1\text{ kg}=1000\text{ g}

🟢 39. IMPORTANT DI SHORTCUT If a value is increased by x%:

\text{New Value} = \text{Old Value}\left(1+\frac{x}{100}\right)

If a value is decreased by x%:

\text{New Value} = \text{Old Value}\left(1-\frac{x}{100}\right)

🟢 40. SUCCESSIVE CHANGES If a value changes successively by a% and b%:

\text{Net Change} = \left(a+b+\frac{ab}{100}\right)\%

when both changes are increases. For two decreases:

\text{Net Decrease} = \left(a+b-\frac{ab}{100}\right)\%

🔴 IMPORTANT KEY POINTS

\text{Read the title and headings before solving.}

\text{Check the units carefully.}

\text{Check the scale of graphs.}

\text{Identify the total before calculating percentages.}

\text{For percentage, use the correct reference value as the denominator.}

\text{For pie charts, remember }360^\circ=100\%.

\text{For averages, Average}=\frac{\text{Total}}{\text{Number of Observations}}.

\text{For ratios, simplify the final ratio whenever possible.}

\text{For differences, subtract the smaller value from the larger value.}

\text{For percentage increase or decrease, compare with the original value.}

\text{Do not confuse total value with average value.}

\text{Use only the data required by the question.}

\text{Check the final answer against the given data.}

Logical Reasoning · Data Arrangements and Blood Relations

Blood Relations

🔵

\text{BLOOD RELATIONS}

🟢 1. BASIC BLOOD RELATIONSHIP Blood relation questions involve identifying the relationship between two people using family connections.

\text{Blood Relation} = \text{Relationship between two members of a family}

Common relationships include:

\text{Father,\ Mother,\ Son,\ Daughter,\ Brother,\ Sister}

\text{Grandfather,\ Grandmother,\ Grandson,\ Granddaughter}

\text{Uncle,\ Aunt,\ Nephew,\ Niece}

\text{Cousin}

🟡 KEY POINT

\text{Always start from the person mentioned in the question and trace the relationship step by step.}

🟢 2. IMMEDIATE FAMILY RELATIONS Father:

\text{Father}=\text{Male parent}

Mother:

\text{Mother}=\text{Female parent}

Son:

\text{Son}=\text{Male child}

Daughter:

\text{Daughter}=\text{Female child}

Brother:

\text{Brother}=\text{Male sibling}

Sister:

\text{Sister}=\text{Female sibling}

🟢 3. PARENT-CHILD RELATIONSHIP If A is the father of B:

A\rightarrow B

If A is the mother of B:

A\rightarrow B

If A is the son of B:

B\rightarrow A

If A is the daughter of B:

B\rightarrow A

🟡 KEY POINT

\text{Parent is one generation above the child.}

🟢 4. SIBLING RELATIONSHIP A brother and sister share at least one parent. If A and B have the same parents:

A\leftrightarrow B

If A is male:

A=\text{Brother}

If A is female:

A=\text{Sister}

🟡 KEY POINT

\text{Brothers and sisters belong to the same generation.}

🟢 5. GRANDFATHER AND GRANDMOTHER A grandfather is the father of one's parent.

\text{Grandfather} = \text{Father of Father or Father of Mother}

A grandmother is the mother of one's parent.

\text{Grandmother} = \text{Mother of Father or Mother of Mother}

🟢 6. GRANDCHILDREN A grandson is the son of one's child.

\text{Grandson} = \text{Son of Son or Son of Daughter}

A granddaughter is the daughter of one's child.

\text{Granddaughter} = \text{Daughter of Son or Daughter of Daughter}

🟢 7. UNCLE An uncle is generally the brother of one's parent.

\text{Uncle} = \text{Brother of Father or Brother of Mother}

In some family structures, the husband of an aunt may also be referred to as an uncle. 🟡 KEY POINT

\text{Trace through the parent first when identifying an uncle.}

🟢 8. AUNT An aunt is generally the sister of one's parent.

\text{Aunt} = \text{Sister of Father or Sister of Mother}

The wife of an uncle may also be referred to as an aunt. 🟢 9. NEPHEW A nephew is the son of one's brother or sister.

\text{Nephew} = \text{Son of Brother or Son of Sister}

🟢 10. NIECE A niece is the daughter of one's brother or sister.

\text{Niece} = \text{Daughter of Brother or Daughter of Sister}

🟢 11. COUSIN A cousin is generally the child of one's uncle or aunt.

\text{Cousin} = \text{Child of Uncle or Aunt}

A male cousin is:

\text{Male Cousin}

A female cousin is:

\text{Female Cousin}

🟢 12. PATERNAL RELATIONS Paternal relationships are related through the father. Father's father:

\text{Paternal Grandfather}

Father's mother:

\text{Paternal Grandmother}

Father's brother:

\text{Paternal Uncle}

Father's sister:

\text{Paternal Aunt}

🟡 KEY POINT

\text{Paternal means related through the father.}

🟢 13. MATERNAL RELATIONS Maternal relationships are related through the mother. Mother's father:

\text{Maternal Grandfather}

Mother's mother:

\text{Maternal Grandmother}

Mother's brother:

\text{Maternal Uncle}

Mother's sister:

\text{Maternal Aunt}

🟡 KEY POINT

\text{Maternal means related through the mother.}

🟢 14. BROTHER'S SON The son of one's brother is:

\text{Brother's Son} = \text{Nephew}

🟢 15. BROTHER'S DAUGHTER The daughter of one's brother is:

\text{Brother's Daughter} = \text{Niece}

🟢 16. SISTER'S SON The son of one's sister is:

\text{Sister's Son} = \text{Nephew}

🟢 17. SISTER'S DAUGHTER The daughter of one's sister is:

\text{Sister's Daughter} = \text{Niece}

🟢 18. FATHER'S BROTHER The brother of one's father is:

\text{Father's Brother} = \text{Uncle}

🟢 19. FATHER'S SISTER The sister of one's father is:

\text{Father's Sister} = \text{Aunt}

🟢 20. MOTHER'S BROTHER The brother of one's mother is:

\text{Mother's Brother} = \text{Maternal Uncle}

🟢 21. MOTHER'S SISTER The sister of one's mother is:

\text{Mother's Sister} = \text{Maternal Aunt}

🟢 22. FATHER'S FATHER The father of one's father is:

\text{Father's Father} = \text{Grandfather}

More specifically:

\text{Father's Father} = \text{Paternal Grandfather}

🟢 23. MOTHER'S FATHER The father of one's mother is:

\text{Mother's Father} = \text{Maternal Grandfather}

🟢 24. FATHER'S MOTHER The mother of one's father is:

\text{Father's Mother} = \text{Paternal Grandmother}

🟢 25. MOTHER'S MOTHER The mother of one's mother is:

\text{Mother's Mother} = \text{Maternal Grandmother}

🟢 26. GENERATION LEVELS Family relationships can be understood using generations. Parents are:

+1\text{ generation}

Children are:

-1\text{ generation}

Grandparents are:

+2\text{ generations}

Grandchildren are:

-2\text{ generations}

Siblings are:

0\text{ generation difference}

🟡 KEY POINT

\text{Use generation levels to simplify complicated family relationships.}

🟢 27. FAMILY TREE A family tree represents relationships using levels. For example:

\text{Grandfather}

\downarrow

\text{Father}

\downarrow

\text{Son}

For siblings:

\text{Father} \rightarrow \begin{cases} \text{Son}\\ \text{Daughter} \end{cases}

🟡 KEY POINT

\text{Draw a family tree when the relationship chain contains several steps.}

🟢 28. SYMBOLIC BLOOD RELATIONS Some questions use symbols to represent relationships. For example:

A+B

may represent:

A\text{ is the father of }B

If:

A-B

represents:

A\text{ is the mother of }B

The meaning of each symbol must be determined from the question. 🔴 IMPORTANT

\text{Never assume the meaning of a symbol without checking the given definitions.}

🟢 29. STATEMENT-BASED BLOOD RELATIONS In statement-based questions, convert each sentence into a relationship. For example:

A\text{ is the brother of }B

means:

A\leftrightarrow B

with A being male. If:

B\text{ is the mother of }C

then:

B\rightarrow C

Therefore, A is the maternal uncle of C. 🟢 30. RELATIONSHIP THROUGH TWO PEOPLE If:

A\text{ is the brother of }B

and:

B\text{ is the mother of }C

then:

A\rightarrow B\rightarrow C

Therefore:

A=\text{Maternal Uncle of }C

🟢 31. RELATIONSHIP THROUGH THREE PEOPLE If:

A\text{ is the father of }B

B\text{ is the sister of }C

then A is also the father of C, assuming B and C are siblings. Therefore:

A=\text{Father of }C

🟢 32. HUSBAND AND WIFE RELATIONSHIP A husband and wife are spouses.

\text{Husband}\leftrightarrow\text{Wife}

They belong to the same generation.

\text{Generation Difference}=0

🟡 KEY POINT

\text{Marriage relationships are not blood relationships, but they may be included in family-relation questions.}

🟢 33. FATHER-IN-LAW The father of one's husband or wife is:

\text{Father-in-law}

Therefore:

\text{Father-in-law} = \text{Father of Spouse}

🟢 34. MOTHER-IN-LAW The mother of one's husband or wife is:

\text{Mother-in-law}

Therefore:

\text{Mother-in-law} = \text{Mother of Spouse}

🟢 35. SON-IN-LAW The husband of one's daughter is:

\text{Son-in-law} = \text{Husband of Daughter}

🟢 36. DAUGHTER-IN-LAW The wife of one's son is:

\text{Daughter-in-law} = \text{Wife of Son}

🟢 37. SIBLING-IN-LAW The sibling of one's spouse is commonly called a brother-in-law or sister-in-law depending on gender.

\text{Brother-in-law} = \text{Male sibling of Spouse}

\text{Sister-in-law} = \text{Female sibling of Spouse}

🟢 38. HOW TO SOLVE BLOOD RELATION QUESTIONS First identify the person whose relationship is being asked.

\text{Step 1: Identify the starting person}

Then trace each relationship.

\text{Step 2: Follow the relationship chain}

Draw the family tree if necessary.

\text{Step 3: Create a simple family tree}

Determine the generation.

\text{Step 4: Identify the generation level}

Finally, identify the exact relationship.

\text{Step 5: Determine the final relationship}

🟡 KEY POINT

\text{Never jump directly to the answer when several relationships are given.}

🟢 39. QUICK RELATIONSHIP CHAIN Father's brother:

\text{Father}\rightarrow\text{Brother} = \text{Uncle}

Mother's sister:

\text{Mother}\rightarrow\text{Sister} = \text{Aunt}

Brother's son:

\text{Brother}\rightarrow\text{Son} = \text{Nephew}

Sister's daughter:

\text{Sister}\rightarrow\text{Daughter} = \text{Niece}

Father's father:

\text{Father}\rightarrow\text{Father} = \text{Grandfather}

Mother's mother:

\text{Mother}\rightarrow\text{Mother} = \text{Grandmother}

🟡 IMPORTANT KEY POINTS

\text{Father's brother}=Uncle

\text{Father's sister}=Aunt

\text{Mother's brother}=Maternal\ Uncle

\text{Mother's sister}=Maternal\ Aunt

\text{Father's father}=Paternal\ Grandfather

\text{Father's mother}=Paternal\ Grandmother

\text{Mother's father}=Maternal\ Grandfather

\text{Mother's mother}=Maternal\ Grandmother

\text{Brother's son}=Nephew

\text{Brother's daughter}=Niece

\text{Sister's son}=Nephew

\text{Sister's daughter}=Niece

\text{Son's son}=Grandson

\text{Son's daughter}=Granddaughter

\text{Daughter's son}=Grandson

\text{Daughter's daughter}=Granddaughter

\text{Parent}=+1\text{ generation}

\text{Grandparent}=+2\text{ generations}

\text{Child}=-1\text{ generation}

\text{Grandchild}=-2\text{ generations}

\text{Sibling}=0\text{ generation difference}

\text{Always trace the relationship step by step.}

\text{Draw a family tree when the relationship chain is complicated.}

Logical Reasoning · Data Arrangements and Blood Relations

Data Arrangements - Formulas, Key Points and formulas

🔵

\text{ DATA ARRANGEMENTS}

🟢 1. BASIC DATA ARRANGEMENT Data arrangement questions require arranging given information according to specific conditions.

\text{Data Arrangement} = \text{Organizing given information according to the conditions}

The information may involve:

\text{Persons, Objects, Places, Numbers, Days, Ranks or Positions}

🟡 KEY POINT

\text{Read all the conditions carefully before starting the arrangement.}

🟢 2. LINEAR ARRANGEMENT In a linear arrangement, people or objects are arranged in a straight line. For example:

A\quad B\quad C\quad D\quad E

If

A

is to the left of

B

:

A\quad B

If

A

is immediately to the right of

B

:

B\quad A

🟡 KEY POINT

\text{Left and right depend on the direction specified in the question.}

🟢 3. POSITION IN A LINE If a person is at position

P

from the left in a row of

N

persons, the position from the right is:

\text{Position from Right} = N-P+1

Similarly:

\text{Position from Left} = N-P+1

if the position from the right is known. 🟡 KEY POINT

\text{Position from opposite side} = \text{Total Persons}-\text{Known Position}+1

🟢 4. NUMBER OF PERSONS BETWEEN TWO POSITIONS If two persons occupy positions

P_1

and

P_2

:

\text{Persons Between} = |P_1-P_2|-1

For example, if two persons are at positions 4 and 9:

|9-4|-1

=5-1

=4

🟡 KEY POINT

\text{Always subtract 1 when finding the number of persons between two positions.}

🟢 5. IMMEDIATE LEFT AND RIGHT If

A

is immediately left of

B

:

A\quad B

If

A

is immediately right of

B

:

B\quad A

The word "immediately" means there is no person or object between them. 🟡 KEY POINT

\text{Immediate left/right means adjacent positions.}

🟢 6. SECOND TO THE LEFT OR RIGHT If

A

is second to the left of

B

, exactly one position lies between them.

A\quad X\quad B

If

A

is second to the right of

B

:

B\quad X\quad A

🟡 KEY POINT

\text{Second position means one person or object lies between them.}

🟢 7. THIRD POSITION If

A

is third to the left of

B

:

A\quad X\quad X\quad B

There are two positions between them. Therefore:

\text{Persons Between}=3-1

=2

🟢 8. ORDER AND RANKING Ranking questions involve the position of a person or object according to a particular order. If a person ranks

P

from the top in a group of

N

persons:

\text{Rank from Bottom} = N-P+1

If a person ranks

P

from the bottom:

\text{Rank from Top} = N-P+1

🟡 KEY POINT

\text{Opposite Rank} = \text{Total Number}-\text{Known Rank}+1

🟢 9. TOTAL NUMBER FROM TWO RANKS If a person's rank from the top is

P

and from the bottom is

Q

:

\text{Total Persons} = P+Q-1

The subtraction of 1 is necessary because the same person is counted in both ranks. 🟡 KEY POINT

\text{Total}= \text{Top Rank}+\text{Bottom Rank}-1

🟢 10. CIRCULAR ARRANGEMENT In a circular arrangement, people or objects are arranged around a circle. For example:

A\quad B\quad C\quad D

may be arranged around a circle. In circular arrangements, relative positions are more important than absolute positions. 🟡 KEY POINT

\text{There is no fixed leftmost or rightmost position in a circle.}

🟢 11. CLOCKWISE AND ANTICLOCKWISE Clockwise means moving in the same direction as the hands of a clock.

\text{Clockwise} = \text{Direction of clock hands}

Anticlockwise means moving in the opposite direction.

\text{Anticlockwise} = \text{Opposite direction to clock hands}

🟡 KEY POINT

\text{Always identify the facing direction before deciding left and right.}

🟢 12. FACING THE CENTRE When people face the centre of a circle:

\text{Left and Right are determined from the person's own perspective.}

For a person facing the centre:

\text{Left side} = \text{Clockwise direction}

\text{Right side} = \text{Anticlockwise direction}

🟡 KEY POINT

\text{For people facing the centre, left is clockwise and right is anticlockwise.}

🟢 13. FACING OUTSIDE When people face away from the centre:

\text{Left side} = \text{Anticlockwise direction}

\text{Right side} = \text{Clockwise direction}

🟡 KEY POINT

\text{For people facing outside, left and right are reversed.}

🟢 14. FIXED POSITION A condition may directly specify a person's position. For example:

A\text{ is at the extreme left.}

Then:

A\quad \_\quad \_\quad \_\quad \_

If

B

is at the extreme right:

A\quad \_\quad \_\quad \_\quad B

🟡 KEY POINT

\text{Place fixed positions first.}

🟢 15. BETWEEN TWO PERSONS If

A

is between

B

and

C

:

B\quad A\quad C

or:

C\quad A\quad B

The exact order depends on additional conditions. 🟡 KEY POINT

\text{"Between"}\text{ does not always mean immediately between.}

🟢 16. ADJACENT POSITIONS Two people are adjacent when they occupy consecutive positions.

A\quad B

There is no person between them. Therefore:

\text{Number of Persons Between}=0

🟢 17. NOT ADJACENT If two persons are not adjacent:

A\quad X\quad B

At least one person or object must be between them.

\text{Persons Between}\geq1

🟢 18. ORDERING BY AGE, HEIGHT OR WEIGHT Data arrangement can involve ordering people according to measurable characteristics. For increasing order:

A<B<C<D

For decreasing order:

D>C>B>A

🟡 KEY POINT

\text{Convert every comparison into a clear order before solving.}

🟢 19. COMPARISON STATEMENTS If

A

is taller than

B

:

A>B

If

C

is shorter than

A

:

C<A

Therefore:

A>C

If:

A>B>C

then:

A>B

B>C

and:

A>C

🟢 20. CONDITIONAL ARRANGEMENT Some questions contain conditions such as:

A\text{ sits to the left of }B.

C\text{ sits immediately right of }D.

E\text{ is not at an extreme position.}

All conditions must be satisfied simultaneously. 🟡 KEY POINT

\text{Do not solve each condition separately without checking the complete arrangement.}

🟢 21. GROUPING ARRANGEMENT Some questions require placing people or objects into groups. For example:

\text{Group 1}:\ A,B,C

\text{Group 2}:\ D,E,F

If

A

and

B

must be together:

(A,B)

can be treated as one unit during the initial arrangement. 🟡 KEY POINT

\text{Combine items that must stay together into a block.}

🟢 22. DAYS AND SCHEDULE ARRANGEMENT Data arrangement may involve assigning activities to different days. For example:

\text{Monday}\rightarrow A

\text{Tuesday}\rightarrow B

\text{Wednesday}\rightarrow C

Conditions may specify before, after, immediately before or immediately after. 🟡 KEY POINT

\text{Before and after conditions should be converted into positional relationships.}

🟢 23. BEFORE AND AFTER If

A

occurs before

B

:

A<B

If

C

occurs after

B

:

B<C

Therefore:

A<B<C

🟡 KEY POINT

\text{Before/after relationships help create an order chain.}

🟢 24. IMMEDIATELY BEFORE AND AFTER If

A

occurs immediately before

B

:

A\quad B

If

C

occurs immediately after

D

:

D\quad C

🟡 KEY POINT

\text{"Immediately before/after"}\text{ means consecutive positions.}

🟢 25. EXTREME POSITIONS In a row, the extreme positions are:

\text{First Position and Last Position}

For

N

positions:

\text{First Position}=1

\text{Last Position}=N

🟡 KEY POINT

\text{Check extreme positions before filling middle positions.}

🟢 26. MIDDLE POSITION If there are

N

positions and

N

is odd, the middle position is:

\text{Middle Position} = \frac{N+1}{2}

For example, for 9 positions:

\frac{9+1}{2}=5

Therefore:

\text{Middle Position}=5

🟢 27. TWO MIDDLE POSITIONS If the number of positions is even, there are two middle positions. For

N

positions:

\text{Middle Positions} = \frac{N}{2} \quad\text{and}\quad \frac{N}{2}+1

For 10 positions:

\frac{10}{2}=5

and:

5+1=6

Therefore:

\text{Middle Positions}=5\text{ and }6

🟢 28. POSITION FROM BOTH ENDS If there are

N

people and a person is at position

P

from the left:

\text{Position from Right} = N-P+1

If the position from the right is

Q

:

\text{Position from Left} = N-Q+1

🟢 29. NUMBER OF PEOPLE BETWEEN TWO PEOPLE If two people are at positions

P

and

Q

:

\text{Number Between} = |P-Q|-1

If they are at positions 3 and 8:

|8-3|-1

=4

Therefore:

\text{Number Between}=4

🟢 30. BEST METHOD TO SOLVE DATA ARRANGEMENT First identify the type of arrangement.

\text{Linear}

\text{Circular}

\text{Ranking}

\text{Grouping}

\text{Scheduling}

Then:

\text{Step 1: Identify fixed information}

\text{Step 2: Place definite positions}

\text{Step 3: Create blocks for linked information}

\text{Step 4: Apply remaining conditions}

\text{Step 5: Check every condition}

\text{Step 6: Answer the question}

🟡 IMPORTANT KEY POINTS

\text{Read every condition carefully.}

\text{Place fixed positions first.}

\text{Use blocks for people who must be together.}

\text{Use position numbers to avoid confusion.}

\text{For opposite rank, use }N-P+1.

\text{For people between two positions, use }|P-Q|-1.

\text{For circular arrangements, identify the facing direction first.}

\text{For centre-facing people, left is clockwise and right is anticlockwise.}

\text{For outside-facing people, left is anticlockwise and right is clockwise.}

\text{Always verify the complete arrangement before selecting the answer.}

Logical Reasoning · Clocks _ Calenders

Basics- Formulas, Key Points and Examples

🔵 CLOCKS AND CALENDARS 🟢 1. BASIC CLOCK CONCEPT A clock has 12 numbers and completes one full revolution in 12 hours.

360^\circ=12\text{ hours}

The minute hand completes one revolution in 60 minutes.

360^\circ=60\text{ minutes}

Therefore:

1\text{ minute}=6^\circ

The hour hand moves 30° in one hour.

1\text{ hour}=30^\circ

The hour hand moves 0.5° in one minute.

1\text{ minute}=0.5^\circ

🟡 KEY POINT

\text{Minute Hand Speed}=6^\circ\text{ per minute}

\text{Hour Hand Speed}=0.5^\circ\text{ per minute}

🟢 2. MOVEMENT OF MINUTE HAND The minute hand moves 6° every minute.

\text{Angle moved by Minute Hand} = 6M

where

M

is the number of minutes. For example, in 20 minutes:

6\times20=120^\circ

🟡 KEY POINT

\text{Minute Hand Angle}=6M

🟢 3. MOVEMENT OF HOUR HAND The hour hand moves 30° in one hour.

\text{Angle moved in 1 hour}=30^\circ

Since one hour contains 60 minutes:

\text{Angle moved in 1 minute} = \frac{30}{60}

=0.5^\circ

Therefore, at

H

hours and

M

minutes:

\text{Hour Hand Angle} = 30H+0.5M

🟢 4. ANGLE BETWEEN THE HANDS OF A CLOCK At

H

hours and

M

minutes:

\text{Hour Hand Angle} = 30H+0.5M

\text{Minute Hand Angle} = 6M

Therefore:

\text{Angle} = \left| 30H+0.5M-6M \right|

Simplifying:

\text{Angle} = |30H-5.5M|

The smaller angle between the hands is:

\text{Smaller Angle} = \min \left( |30H-5.5M|, 360-|30H-5.5M| \right)

🔴 IMPORTANT

\text{If the calculated angle is greater than }180^\circ, \text{ subtract it from }360^\circ.

\text{Smaller Angle} = 360^\circ-\text{Larger Angle}

🟢 5. RIGHT ANGLE BETWEEN CLOCK HANDS A right angle is:

90^\circ

Therefore, the clock hands are perpendicular when:

|30H-5.5M|=90

🟡 KEY POINT

\text{Right Angle}=90^\circ

🟢 6. STRAIGHT ANGLE BETWEEN CLOCK HANDS A straight angle is:

180^\circ

Therefore, the clock hands are opposite when:

|30H-5.5M|=180

🟡 KEY POINT

\text{Straight Angle}=180^\circ

🟢 7. COINCIDENCE OF CLOCK HANDS The hands coincide when both hands are at the same position. Therefore:

30H+0.5M=6M

Simplifying:

30H=5.5M

Therefore:

M=\frac{30H}{5.5}

M=\frac{60H}{11}

The hands coincide approximately every:

\frac{720}{11}\text{ minutes}

Therefore:

\frac{720}{11} = 65\frac{5}{11}\text{ minutes}

🔴 IMPORTANT

\text{The hour and minute hands coincide 11 times in 12 hours.}

🟢 8. OPPOSITE POSITIONS OF CLOCK HANDS The hands are opposite when the angle between them is:

180^\circ

Therefore:

|30H-5.5M|=180

🟢 9. CLOCK HANDS AT A GIVEN ANGLE If the angle between the hands is

\theta

:

|30H-5.5M|=\theta

For a

60^\circ

angle:

|30H-5.5M|=60

For a

120^\circ

angle:

|30H-5.5M|=120

For a

90^\circ

angle:

|30H-5.5M|=90

🟢 10. ANGLE MOVED BY THE MINUTE HAND The minute hand completes:

360^\circ

in 60 minutes. Therefore:

\text{Angle per minute} = \frac{360}{60}

=6^\circ

Hence:

\text{Angle in }M\text{ minutes} = 6M

🟢 11. ANGLE MOVED BY THE HOUR HAND The hour hand completes:

360^\circ

in 12 hours. Therefore:

\text{Angle per hour} = \frac{360}{12}

=30^\circ

Since one hour contains 60 minutes:

\text{Angle per minute} = \frac{30}{60}

=0.5^\circ

🟢 12. CLOCK GAINING TIME If a clock gains time, it moves faster than the correct clock.

\text{Gain per hour} = \frac{\text{Total Gain}}{\text{Total Time in Hours}}

🟡 KEY POINT

\text{A fast clock shows more time than the actual time.}

🟢 13. CLOCK LOSING TIME If a clock loses time, it moves slower than the correct clock.

\text{Loss per hour} = \frac{\text{Total Loss}}{\text{Total Time in Hours}}

🟡 KEY POINT

\text{A slow clock shows less time than the actual time.}

🟢 14. RELATIVE GAIN AND LOSS OF CLOCKS If one clock gains time and another clock loses time, their relative difference increases.

\text{Relative Difference} = \text{Gain of First Clock} + \text{Loss of Second Clock}

If both clocks gain or both clocks lose time:

\text{Relative Difference} = \left| \text{Rate of First Clock} - \text{Rate of Second Clock} \right|

🟡 KEY POINT

\text{Always compare the rates of the two clocks when solving relative clock problems.}

🟢 15. BASIC CALENDAR CONCEPT A calendar is based on days, weeks, months and years. One week contains:

7\text{ days}

A normal year contains:

365\text{ days}

A leap year contains:

366\text{ days}

🟢 16. ODD DAYS Odd days are the number of days left after complete weeks are removed. Since:

1\text{ week}=7\text{ days}

The number of odd days is the remainder when the number of days is divided by 7. For 365 days:

365=52\times7+1

Therefore:

\text{Odd Days}=1

For 366 days:

366=52\times7+2

Therefore:

\text{Odd Days}=2

🟢 17. NORMAL YEAR A normal year has:

365\text{ days}

Therefore:

365=52\times7+1

Hence:

\text{Normal Year}=1\text{ Odd Day}

🟢 18. LEAP YEAR A leap year has:

366\text{ days}

Therefore:

366=52\times7+2

Hence:

\text{Leap Year}=2\text{ Odd Days}

🟢 19. LEAP YEAR RULE A year is generally a leap year if it is divisible by 4. For example:

2024\div4=506

Therefore:

2024\text{ is a leap year.}

However, century years must also be divisible by 400. For example:

2000\div400=5

Therefore:

2000\text{ is a leap year.}

But:

1900\div400=4.75

Therefore:

1900\text{ is not a leap year.}

🔴 IMPORTANT

\text{A century year is a leap year only if it is divisible by 400.}

🟢 20. DAYS IN DIFFERENT MONTHS The number of days in each month is:

\text{January}=31

\text{February}=28

\text{March}=31

\text{April}=30

\text{May}=31

\text{June}=30

\text{July}=31

\text{August}=31

\text{September}=30

\text{October}=31

\text{November}=30

\text{December}=31

In a leap year:

\text{February}=29

🟢 21. DAYS IN A WEEK The seven days of the week are:

\text{Sunday}

\text{Monday}

\text{Tuesday}

\text{Wednesday}

\text{Thursday}

\text{Friday}

\text{Saturday}

🟡 KEY POINT

1\text{ week}=7\text{ days}

🟢 22. DAY AFTER A GIVEN NUMBER OF DAYS If today is a particular day, the day after

N

days depends on the remainder when

N

is divided by 7.

N=7q+r

where

r

is the number of odd days. Therefore:

\text{Required Day} = \text{Starting Day}+r

🟡 KEY POINT

\text{For calendar problems, divide the number of days by 7 and use the remainder.}

🟢 23. DAY BEFORE A GIVEN NUMBER OF DAYS If we need to find the day before a given date, subtract the odd days.

N=7q+r

Therefore:

\text{Required Day} = \text{Starting Day}-r

🟢 24. DAY OF THE WEEK To determine the day of the week for a particular date, calculate the total number of odd days from a known reference date. The calculation generally involves:

\text{Odd Days} = \text{Odd Days in Complete Years} + \text{Odd Days in Complete Months} + \text{Odd Days in Remaining Days}

Then:

\text{Required Day} = \text{Reference Day}+\text{Total Odd Days}

The final result is reduced using:

\text{Total Odd Days}\bmod7

🟢 25. ODD DAYS IN COMPLETE YEARS For a group of years:

\text{Total Days} = 365(\text{Number of Normal Years}) + 366(\text{Number of Leap Years})

Then:

\text{Odd Days} = \text{Total Days}\bmod7

🟢 26. ODD DAYS IN COMPLETE MONTHS The number of days in complete months is added before the required date. For a normal year:

\text{Days in February}=28

For a leap year:

\text{Days in February}=29

Then:

\text{Odd Days} = \text{Total Days}\bmod7

🟢 27. NUMBER OF ODD DAYS BETWEEN TWO DATES To find the number of days between two dates:

\text{Total Days} = \text{Days in Complete Years} + \text{Days in Complete Months} + \text{Remaining Days}

Then:

\text{Odd Days} = \text{Total Days}\bmod7

🟢 28. SAME CALENDAR YEAR Two years can have the same calendar when their starting day is the same and the total odd days between them is a multiple of 7.

\text{Total Odd Days}\equiv0\pmod7

Therefore:

\text{Starting Day of Year 1} = \text{Starting Day of Year 2}

🟡 KEY POINT

\text{Leap years affect the starting day of the following year.}

🟢 29. CALENDAR AFTER A NORMAL YEAR A normal year has one odd day. Therefore, if a normal year starts on a particular day:

\text{Next Year Start} = \text{Starting Day}+1

For example, if a normal year starts on Monday:

\text{Next Year Start}=Tuesday

🟢 30. CALENDAR AFTER A LEAP YEAR A leap year has two odd days. Therefore:

\text{Next Year Start} = \text{Starting Day}+2

For example, if a leap year starts on Monday:

\text{Next Year Start}=Wednesday

🟢 31. IMPORTANT CLOCK FORMULAS FOR REVISION Minute hand angle:

\text{Minute Hand Angle}=6M

Hour hand angle:

\text{Hour Hand Angle}=30H+0.5M

Angle between hands:

\text{Angle}=|30H-5.5M|

Smaller angle:

\text{Smaller Angle} = \min \left( |30H-5.5M|, 360-|30H-5.5M| \right)

Right angle:

|30H-5.5M|=90

Straight angle:

|30H-5.5M|=180

Coincidence:

M=\frac{60H}{11}

🟢 32. IMPORTANT CALENDAR FORMULAS FOR REVISION Number of days in a normal year:

365=52\times7+1

Therefore:

\text{Normal Year}=1\text{ Odd Day}

Number of days in a leap year:

366=52\times7+2

Therefore:

\text{Leap Year}=2\text{ Odd Days}

Odd days:

\text{Odd Days} = \text{Number of Days}\bmod7

Day after

N

days:

N=7q+r

\text{Required Day} = \text{Starting Day}+r

Day before

N

days:

N=7q+r

\text{Required Day} = \text{Starting Day}-r

🟡 KEY POINTS

1\text{ hour}=60\text{ minutes}

1\text{ minute}=60\text{ seconds}

1\text{ week}=7\text{ days}

1\text{ normal year}=365\text{ days}

1\text{ leap year}=366\text{ days}

1\text{ normal year}=1\text{ odd day}

1\text{ leap year}=2\text{ odd days}

\text{Minute hand moves }6^\circ\text{ per minute.}

\text{Hour hand moves }0.5^\circ\text{ per minute.}

\text{Use }|30H-5.5M|\text{ to calculate the angle between clock hands.}

\text{For calendar problems, reduce the number of days using division by 7.}

\text{A leap year is divisible by 4, except century years.}

\text{A century year is a leap year only when it is divisible by 400.}

\text{Always check whether February has 28 or 29 days.}

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Verbal Ability

Interview Preparation

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UGC-CSIR

Mock Tests

Built for consistent, measurable progress

Structured Learning Paths

Topic-wise Practice

Performance Analytics

Mock Tests & Assessments

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Latest Concepts

Venn Diagram - Formula, Key Point and Examples

Data Sufficiency - Formulas, Key Points and Examples

Data Interpretation - Formulas, Key Points and Examples

Blood Relations

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Basics- Formulas, Key Points and Examples

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