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Logical Reasoning · Data Interpretation and Data Sufficiency

Data Sufficiency - Formulas, Key Points and Examples

Explanation

🔵
DATA SUFFICIENCY\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.
Data Sufficiency=Checking whether the given data is sufficient to answer the question \text{Data Sufficiency} = \text{Checking whether the given data is sufficient to answer the question}
🟡 KEY POINT
Sufficient Data≠Necessarily Calculating the Final Answer \text{Sufficient Data}\neq\text{Necessarily Calculating the Final Answer}
🟢 2. STATEMENT-BASED DATA SUFFICIENCY A typical question contains:
Question+Statement I+Statement II \text{Question}+\text{Statement I}+\text{Statement II}
You must determine whether:
Statement I alone is sufficient \text{Statement I alone is sufficient}
or:
Statement II alone is sufficient \text{Statement II alone is sufficient}
or:
Both statements together are sufficient \text{Both statements together are sufficient}
or:
Even both statements together are insufficient \text{Even both statements together are insufficient}
🟢 3. MAIN RULE Always test the statements separately first.
Step 1: Test Statement I alone \text{Step 1: Test Statement I alone}
Step 2: Test Statement II alone \text{Step 2: Test Statement II alone}
Step 3: If neither is sufficient, test Statements I and II together \text{Step 3: If neither is sufficient, test Statements I and II together}
🟡 KEY POINT
Do not combine the statements immediately \text{Do not combine the statements immediately}
🟢 4. STATEMENT I ALONE If Statement I gives enough information to determine a unique answer:
Statement I alone is sufficient \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:
Statement II alone is sufficient \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.
Statement I alone=Insufficient \text{Statement I alone}=\text{Insufficient}
Statement II alone=Insufficient \text{Statement II alone}=\text{Insufficient}
But:
Statement I+Statement II=Sufficient \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:
Statement I+Statement II=Insufficient \text{Statement I}+\text{Statement II}=\text{Insufficient}
🟡 KEY POINT
Multiple Possible Answers⇒Insufficient Data \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 x=10
then the value of \(x\) is uniquely determined. Therefore:
Sufficient \text{Sufficient}
But if:
x>5 x>5
then many values are possible. Therefore:
Insufficient \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:
YES \text{YES}
or:
NO \text{NO}
If both possibilities remain:
YES or NO \text{YES or NO}
then:
Insufficient \text{Insufficient}
🟢 10. VALUE-BASED QUESTIONS Suppose the question asks:
What is the value of x? \text{What is the value of }x?
The data is sufficient only if \(x\) can be uniquely determined. For example:
x+5=12 x+5=12
Therefore:
x=7 x=7
Hence:
Sufficient \text{Sufficient}
🟢 11. AGE PROBLEMS For age questions, identify the number of unknown ages and the relationships between them. For example:
A+B=40 A+B=40
This alone does not determine \(A\) and \(B\) individually. Therefore:
Insufficient \text{Insufficient}
If another independent equation is given:
A−B=10 A-B=10
then:
A+B=40 A+B=40
A−B=10 A-B=10
Adding both equations:
2A=50 2A=50
A=25 A=25
Therefore:
B=15 B=15
Hence:
Sufficient \text{Sufficient}
🟢 12. NUMBER PROBLEMS If the question asks for the value of a number and a statement gives:
x=25 x=25
then:
Statement is Sufficient \text{Statement is Sufficient}
But if the statement only gives:
x>10 x>10
then:
Statement is Insufficient \text{Statement is Insufficient}
🟢 13. EQUATION-BASED SUFFICIENCY One equation with two unknowns is generally insufficient. For example:
x+y=20 x+y=20
There are many possible values of \(x\) and \(y\). Therefore:
Insufficient \text{Insufficient}
Two independent equations may be sufficient:
x+y=20 x+y=20
x−y=4 x-y=4
Adding the equations:
2x=24 2x=24
x=12 x=12
Therefore:
y=8 y=8
Hence:
Sufficient \text{Sufficient}
🟢 14. INEQUALITY An inequality may or may not provide a unique answer. For example:
x>10 x>10
does not determine a unique value. Therefore:
Insufficient \text{Insufficient}
But if the question asks whether \(x\) is positive:
x>10 x>10
then:
x>0 x>0
Therefore:
Sufficient \text{Sufficient}
🟢 15. RATIO PROBLEMS A ratio alone may be insufficient if the actual values are required. For example:
A:B=2:3 A:B=2:3
The actual values could be:
A=2,B=3 A=2,\quad B=3
or:
A=4,B=6 A=4,\quad B=6
or:
A=20,B=30 A=20,\quad B=30
Therefore, the actual values cannot be uniquely determined.
Insufficient \text{Insufficient}
🟢 16. RATIO WITH TOTAL If:
A:B=2:3 A:B=2:3
and:
A+B=50 A+B=50
then:
2x+3x=50 2x+3x=50
5x=50 5x=50
x=10 x=10
Therefore:
A=20 A=20
B=30 B=30
Hence:
Sufficient \text{Sufficient}
🟢 17. PERCENTAGE PROBLEMS If the question asks for a percentage and the required part and whole are known:
Percentage=PartWhole×100 \text{Percentage} = \frac{\text{Part}}{\text{Whole}}\times100
If both the part and whole can be determined:
Sufficient \text{Sufficient}
If either the part or whole cannot be determined:
Insufficient \text{Insufficient}
🟢 18. AVERAGE PROBLEMS If the number of observations and total are known:
Average=TotalNumber of Observations \text{Average} = \frac{\text{Total}}{\text{Number of Observations}}
Therefore:
Total+Number of Observations⇒Sufficient \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:
Area=l×b \text{Area}=l\times b
If both length and breadth are known:
l,b⇒Area l,\quad b\Rightarrow\text{Area}
Hence:
Sufficient \text{Sufficient}
🟢 20. TRIANGLE PROBLEMS The sum of the angles of a triangle is:
A+B+C=180∘ A+B+C=180^\circ
If two angles are known, the third angle can be determined. For example:
A=60∘ A=60^\circ
B=70∘ B=70^\circ
Therefore:
C=180∘−60∘−70∘ C=180^\circ-60^\circ-70^\circ
C=50∘ C=50^\circ
Hence:
Sufficient \text{Sufficient}
🟢 21. SPEED, TIME AND DISTANCE The basic relationship is:
Distance=Speed×Time \text{Distance} = \text{Speed}\times\text{Time}
Therefore:
d=st d=st
If speed and time are known:
s,t⇒d s,\quad t\Rightarrow d
Hence:
Sufficient \text{Sufficient}
🟢 22. TIME AND WORK The basic relationship is:
Work=Rate×Time \text{Work} = \text{Rate}\times\text{Time}
If the rate and time are known:
W=RT W=RT
Therefore:
R,T⇒W R,\quad T\Rightarrow W
Hence:
Sufficient \text{Sufficient}
🟢 23. PROFIT AND LOSS The basic relationships are:
Profit=SP−CP \text{Profit}=SP-CP
and:
Loss=CP−SP \text{Loss}=CP-SP
To determine profit or loss, the required values must be known or uniquely derivable. 🟡 KEY POINT
Do Not Assume a Missing Value \text{Do Not Assume a Missing Value}
🟢 24. DATA SUFFICIENCY VS DATA INTERPRETATION Data Interpretation asks you to calculate or interpret information.
Data Interpretation→Find the Answer \text{Data Interpretation}\rightarrow\text{Find the Answer}
Data Sufficiency asks whether the information is enough.
Data Sufficiency→Check Whether the Answer Can Be Determined \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:
Can the Required Answer Be Uniquely Determined? \text{Can the Required Answer Be Uniquely Determined?}
If yes:
Sufficient \text{Sufficient}
If no:
Insufficient \text{Insufficient}
🟢 26. INDEPENDENT INFORMATION Two statements are useful together when they provide independent information. For example:
x+y=20 x+y=20
and:
x−y=4 x-y=4
These are independent equations and determine \(x\) and \(y\). Therefore:
Together They Are Sufficient \text{Together They Are Sufficient}
🟢 27. REDUNDANT INFORMATION Sometimes both statements provide essentially the same information. For example:
x=10 x=10
and:
2x=20 2x=20
Statement I alone is sufficient. Statement II alone is also sufficient. Therefore:
Either Statement Alone Is Sufficient \text{Either Statement Alone Is Sufficient}
🟢 28. COMMON ANSWER PATTERN Many aptitude tests use answer choices such as:
A. Statement I alone is sufficient \text{A. Statement I alone is sufficient}
B. Statement II alone is sufficient \text{B. Statement II alone is sufficient}
C. Both statements together are sufficient \text{C. Both statements together are sufficient}
D. Either statement alone is sufficient \text{D. Either statement alone is sufficient}
E. Even both statements together are insufficient \text{E. Even both statements together are insufficient}
🟡 KEY POINT
Follow the Exact Answer-Choice Convention Given in the Examination \text{Follow the Exact Answer-Choice Convention Given in the Examination}
🔴 IMPORTANT KEY POINTS
Test Statement I Alone First \text{Test Statement I Alone First}
Test Statement II Alone Next \text{Test Statement II Alone Next}
Combine Them Only When Necessary \text{Combine Them Only When Necessary}
Sufficient Means a Unique Answer Can Be Determined \text{Sufficient Means a Unique Answer Can Be Determined}
More Than One Possible Answer Means Insufficient \text{More Than One Possible Answer Means Insufficient}
Do Not Assume Information That Is Not Given \text{Do Not Assume Information That Is Not Given}
Do Not Solve More Than Necessary \text{Do Not Solve More Than Necessary}
A Statement Can Be Sufficient Even If the Actual Answer Is Not Calculated \text{A Statement Can Be Sufficient Even If the Actual Answer Is Not Calculated}
For YES/NO Questions, One Definite Answer Is Sufficient \text{For YES/NO Questions, One Definite Answer Is Sufficient}
Check Each Statement Independently Before Combining Them \text{Check Each Statement Independently Before Combining Them}

Example

🔵
DATA SUFFICIENCY - Solved Examples\text{DATA SUFFICIENCY - Solved Examples}
🟢 EXAMPLE 1: SIMPLE NUMBER QUESTION: What is the value of x? Statement I: x + 5 = 12. Statement II: x is a positive integer. SOLUTION: Using Statement I:
x+5=12 x+5=12
x=7 x=7
Therefore, Statement I alone determines the value of x.
Statement I alone is sufficient \text{Statement I alone is sufficient}
🔵 ANSWER:
Statement I alone is sufficient \text{Statement I alone is sufficient}
🟢 EXAMPLE 2: AGE QUESTION: What is the age of A? Statement I: A is 5 years older than B. Statement II: The sum of their ages is 35 years. SOLUTION: Using Statement I alone:
A=B+5 A=B+5
The individual ages cannot be determined. Therefore:
Statement I alone is insufficient \text{Statement I alone is insufficient}
Using Statement II alone:
A+B=35 A+B=35
The individual ages cannot be determined. Therefore:
Statement II alone is insufficient \text{Statement II alone is insufficient}
Using both statements:
A=B+5 A=B+5
A+B=35 A+B=35
Substituting:
B+5+B=35 B+5+B=35
2B=30 2B=30
B=15 B=15
Therefore:
A=20 A=20
Hence, both statements together are sufficient. 🔵 ANSWER:
Both statements together are sufficient \text{Both statements together are sufficient}
🟢 EXAMPLE 3: NUMBER QUESTION: Is x an even number? Statement I:
x=24 x=24
Statement II:
x>20 x>20
SOLUTION: Using Statement I:
x=24 x=24
Since 24 is even:
Answer = YES \text{Answer = YES}
Therefore:
Statement I alone is sufficient \text{Statement I alone is sufficient}
Using Statement II:
x>20 x>20
x could be 21, 22, 23, 24, etc. Therefore, x may be odd or even.
Statement II alone is insufficient \text{Statement II alone is insufficient}
🔵 ANSWER:
Statement I alone is sufficient \text{Statement I alone is sufficient}
🟢 EXAMPLE 4: RATIO QUESTION: What is the value of A? Statement I:
A:B=2:3 A:B=2:3
Statement II:
A+B=50 A+B=50
SOLUTION: Using Statement I alone:
A:B=2:3 A:B=2:3
Let:
A=2x,B=3x A=2x,\quad B=3x
The value of x is unknown. Therefore:
Statement I alone is insufficient \text{Statement I alone is insufficient}
Using Statement II alone:
A+B=50 A+B=50
The individual values of A and B are unknown. Therefore:
Statement II alone is insufficient \text{Statement II alone is insufficient}
Using both statements:
2x+3x=50 2x+3x=50
5x=50 5x=50
x=10 x=10
Therefore:
A=2(10) A=2(10)
A=20 A=20
Hence:
Both statements together are sufficient \text{Both statements together are sufficient}
🔵 ANSWER:
A=20 A=20
🟢 EXAMPLE 5: AVERAGE QUESTION: What is the average of five numbers? Statement I: Their sum is 250. Statement II: There are five numbers. SOLUTION: Using Statement I alone:
Sum=250 \text{Sum}=250
The number of observations is not known. Therefore:
Statement I alone is insufficient \text{Statement I alone is insufficient}
Using Statement II alone:
Number of observations=5 \text{Number of observations}=5
The total is not known. Therefore:
Statement II alone is insufficient \text{Statement II alone is insufficient}
Using both statements:
Average=SumNumber of Observations \text{Average} = \frac{\text{Sum}}{\text{Number of Observations}}
=2505 = \frac{250}{5}
=50 =50
Therefore:
Both statements together are sufficient \text{Both statements together are sufficient}
🔵 ANSWER:
50 50
🟢 EXAMPLE 6: PERCENTAGE QUESTION: What is 20% of x? Statement I:
x=500 x=500
Statement II:
x>400 x>400
SOLUTION: Using Statement I:
x=500 x=500
Therefore:
20% of x=20100×500 20\%\text{ of }x = \frac{20}{100}\times500
=100 =100
Hence:
Statement I alone is sufficient \text{Statement I alone is sufficient}
Using Statement II:
x>400 x>400
The exact value of x is unknown. Therefore:
Statement II alone is insufficient \text{Statement II alone is insufficient}
🔵 ANSWER:
Statement I alone is sufficient \text{Statement I alone is sufficient}
🟢 EXAMPLE 7: PROFIT QUESTION: What is the profit earned on an article? Statement I: The cost price is Rs. 800. Statement II: The selling price is Rs. 950. SOLUTION: Using Statement I alone:
CP=800 CP=800
The selling price is unknown. Therefore:
Statement I alone is insufficient \text{Statement I alone is insufficient}
Using Statement II alone:
SP=950 SP=950
The cost price is unknown. Therefore:
Statement II alone is insufficient \text{Statement II alone is insufficient}
Using both statements:
Profit=SP−CP \text{Profit}=SP-CP
=950−800 =950-800
=150 =150
Hence:
Both statements together are sufficient \text{Both statements together are sufficient}
🔵 ANSWER:
Rs. 150 Rs.\ 150
🟢 EXAMPLE 8: SPEED AND TIME QUESTION: What is the distance travelled by a car? Statement I: The speed of the car is 60 km/h. Statement II: The car travels for 4 hours. SOLUTION: Using Statement I alone:
Speed=60 km/h \text{Speed}=60\text{ km/h}
Time is unknown. Therefore:
Statement I alone is insufficient \text{Statement I alone is insufficient}
Using Statement II alone:
Time=4 hours \text{Time}=4\text{ hours}
Speed is unknown. Therefore:
Statement II alone is insufficient \text{Statement II alone is insufficient}
Using both statements:
Distance=Speed×Time \text{Distance} = \text{Speed}\times\text{Time}
=60×4 =60\times4
=240 km =240\text{ km}
Hence:
Both statements together are sufficient \text{Both statements together are sufficient}
🔵 ANSWER:
240 km 240\text{ km}
🟢 EXAMPLE 9: TIME AND WORK QUESTION: How many days will A take to complete a work? Statement I: A can complete the work at a rate of 10% per day. Statement II: The work is 100% complete. SOLUTION: Using Statement I:
Work Rate=10% per day \text{Work Rate}=10\%\text{ per day}
Using Statement II:
Total Work=100% \text{Total Work}=100\%
Using both:
Time=Total WorkWork Rate \text{Time} = \frac{\text{Total Work}}{\text{Work Rate}}
=10010 = \frac{100}{10}
=10 days =10\text{ days}
Hence:
Both statements together are sufficient \text{Both statements together are sufficient}
🔵 ANSWER:
10 days 10\text{ days}
🟢 EXAMPLE 10: GEOMETRY QUESTION: What is the area of a rectangle? Statement I: Its length is 12 cm. Statement II: Its breadth is 8 cm. SOLUTION: Using Statement I alone:
l=12 cm l=12\text{ cm}
Breadth is unknown. Therefore:
Statement I alone is insufficient \text{Statement I alone is insufficient}
Using Statement II alone:
b=8 cm b=8\text{ cm}
Length is unknown. Therefore:
Statement II alone is insufficient \text{Statement II alone is insufficient}
Using both statements:
Area=l×b \text{Area}=l\times b
=12×8 =12\times8
=96 cm2 =96\text{ cm}^2
Hence:
Both statements together are sufficient \text{Both statements together are sufficient}
🔵 ANSWER:
96 cm2 96\text{ cm}^2
🟢 EXAMPLE 11: TRIANGLE QUESTION: What is the third angle of a triangle? Statement I:
A=60∘ A=60^\circ
Statement II:
B=80∘ B=80^\circ
SOLUTION: The sum of the angles of a triangle is:
A+B+C=180∘ A+B+C=180^\circ
Using Statement I alone:
A=60∘ A=60^\circ
B is unknown. Therefore:
Statement I alone is insufficient \text{Statement I alone is insufficient}
Using Statement II alone:
B=80∘ B=80^\circ
A is unknown. Therefore:
Statement II alone is insufficient \text{Statement II alone is insufficient}
Using both:
C=180∘−60∘−80∘ C=180^\circ-60^\circ-80^\circ
C=40∘ C=40^\circ
Hence:
Both statements together are sufficient \text{Both statements together are sufficient}
🔵 ANSWER:
40∘ 40^\circ
🟢 EXAMPLE 12: SQUARE QUESTION: What is the perimeter of a square? Statement I: The side of the square is 15 cm. Statement II: The area of the square is 225 cm². SOLUTION: Using Statement I:
Side=15 cm \text{Side}=15\text{ cm}
Therefore:
Perimeter=4×15 \text{Perimeter}=4\times15
=60 cm =60\text{ cm}
Hence:
Statement I alone is sufficient \text{Statement I alone is sufficient}
Using Statement II:
Area=225 cm2 \text{Area}=225\text{ cm}^2
For a square:
Side=225 \text{Side}=\sqrt{225}
=15 cm =15\text{ cm}
Therefore:
Perimeter=4×15 \text{Perimeter}=4\times15
=60 cm =60\text{ cm}
Hence:
Statement II alone is sufficient \text{Statement II alone is sufficient}
🔵 ANSWER:
Either statement alone is sufficient \text{Either statement alone is sufficient}
🟢 EXAMPLE 13: YES OR NO QUESTION: Is A greater than B? Statement I:
A=80 A=80
Statement II:
B=60 B=60
SOLUTION: Using Statement I alone:
A=80 A=80
The value of B is unknown. Therefore:
Statement I alone is insufficient \text{Statement I alone is insufficient}
Using Statement II alone:
B=60 B=60
The value of A is unknown. Therefore:
Statement II alone is insufficient \text{Statement II alone is insufficient}
Using both:
A=80 A=80
B=60 B=60
Therefore:
A>B A>B
Hence:
Both statements together are sufficient \text{Both statements together are sufficient}
🔵 ANSWER:
Yes, A is greater than B \text{Yes, A is greater than B}
🟢 EXAMPLE 14: NUMBER DIVISIBILITY QUESTION: Is n divisible by 6? Statement I:
n=42 n=42
Statement II:
n is divisible by 3 n\text{ is divisible by }3
SOLUTION: Using Statement I:
n=42 n=42
Since:
42÷6=7 42\div6=7
Therefore:
n is divisible by 6 n\text{ is divisible by }6
Hence:
Statement I alone is sufficient \text{Statement I alone is sufficient}
Using Statement II alone:
n is divisible by 3 n\text{ is divisible by }3
But n may be 3, 9, 15, 21, etc. Therefore:
Statement II alone is insufficient \text{Statement II alone is insufficient}
🔵 ANSWER:
Statement I alone is sufficient \text{Statement I alone is sufficient}
🟢 EXAMPLE 15: AGE DIFFERENCE QUESTION: What is the age of A? Statement I: A is twice as old as B. Statement II: The sum of their ages is 36 years. SOLUTION: Using Statement I:
A=2B A=2B
The exact ages cannot be determined. Therefore:
Statement I alone is insufficient \text{Statement I alone is insufficient}
Using Statement II:
A+B=36 A+B=36
The individual ages cannot be determined. Therefore:
Statement II alone is insufficient \text{Statement II alone is insufficient}
Using both:
A=2B A=2B
A+B=36 A+B=36
Substituting:
2B+B=36 2B+B=36
3B=36 3B=36
B=12 B=12
Therefore:
A=24 A=24
Hence:
Both statements together are sufficient \text{Both statements together are sufficient}
🔵 ANSWER:
24 years 24\text{ years}
🟢 EXAMPLE 16: DATA SUFFICIENCY WITH REDUNDANT INFORMATION QUESTION: What is the value of x? Statement I:
x=15 x=15
Statement II:
2x=30 2x=30
SOLUTION: Using Statement I:
x=15 x=15
Therefore:
Statement I alone is sufficient \text{Statement I alone is sufficient}
Using Statement II:
2x=30 2x=30
x=15 x=15
Therefore:
Statement II alone is sufficient \text{Statement II alone is sufficient}
Hence:
Either statement alone is sufficient \text{Either statement alone is sufficient}
🔵 ANSWER:
x=15 x=15
🟢 EXAMPLE 17: INEQUALITY QUESTION: Is x positive? Statement I:
x>10 x>10
Statement II:
x=5 x=5
SOLUTION: Using Statement I:
x>10 x>10
Therefore:
x>0 x>0
Hence:
Statement I alone is sufficient \text{Statement I alone is sufficient}
Using Statement II:
x=5 x=5
Therefore:
x>0 x>0
Hence:
Statement II alone is sufficient \text{Statement II alone is sufficient}
🔵 ANSWER:
Either statement alone is sufficient \text{Either statement alone is sufficient}
🟢 EXAMPLE 18: INSUFFICIENT EVEN TOGETHER QUESTION: What is the value of x? Statement I:
x+y=20 x+y=20
Statement II:
x>5 x>5
SOLUTION: Using Statement I:
x+y=20 x+y=20
There are many possible values of x and y. Therefore:
Statement I alone is insufficient \text{Statement I alone is insufficient}
Using Statement II:
x>5 x>5
The exact value of x is unknown. Therefore:
Statement II alone is insufficient \text{Statement II alone is insufficient}
Using both:
x+y=20 x+y=20
and:
x>5 x>5
Possible values include:
x=6,y=14 x=6,\quad y=14
and:
x=10,y=10 x=10,\quad y=10
Therefore, x is not uniquely determined. Hence:
Even both statements together are insufficient \text{Even both statements together are insufficient}
🔵 ANSWER:
Insufficient Data \text{Insufficient Data}
🟢 EXAMPLE 19: PERCENTAGE AND TOTAL QUESTION: What is the number of students who passed an examination? Statement I: 80% of the students passed. Statement II: There are 500 students in total. SOLUTION: Using Statement I alone:
Passed=80% of Total \text{Passed}=80\%\text{ of Total}
The total number of students is unknown. Therefore:
Statement I alone is insufficient \text{Statement I alone is insufficient}
Using Statement II alone:
Total=500 \text{Total}=500
The percentage who passed is unknown. Therefore:
Statement II alone is insufficient \text{Statement II alone is insufficient}
Using both:
Passed=80100×500 \text{Passed} = \frac{80}{100}\times500
=400 =400
Hence:
Both statements together are sufficient \text{Both statements together are sufficient}
🔵 ANSWER:
400 students 400\text{ students}
🟢 EXAMPLE 20: COMPOUND DATA SUFFICIENCY QUESTION: What is the average of A, B and C? Statement I:
A+B+C=90 A+B+C=90
Statement II:
There are 3 values \text{There are 3 values}
SOLUTION: Using Statement I:
A+B+C=90 A+B+C=90
The number of values is understood from the question. Therefore:
Average=903 \text{Average} = \frac{90}{3}
=30 =30
Hence:
Statement I alone is sufficient \text{Statement I alone is sufficient}
Statement II does not provide the total. Therefore:
Statement II alone is insufficient \text{Statement II alone is insufficient}
🔵 ANSWER:
Statement I alone is sufficient \text{Statement I alone is sufficient}