Data Sufficiency

Data Sufficiency: Complete Beginner Guide with Easy Examples

Data sufficiency means deciding whether the given information is enough to answer a target clearly.

The word data means information. The word sufficient means enough. Therefore, data sufficiency simply asks:

Do we have enough information to give one definite answer?

This topic is different from ordinary calculation. In an ordinary problem, the main aim is to calculate the answer. In data sufficiency, the main aim is to judge the information. We calculate only as much as needed to decide whether the information is enough.

Suppose the target is to find $x$.

• Statement I says $x+7=19$.
• Statement II says $x$ is an even number below $20$.

Statement I gives $x=12$, so it is sufficient. Statement II allows many values such as $2,4,6,8$ and so on, so it is not sufficient.

The important word is definite. Information is sufficient only when every allowed case gives the same required answer.

Basics: The Three Parts of Data Sufficiency

A data sufficiency set usually has three parts.

1. The target

The target tells us what must be found or decided.

Examples:

• Find the value of $x$.
• Find the area of a rectangle.
• Decide whether $n$ is divisible by $6$.
• Decide whether $x>3$.
• Identify who sits second from the left.

The target controls what information is needed. The same statement may be enough for one target but not enough for another.

For example, $x^2=49$ is not enough to find the exact value of $x$, because $x$ can be $7$ or $-7$. However, it is enough to decide whether $x^2$ is positive, because both possible values make $x^2$ positive.

2. The statements

The statements provide pieces of information. They are normally labelled Statement I and Statement II.

Each statement must first be tested by itself. If neither works alone, then they may be combined.

3. The answer choices

The choices do not usually ask for the calculated number. They ask which statement or group of statements is sufficient.

A common four-choice format is:

• A: Statement I alone is sufficient, but Statement II alone is not.
• B: Statement II alone is sufficient, but Statement I alone is not.
• C: Both statements together are sufficient, but neither alone is sufficient.
• D: Even both statements together are not sufficient.

Some exams use a fifth case:

• E: Each statement alone is sufficient.

The letters can change from one exam to another. Always read the given answer key before using a memorised option letter.

What “Enough Information” Really Means

Information is sufficient when it fixes the required answer.

Imagine that a statement creates a box of possible cases.

• If every possible case gives the same target answer, the statement is sufficient.
• If two possible cases give different target answers, the statement is not sufficient.

Example: The target is to find $y$.

Statement I: $y$ is a multiple of $5$.

Possible values include $5,10,15,20$ and many more. Since the exact value changes, Statement I is not sufficient.

Statement II: $y-6=11$.

This gives:

$$y=11+6=17$$

Only one value is possible, so Statement II is sufficient.

Exact-Value Targets and Yes-or-No Targets

This difference is one of the most important ideas in data sufficiency.

Exact-value target

If the target says “find $x$,” “find the age,” or “find the area,” we normally need one exact value.

If $x$ can be $6$ or $-6$, the exact value is not fixed. The information is not sufficient.

Yes-or-no target

If the target says “is $x$ positive?” or “is $n$ divisible by $6$?”, we do not always need the exact value. We only need the answer to be definitely yes or definitely no.

For example, suppose the target is:

Is $x$ greater than $3$?

If a statement says $x>10$, we do not know the exact value of $x$. It could be $11$, $20$ or $100$. Still, every allowed value is greater than $3$. Therefore, the answer is definitely yes, and the statement is sufficient.

If a statement says $x<0$, the answer is definitely no. That is also sufficient.

Sufficiency does not mean the answer must be yes. A definite no is just as sufficient as a definite yes.

The Correct Order for Testing Statements

Use this order every time.

Step 1: Write exactly what must be known

If the target is the average, write:

$$\text{Average}=\frac{\text{total}}{\text{number of items}}$$

This shows that both the total and the count are needed.

Step 2: Test Statement I alone

Use only Statement I and facts stated in the main target. Do not secretly use Statement II.

Ask whether the required answer is fixed.

Step 3: Test Statement II alone

Forget the work from Statement I. Start fresh with only Statement II and the main target.

Step 4: Combine only if both fail alone

If neither statement is sufficient by itself, use them together.

Step 5: Match the result to the answer format

Decide whether the result is I alone, II alone, both together, neither even together, or each alone.

This fixed order prevents the most common mistake: mixing the two statements too early.

The Five Possible Sufficiency Results

Statement I alone is sufficient

Statement I fixes the target. Statement II does not.

Statement II alone is sufficient

Statement II fixes the target. Statement I does not.

Both statements are needed together

Each statement leaves something missing, but the missing pieces complete each other.

Even both statements are not sufficient

After combining them, more than one target answer is still possible.

Each statement alone is sufficient

Statement I fixes the target, and Statement II also fixes it independently. This case is used only when the given answer format includes it.

One Statement Alone: Direct Exact Information

The simplest sufficient statement directly gives the requested value or gives one equation that fixes it.

Example: Find $x$.

Statement I: $x+7=19$.

Statement II: $x$ is an even number below $20$.

What we need: One exact value of $x$.

Checking Statement I alone:

$$x+7=19$$

Subtract $7$ from both sides:

$$x=12$$

Statement I gives one exact value. It is sufficient.

Checking Statement II alone: Even positive numbers below $20$ include $2,4,6,8,10,12,14,16$ and $18$. More than one value is possible. It is not sufficient.

Sufficiency result: Statement I alone is sufficient.

Why: The target is fixed by Statement I, while Statement II gives only a group of possible values.

Comparison Is Not an Exact Value

A comparison such as older, taller, greater or cheaper usually does not give an exact number.

Example: Find Riya's present age.

Statement I: Riya is $24$ years old.

Statement II: Riya is older than Neha.

What we need: Riya's exact present age.

Checking Statement I alone: It directly gives the age as $24$. It is sufficient.

Checking Statement II alone: Riya may be $15$, $20$, $24$ or many other ages, as long as Neha is younger. The exact age is not fixed. It is not sufficient.

Sufficiency result: Statement I alone is sufficient.

Meaning: A comparison gives order, not an exact measurement.

Both Statements Give Two Missing Pieces

Many targets depend on two separate values. One statement gives the first value and the other gives the second.

Example: Find $a+b$.

Statement I: $a=14$.

Statement II: $b=23$.

What we need: Enough information to fix the sum.

Checking Statement I alone: We know $a$, but $b$ can change. The sum is not fixed.

Checking Statement II alone: We know $b$, but $a$ can change. The sum is not fixed.

Using both statements:

$$a+b=14+23=37$$

Sufficiency result: Both statements together are sufficient, but neither alone is sufficient.

Meaning: Each statement supplies one missing part of the target.

Divisibility Conditions

Some properties need more than one condition.

A number is divisible by $6$ exactly when it is divisible by both $2$ and $3$.

Example: Decide whether $n$ is divisible by $6$.

Statement I: $n$ is divisible by $2$.

Statement II: $n$ is divisible by $3$.

What we need: Both divisibility conditions.

Checking Statement I alone: A number such as $10$ is divisible by $2$ but not by $6$. A number such as $12$ is divisible by both. The answer can change, so Statement I is not sufficient.

Checking Statement II alone: A number such as $9$ is divisible by $3$ but not by $6$. A number such as $12$ is divisible by $6$. Again, the answer can change. Statement II is not sufficient.

Using both statements: $n$ has factors $2$ and $3$. Since $2$ and $3$ have no common factor except $1$, their product $6$ divides $n$.

Sufficiency result: Both statements together are sufficient.

Meaning: Each statement supplies one necessary condition.

Square Equations and the Sign Problem

When a square is given, remember the positive and negative roots.

$$x^2=36$$

does not mean only $x=6$. It means:

$$x=6\text{or}x=-6$$

because both values have square $36$.

Example: Find $x$.

Statement I: $x^2=36$.

Statement II: $x$ is positive.

What we need: One exact value of $x$.

Checking Statement I alone: Two values, $6$ and $-6$, are possible. It is not sufficient.

Checking Statement II alone: There are many positive values. It is not sufficient.

Using both statements: Statement I narrows $x$ to $6$ or $-6$. Statement II removes the negative value. Therefore:

$$x=6$$

Sufficiency result: Both statements together are sufficient.

Meaning: One statement gives a small set of possibilities, and the other selects one value from that set.

Absolute Value and Two Possibilities

Absolute value means distance from zero, so it cannot be negative.

If:

$$|x-3|=5$$

then $x-3$ can be $5$ or $-5$.

Example: Decide whether $x>3$.

Statement I: $|x-3|=5$.

Statement II: $x$ is positive.

What we need: A definite yes or no.

Checking Statement I alone:

$$x-3=5\Rightarrow x=8$$

or:

$$x-3=-5\Rightarrow x=-2$$

For $x=8$, the answer is yes. For $x=-2$, the answer is no. Statement I is not sufficient.

Checking Statement II alone: A positive value could be $2$ or $8$. One is below $3$, and one is above $3$. Statement II is not sufficient.

Using both statements: Of the two values from Statement I, only $8$ is positive. Therefore $x>3$ is definitely true.

Sufficiency result: Both statements together are sufficient.

Two Equations and Two Unknowns

One equation with two unknown values often allows many pairs. Two independent equations can fix both values.

The word independent means the second equation gives new information. If it is only a copy or multiple of the first equation, it may not help.

Example: Find $x$.

Statement I: $2x+y=19$.

Statement II: $x-y=2$.

What we need: One exact value of $x$.

Checking Statement I alone: If $x=5$, then $y=9$. If $x=7$, then $y=5$. Many pairs work, so $x$ is not fixed.

Checking Statement II alone: If $x=5$, then $y=3$. If $x=7$, then $y=5$. Again, many pairs work.

Using both statements: Add the equations:

$$(2x+y)+(x-y)=19+2$$

The $y$ terms cancel:

$$3x=21$$

Therefore:

$$x=7$$

Sufficiency result: Both statements together are sufficient.

Knowing the Target Expression Can Be Enough

We do not always need every individual variable. We only need the target.

Suppose the target is $x+y$, and a statement directly says $x+y=20$. That statement is sufficient even though $x$ and $y$ are not individually known.

This idea is useful in area, average and algebra examples. Do not calculate extra values when the required expression is already fixed.

When Both Statements Still Fail

Combining two statements does not automatically make them sufficient.

Example: Find the average of $a,b,c$.

Statement I: $a+b=30$.

Statement II: $b+c=42$.

What we need:

$$\text{Average}=\frac{a+b+c}{3}$$

So the sum $a+b+c$ must be fixed.

Checking Statement I alone: $c$ is missing. It is not sufficient.

Checking Statement II alone: $a$ is missing. It is not sufficient.

Using both statements: Add them:

$$a+2b+c=72$$

Therefore:

$$a+b+c=72-b$$

The result still depends on $b$.

For example:

• If $b=10$, then $a=20$, $c=32$, and the sum is $62$.
• If $b=12$, then $a=18$, $c=30$, and the sum is $60$.

Different averages are possible.

Sufficiency result: Even both statements together are not sufficient.

Meaning: Two equations do not always fix the requested expression.

Counterexamples: The Fastest Proof of Insufficiency

A counterexample is one allowed case that shows a claim is not always fixed. For data sufficiency, two allowed cases with different target answers prove that the information is not sufficient.

Suppose the target is: Is $x>3$?

Statement: $x$ is positive.

• $x=2$ is positive, but the answer is no.
• $x=5$ is positive, and the answer is yes.

Because both values follow the statement but give different answers, the statement is not sufficient.

Counterexamples are especially useful with inequalities, divisibility, age comparisons and seating arrangements.

Geometry: First List What the Formula Needs

Before testing statements, write the formula and mark the required inputs.

Area of a square

$$\text{Area}=s^2$$

One exact side length is enough.

Example: Find the area of a square.

Statement I: Its side is $13$ cm.

Statement II: Its perimeter is more than $40$ cm.

Checking Statement I alone:

$$\text{Area}={13}^2=169{}^{\text{cm}}$$

It is sufficient.

Checking Statement II alone:

$$4s>40\Rightarrow s>10$$

The side may be $11$, $12$, $13$ and many other values. Different sides give different areas. It is not sufficient.

Sufficiency result: Statement I alone is sufficient.

Perimeter of a rectangle

$$P=2(l+b)$$

If one statement gives $l$ and the other gives $b$, both are normally needed.

Example: Find the perimeter of a rectangle.

Statement I: $l=18$ cm.

Statement II: $b=11$ cm.

Statement I lacks breadth. Statement II lacks length. Together:

$$P=2(18+11)=58\text{cm}$$

Sufficiency result: Both statements together are sufficient.

Area from diagonal and perimeter

Sometimes we can find the target without finding length and breadth separately.

Example: Find the area of a rectangle.

Statement I: Its diagonal is $13$ cm.

Statement II: Its perimeter is $34$ cm.

Checking Statement I alone: By the Pythagorean relation:

$$l^2+b^2={13}^2=169$$

This does not fix $lb$, so it does not fix area.

Checking Statement II alone:

$$2(l+b)=34\Rightarrow l+b=17$$

Many length-breadth pairs have sum $17$, so the area is not fixed.

Using both statements:

$${(l+b)}^2=l^2+b^2+2lb$$

Substitute the known values:

$${17}^2=169+2lb$$ $$289-169=2lb$$ $$120=2lb\Rightarrow lb=60$$

Since area is $lb$, the area is fixed at $60{}^{\text{cm}}$.

Sufficiency result: Both statements together are sufficient.

Notice that we did not need to find $l$ and $b$ separately.

Area of a sector

$$\text{Sector area}=\frac{\theta }{{360}^{°}}\pi r^2$$

The radius $r$ and central angle $\theta$ are needed. A value for $\pi$ may be stated when a numerical form is expected.

Example: Find the area of a sector.

Statement I: $r=14$ cm.

Statement II: $\theta ={90}^{°}$ and use $\pi =\frac{22}{7}$.

Neither statement supplies all the needed inputs. Together:

$$\text{Area}=\frac{90}{360}\times \frac{22}{7}\times {14}^2=154{}^{\text{cm}}$$

Sufficiency result: Both statements together are sufficient.

Ages and Ratios

An age ratio gives relative values, not actual ages. A second fact such as a sum, difference or future ratio can fix the scale.

Ratio and total age

Example: Find the son's present age.

Statement I: The father's age is three times the son's age.

Statement II: Their ages add to $64$.

Checking Statement I alone: Let the son's age be $x$. The father's age is $3x$. Many values of $x$ are possible.

Checking Statement II alone: Many age pairs add to $64$.

Using both statements:

$$x+3x=64$$ $$4x=64\Rightarrow x=16$$

Sufficiency result: Both statements together are sufficient.

Present ratio and future ratio

Example: Find Meera's present age.

Statement I: Meera and Tina are in the ratio $4:5$.

Statement II: After $6$ years, their ages will be in the ratio $5:6$.

Checking each statement alone: Either ratio by itself can fit many actual ages.

Using both statements: Write present ages as $4x$ and $5x$. After $6$ years:

$$\frac{4x+6}{5x+6}=\frac{5}{6}$$

Cross multiply:

$$6(4x+6)=5(5x+6)$$ $$24x+36=25x+30$$ $$x=6$$

Meera's age is:

$$4x=24$$

Sufficiency result: Both statements together are sufficient.

Average and Weighted Average

An average needs a total and a count:

$$\text{Average}=\frac{\text{Total}}{\text{Count}}$$

If an average and count are known, the total can be recovered:

$$\text{Total}=\text{Average}\times \text{Count}$$

Example: Find the average score of a group.

Statement I: The total score is $480$.

Statement II: There are $12$ students.

Statement I lacks the count. Statement II lacks the total. Together:

$$\text{Average}=\frac{480}{12}=40$$

Sufficiency result: Both statements together are sufficient.

Combining two group averages

Example: Find the average salary of a team containing only developers and testers.

Statement I: Six developers have average salary ₹50,000.

Statement II: Four testers have average salary ₹40,000.

Checking Statement I alone: The tester total is missing.

Checking Statement II alone: The developer total is missing.

Using both statements:

$$\text{Developer total}=6\times 50,000=300,000$$ $$\text{Tester total}=4\times 40,000=160,000$$ $$\text{Team average}=\frac{300,000+160,000}{6+4}=46,000$$

Sufficiency result: Both statements together are sufficient.

Do not directly average ₹50,000 and ₹40,000. The two groups have different numbers of people.

Percentage, Discount and Profit

Percentage targets often need a base value and a percentage rate.

One discount

Example: Find the selling price of a shirt.

Statement I: Marked price is ₹800.

Statement II: Discount is $15\%$.

Neither statement works alone. Together:

$$\text{Selling price}=800(1-\frac{15}{100})$$ $$=800\times 0.85=\text{Rs.}680$$

Sufficiency result: Both statements together are sufficient.

Successive discounts

The second discount is applied to the already reduced price, not to the original price.

Example: Find the price after two successive discounts.

Statement I: Marked price is ₹1,000.

Statement II: The discounts are $10\%$ and $20\%$.

Neither statement works alone. Together, after the first discount:

$$1000\times 0.90=\text{Rs.}900$$

After the second discount:

$$900\times 0.80=\text{Rs.}720$$

Sufficiency result: Both statements together are sufficient.

Cost price from selling price and profit

Example: Find the cost price.

Statement I: An item is sold for ₹720 at a profit of $20\%$.

Statement II: The item is sold above cost price.

Checking Statement I alone: It gives both selling price and profit rate:

$$\text{Cost price}=\frac{720\times 100}{120}=\text{Rs.}600$$

Statement I is sufficient.

Checking Statement II alone: It tells us only that some profit exists. The exact cost price can vary. It is not sufficient.

Sufficiency result: Statement I alone is sufficient.

Simple Interest

Simple interest uses:

$$SI=\frac{PRT}{100}$$

To find rate $R$:

$$R=\frac{100SI}{PT}$$

Example: Find the annual simple-interest rate.

Statement I: Principal is ₹8,000 and time is $3$ years.

Statement II: Simple interest is ₹1,920.

Statement I lacks interest. Statement II lacks principal and time. Together:

$$R=\frac{100\times 1920}{8000\times 3}=8\%$$

Sufficiency result: Both statements together are sufficient.

Time and Work or Pipes

If a person finishes one whole job in $T$ days, the person's one-day work rate is:

$$\frac{1}{T}$$

For two people working together:

$$r_A+r_B=r_{A+B}$$

Example: Find the time A needs to finish a job alone.

Statement I: A and B together finish it in $12$ days.

Statement II: B alone finishes it in $20$ days.

Checking Statement I alone: We know only the combined rate $\frac{1}{12}$.

Checking Statement II alone: We know only B's rate $\frac{1}{20}$.

Using both statements:

$$r_A=\frac{1}{12}-\frac{1}{20}$$

The least common denominator is $60$:

$$r_A=\frac{5}{60}-\frac{3}{60}=\frac{2}{60}=\frac{1}{30}$$

If A completes $\frac{1}{30}$ of the job per day, A needs $30$ days for the whole job.

Sufficiency result: Both statements together are sufficient.

The same rate idea works for two inlet pipes filling a tank.

Speed, Distance and Time

The basic relation is:

$$\text{Speed}=\frac{\text{Distance}}{\text{Time}}$$

Example: Find a cyclist's speed.

Statement I: The cyclist covers $45$ km.

Statement II: A walker travels at $5$ km/h, and the cyclist takes $3$ hours less than the walker over the same distance.

Checking Statement I alone: Distance without time does not fix speed.

Checking Statement II alone: Without the common distance, the walker's time and cyclist's time are not fixed.

Using both statements: The walker's time is:

$$\frac{45}{5}=9\text{hours}$$

The cyclist takes:

$$9-3=6\text{hours}$$

Therefore:

$$\text{Cyclist's speed}=\frac{45}{6}=7.5\text{km/h}$$

Sufficiency result: Both statements together are sufficient.

Boats and Streams

Let downstream speed be $D$ and upstream speed be $U$.

The speed in still water is:

$$b=\frac{D+U}{2}$$

Example: Find the boat's speed in still water.

Statement I: Downstream speed is $18$ km/h.

Statement II: Upstream speed is $12$ km/h.

Neither speed alone is enough. Together:

$$b=\frac{18+12}{2}=15\text{km/h}$$

Sufficiency result: Both statements together are sufficient.

Mixtures and Concentration

To find the amount of one part in a mixture, we normally need the total amount and its fraction or percentage.

Example: Find the amount of sugar in a mixture.

Statement I: The mixture weighs $40$ kg.

Statement II: Sugar is $15\%$ of the mixture.

Neither statement alone gives the sugar amount. Together:

$$\text{Sugar}=\frac{15}{100}\times 40=6\text{kg}$$

Sufficiency result: Both statements together are sufficient.

Removal and replacement

When some pure liquid is removed and replaced by another liquid, we must know the original quantity and the removed quantity.

Example: A container initially has $40$ litres of milk. Ten litres are removed and replaced with water. Find the final milk percentage.

Milk left:

$$40-10=30\text{litres}$$

The total returns to $40$ litres after replacement. Therefore:

$$\text{Milk percentage}=\frac{30}{40}\times 100=75\%$$

The original amount alone is not enough. The removed amount alone is not enough. Together they are sufficient.

Quadratic Roots and an Ordering Clue

A quadratic equation may give two roots without telling which root belongs to which named variable.

Example: Find $x$.

Statement I: $x$ and $y$ are the roots of:

$$t^2-9t+20=0$$

Statement II: $x>y$.

Checking Statement I alone: Factor:

$$(t-4)(t-5)=0$$

The roots are $4$ and $5$, but $x$ could be either one. It is not sufficient.

Checking Statement II alone: It gives order but no values. It is not sufficient.

Using both statements: The larger root is $5$, so $x=5$ and $y=4$.

Sufficiency result: Both statements together are sufficient.

Trains and Relative Speed

When a train crosses a pole, the distance covered equals the train's own length.

When two trains completely cross each other, the distance equals the sum of their lengths. In opposite directions, relative speed is the sum of their speeds.

Example: Find the length of Train A.

Statement I: Train A crosses a pole in $15$ seconds.

Statement II: Train A crosses an opposite-moving Train B of length $180$ m in $12$ seconds. Train B moves at $20$ m/s.

Checking Statement I alone: If Train A's speed is $v$ m/s, its length is $15v$. Since $v$ is unknown, length is not fixed.

Checking Statement II alone: Both Train A's speed and length are unknown. It is not sufficient.

Using both statements:

Let Train A's speed be $v$ m/s. Its length from Statement I is $15v$ metres.

During the crossing:

$$\text{Total distance}=15v+180$$ $$\text{Relative speed}=v+20$$

Use distance $=$ speed $\times$ time:

$$15v+180=12(v+20)$$ $$15v+180=12v+240$$ $$3v=60\Rightarrow v=20$$

Therefore, Train A's length is:

$$15\times 20=300\text{m}$$

Sufficiency result: Both statements together are sufficient.

Probability Information

Probability needs a clear experiment and enough information about how likely its outcomes are.

Example: Find the probability of at least one head in two tosses.

Statement I: A fair coin is tossed twice.

Statement II: At least one head means one head or two heads.

Checking Statement I alone: A fair coin tossed twice has equally likely outcomes:

$$HH,HT,TH,TT$$

Three contain at least one head, so:

$$P\left(\text{at least one head}\right)=\frac{3}{4}$$

Statement I is sufficient.

Checking Statement II alone: It explains the event but gives no experiment, number of tosses or fairness. It is not sufficient.

Sufficiency result: Statement I alone is sufficient.

The meaning of a phrase may already be part of normal mathematics. A statement that only repeats a definition may add no useful data.

Sets and “Exactly One”

For two groups $T$ and $C$:

$$\text{Exactly one}=\left|T\right|+\left|C\right|-2|T\cap C|$$

The overlap is subtracted twice because people in both groups were counted once in each total, but “exactly one” must exclude them completely.

Example: Find how many students like exactly one of tea and coffee.

Statement I: $20$ like tea, $18$ like coffee and $8$ like both.

Statement II: Every student likes at least one of the two drinks.

Checking Statement I alone:

$$20+18-2\left(8\right)=22$$

It fixes the target and is sufficient.

Checking Statement II alone: It gives no counts. It is not sufficient.

Sufficiency result: Statement I alone is sufficient.

The number who like neither is not needed because the target asks only about exactly one.

Permutation Information

For arrangements with vowels together, we need to know how many vowel units and consonant units are being arranged, and whether any letters repeat.

For example, MATRIX has six distinct letters and two vowels, $A$ and $I$. Treat the two vowels as one block. Then there are five units: the vowel block and four consonants.

The five units can be arranged in $5!$ ways. The vowels inside their block can be arranged in $2!$ ways:

$$5!\times 2!=240$$

In a data sufficiency setting, use all facts explicitly given in the main stem as well as each tested statement. Do not ignore visible or stated facts unless the directions clearly tell you to treat them as unknown. If a test supplies the number of vowels in one statement and distinctness in another, check whether the main stem itself already makes either fact available.

This is why reading the exact directions matters. Sufficiency must be judged from the allowed information, not from an artificial refusal to use facts already given.

Seating Arrangement Sufficiency

For arrangements, a statement is sufficient only when the requested position or person is the same in every valid arrangement.

Example: Four people A, B, C and D sit in a row. Find who sits second from the left.

Statement I: A sits immediately left of C, and B is at the left end.

Statement II: D is not next to B.

Checking Statement I alone: B must be first. The block $AC$ can fit in two ways:

$$BACD$$

or:

$$BDAC$$

The second person is A in the first arrangement and D in the second. Statement I is not sufficient.

Checking Statement II alone: Many arrangements satisfy D not being beside B. It is not sufficient.

Using both statements: The second arrangement puts D next to B, so it is removed. Only:

$$BACD$$

remains. A is second from the left.

Sufficiency result: Both statements together are sufficient.

You do not need the entire arrangement if the requested seat is fixed in every remaining case.

Blood Relation Sufficiency

Draw a small family link instead of trying to hold every relation in your head.

Example: Find the relationship between K and M.

Statement I: K is the only daughter of L.

Statement II: L is the mother of M.

Checking Statement I alone: It connects K with L but gives no link to M.

Checking Statement II alone: It connects L with M but gives no link to K.

Using both statements: L is mother of both K and M. K is L's only daughter, so M cannot be another daughter. M is L's son. Therefore, K is M's sister.

Sufficiency result: Both statements together are sufficient.

Words such as only, son, daughter, mother and father give gender information. Do not add gender from a person's name alone.

Information That Looks Useful but Is Not Needed

Data sufficiency often includes extra facts. A fact may be true and still be unnecessary for the target.

For example, to find the number who like exactly one of two drinks, the counts of each group and their overlap are enough. Knowing that nobody likes neither drink does not change that target.

Ask one simple question:

If I remove this fact, is the target still fixed?

If yes, that fact is not needed.

Common Mistakes

Solving for a number before judging sufficiency

Calculate only until the target is clearly fixed or clearly not fixed. Long calculation can waste time.

Combining statements immediately

Always test Statement I alone and Statement II alone before using both together.

Carrying information from Statement I into Statement II

When testing Statement II alone, start fresh. Use only the main stem and Statement II.

Thinking “not exact” always means insufficient

For a yes-or-no target, a range can be sufficient if every value gives the same yes or no answer.

Thinking a definite no is insufficient

A definite no fully answers a yes-or-no target. It is sufficient.

Forgetting positive and negative roots

From $x^2=a^2$, both $x=a$ and $x=-a$ may be possible unless another condition fixes the sign.

Removing absolute value as only one case

From $\left|u\right|=k$, use $u=k$ and $u=-k$ when $k>0$.

Counting equations instead of checking information

Two equations can still describe the same relation. They must provide new independent information.

Finding every variable when only one expression is needed

If the target is $l\times b$, finding the product is enough. The individual sides may remain unknown.

Using unstated assumptions

Do not assume a number is positive, a coin is fair, letters are distinct or people are different unless the stem, statements or normal definition guarantees it.

Ignoring information already in the main stem

Facts in the main stem are available while testing either statement. Do not pretend they are unknown.

Memorising option letters without reading the key

The meaning of A, B, C, D and E can change between exams.

Believing both statements must be useful

One statement may be sufficient while the other adds nothing. Judge each by the target.

Safe Shortcuts

• Write the target at the top before reading the statements.
• Mark whether the target needs an exact value or only a definite yes or no.
• Test Statement I alone, reset, then test Statement II alone.
• Combine the statements only after both fail separately.
• Use two quick counterexamples to prove that a statement is not sufficient.
• For any required calculation, list only the inputs needed for the target and see which statement supplies each one.
• If a statement leaves two roots or two arrangements, check whether both give the same target answer.
• Do not find every unknown when the target expression itself can be fixed.
• In algebra, check whether two equations are truly independent.
• In seating, list all valid small arrangements and see whether the requested position changes.
• In blood relations, draw links and mark gender only when it is given.
• Match the result to the printed answer format only at the end.

A shortcut is safe only when it protects the independence of the statements and the exact target. It should make the sufficiency check faster, not replace it.

Frequently Asked Questions

What is data sufficiency?

Data sufficiency is the process of deciding whether given statements contain enough information to answer a target definitely.

Do I need to calculate the final numerical answer?

Not always. Calculate only enough to prove that the target has one fixed answer. Sometimes simply showing that one unique answer exists is enough.

What does sufficient mean in reasoning?

Sufficient means enough to settle the target. Every case allowed by the information must give the same required answer.

How should I test two data sufficiency statements?

Test Statement I alone, then Statement II alone, and combine them only if both fail separately.

Can I use Statement I while checking Statement II?

No. Each statement must first be checked independently using only the main stem and that statement.

Is a range of values always insufficient?

No. A range is insufficient for an exact-value target if the answer changes. For a yes-or-no target, it can be sufficient when every value gives the same result.

Does a definite no count as sufficient?

Yes. A definite no answers a yes-or-no target completely.

Why is $x^2=36$ not enough to find x?

Because both $6$ and $-6$ have square $36$. Another clue is needed to choose the sign.

How do I handle absolute value in data sufficiency?

Write both branches. If $\left|u\right|=k$ and $k>0$, then $u=k$ or $u=-k$. Check the target for both branches.

Are two equations always enough for two unknowns?

No. The equations must provide different information. If one is just a multiple of the other, many pairs may remain possible.

How can I prove a statement is insufficient quickly?

Find two cases that follow the statement but give different target answers. These are counterexamples.

Can the target be fixed even when individual values are unknown?

Yes. If only $x+y$ is required and a statement fixes $x+y$, the target is sufficient even if $x$ and $y$ are not known separately.

What facts can I use while testing a statement?

Use the main stem, normal mathematical definitions and the statement being tested. Do not use the other statement or add unsupported assumptions.

Why must I read the option instructions every time?

Different exams map the sufficiency cases to different letters. A memorised option letter can therefore give the wrong response.

What topics can appear inside data sufficiency?

It can use algebra, arithmetic, ages, ratios, percentages, averages, geometry, time and work, speed, trains, probability, sets, seating arrangements, blood relations and many other topics.

What is the most important data sufficiency habit?

Focus on the target and keep the statements separate until both have been tested alone.

Summary

Data sufficiency is about judging information, not doing the longest possible calculation. First understand the target. Decide whether it needs one exact value or only a definite yes or no. Then test Statement I alone, test Statement II alone, and combine them only when neither works independently.

One fixed target answer means sufficient. Two allowed cases with different target answers mean insufficient. A definite yes and a definite no are both complete answers to a yes-or-no target.

Use formulas to list the information needed, counterexamples to show missing information, and small diagrams for seating or family relations. Always use facts in the main stem, avoid unstated assumptions, and read the answer-choice meaning before selecting a letter.