Data Interpretation - MCQ Practice Questions

Understanding:

We need to determine whether a unique two-digit number can be found.

Formula:

Let the two-digit number be $\overline{ab}$, i.e., $10a+b$, where $a$ is the tens digit and $b$ is the units digit.

Step 1: Analyse Statement I alone

From Statement I: $a+b=9$.

Possible pairs: $(1,8),(2,7),(3,6),(4,5),(5,4),(6,3),(7,2),(8,1),(9,0)$.

Multiple solutions exist, so Statement I alone is not sufficient.

Step 2: Analyse Statement II alone

From Statement II:

Possible pairs: $(4,1),(5,2),(6,3),(7,4),(8,5),(9,6)$.

Multiple solutions exist, so Statement II alone is not sufficient.

Step 3: Combine both statements

From I: $a+b=9$

From II: $a-b=3$

The unique number is $63$. Both statements together are sufficient.

Answer:

Both statements together uniquely determine the number as 63, but neither alone is sufficient.

Quick Tip:

In data sufficiency, always check if Statement I alone gives a unique answer before testing combinations. A system of two equations with two unknowns typically requires both.

Understanding:

We need to determine if $n$ is even.

Formula:

An integer is even if it is divisible by 2. We check each statement for sufficiency.

Step 1: Analyse Statement I alone

If $n$ is even, say $n=2k$: $(2k-1)(2k+1)$ is a product of two consecutive odd numbers, which is odd — not divisible by 4.

If $n$ is odd, say $n=2k+1$: $(2k)(2k+2)=4k(k+1)$, which is always divisible by 4.

So $4\mid (n^2-1)$ if and only if $n$ is odd. Statement I tells us $n$ is odd (i.e., not even). This is a definitive answer — Statement I alone is sufficient.

Step 2: Analyse Statement II alone

$n+5$ is odd $\Rightarrow$ $n$ is even (since odd $-$ odd $=$ even, and $5$ is odd, so $n$ must be even for the sum to be odd).

This directly tells us $n$ is even. Statement II alone is sufficient.

Step 3: Conclusion

Both statements independently give a definitive (though opposite in this case) answer about the parity of $n$. Each statement alone is sufficient.

Answer:

Either statement alone is sufficient to answer the question.

Quick Tip:

For parity questions, translating algebraic conditions (like divisibility) into odd/even properties quickly reveals sufficiency.

Understanding:

We need to find the area of a rectangle.

Formula:

Step 1: Analyse Statement I alone

From $2(l+b)=40$, we get $l+b=20$.

This gives infinitely many $(l,b)$ pairs: $(11,9),(12,8)$, etc. — not sufficient.

Step 2: Analyse Statement II alone

$l=b+4$ gives one equation in two unknowns — infinitely many solutions. Not sufficient.

Step 3: Combine both statements

Adding: $2l=24\Rightarrow l=12\text{ cm}$, $b=8\text{ cm}$.

A unique area is obtained.

Answer:

Both statements together are sufficient to find the area; neither alone is sufficient.

Quick Tip:

Perimeter gives $l+b$; the extra relation gives $l-b$. Together they form a solvable system — a classic data sufficiency pattern.

Understanding:

We need to find the number of students who play neither sport in a class of 60.

Formula:

By the inclusion-exclusion principle:

Step 1: Analyse Statement I alone

We know $|C|+|F|=65$, but without $|C\cap F|$, we cannot find $|C\cup F|$. Not sufficient.

Step 2: Analyse Statement II alone

Knowing only $|C\cap F|=15$ without $|C|$ and $|F|$ is insufficient.

Step 3: Combine both statements

Answer:

Both statements together give a unique answer; neither alone is sufficient.

Quick Tip:

For set-based sufficiency questions, inclusion-exclusion needs three values: $|A|$, $|B|$, and $|A\cap B|$. Any two of these three is typically insufficient.

Understanding:

We need a unique value of $x$.

Formula:

Step 1: Analyse Statement I alone

$x^2=49$ gives $x=7$ or $x=-7$. Two solutions — not sufficient for a unique value.

Step 2: Analyse Statement II alone

$x>0$ gives infinitely many positive values. Not sufficient.

Step 3: Combine both statements

From Statement I: $x\in \{7,-7\}$.

Applying Statement II ($x>0$): only $x=7$ qualifies.

Answer:

Both statements together uniquely determine $x$; neither alone is sufficient.

Quick Tip:

Whenever a quadratic equation gives two roots (positive and negative), a sign condition from a second statement is typically what makes the answer unique — a very common data sufficiency pattern.

Understanding:

We need to find the profit percentage.

Formula:

Step 1: Analyse Statement I alone

Knowing only $\text{CP}=₹500$ without the profit amount is insufficient to compute profit percentage.

Step 2: Analyse Statement II alone

Knowing only $\text{Profit}=₹125$ without the cost price is insufficient.

Step 3: Combine both statements

Answer:

Both statements together are sufficient; neither alone is sufficient.

Quick Tip:

Profit percentage always needs both the profit amount and the cost price. If a statement gives only one of these, it is never independently sufficient.

Understanding:

We need to determine if $\frac{p}{q}$ (in lowest terms) is a terminating decimal.

Formula:

A fraction $\frac{p}{q}$ in its lowest terms is a terminating decimal if and only if the denominator $q$ has no prime factors other than 2 and 5:

Step 1: Analyse Statement I alone

Factorise $q=40$:

The only prime factors are 2 and 5, so $\frac{p}{40}$ in lowest terms is always a terminating decimal, regardless of the value of $p$. Statement I alone is sufficient.

Step 2: Analyse Statement II alone

Knowing $p=3$ tells us nothing about the denominator's prime factors. For example, $\frac{3}{4}$ terminates but $\frac{3}{7}$ does not. Not sufficient.

Answer:

Statement I alone is sufficient to determine that the fraction is a terminating decimal.

Quick Tip:

For terminating decimal questions, the numerator is irrelevant — only the prime factorisation of the denominator (after full reduction) matters. If $q$ only has factors of 2 and 5, the answer is always yes.

Understanding:

We need to find the average speed for the entire journey where the two halves are covered at different speeds.

Formula:

When equal distances are covered at two different speeds $v_1$ and $v_2$, the average speed is the harmonic mean:

Step 1: Analyse Statement I alone

Only the speed for the first half is known. Without the second half's speed, average speed cannot be determined. Not sufficient.

Step 2: Analyse Statement II alone

Only the speed for the second half is known. Not sufficient.

Step 3: Combine both statements

Answer:

Both statements together uniquely determine the average speed; neither alone is sufficient.

Quick Tip:

For equal-distance two-speed problems, always use the harmonic mean formula. The arithmetic mean $\frac{60+90}{2}=75$ is a common wrong answer — the correct value is always less.

Understanding:

We need to find the number of days A alone takes to complete the work.

Formula:

Using the work-rate principle:

where $A$ and $B$ represent the number of days each person takes alone.

Step 1: Analyse Statement I alone

$\frac{1}{A}+\frac{1}{B}=\frac{1}{12}$. Without knowing $B$, we cannot find $A$. Not sufficient.

Step 2: Analyse Statement II alone

Only $\frac{1}{B}=\frac{1}{30}$ is known. Without $A+B$ combined rate, we cannot find $A$. Not sufficient.

Step 3: Combine both statements

So A alone completes the work in $20$ days.

Answer:

Both statements together uniquely determine the answer; neither alone is sufficient.

Quick Tip:

In work problems, the combined rate minus one person's rate gives the other person's rate. Always convert days to rates ($\frac{1}{\text{days}}$) before adding or subtracting.

Understanding:

We need to find the compound interest earned after 2 years.

Formula:

Step 1: Analyse Statement I alone

Knowing only $r=10\%$ without the principal $P$ is insufficient to calculate CI. Not sufficient.

Step 2: Analyse Statement II alone

From SI $=800$: $\frac{P\times r\times 2}{100}=800$, but without $r$, we cannot find $P$. Not sufficient.

Step 3: Combine both statements

From Statement II with $r=10\%$, $t=2$:

Now compute CI:

Answer:

Both statements together are sufficient; neither alone is sufficient.

Quick Tip:

A useful shortcut: for 2 years, $\text{CI}-\text{SI}=\frac{{\text{SI}}^2}{4P}$, or equivalently $\text{CI}=\text{SI}+\frac{r\times {\text{SI}}_1}{100}$ where ${\text{SI}}_1$ is the first year's interest. Here, $\text{CI}=800+\frac{10\%\times 400}{1}=800+40=840$.