Lesson 29 of 35 · Data Insights: data sufficiency

Data sufficiency: yes/no questions

Settle yes/no data sufficiency questions by testing well-chosen numbers, rephrasing the stem, and remembering that a certain "no" is sufficient too.

About 22 minutes

In this lesson

State when a yes/no question is answered sufficiently, including by a definite "no".
Rephrase a yes/no stem into a simpler condition before testing statements.
Choose test numbers, including negatives, zero and fractions, that expose an insufficient statement.

The idea

Sufficient means a definite answer, either way

Many data sufficiency stems ask a yes/no question: "Is $x > 1$ ?", "Is $n$ odd?", "Is $a$ greater than $b$ ?" A statement is sufficient if it forces the same answer in every case it allows: always yes or always no.

That second half is the one people forget. If a statement shows that $z$ is less than 8, then the question "Is $z > 10$ ?" is answered: no, definitely. A definite "no" is just as sufficient as a definite "yes". A statement is not sufficient only when it allows at least one case answering yes and at least one answering no.

Cases the statement allows	Verdict
all yes	sufficient
all no	sufficient
some yes, some no	not sufficient

Rephrase the stem first

A yes/no stem can often be rewritten as a simpler condition. Doing that once saves you from repeating the algebra with every statement.

"Is $x^2 > x$ ?" Factor: $x^2 - x = x(x - 1) > 0$ holds when $x > 1$ or $x < 0$ . So the question becomes "Is $x$ outside the interval from 0 to 1?"
"Is $\frac{a}{b} > 0$ ?" is the same as "Do $a$ and $b$ have the same sign?"
"Is $k$ odd?" for an integer $k$ is a question about parity, and statements like " $k + 4$ is odd" translate immediately ( $k$ is odd, since adding an even number keeps parity).

Be careful to rephrase without assuming anything the stem does not give. If $x$ might be negative, "Is $x^2 > x$ ?" cannot be divided by $x$ .

Test numbers on purpose

For yes/no questions, the most reliable method is to try to make a statement give both answers. Pick numbers that satisfy the statement and see what the stem says. If you find a yes and a no, the statement is not sufficient and you can move on.

Do not stop at the first easy number. Run through this list, keeping only numbers the statement and stem allow:

Kind of number	Example	What it catches
positive integer	2, 3	the obvious case
zero	0	products and squares behave oddly
negative integer	$-2$	squares and even powers hide signs
fraction between 0 and 1	$\frac{1}{2}$	squares get smaller, reciprocals larger
negative fraction	$-\frac{1}{2}$	both effects at once
the boundary itself	1, if the stem says $> 1$	strict versus non-strict

If the stem says the numbers are integers, or positive, the list shrinks; that restriction is often what makes a statement sufficient.

When to stop testing

Testing can prove "not sufficient" (one yes and one no) but cannot by itself prove "sufficient": three yeses in a row might be luck. When your tests keep giving the same answer, look for the reason. " $x^3 > 1$ means $x > 1$ , because cubing keeps order" is a reason. " $k + 4$ odd means $k$ odd" is a reason. If you find one, the statement is sufficient.

Combining in yes/no questions

When neither statement is sufficient alone, combine them and repeat the search: look for one case satisfying both statements that answers yes and another that answers no. The cases you found earlier are a good start: keep those that satisfy both statements and see whether they still disagree. Negative numbers often survive the combination, so try them before concluding that the pair is sufficient.

Worked examples

Example 1: A square hides the sign, a cube does not

Is $x > 1$ ?

(1) $x^2 > 1$

(2) $x^3 > 1$

Statement (1): $x = 2$ satisfies it and answers yes; $x = -2$ satisfies it ( $4 > 1$ ) and answers no. Not sufficient.
Statement (2): cubing keeps the order of real numbers, so $x^3 > 1$ exactly when $x > 1$ . Every case answers yes. Sufficient.

Answer

Statement (2) alone is sufficient, but statement (1) alone is not. Testing only positive numbers makes statement (1) look sufficient too, which leads to "each alone".

Example 2: A statement that is always true says nothing

$k$ is an integer. Is $k$ odd?

(1) $k + 4$ is odd.

(2) $k^2 + k$ is even.

Statement (1): adding the even number 4 does not change parity, so $k$ has the same parity as $k + 4$ : odd. Always yes. Sufficient.
Statement (2): $k^2 + k = k(k + 1)$ , a product of two consecutive integers, one of which is even. So it is even for every integer: $k = 2$ gives 6 (answer no), $k = 3$ gives 12 (answer yes). Not sufficient.

Answer

Statement (1) alone is sufficient, but statement (2) alone is not. A statement that holds for every integer cannot tell you anything about $k$ .

Example 3: A definite "no" is sufficient

Is $z > 10$ ?

(1) $z < 8$

(2) $z^2 < 100$

Statement (1): every allowed $z$ is below 8, so the answer is always no. Sufficient.
Statement (2): $z^2 < 100$ means $-10 < z < 10$ . Every allowed $z$ is below 10, so the answer is always no. Sufficient.

Answer

Each statement alone is sufficient. Calling them insufficient because they never produce a yes is the trap.

Common traps

Treating "no" as insufficient
A statement that always gives "no" answers the question. Only a mix of yes and no makes a statement insufficient.
Testing only positive integers
Unless the stem restricts $x$ , try $0$ , a negative number and a fraction between 0 and 1. Squares, reciprocals and products change behaviour exactly there.
Multiplying through by a variable
From $\frac{a}{b} > 1$ you cannot conclude $a > b$ unless you know $b$ is positive: $a = -3, b = -1$ gives $\frac{a}{b} = 3$ but $a < b$ .
Taking a few agreeing cases as proof
Three yeses in a row do not prove sufficiency; find the reason, or keep looking for a case that answers no.

Check yourself

Answer each question before you open its answer.

Question 1: Is x squared greater than x?

Data Sufficiency

Is $x^2 > x$ ?

(1) $x > \frac{1}{2}$

(2) $x^2 > 1$

A.Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
B.Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
C.Both statements together are sufficient, but neither statement alone is sufficient.
D.Each statement alone is sufficient.
E.Statements (1) and (2) together are not sufficient.

Show the answer

Answer: B. Statement (2) alone is sufficient, but statement (1) alone is not sufficient.

Working

Rephrase: $x^2 > x$ means $x(x - 1) > 0$ , which is true when $x > 1$ or $x < 0$ . Statement (1) allows $x = 2$ (yes) and $x = \frac{3}{4}$ or $x = 1$ (no). Not sufficient. Statement (2): $x^2 > 1$ means $x > 1$ or $x < -1$ . Both pieces lie inside the rephrased condition ( $x = 2$ gives $4 > 2$ ; $x = -2$ gives $4 > -2$ ), so the answer is always yes. Sufficient. Statement (2) alone is sufficient, but statement (1) alone is not.

Why the other options are wrong

A. Statement (1) allows $x = 2$ ( $4 > 2$ , yes) and $x = \frac{3}{4}$ ( $\frac{9}{16}$ is less than $\frac{3}{4}$ , no). Testing only whole numbers hides the fractions.
C. Statement (2) is already sufficient on its own, so combining is not needed. Treating $x^2 > 1$ as insufficient because it hides the sign misses that both signs answer yes.
D. Statement (1) gives both answers: $x = 1$ gives $1 > 1$ , which is false, while $x = 2$ gives yes.
E. Statement (2) alone answers the question: $x^2 > 1$ means $x > 1$ or $x < -1$ , and every such $x$ has $x^2 > x$ .

What carries over

Rephrase the stem into a condition on $x$ , then test fractions and boundary values as well as integers.

Question 2: Divisible by 6?

Data Sufficiency

$n$ is a positive integer. Is $n$ divisible by 6?

(1) $n$ is divisible by 3.

(2) $n$ is divisible by 4.

A.Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
B.Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
C.Both statements together are sufficient, but neither statement alone is sufficient.
D.Each statement alone is sufficient.
E.Statements (1) and (2) together are not sufficient.

Show the answer

Answer: C. Both statements together are sufficient, but neither statement alone is sufficient.

Working

Statement (1) alone: $n = 3$ answers no, $n = 6$ yes. Not sufficient. Statement (2) alone: $n = 4$ no, $n = 12$ yes. Not sufficient. Together: divisible by 3 and by 4, which share no prime factor, means divisible by 12, and $12 = 6 \times 2$ , so the answer is always yes. Both together are sufficient, but neither alone is.

Why the other options are wrong

A. Statement (1) allows $n = 3$ (not divisible by 6) and $n = 6$ (divisible).
B. Statement (2) allows $n = 4$ (not divisible by 6) and $n = 12$ (divisible).
D. Each statement alone allows both a yes and a no: 3 and 6 for statement (1), 4 and 12 for statement (2).
E. Together $n$ is divisible by both 3 and 4, hence by 12, and every multiple of 12 is a multiple of 6: always yes.

What carries over

Divisibility by $6$ needs a factor 2 and a factor 3; look for where each statement supplies them.

Question 3: Is a greater than b?

Data Sufficiency

Is $a > b$ ?

(1) $a^2 > b^2$

(2) $\frac{a}{b} > 1$

A.Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
B.Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
C.Both statements together are sufficient, but neither statement alone is sufficient.
D.Each statement alone is sufficient.
E.Statements (1) and (2) together are not sufficient.

Show the answer

Answer: E. Statements (1) and (2) together are not sufficient.

Working

Statement (1): $a = 3, b = 1$ gives yes; $a = -3, b = 1$ gives no. Not sufficient. Statement (2): $a = 3, b = 1$ gives yes; $a = -3, b = -1$ gives $\frac{a}{b} = 3 > 1$ but $-3 < -1$ , no. Not sufficient. Together: $a = 3, b = 1$ satisfies both and gives yes; $a = -3, b = -1$ satisfies both ( $9 > 1$ , $3 > 1$ ) and gives no. Not sufficient. The statements together are not sufficient.

Why the other options are wrong

A. Statement (1) allows $a = 3, b = 1$ (yes) and $a = -3, b = 1$ (no): a larger square does not mean a larger number.
B. This multiplies $\frac{a}{b} > 1$ by $b$ as if $b$ were positive. With $a = -3, b = -1$ , $\frac{a}{b} = 3 > 1$ but $a < b$ .
C. The pair $a = -3, b = -1$ satisfies both statements ( $9 > 1$ and $3 > 1$ ) and answers no, while $a = 3, b = 1$ answers yes.
D. Each statement allows a "no" case once negative values are tried: $a = -3, b = 1$ for statement (1) and $a = -3, b = -1$ for statement (2).

What carries over

When combining, carry the negative cases forward; they often satisfy both statements.

Key points

A yes/no statement is sufficient if it always gives yes or always gives no.
Rephrase the stem into its simplest condition before testing statements.
Test zero, negatives and fractions, not only positive integers.
Testing proves "not sufficient"; for "sufficient", find the reason.

All questions here are original and written for UniExamer; none comes from the test maker.

Written by the UniExamer team with AI assistance. Every worked example and check question was re-solved by an independent checker before publication. No subject expert has reviewed this lesson yet. If you find a mistake, write to support@uniexamer.com.

In this lesson

The idea

Sufficient means a definite answer, either way

Rephrase the stem first

Test numbers on purpose

When to stop testing

Combining in yes/no questions

Worked examples

Example 1: A square hides the sign, a cube does not

Example 2: A statement that is always true says nothing

Example 3: A definite "no" is sufficient

Common traps

Treating "no" as insufficient

Testing only positive integers

Multiplying through by a variable

Taking a few agreeing cases as proof

Check yourself

Question 1: Is x squared greater than x?

Working

Why the other options are wrong

What carries over

Question 2: Divisible by 6?

Working

Why the other options are wrong

What carries over

Question 3: Is a greater than b?

Working

Why the other options are wrong

What carries over

Key points

Practise this topic