Lesson 30 of 35 · Data Insights: data sufficiency

Data sufficiency traps

Avoid the data sufficiency errors that cost the most: leaking one statement into the other, unstated assumptions, hidden integer limits and ranges.

About 22 minutes

In this lesson

Keep each statement separate when testing it alone.
Use only what the stem and statements state, not what a story or figure seems to suggest.
Recognise when an integer condition makes a single equation sufficient.
Tell a unique value from a range of values.

The idea

Most errors are not arithmetic

Data sufficiency rarely requires long calculation, so most lost marks come from reasoning slips: using information you were not given, or not using information you were. This lesson names the common ones and shows how to check for each.

Leaking statement (1) into statement (2)

After working through statement (1), its content stays in your head. When you then test statement (2) "alone", it is easy to use what (1) told you without noticing. Suppose the stem asks "Is $k > 5$ ?", (1) says $k^2 = 49$ and (2) says $k > 0$ . Having just found $k = \pm 7$ from (1), you may read (2) and think "so $k = 7$ , yes". But (2) alone allows $k = 1$ , which answers no.

A practical defence: when you turn to statement (2), cover statement (1) with your hand or say to yourself "new question". Then write down two cases for (2) that do not depend on (1).

Assuming what is not stated

Only the stem and the statements count. Do not assume:

that a number is an integer, unless the stem says so;
that it is positive, unless the stem says so, or it is a count or a length;
that two variables are different, unless stated;
that a figure is drawn to scale: treat any diagram as a sketch of the relationships described, not as a measurement. An angle that looks like a right angle, or two lines that look parallel, are facts only if the text says so.

Natural restrictions do apply: a number of people is a non-negative integer, a length is positive. Those are facts the situation states, not assumptions. The skill is to tell the two apart.

Hidden integer limits

One equation in two unknowns is usually not sufficient. But when the unknowns must be non-negative integers, as counts of items are, one equation may allow only a single solution. Suppose mugs cost $\$7$ and plates $\$4$ , and someone spent $\$30$ on them. Test the number of mugs: 0 leaves $\$30$ (not a multiple of 4), 1 leaves 23, 2 leaves 16 (four plates), 3 leaves 9, 4 leaves 2. Only two mugs works. One equation, one answer: sufficient.

The same purchase with a total of $\$43$ allows two answers (1 mug and 9 plates, or 5 mugs and 2 plates). So you cannot decide in general; you have to list the integer solutions each time. That listing is fast because the possible counts are small.

A unique value versus a range

A value question needs exactly one value. A statement that narrows a variable to a range, such as $4 < x < 6$ , is not sufficient for "What is $x$ ?" unless something else (for example, " $x$ is an integer") collapses the range to one value. Equally, two ranges combined are usually still a range. Read the stem: "What is $x$ ?" needs a point; "Is $x > 3$ ?" only needs the range to sit wholly on one side of 3.

The tempting "together"

When neither statement looks sufficient at a glance, combining them feels natural, and "both together" is often chosen too quickly. Two failures follow:

One statement was sufficient all along. Typical cases: a statement gives the required combination directly ( $3x = 5y$ gives $\frac{x}{y}$ ), or an integer condition makes one equation enough.
Together is still not enough. The statements repeat each other, or a negative case satisfies both.

Before choosing "both together", ask whether either statement alone was really insufficient; prove it with two cases. After combining, ask whether a case with the other answer still survives.

A last routine check

Before you confirm an answer: did you test each statement truly alone? Did you use any fact that is neither in the stem nor in a statement? Did you allow negatives, zero and fractions wherever the stem allows them? Three questions, ten seconds.

Worked examples

Example 1: An integer limit decides it

Mia bought only mugs, at $\$7$ each, and plates, at $\$4$ each. How many mugs did she buy?

(1) She spent $\$43$ in total.

(2) She bought 7 items in total.

Statement (1): $7m + 4p = 43$ with $m$ and $p$ non-negative integers. Try $m = 0$ to 6: the leftover $43 - 7m$ is $43, 36, 29, 22, 15, 8, 1$ , and only 36 and 8 are multiples of 4. So $(m, p) = (1, 9)$ or $(5, 2)$ . Two answers: not sufficient.
Statement (2): $m + p = 7$ allows any split from 0 to 7 mugs. Not sufficient.
Together: of the two cases from statement (1), only $(5, 2)$ has 7 items ( $(1, 9)$ has 10). One answer: sufficient. Check: $5 \times 7 + 2 \times 4 = 35 + 8 = 43$ .

Answer

Both statements together are sufficient, but neither alone is. Had the total in statement (1) been $\$30$ , only $(2, 4)$ would work and statement (1) alone would have been sufficient.

Example 2: Keeping statement (2) alone

Is $x$ positive?

(1) $xy > 0$

(2) $y > 0$

Statement (1): $x = 1, y = 1$ gives yes; $x = -1, y = -1$ gives no. Not sufficient.
Statement (2), truly alone: it says nothing about $x$ . $x = 1$ and $x = -1$ are both allowed. Not sufficient. (If you carried $xy > 0$ over, you would wrongly call it sufficient.)
Together: $xy > 0$ means $x$ and $y$ have the same sign, and $y$ is positive, so $x$ is positive. Always yes. Sufficient.

Answer

Both statements together are sufficient, but neither alone is. Choosing "statement (2) alone" is the leak from statement (1).

Example 3: One statement was enough

$y$ is not zero. What is the value of $\frac{x}{y}$ ?

(1) $3x = 5y$

(2) $x + y = 16$

Statement (1): divide both sides by $3y$ , allowed because $y \neq 0$ : $\frac{x}{y} = \frac{5}{3}$ . One value. Sufficient, even though $x$ and $y$ are unknown.
Statement (2): $x = 8, y = 8$ gives 1; $x = 10, y = 6$ gives $\frac{5}{3}$ . Not sufficient.

Answer

Statement (1) alone is sufficient, but statement (2) alone is not. Choosing "both together", because together they give $x = 10$ and $y = 6$ , is the tempting-together trap.

Common traps

The leak from statement (1)
When testing statement (2) alone, write two cases that satisfy statement (2) only. If your cases also happen to satisfy statement (1), you may be leaking it.
Assuming integers or positive values
" $4 < x < 6$ " pins $x = 5$ only if $x$ is an integer. Unless the stem says so, $x = 4.5$ is allowed.
Trusting a figure
Do not assume a figure is drawn to scale. Use only the lengths, angles and relations the text states; an angle that looks like $90^\circ$ is not known to be one.
Ignoring a natural integer limit
Counts of people or items are non-negative integers even if the stem does not repeat it. That can make a single equation sufficient; list the integer solutions before deciding.
Choosing "together" by reflex
Before combining, prove each statement insufficient with two cases. After combining, look for a surviving case with the other answer.

Check yourself

Answer each question before you open its answer.

Question 1: An integer between 3 and 5

Data Sufficiency

$n$ is an integer. What is the value of $n$ ?

(1) $3 < n < 5$

(2) $n^2 = 16$

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

Working

Statement (1): the only integer strictly between 3 and 5 is 4. Sufficient. Statement (2): $n = 4$ or $n = -4$ . Not sufficient. Statement (1) alone is sufficient, but statement (2) alone is not.

Why the other options are wrong

B. Statement (2) allows $n = 4$ and $n = -4$ ; squaring hides the sign.
C. Statement (1) already gives $n = 4$ , because the stem says $n$ is an integer and 4 is the only integer strictly between 3 and 5.
D. Statement (2) alone leaves two values, 4 and $-4$ .
E. This ignores the stem's condition that $n$ is an integer; with it, statement (1) leaves only 4.

What carries over

An integer condition in the stem can turn a range into a single value.

Question 2: Is k greater than 5?

Data Sufficiency

Is $k > 5$ ?

(1) $k^2 = 49$

(2) $k > 0$

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): $k = 7$ (yes) or $k = -7$ (no). Not sufficient. Statement (2), alone: $k = 1$ (no) and $k = 6$ (yes). Not sufficient. Together: $k^2 = 49$ and $k > 0$ give $k = 7$ , and $7 > 5$ . Sufficient. Both together are sufficient, but neither alone is.

Why the other options are wrong

A. Statement (1) allows $k = 7$ (yes) and $k = -7$ (no).
B. This carries $k^2 = 49$ over from statement (1). Alone, statement (2) allows $k = 1$ , which answers no.
D. Neither statement alone forces an answer: $k = -7$ satisfies statement (1) with a no, and $k = 1$ satisfies statement (2) with a no, while $k = 7$ gives yes for both.
E. Together $k$ is the positive root of 49, which is 7, and $7 > 5$ : always yes.

What carries over

Test statement (2) with cases chosen from statement (2) only.

Question 3: A value or a range?

Data Sufficiency

What is the value of $x$ ?

(1) $4 < x < 6$

(2) $x^2 > 20$

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): $x = 4.5$ and $x = 5$ both satisfy it. Not sufficient. Statement (2): $x = 5$ and $x = 6$ both satisfy it. Not sufficient. Together: $x$ must exceed $\sqrt{20} \approx 4.47$ and be below 6; $x = 4.5$ ( $4.5^2 = 20.25 > 20$ ) and $x = 5$ both qualify. Still not one value. The statements together are not sufficient.

Why the other options are wrong

A. This assumes $x$ is an integer, which would give $x = 5$ . The stem does not say so: $x = 4.5$ and $x = 5$ are both allowed.
B. Statement (2) describes a range, $x > \sqrt{20}$ or $x < -\sqrt{20}$ , with infinitely many values.
C. Together $x$ lies between $\sqrt{20} \approx 4.47$ and 6, still a range: $4.5$ , $5$ and $5.5$ all satisfy both statements.
D. Each statement gives a range of values, not a single value; neither is sufficient alone.

What carries over

Two ranges combined are usually still a range; a value question needs a single point.

Key points

Test statement (2) with cases taken from statement (2) alone.
Use only stated facts: no assumed integers, signs or measurements from a figure.
When unknowns are counts, list the integer solutions; one equation may be enough.
A value question needs one value; a range is not sufficient.
Before choosing "both together", prove each statement insufficient; after combining, look for a surviving counter-case.

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

Most errors are not arithmetic

Leaking statement (1) into statement (2)

Assuming what is not stated

Hidden integer limits

A unique value versus a range

The tempting "together"

A last routine check

Worked examples

Example 1: An integer limit decides it

Example 2: Keeping statement (2) alone

Example 3: One statement was enough

Common traps

The leak from statement (1)

Assuming integers or positive values

Trusting a figure

Ignoring a natural integer limit

Choosing "together" by reflex

Check yourself

Question 1: An integer between 3 and 5

Working

Why the other options are wrong

What carries over

Question 2: Is k greater than 5?

Working

Why the other options are wrong

What carries over

Question 3: A value or a range?

Working

Why the other options are wrong

What carries over

Key points

Practise this topic