Data Sufficiency: Is There Enough Information?

A reliable method

Identify the exact target and all given restrictions.
Test the first statement alone.
Reset and test the second statement alone.
Combine only if necessary; check whether the target has one possible value.

Example 1

What is \(x\)? Statement 1: \(x + y - 12=0\). Statement 2: \(x - y - 4=0\). Here x and y are real. Decide which statements determine one unique value of the requested quantity.

Test statement 1 alone: the target becomes \(12 - y\), so it still depends on an unrestricted quantity.
Reset before testing statement 2 alone: the target becomes \(y + 4\), so it still varies.
Together the target becomes \(8\). both together are sufficient, but neither alone. A unique target can be determined even if every variable is not separately determined.

Answer: both together are sufficient, but neither alone

Example 2

What is \(x\)? Statement 1: \(x - 7=0\). Statement 2: \(y - 3=0\). Here x and y are real. Decide which statements determine one unique value of the requested quantity.

Test statement 1 alone: the target becomes \(7\), so it is uniquely determined.
Reset before testing statement 2 alone: the target becomes \(x\), so it still varies.
Together the target becomes \(7\). statement 1 alone is sufficient. A unique target can be determined even if every variable is not separately determined.

Answer: statement 1 alone is sufficient

Common mistakes

Do not carry information from statement 1 into a test of statement 2 alone.
A target expression can be determined even when its individual variables vary.

Practice without the answer first

What is \(x + y\)? Statement 1: \(x + y - 10=0\). Statement 2: \(2 x + 2 y - 20=0\). Here x and y are real. Decide which statements determine one unique value of the requested quantity.
What is \(y\)? Statement 1: \(x - 3=0\). Statement 2: \(x - 3=0\). Here x and y are real. Decide which statements determine one unique value of the requested quantity.
What is \(x\)? Statement 1: \(x - y - 2=0\). Statement 2: \(x + y - 8=0\). Here x and y are real. Decide which statements determine one unique value of the requested quantity.

Answers and step-by-step explanations

Practice 1

Test statement 1 alone: the target becomes \(10\), so it is uniquely determined.
Reset before testing statement 2 alone: the target becomes \(10\), so it is uniquely determined.
Together the target becomes \(10\). each statement alone is sufficient. A unique target can be determined even if every variable is not separately determined.

Answer: each statement alone is sufficient

Practice 2

Test statement 1 alone: the target becomes \(y\), so it still depends on an unrestricted quantity.
Reset before testing statement 2 alone: the target becomes \(y\), so it still varies.
Together the target becomes \(y\). even both together are insufficient. A unique target can be determined even if every variable is not separately determined.

Answer: even both together are insufficient

Practice 3

Test statement 1 alone: the target becomes \(y + 2\), so it still depends on an unrestricted quantity.
Reset before testing statement 2 alone: the target becomes \(8 - y\), so it still varies.
Together the target becomes \(5\). both together are sufficient, but neither alone. A unique target can be determined even if every variable is not separately determined.