Mathematical Reasoning · Quick Reference

Analytical Thinking Cheat Sheet

Mathematical Reasoning · Lesson 3/4

In one line: analytical thinking is about breaking a complex problem into manageable pieces and figuring out exactly what information you need — and whether you already have enough of it.

Key Ideas

1Data Sufficiency. Determining whether the information given is enough to answer a question, without necessarily solving the problem itself.

2Breaking Down Complex Problems. Splitting a multi-step problem into smaller, simpler sub-problems that can each be solved individually and then combined.

3Working Backwards. Starting from the desired end result and reasoning back to figure out the necessary starting conditions — useful for problems describing a final outcome.

4Elimination Strategy. Ruling out impossible or inconsistent options one at a time, narrowing down to the correct answer, especially useful in puzzle-style problems.

5Assumption Checking. Identifying any unstated assumptions a problem relies on, and verifying whether they're reasonable before proceeding.

Worked Examples

Question: What is x? Statement 1: x + 5 = 12. Is Statement 1 alone sufficient to find x?

Yes, Statement 1 alone is sufficient (x=7)

Question: Is x positive? Statement 1: x^2 = 9. Is Statement 1 alone sufficient?

No, Statement 1 alone is not sufficient (x could be 3 or -3)

A number, after having 8 added to it and then being doubled, equals 30. Work backwards to find the original number.

The original number is 7

Practice Yourself

Question: What is y? Statement: 2y = 16. Sufficient?

Yes, y=8

Question: Is n even? Statement: n is divisible by 3. Sufficient?

No, n could be even (6) or odd (9)

A number is tripled, then 4 is subtracted, giving 11. Work backwards to find the original number.

(11+4)/3 = 5

What does 'data sufficiency' ask you to determine?

Whether the given information is enough to answer the question, without necessarily solving it fully