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📘 Lesson 3 of 4 · Mathematical Reasoning

🧠 Analytical Thinking

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.

Course progress: 75%

01 Key Concepts

Data Sufficiency

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

Breaking Down Complex Problems

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

Working Backwards

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

Elimination Strategy

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

Assumption Checking

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

02 Solved Examples

Example 1 Question: What is x? Statement 1: x + 5 = 12. Is Statement 1 alone sufficient to find x?
  1. Statement 1 gives a single equation with one unknown: x + 5 = 12.
  2. Solving: x = 12 - 5 = 7.
  3. This uniquely determines x.
Answer: Yes, Statement 1 alone is sufficient (x=7)
Example 2 Question: Is x positive? Statement 1: x^2 = 9. Is Statement 1 alone sufficient?
  1. x^2 = 9 means x = 3 or x = -3.
  2. Since x could be either positive or negative, we cannot determine for certain whether x is positive.
Answer: No, Statement 1 alone is not sufficient (x could be 3 or -3)
Example 3 A number, after having 8 added to it and then being doubled, equals 30. Work backwards to find the original number.
  1. Reverse the last step (doubling): 30 / 2 = 15.
  2. Reverse the first step (adding 8): 15 - 8.
Answer: The original number is 7

03 Practice Questions

1Question: What is y? Statement: 2y = 16. Sufficient?
Yes, y=8
2Question: Is n even? Statement: n is divisible by 3. Sufficient?
No, n could be even (6) or odd (9)
3A number is tripled, then 4 is subtracted, giving 11. Work backwards to find the original number.
(11+4)/3 = 5
4What does 'data sufficiency' ask you to determine?
Whether the given information is enough to answer the question, without necessarily solving it fully
5Why is 'working backwards' useful for problems describing a final result?
It lets you reverse each step from the known outcome to find the unknown starting point

📄 Analytical Thinking — Downloadable Worksheet

10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.

🗒 Cheat Sheet ⬇ Download PDF 🖥 Interactive Worksheet