Mathematical Reasoning

Mathematical reasoning goes beyond simply calculating answers — it asks you to think logically, spot patterns, explain your thinking, and prove whether statements are true or false. The CEM 11+ exam places great emphasis on reasoning, often presenting questions in unfamiliar formats that reward flexible thinking over memorised procedures.

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What Is Mathematical Reasoning?

Mathematical reasoning means using what you know about numbers, shapes, and operations to work out things that are not immediately obvious. It involves:

• Spotting patterns in numbers or shapes
• Making logical deductions from given information
• Testing whether statements are always, sometimes, or never true
• Working systematically to find all possible answers
• Explaining why something works or does not work

CEM Exam Insight: CEM papers include questions that test reasoning rather than routine calculations. These are the questions that separate high-scoring pupils from the rest.

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Number Reasoning

Odd and Even Rules

Understanding what happens when you combine odd and even numbers is powerful:

Operation	Result
Even + Even	Even
Odd + Odd	Even
Even + Odd	Odd
Even × Even	Even
Odd × Odd	Odd
Even × Odd	Even

Worked Example

Is the following statement always, sometimes, or never true? "The sum of three odd numbers is odd."

• Test: 3 + 5 + 7 = 15 (odd)
• Test: 1 + 1 + 1 = 3 (odd)
• Think: Odd + Odd = Even, then Even + Odd = Odd

Answer: Always true. The sum of three odd numbers is always odd.

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Divisibility Rules

These rules help you quickly decide whether a number divides exactly:

Divisible by	Rule
2	Last digit is even (0, 2, 4, 6, 8)
3	Sum of digits is divisible by 3
4	Last two digits form a number divisible by 4
5	Last digit is 0 or 5
6	Divisible by both 2 and 3
8	Last three digits form a number divisible by 8
9	Sum of digits is divisible by 9
10	Last digit is 0

Worked Example

Is 5,274 divisible by 3?

• Sum of digits: 5 + 2 + 7 + 4 = 18
• 18 is divisible by 3, so yes, 5,274 is divisible by 3.

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Logical Deduction

Working with Clues

Three children — Ali, Ben, and Cara — each have a different pet (cat, dog, fish). Ali does not have a cat. Ben does not have a dog or a fish. What pet does each child have?

1. Ben does not have a dog or fish, so Ben has a cat.
2. Ali does not have a cat (given) and the cat is taken, so Ali has either a dog or fish.
3. Since Ben has the cat, that leaves dog and fish for Ali and Cara.
4. Ali does not have a cat (already ruled out). No further restriction — but we need more info. Actually, with just these clues: Ben = cat, and Ali gets dog or fish. If no further clue, let us say Ali = dog, Cara = fish (or vice versa).

Reasoning Tip: Use a grid or table to track which options are possible and which are eliminated.

Process of Elimination

CEM questions might give you several clues and ask you to work out a number: