Algebra and Sequences

Algebra is the part of maths where we use letters to stand for unknown numbers. In the CEM 11+ exam, algebra questions can appear in many forms — from simple "find the missing number" puzzles to function machines and number sequences. CEM papers sometimes present algebra in unfamiliar layouts, so it is important to understand the underlying ideas, not just memorise methods.

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Using Letters for Numbers

In algebra, letters like a, b, n, and x represent numbers we do not know yet:

• 3a means 3 × a
• ab means a × b
• a² means a × a
• 2a + 5 means "double a, then add 5"

CEM Tip: CEM questions may not always use x. They might use shapes, blanks, or other symbols to represent the unknown. The method is exactly the same.

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Simplifying Expressions

Collect like terms (terms with the same letter) together:

Worked Example

Simplify: 3a + 5b + 2a - b

• Collect the a terms: 3a + 2a = 5a
• Collect the b terms: 5b - b = 4b

Answer: 5a + 4b

More Examples

Expression	Simplified
4x + 3y + 2x - y	6x + 2y
7m - 3m + 2n + 5n	4m + 7n
10p - 4p - 3q + q	6p - 2q

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Substitution

Replace the letter with a number and calculate:

If a = 4 and b = 3, find the value of 2a + 3b - 1.

• 2 × 4 + 3 × 3 - 1
• = 8 + 9 - 1
• = 16

Worked Example with Powers

If x = 5, find the value of x² + 3x - 2.

• 5² + 3 × 5 - 2
• = 25 + 15 - 2
• = 38

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Solving Simple Equations

An equation has an equals sign. To solve it, find the value of the unknown letter.

Method: Do the Same to Both Sides

Solve: 2x + 5 = 19

1. Subtract 5 from both sides: 2x = 14
2. Divide both sides by 2: x = 7

Check: 2 × 7 + 5 = 14 + 5 = 19

Worked Example

Solve: 3n - 4 = 17

1. Add 4 to both sides: 3n = 21
2. Divide by 3: n = 7

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"Think of a Number" Problems

These are a CEM favourite! You need to work backwards.

Tom thinks of a number, doubles it, and adds 9. His answer is 31. What number did Tom think of?

• Work backwards: 31 - 9 = 22, then 22 ÷ 2 = 11

Check: 11 × 2 + 9 = 22 + 9 = 31

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Function Machines

A function machine takes an input, applies operations, and produces an output.

Input → [× 3] → [+ 2] → Output