Quantitative reasoning tips for Common Entrance (with worked examples)

Quantitative reasoning (also called quantitative aptitude) tests how well a pupil thinks with numbers: spotting patterns, working out rules and solving puzzles. It's one of the papers in the National Common Entrance Examination and in many private secondary school entrance tests.

It is not the same as Mathematics. A pupil can be very good at sums and still be puzzled by a quantitative reasoning paper if they've never seen the question types before. But once they recognise each type and know a strategy, scores rise quickly.

The one skill that helps with everything

Know the times tables up to 12 × 12 instantly. Almost every quantitative reasoning question depends on spotting multiplication patterns quickly. A pupil who has to work out 7 × 8 on paper loses valuable seconds on every question.

It also helps to recognise:

• Square numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144
• Cube numbers: 1, 8, 27, 64, 125
• Prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 …

Type 1: Number sequences

Format: find the next (or missing) number in a pattern.

Strategy: check the differences

Write the difference between each pair of numbers. Most sequences are built from a pattern in the differences.

Example 1: 3, 7, 11, 15, ___

Differences: +4, +4, +4. Next: 15 + 4 = 19.

Example 2: 2, 5, 10, 17, 26, ___

Differences: +3, +5, +7, +9. They go up by 2, so the next difference is +11. Next: 26 + 11 = 37.

Example 3: 3, 6, 12, 24, ___

Differences: +3, +6, +12, which double each time. Or notice that each number is multiplied by 2. Next: 24 × 2 = 48.

Example 4: 1, 4, 9, 16, ___

These are square numbers: 1², 2², 3², 4². Next: 5² = 25.

Example 5: 1, 1, 2, 3, 5, 8, ___

Each number is the sum of the two before it. Next: 5 + 8 = 13.

Example 6: 2, 10, 4, 20, 6, 30, ___

Two sequences are mixed together: 2, 4, 6 … and 10, 20, 30 … Next: 8.

If differences don't work, try:

• multiplying or dividing,
• square or cube numbers,
• adding the two previous numbers,
• two sequences mixed together.

Type 2: Number analogies

Format: A is to B as C is to ?

Example 1: 3 is to 9 as 5 is to ___

Find the rule: 3 → 9 could be ×3 or squared (3² = 9). Check with 5: ×3 gives 15; squared gives 25. Look at the options to see which rule the question intends. If 25 is an option and 15 isn't, the rule is squaring.

Example 2: 4 : 12 :: 7 : ___

4 × 3 = 12, so 7 × 3 = 21.

Strategy: find the rule in the first pair, check it makes sense, then apply it to the second. If more than one rule works, use the options to decide.

Type 3: Function machines (input and output)

A number goes into a "machine", which applies a rule, and another number comes out.

Example:

Try a rule: ×2 + 1. Check: 2 × 2 + 1 = 5 ✓; 4 × 2 + 1 = 9 ✓; 6 × 2 + 1 = 13 ✓.

So for 8: 8 × 2 + 1 = 17.

Strategy: test your rule on every pair given, not just the first. A rule that works for one pair may fail for the others.

Type 4: Missing numbers in operations

Example: 6 × ___ = 42

42 ÷ 6 = 7.

Example: ___ − 15 = 28

28 + 15 = 43.

Strategy: use the inverse (opposite) operation. To undo multiplication, divide; to undo subtraction, add.

Type 5: Number grids and patterns

Example:

Each row is the first number, then ×2, then ×3. So the missing number is 4 × 3 = 12.

Strategy: look across rows and down columns. The pattern might be in either direction.

Magic squares: every row, column and diagonal adds up to the same total. Find a complete line to get the total, then work out the missing numbers.

Type 6: Odd one out with numbers

Example: Which number does not belong? 4, 9, 16, 20, 25

4, 9, 16 and 25 are square numbers. 20 is not. Answer: 20.

Example: 3, 5, 7, 9, 11

All are odd, but 3, 5, 7 and 11 are prime; 9 is not (9 = 3 × 3). Answer: 9.

Strategy: test common properties: odd or even, prime, square, multiples of a number.

Type 7: Word problems and puzzles

Age problems

Example: Ada is 3 times as old as her brother. If her brother is 4, how old will Ada be in 5 years?

Ada now: 3 × 4 = 12. In 5 years: 12 + 5 = 17.

Sharing (ratio) problems

Example: Share 20 sweets between Tolu and Emeka in the ratio 3:2.

Total parts: 3 + 2 = 5. One part: 20 ÷ 5 = 4. Tolu: 3 × 4 = 12. Emeka: 2 × 4 = 8.

Money problems

Example: A pencil costs ₦50 and a ruler costs ₦120. How much do 3 pencils and 2 rulers cost?

3 × 50 = 150; 2 × 120 = 240; total = ₦390.

Time problems

Example: A lesson starts at 9:45 am and lasts 1 hour 30 minutes. When does it end?

9:45 + 1 hour = 10:45; + 30 minutes = 11:15 am.

Strategy: underline the question, break it into small steps, and write down each step.

Type 8: Shape and measurement puzzles

Example: A square has a perimeter of 24 cm. What is its area?

Each side: 24 ÷ 4 = 6 cm. Area: 6 × 6 = 36 cm².

Example: How many faces does a cube have? 6.

Learn the faces, edges and vertices (corners) of common 3D shapes:

General exam tips for pupils

1. Read every question carefully. Check whether it asks for the next number, the missing number or the odd one out.
2. Write your working down. Differences, rules and steps are easier to check on paper.
3. Check your answer fits the whole pattern, not just the last step.
4. Don't get stuck. If a question takes too long, mark it, guess, and come back if time allows.
5. Never leave a question blank.

Tips for parents

• Make times tables automatic. A few minutes of daily practice, through games, songs or quick-fire questions, pays off across the whole paper.
• Introduce one question type at a time, with several examples, before mixing types.
• Ask your child to explain the rule they found. Saying it aloud builds real understanding.
• Use everyday life: let your child calculate shopping totals, change, cooking quantities and journey times.
• Keep it encouraging. Puzzles should feel like a challenge, not a punishment.

Practise with Levia

Levia's Quantitative Reasoning subject has five levels: Number Patterns and Sequences, Number Machines and Operations, Fraction and Ratio Puzzles, Shape, Space and Measure Puzzles, and Mixed Reasoning Challenges. Every question explains the rule or pattern in child-friendly language. Try Level 1 free; no account needed.