Mental Maths

Why Mental Maths Matters

Strong mental maths skills save time, help you spot errors in calculator answers, and build genuine number confidence. The key is choosing the right strategy for each type of calculation rather than working everything out the long way.

Key Terms

• Compensation — round a number to make it easier, then adjust at the end
• Partitioning — split numbers into hundreds, tens, and units to calculate each part separately
• Estimation — finding an approximate answer quickly using rounded numbers
• Compatible numbers — pairs of numbers that are easy to work with together (e.g. 25 and 4)

Strategy 1: Rounding and Adjusting (Compensation)

Round one number to a nearby convenient value, calculate, then add or subtract the adjustment:

• 99 × 7 = 100 × 7 - 1 × 7 = 700 - 7 = 693
• 48 + 39 = 48 + 40 - 1 = 87
• 201 × 5 = 200 × 5 + 1 × 5 = 1,000 + 5 = 1,005

This works especially well when a number is close to a round number (e.g., 99, 201, 49).

Strategy 2: Doubling and Halving

For multiplication, double one factor and halve the other — the product stays the same:

• 16 × 25 → double 25 and halve 16: 8 × 50 = 4 × 100 = 400
• 14 × 5 → halve 14 and double 5: 7 × 10 = 70

Halving twice is useful for dividing by 4:

• 64 ÷ 4 → 64 ÷ 2 = 32, then 32 ÷ 2 = 16
• 92 ÷ 4 → 46 ÷ 2 = 23

Strategy 3: Partitioning

Break numbers into their place value components, then add or multiply each part:

Addition: 34 + 58 = (30 + 50) + (4 + 8) = 80 + 12 = 92

Multiplication: 7 × 34 = 7 × 30 + 7 × 4 = 210 + 28 = 238

This mirrors how long multiplication works, but done entirely in your head.

Strategy 4: Using Known Facts

Build on multiplication facts you already know:

• 7 × 8 = 56, so:
  • 7 × 80 = 560
  • 70 × 8 = 560
  • 7 × 800 = 5,600
  • 0.7 × 8 = 5.6

This technique — scaling known facts by powers of 10 — is one of the most powerful mental tools.

Strategy 5: Finding Percentages Without a Calculator

Find 10%, then scale up or down:

• 15% of 80: 10% = 8, 5% = 4, so 15% = 8 + 4 = 12
• 30% of 70: 10% = 7, so 30% = 7 × 3 = 21

Useful fraction-percentage equivalents:

• 50% = 1/2: divide by 2
• 25% = 1/4: divide by 4
• 10% = 1/10: divide by 10
• 5% = 1/20: find 10% then halve it
• 1% = 1/100: divide by 100

Worked Example: 25% of 360 = 360 ÷ 4 = 90

Quick Estimation

Round all numbers to 1 significant figure before calculating. This gives a fast approximate answer useful for checking:

• 49 × 21 ≈ 50 × 20 = 1,000
• 312 + 489 ≈ 300 + 500 = 800
• 7.8 × 3.1 ≈ 8 × 3 = 24

When to estimate: when you need a sanity check, when you do not need an exact answer, or when spotting whether a calculator result is reasonable.

Compatible Numbers

Compatible numbers are pairs or groups that combine easily:

• 37 + 63 = 100 (they sum to a round number)
• 25 × 4 = 100
• 125 × 8 = 1,000

Look for these combinations in longer calculations — they can simplify things dramatically.

Common Mistakes

• Mistake: applying compensation but forgetting to adjust at the end. Fix: write a small note — "rounded up 1, so subtract 1 at end."
• Mistake: partitioning incorrectly (e.g. 34 + 58 = 80 + 12 ≠ 81). Fix: check the tens and units separately.
• Mistake: finding 10% and then doubling it to get 20% — but forgetting this gives 20%, not 25%. Fix: 25% = 10% + 10% + 5%, or simply divide by 4.
• Mistake: rounding 49 to 40 instead of 50. Fix: round to the nearest 10: 49 rounds to 50.

Tips and Tricks

• Practice times tables to at least 12 × 12 — they are the building blocks for all other strategies.
• For dividing by 5, multiply by 2 then divide by 10: 85 ÷ 5 = 170 ÷ 10 = 17.
• To multiply by 9, multiply by 10 and subtract the number: 9 × 7 = 70 - 7 = 63.
• Estimation is not guessing — it is a disciplined approach using rounded values to get a reliable ballpark.