Every maths teacher says "show your working" — but the fastest mathematicians work largely in their heads. Mental maths is not about showing off; it is about thinking faster, catching errors by estimation, and handling the parts of a calculation that do not need paper. Here are 15 techniques in order from simplest to most powerful.
Beginner Techniques (Class 1-5)
1. Count-On Method
For small additions (adding 1, 2, or 3): say the larger number, count on by the smaller. For 7+3: say "7", count "8, 9, 10". Answer: 10. Essential foundation for all later techniques.
2. Make-10 Strategy
Bridge through 10: 8+7 = 8+(2+5) = (8+2)+5 = 10+5 = 15. Once you memorise "bonds to 10" (1+9, 2+8, 3+7, 4+6, 5+5), this becomes instant.
3. Doubles and Near-Doubles
Memorise doubles up to 20: 6+6=12, 7+7=14... Then for near-doubles: 6+7=12+1=13. One memorisation unlocks a dozen calculations.
4. Skip Counting for Multiplication
Count in multiples: ×2=count in 2s, ×5=count in 5s (ends in 0 or 5), ×10=add a zero. The 5 and 10 tables are the easiest; learn these first.
Intermediate Techniques (Class 6-9)
5. Compensation Method
Round one number to a multiple of 10, calculate, then adjust. 47+29 = 47+30−1 = 77−1 = 76. Or: 63×9 = 63×10−63 = 630−63 = 567. Works for both addition and multiplication.
6. Doubling and Halving
Product is unchanged if you double one factor and halve the other. 25×16 = 50×8 = 100×4 = 400. Keep going until one factor is a power of 10.
7. Multiplying by 25, 50, 75
×25 = ×100÷4. ×50 = ×100÷2. ×75 = ×100×3÷4. Example: 48×25 = 48×100÷4 = 4800÷4 = 1200.
8. Adding Large Numbers Left to Right
Standard addition goes right to left. Mental addition goes left to right (which is how we read numbers). 473+328: 400+300=700, 70+20=90, 3+8=11. Total: 700+90+11=801. Each step is simple because you work with hundreds, then tens, then units.
9. Estimating by Rounding
Before any calculation, estimate: round each number to the nearest 10 or 100 and compute. This gives you a "sanity check" for your answer. If your estimate is 600 and your exact answer is 6000, you have an error.
Advanced Techniques (Class 10+)
10. Squaring Numbers Near 50
(50+d)² = 2500 + 100d + d². For 53²: d=3. 2500+300+9 = 2809. For 47²: d=−3. 2500−300+9 = 2209.
11. Multiplying Numbers Near 100 (Nikhilam)
97×96: complements are 3 and 4. Left part: 97−4 = 93. Right part: 3×4 = 12. Answer: 9312. (This is Vedic Nikhilam sutra.)
12. Casting Out Nines (Error Check)
Find the digital root of your answer and of the expected result. If they do not match, there is an arithmetic error. 347×8=2776. Check: digital root of 347 = 3+4+7=14→5. Digital root of 8 = 8. 5×8=40→4. Digital root of 2776 = 2+7+7+6=22→4. Match — calculation is likely correct.
13. Dividing by 5
Dividing by 5 = multiplying by 2 then dividing by 10 (shift decimal one left). 340÷5 = 680÷10 = 68. 1375÷5 = 2750÷10 = 275.
14. Square of Any 2-Digit Number
ab² = (a×10)² + 2(a×10)(b) + b² = a²×100 + 20ab + b². For 67²: 6²×100=3600, 20×6×7=840, 7²=49. Total=3600+840+49=4489.
15. Percentage Calculations
1% = divide by 100. 10% = divide by 10. 5% = half of 10%. 15% = 10% + 5%. 25% = quarter. For 17.5% of 840: 10%=84, 5%=42, 2.5%=21. 17.5%=84+42+21=147. No formula needed.
The MathVis "Alternative Methods" section for every arithmetic formula includes specific mental maths techniques. Browse the formula library to find the mental approach for any topic you are studying.