Mental Math Strategies

Addition: make tens and split numbers

8 + 7 = 10 + 5 = 15

47 + 26 = 47 + 20 + 6 = 73

Move 2 from 7 to 8, leaving 5. Moving the same amount between the addends preserves their total. For larger numbers, split by place value: add 20 to 47 to make 67, then add the remaining 6 to make 73.

Common mistake: Do not add the full 7 after moving 2 to make ten. Only 5 remains.

Try three questions, then check

• 9 + 6 = 10 + 5 = 15
• 38 + 24 = 38 + 20 + 4 = 62
• 56 + 17 = 56 + 10 + 7 = 73

Subtraction: round and compensate

83 − 29 = 83 − 30 + 1 = 54

Subtract the nearby round number 30 first. You subtracted one too many, so add 1 back to preserve the original difference.

Common mistake: Add back the extra amount. Subtracting another 1 would move you farther from the answer.

Try three questions, then check

• 72 − 19 = 72 − 20 + 1 = 53
• 91 − 48 = 91 − 50 + 2 = 43
• 64 − 28 = 64 − 30 + 2 = 36

Multiplication: break apart or double and halve

23 × 6 = 120 + 18 = 138

25 × 16 = 50 × 8 = 400

Distribution lets you multiply each part: (20 + 3) × 6 = 20 × 6 + 3 × 6. Alternatively, doubling one factor while halving the other preserves the product because the factors of 2 cancel. Halve an even factor to keep the arithmetic simple.

Common mistake: Multiply both parts when distributing. Doubling both factors would multiply the answer by four.

Try three questions, then check

• 32 × 4 = 120 + 8 = 128
• 25 × 24 = 50 × 12 = 600
• 19 × 7 = 140 − 7 = 133

Division: split into known multiples

156 ÷ 6 = 120 ÷ 6 + 36 ÷ 6 = 26

Check: 26 × 6 = 156

Split the dividend into multiples of the same divisor. Here 120 ÷ 6 = 20 and 36 ÷ 6 = 6, so add the quotients to get 26. Multiplying the quotient by the divisor checks an exact answer. For a remainder, also add that remainder back.

Common mistake: Divide every part by the same divisor. Do not divide the divisor into pieces as well.

Try three questions, then check

• 84 ÷ 7 = 70 ÷ 7 + 14 ÷ 7 = 12
• 144 ÷ 8 = 80 ÷ 8 + 64 ÷ 8 = 18
• 195 ÷ 5 = 150 ÷ 5 + 45 ÷ 5 = 39

Squares: use a nearby round number

39² = (40 − 1)² = 1600 − 80 + 1 = 1521

Use (a − b)² = a² − 2ab + b². With a = 40 and b = 1, the middle term is 2 × 40 × 1 = 80. It accounts for both cross-products when multiplying (40 − 1) by itself. This is not 40² − 1².

Common mistake: Keep the middle term and add the final square, even when the nearby number is larger.

Try three questions, then check

• 29² = 900 − 60 + 1 = 841
• 51² = 2500 + 100 + 1 = 2601
• 48² = 2500 − 200 + 4 = 2304

Percentages: start with 10% and 5%

15% of 80 = 8 + 4 = 12

Ten percent is one tenth: divide 80 by 10 to get 8. Five percent is half of ten percent, so halve 8 to get 4. Add them for 15 percent of the same whole.

Common mistake: Keep the original whole fixed. Five percent of 80 is 4, not five percent of the intermediate answer 8.

Try three questions, then check

• 15% of 60 = 6 + 3 = 9
• 25% of 120 = 12 + 12 + 6 = 30
• 35% of 200 = 20 + 20 + 20 + 10 = 70