Arithmetic you do in your head, on sight.

7 × 8 without tables

Every number from 5 to 9 sits just below 10. Write how far below in a circle, cross-subtract, then multiply the circles. This single move replaces the upper half of the times tables.

Add the neighbour

Write the number with a zero in front and behind, then replace every digit by itself plus its right-hand neighbour. Carry when a pair exceeds nine.

× 5, × 25, × 15

Awkward multipliers are friendly ones in disguise: × 5 is × 10 ÷ 2, × 25 is × 100 ÷ 4, × 15 is × 10 + half again.

96 × 97 in one line

The same circles work with a reference of 100. Because the circles are small, the whole product falls out in two chunks: hundreds from the cross-subtraction, and the last two digits from the circles.

106 × 104

Above the reference number the circles are positive, so you cross-add instead of cross-subtract. Nothing else changes.

98 × 134

When one number is above the reference and the other below, the circle product is negative — so the last move is a subtraction. This is the only place beginners slip.

42 × 48 with reference 40

Any multiple of ten can be the reference number. Pick the nearest one to both numbers and the circles stay tiny — this covers all of the two-digit multiplications the 100-reference handles badly.

994 × 1013

The circles method scales. With a reference of 1000 you multiply three-digit numbers faster than you can write them down.

9 985 × 10 021

Four-digit numbers clustered around 10 000 are the showpiece of the method: two small circles, one cross-add, and a product of eight digits appears.

3 826 × 47 by splitting

When no reference number is close, split the smaller number into pieces you can multiply on sight, work left to right, and add as you go. You only ever hold one running total.

23 × 79 with two references

Numbers like 23 and 79 sit nowhere near a shared reference. Give each one its own reference instead: leave one factor alone, round the other to the nearest ten, multiply, then correct by the small amount you moved.

85² instantly

Take the tens digit, multiply it by the next number up, and stick 25 on the end. That is the whole method.

97² and 106²

A square is just a multiplication where both circles are equal, so the circle product is always a perfect square — the easiest possible last step.

Add the way you read

Adding from the left keeps a single running total and gives you the leading digits first — which is what you usually care about. Add the thousands, then hundreds, then tens, then units.

8 000 − 3 467 with no borrowing

Round the number you are subtracting up to a friendly figure, subtract that, then give back the difference. No borrowing, ever.

3 528 ÷ 24

Split the divisor into two easy factors and divide twice. Two short divisions beat one long one every time.

2/3 + 1/4 in one move

You never need to hunt for a lowest common denominator. Multiply crosswise upwards for the top, multiply the bottoms for the bottom, then reduce once at the end.

5/6 − 1/4

Subtraction uses the identical crosswise pattern — the only change is a minus sign between the two cross-products. Keep the order: first fraction's numerator leads.

3/8 × 4/9

Multiplying fractions is the easy case: straight across. The mental trick is cancelling any top against any bottom first, so you only ever multiply small numbers.

3/4 ÷ 2/5

Flip the second fraction and multiply. Dividing by 2/5 is multiplying by 5/2 — dividing by a number smaller than one makes the answer bigger, which is the sanity check to keep in mind.

35% of 240 in two steps

Every percentage is built from two blocks you can read off instantly: 10% is the number with the decimal point moved one place left, 1% two places. Count how many of each you need and add.

8% of 350 = 350% of 8

x% of y always equals y% of x. When one side is awkward, swap them — 8% of 350 looks like work, 350% of 8 is 3.5 × 8 = 28 on sight.

240 − 15% in one multiply

Never compute the part and then subtract if you can help it. A 15% discount leaves 85% — one multiplication by 0.85 finishes the job, and the same trick handles tax, tips and interest.

12 is what % of 80?

Going backwards is one division, not a formula. Work out 1% of the whole, then see how many of those the part contains — that count is the percentage.

Casting out nines

Add the digits of a number until one digit is left. Digit sums survive multiplication and addition, so they catch almost every slip you can make. Drill: give the digit sum of the number shown.