Multiply without long multiplication

Long multiplication writes every partial product and adds them at the end. Criss-cross does the same sums in your head, one answer digit at a time, right to left, so you only ever write the answer.

Start with this

Long multiplication writes every partial product and adds them at the end. The shortcuts below do exactly the same sums, just in a smarter order, so you write only the answer.

Every quantitative question ends in arithmetic. In a 60-minute paper with 30 questions you get 2 minutes each. Saving 20 seconds per multiplication is 5 extra minutes, which is 2 or 3 extra questions attempted. These are the speed tricks students asked for.

Why the tricks work: multiplication is area

Worked example

43 × 27 using the four pieces.

1. 40 × 20 = 800.
   Why: Tens × tens.
2. 3 × 20 = 60 and 40 × 7 = 280, together 340.
   Why: The two cross pieces: this is the 'middle' of criss-cross.
3. 3 × 7 = 21.
   Why: Units × units.
4. 800 + 340 + 21 = 1161.

Answer: 1161

Criss-cross: any two 2-digit numbers

Work right to left, writing one digit at a time and carrying the rest.

Right column: units × units. Middle column: the two cross products added (first digit of one × second digit of the other, both ways). Left column: tens × tens. Add each carry to the next column.

With practice this takes under 10 seconds. The animated visual further down the page shows it step by step.

Criss-cross: any two 2-digit numbers: the rules

Worked example

67 × 84

1. Right: 7 × 4 = 28. Write 8, carry 2.
   Why: Only one digit goes in each column.
2. Middle: 6 × 4 + 7 × 8 = 24 + 56 = 80, plus carry 2 = 82. Write 2, carry 8.
   Why: The two cross products are the middle pieces of the area picture.
3. Left: 6 × 8 = 48, plus carry 8 = 56. Write 56.
   Why: The last column takes the whole number.
4. Answer 5628.

Answer: 5628

Worked example

38 × 76

1. Right: 8 × 6 = 48. Write 8, carry 4.
2. Middle: 3 × 6 + 8 × 7 = 18 + 56 = 74, + 4 = 78. Write 8, carry 7.
   Why: Always add the carry after the cross products.
3. Left: 3 × 7 = 21, + 7 = 28.
4. Answer 2888.

Answer: 2888

Exam tip

Numbers close to 10, 100 or 1000 (base method)

When both numbers sit just below or just above a round number (the base), work with how far each is from the base (the deviation).

Left part: one number plus the OTHER number's deviation (both ways give the same answer). Right part: the two deviations multiplied, written with as many digits as the base has zeros.

Below the base the deviation is negative (97 is -3 from 100). Above, it is positive (104 is +4).

Numbers close to 10, 100 or 1000 (base method): the rules

Worked example

96 × 89

1. Base 100. Deviations: 96 is -4, 89 is -11.
   Why: Both are below 100.
2. Left: 96 - 11 = 85.
   Why: One number plus the other's deviation.
3. Right: (-4) × (-11) = 44, two digits.
   Why: Base 100 has two zeros, so the right part has two digits.
4. Answer 8544.

Answer: 8544

Worked example

112 × 107

1. Base 100. Deviations +12 and +7.
2. Left: 112 + 7 = 119. Right: 12 × 7 = 84.
3. Answer 11984.

Answer: 11984

Exam tip

Quick multipliers: x 5, x 25, x 125, x 9, x 99, x 11

x 5 is x 10 then halve. x 25 is x 100 then divide by 4. x 125 is x 1000 then divide by 8. Halving is easier than multiplying.

x 9 is x 10 minus the number. x 99 is x 100 minus the number: 47 × 99 = 4700 - 47 = 4653.

x 11 for a 2-digit number: write the first digit, then the sum of the two digits, then the last digit. 53 × 11 = 5 | 8 | 3 = 583. If the middle sum is 10 or more, carry: 78 × 11 = 7 | 15 | 8 → 858.

Quick multipliers: x 5, x 25, x 125, x 9, x 99, x 11: reference

Worked example

264 × 25

1. 264 × 100 = 26400.
   Why: 25 is a quarter of 100.
2. 26400 / 4 = 6600.

Answer: 6600

Special pairs: same tens and units that add to 10, and (a + b)(a - b)

If two numbers have the same tens digit and their units add to 10 (63 and 67, 42 and 48), then: left part = tens × (tens + 1), right part = units × units (always two digits). 63 × 67 = 6 × 7 | 3 × 7 = 42 | 21 = 4221.

If two numbers sit the same distance either side of a round number, use (a + b)(a - b) = a² - b². 53 × 47 = 50² - 3² = 2500 - 9 = 2491.

Worked example

84 × 86

1. Same tens 8, units 4 + 6 = 10.
   Why: The pattern fits.
2. Left 8 × 9 = 72. Right 4 × 6 = 24.
3. 7224.

Answer: 7224

Worked example

98 × 102

1. Both are 2 away from 100.
   Why: Equal distance either side.
2. 100² - 2² = 10000 - 4 = 9996.

Answer: 9996

Three-digit criss-cross

The same idea with five columns. For abc × def, the columns from the right are: c f | b f + c e | a f + b e + c d | a e + b d | a d.

The middle column has three products. For a 2-digit × 3-digit, put a 0 in front of the short number and use the same five columns: the zeros kill half the products.

Worked example

234 × 567

1. c × f = 4 × 7 = 28. Write 8, carry 2.
2. b × f + c × e = 3 × 7 + 4 × 6 = 45, + 2 = 47. Write 7, carry 4.
3. a × f + b × e + c × d = 14 + 18 + 20 = 52, + 4 = 56. Write 6, carry 5.
   Why: The widest column: three pairs whose places add up to the same total.
4. a × e + b × d = 12 + 15 = 27, + 5 = 32. Write 2, carry 3.
5. a × d = 10, + 3 = 13.
6. Answer 1,32,678.

Answer: 132678

Exam tip

3-digit × 3-digit, five columns: 234 × 567

2-digit × 3-digit: 45 × 312 (pad to 045)

One above, one below: 103 × 98

Base 1000: 996 × 993

Why 58 × 42 = 50² - 8² (areas)

Shortcut: Criss-cross, 2 × 2

Use it when: Any two 2-digit numbers. Target: under 10 seconds.

1. Units × units gives the last digit (carry the tens).
2. Cross: first-of-top × second-of-bottom + second-of-top × first-of-bottom, plus the carry.
3. Tens × tens plus the carry gives the front.

Criss-cross, 2 × 2: try it

43 × 27

1. Units: 3 × 7 = 21. Write 1, carry 2.
2. Cross: 4 × 7 + 3 × 2 = 34, plus 2 = 36. Write 6, carry 3.
3. Tens: 4 × 2 = 8, plus 3 = 11. Write 11.

Answer: 1161

Shortcut: Criss-cross, 3 × 3

Use it when: Any two 3-digit numbers. Five columns instead of three.

1. Label abc × def.
2. Column 1: c*f.
3. Column 2: b*f + c*e.
4. Column 3: a*f + b*e + c*d (the big one, three products).
5. Column 4: a*e + b*d.
6. Column 5: a*d. Carry left at every step.

Criss-cross, 3 × 3: try it

234 × 567

1. c*f = 4×7 = 28 → write 8, carry 2.
2. b*f + c*e = 21 + 24 = 45, +2 = 47 → write 7, carry 4.
3. a*f + b*e + c*d = 14 + 18 + 20 = 52, +4 = 56 → write 6, carry 5.
4. a*e + b*d = 12 + 15 = 27, +5 = 32 → write 2, carry 3.
5. a*d = 10, +3 = 13 → write 13.

Answer: 1,32,678

Shortcut: 2-digit × 3-digit

Use it when: Pad the short number with a leading zero and use the 3×3 columns.

1. 45 × 312 becomes 045 × 312.
2. Every product with the leading 0 vanishes, so it is quicker than a real 3×3.

2-digit × 3-digit: try it

45 × 312

1. c*f = 5×2 = 10 → 0, carry 1.
2. b*f + c*e = 4×2 + 5×1 = 13, +1 = 14 → 4, carry 1.
3. a*f + b*e + c*d = 0 + 4 + 15 = 19, +1 = 20 → 0, carry 2.
4. a*e + b*d = 0 + 12 = 12, +2 = 14 → 4, carry 1.
5. a*d = 0, +1 = 1.

Answer: 14,040

Shortcut: Base 100 (both below)

Use it when: Both numbers between about 85 and 99.

1. Deviations from 100: 97 is -3, 96 is -4.
2. Left = 97 - 4 (or 96 - 3) = 93.
3. Right = (-3)(-4) = 12, written as 2 digits.
4. Answer 93|12.

Base 100 (both below): try it

97 × 96

1. -3 and -4.
2. Left 93, right 12.

Answer: 9312

Shortcut: Base 100 (one above, one below)

Use it when: Numbers straddle 100.

1. 103 is +3, 98 is -2.
2. Left = 103 - 2 = 101 → 10100.
3. Right = (+3)(-2) = -6, so subtract.
4. 10100 - 6.

Base 100 (one above, one below): try it

103 × 98

1. 101|-06
2. = 10100 - 6

Answer: 10094

Shortcut: Base 1000

Use it when: Numbers near 1000. Right part must be 3 digits.

1. 996 is -4, 993 is -7.
2. Left = 996 - 7 = 989.
3. Right = 28, padded to 028.

Base 1000: try it

996 × 993

1. 989|028

Answer: 9,89,028

Shortcut: Working base 50

Use it when: Numbers near 50: use 100 as the reference, then halve.

1. 48 is -2, 47 is -3 from 50.
2. Cross: 48 - 3 = 45.
3. 45 × 50 = 2250 (half of 4500).
4. Add (-2)(-3) = 6.

Working base 50: try it

48 × 47

1. 45 × 50 = 2250
2. + 6

Answer: 2256

Shortcut: Times 11

Use it when: Instant for any number.

1. Keep the outer digits, add neighbours in between.

Times 11: try it

352 × 11

1. 3 | 3+5 | 5+2 | 2
2. 3 | 8 | 7 | 2

Answer: 3872

Common mistakes

Common mistakes

Common mistakes

Your turn medium

78 × 64

1. Units 8×4 = 32 → 2, c3.
2. Cross 7×4 + 8×6 = 76, +3 = 79 → 9, c7.
3. 7×6 = 42, +7 = 49.
4. 4992.

Answer: 4992

Your turn medium

89 × 76

1. 9×6 = 54 → 4, c5.
2. 8×6 + 9×7 = 111, +5 = 116 → 6, c11.
3. 8×7 = 56 + 11 = 67.
4. 6764.

Answer: 6764

Your turn hard

347 × 256

1. 7×6 = 42 → 2 c4.
2. 4×6 + 7×5 = 59 +4 = 63 → 3 c6.
3. 3×6 + 4×5 + 7×2 = 52 +6 = 58 → 8 c5.
4. 3×5 + 4×2 = 23 +5 = 28 → 8 c2.
5. 3×2 = 6 +2 = 8.
6. 88832.

Answer: 88,832

Your turn hard

623 × 418

1. 3×8 = 24 → 4 c2.
2. 2×8 + 3×1 = 19 +2 = 21 → 1 c2.
3. 6×8 + 2×1 + 3×4 = 62 +2 = 64 → 4 c6.
4. 6×1 + 2×4 = 14 +6 = 20 → 0 c2.
5. 6×4 = 24 +2 = 26.
6. 260414.

Answer: 2,60,414

Recap

• explain criss-cross with the four-piece area picture
• multiply any two 2-digit numbers in under 15 seconds
• use the base method for numbers near 100 or 1000
• pick the fastest trick for × 5, 25, 125, 9, 99, 11

Now practise

25 questions with full solutions.

Open practice Back to the lesson