Lessons / Speed maths
Speed maths

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.

Practise (25 questions) Present as slidesCheat sheet: Squares, cubes and powers

Why this topic matters

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

43 × 27 is the area of a rectangle 43 wide and 27 tall. Split 43 into 40 + 3 and 27 into 20 + 7 and the rectangle falls into four pieces: 40 × 20, 3 × 20, 40 × 7 and 3 × 7.

Every multiplication trick is just a quick way to add those four pieces. Criss-cross adds the two middle pieces together in one step. Once you see the picture, you do not have to trust the trick, you can see why it must be right.

4032078006028021
43 × 27: four pieces. Tens × tens, two cross pieces (the 'middle'), units × units. Total 1161
Example 143 × 27 using the four pieces.
  1. 40 × 20 = 800.Tens × tens.
  2. 3 × 20 = 60 and 40 × 7 = 280, together 340.The two cross pieces: this is the 'middle' of criss-cross.
  3. 3 × 7 = 21.Units × units.
  4. 800 + 340 + 21 = 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.

ab × cd
right: b × d | middle: a × d + b × c | left: a × c (carry leftwards)
Example 167 × 84
  1. Right: 7 × 4 = 28. Write 8, carry 2.Only one digit goes in each column.
  2. Middle: 6 × 4 + 7 × 8 = 24 + 56 = 80, plus carry 2 = 82. Write 2, carry 8.The two cross products are the middle pieces of the area picture.
  3. Left: 6 × 8 = 48, plus carry 8 = 56. Write 56.The last column takes the whole number.
  4. Answer 5628.
Example 238 × 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.Always add the carry after the cross products.
  3. Left: 3 × 7 = 21, + 7 = 28.
  4. Answer 2888.
Exam tipSay the carry out loud ('write 8, carry 4'). Dropped carries cause most mistakes, not wrong products.

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).

97 × 96 (base 100)
deviations -3 and -4. Left 97 - 4 = 93. Right (-3)(-4) = 12. Answer 9312.
104 × 108
+4 and +8. Left 104 + 8 = 112. Right 32. Answer 11232.
103 × 98
+3 and -2. Left 103 - 2 = 101. Right -6: borrow 1 hundred: 10100 - 6 = 10094.
996 × 993 (base 1000)
-4 and -7. Left 989. Right 028 (three digits). Answer 989028.
Example 196 × 89
  1. Base 100. Deviations: 96 is -4, 89 is -11.Both are below 100.
  2. Left: 96 - 11 = 85.One number plus the other's deviation.
  3. Right: (-4) × (-11) = 44, two digits.Base 100 has two zeros, so the right part has two digits.
  4. Answer 8544.
Example 2112 × 107
  1. Base 100. Deviations +12 and +7.
  2. Left: 112 + 7 = 119. Right: 12 × 7 = 84.
  3. Answer 11984.
Exam tipIf the right part has more digits than the base has zeros, carry the extra digit into the left part: 88 × 88 → 76 | 144 → 7744.

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.

Multiply byDo insteadExample
5x 10, then / 286 × 5 = 860 / 2 = 430
25x 100, then / 464 × 25 = 6400 / 4 = 1600
125x 1000, then / 848 × 125 = 48000 / 8 = 6000
9x 10, then minus the number73 × 9 = 730 - 73 = 657
99x 100, then minus the number58 × 99 = 5800 - 58 = 5742
11split and add neighbours72 × 11 = 7 | 9 | 2 = 792
15x 10, then add half of that44 × 15 = 440 + 220 = 660
50x 100, then / 237 × 50 = 3700 / 2 = 1850
Example 1264 × 25
  1. 264 × 100 = 26400.25 is a quarter of 100.
  2. 26400 / 4 = 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.

Example 184 × 86
  1. Same tens 8, units 4 + 6 = 10.The pattern fits.
  2. Left 8 × 9 = 72. Right 4 × 6 = 24.
  3. 7224.
Example 298 × 102
  1. Both are 2 away from 100.Equal distance either side.
  2. 1002 - 22 = 10000 - 4 = 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.

Example 1234 × 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.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.
Exam tipCheck a 3-digit product with digit sums (see the checks chapter) before you mark the option.

Formula summary

2-digit × 2-digit (ab × cd)
Right: b*d. Middle: a*d + b*c. Left: a*c. Write one digit, carry the rest.
3-digit × 3-digit (abc × def)
Five columns, right to left: cf | bf+ce | af+be+cd | ae+bd | ad.
2-digit × 3-digit
Put a zero in front: 45 becomes 045, then use the 3×3 pattern. The zero kills half the products.
Near a base (100, 1000)
Write each number as base +/- deviation. Left part = one number + other's deviation. Right part = product of deviations, padded to as many digits as the base has zeros.
Same tens, units add to 10
63 × 67: tens × (tens+1) | units × units → 6×7 | 3×7 = 4221.
(a+b)(a-b) = a2 - b2
53 × 47 = 502 - 32 = 2491. Use it when two numbers sit equally either side of a round number.
x 5, x 25, x 125
x5 = x10 / 2. x25 = x100 / 4. x125 = x1000 / 8.
x 9, x 99, x 999
n × 99 = n × 100 - n.
x 11
Write the first digit, then each pair of neighbours added, then the last digit. Carry when a sum passes 9.

Watch it step by step

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)

Shortcuts and tricks

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.
Example43 × 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.

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.
Example234 × 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.

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.
Example45 × 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.

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.
Example97 × 96
  1. -3 and -4.
  2. Left 93, right 12.

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.
Example103 × 98
  1. 101|-06
  2. = 10100 - 6

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.
Example996 × 993
  1. 989|028

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.
Example48 × 47
  1. 45 × 50 = 2250
  2. + 6

Times 11

Use it when: Instant for any number.

  1. Keep the outer digits, add neighbours in between.
Example352 × 11
  1. 3 | 3+5 | 5+2 | 2
  2. 3 | 8 | 7 | 2

Common mistakes

Forgetting the carry into the middle column.
Say the carry out loud: 'write 1, carry 2'. Most errors are dropped carries, not bad products.
Writing 93|6 for a base-100 product with a 1-digit right part.
Pad the right part to as many digits as the base has zeros: 06, 006.
Forgetting to add the carry in the middle column.
Say 'write, carry' at every column.
Writing the base-method right part with the wrong number of digits (97 × 98 = 95 | 6 → 956).
Base 100 needs two digits: 95 | 06 = 9506.
Using same-tens-units-add-to-10 on 63 × 68.
Check the pattern first: 3 + 8 is not 10. Use criss-cross instead.

Before you move on, you should be able to...

  • 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

Practice questions (25)

Try each one before opening the solution. Or practise them one by one so your score is saved.

easy 76 × 74
  1. Same tens, units 6+4 = 10.
  2. 7×8 = 56 | 6×4 = 24.
  3. 5624.

Answer: 5624

easy 62 × 58
  1. 602 - 22 = 3600 - 4.

Answer: 3596

easy 4868 × 25
  1. 486800 / 4 = 121700.

Answer: 1,21,700

easy 736 × 125
  1. 736000 / 8 = 92000.

Answer: 92,000

easy 857 × 99
  1. 85700 - 857 = 84843.

Answer: 84,843

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

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

medium 994 × 988
  1. -6 and -12.
  2. Left 994 - 12 = 982.
  3. Right 6 × 12 = 72 → 072.
  4. 982072.

Answer: 9,82,072

medium 112 × 109
  1. +12 and +9.
  2. Left 112 + 9 = 121.
  3. Right 108.
  4. 121|08 → 12208.

Answer: 12,208

medium 4789 × 11
  1. 4 | 4+7 | 7+8 | 8+9 | 9 → 4 | 11 | 15 | 17 | 9.
  2. Carry right to left: 9, 17 → 7 c1, 15+1 = 16 → 6 c1, 11+1 = 12 → 2 c1, 4+1 = 5.
  3. 52679.

Answer: 52,679

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

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

hard 58 × 473
  1. Pad: 058 × 473.
  2. 8×3 = 24 → 4 c2.
  3. 5×3 + 8×7 = 71 +2 = 73 → 3 c7.
  4. 0 + 5×7 + 8×4 = 67 +7 = 74 → 4 c7.
  5. 0 + 5×4 = 20 +7 = 27.
  6. 27434.

Answer: 27,434

hard 104 × 93
  1. +4 and -7.
  2. Left 104 - 7 = 97 → 9700.
  3. Right -28.
  4. 9700 - 28 = 9672.

Answer: 9672

hard 1234 × 5678 (hint: split 12|34 and 56|78 and criss-cross the pairs as if they were digits)
  1. Treat 12|34 × 56|78 as 2-digit × 2-digit in base 100.
  2. 34 × 78 = 2652 → write 52, carry 26.
  3. 12×78 + 34×56 = 936 + 1904 = 2840, +26 = 2866 → write 66, carry 28.
  4. 12 × 56 = 672, +28 = 700.
  5. 700|66|52 → 7006652.

Answer: 70,06,652

hard 867 × 934
  1. 7×4 = 28 → write 8, carry 2
  2. 7×3 + 6×4 + carry 2 = 47 → write 7, carry 4
  3. 7×9 + 6×3 + 8×4 + carry 4 = 117 → write 7, carry 11
  4. 6×9 + 8×3 + carry 11 = 89 → write 9, carry 8
  5. 8×9 + carry 8 = 80 → write 80
  6. Read the written digits right to left: 809,778.

Answer: 809,778

hard 709 × 386
  1. 9×6 = 54 → write 4, carry 5
  2. 9×8 + 0×6 + carry 5 = 77 → write 7, carry 7
  3. 9×3 + 0×8 + 7×6 + carry 7 = 76 → write 6, carry 7
  4. 0×3 + 7×8 + carry 7 = 63 → write 3, carry 6
  5. 7×3 + carry 6 = 27 → write 27
  6. Read the written digits right to left: 273,674.

Answer: 273,674

hard 47 × 865 (2-digit × 3-digit)
  1. Pad the shorter number with zeros: 047 × 865.
  2. 7×5 = 35 → write 5, carry 3
  3. 7×6 + 4×5 + carry 3 = 65 → write 5, carry 6
  4. 7×8 + 4×6 + 0×5 + carry 6 = 86 → write 6, carry 8
  5. 4×8 + 0×6 + carry 8 = 40 → write 0, carry 4
  6. 0×8 + carry 4 = 4 → write 4
  7. Read the written digits right to left: 40,655.

Answer: 40,655

hard 9996 × 9989 (base 10000)
  1. Deviations -4 and -11.
  2. Left 9996 - 11 = 9985.
  3. Right 44, padded to 4 digits: 0044.
  4. 99850044.

Answer: 9,98,50,044

hard 1008 × 1012 (base 1000)
  1. +8 and +12.
  2. Left 1020.
  3. Right 96 → 096.
  4. 1020096.

Answer: 10,20,096

hard 998 × 1003
  1. -2 and +3.
  2. Left 998 + 3 = 1001 → 1001000.
  3. Right -6.
  4. 1001000 - 6.

Answer: 10,00,994

hard 87 × 83
  1. Same tens, units 7 + 3 = 10.
  2. 8 × 9 = 72 | 7 × 3 = 21.

Answer: 7221

hard 12345 × 11
  1. 1 | 1+2 | 2+3 | 3+4 | 4+5 | 5 → 1 3 5 7 9 5.

Answer: 1,35,795

hard 999 × 457
  1. 457000 - 457.

Answer: 4,56,543

hard 48 × 52 × 25
  1. 48 × 52 = 502 - 22 = 2496.
  2. x25 = x100/4: 249600/4 = 62400.

Answer: 62,400

Class plan (2 hours, for trainers)

A tested order for teaching this topic in one 2-hour session. Present mode follows the same order.

10 minRace: two students multiply 43 × 27, one long way, one watching you do criss-cross.
15 minThe area picture. Draw it for 43 × 27 and point out the middle pieces.
20 minCriss-cross 2 × 2: teach, then 8 drills in pairs.
20 minBase method: both below, both above, mixed, base 1000.
15 minQuick multipliers table and x 11. Students make their own examples.
10 minSpecial pairs: same tens, and (a+b)(a-b).
10 min3 × 3 criss-cross with the animated visual.
15 minSpeed drill sheet (PDF) against the clock.
5 minRecap: which trick for which pair of numbers.
← Where these shortcuts come fromSquares, cubes and roots in seconds →
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