Tables & Speed Tricks
Everything worth memorising for fast calculation: tables 1–30, squares 1–100, cubes 1–50, roots, fractions, factorials, powers, primes and logs, plus the speed tricks. Every number on this page is computed by a program, not typed. Print it for a pocket PDF.
1. Multiplication Tables 1 to 30
Tables are the base of every fast calculation: percentages, ratios, time and work and division all fall back on them. Know 1 to 20 cold, and 21 to 30 well enough to write them out in seconds.
Like knowing your friends' phone numbers by heart instead of opening the contacts every time: the lookup is what slows you down.
Split the number: 17 × 8 = 10 × 8 + 7 × 8 = 80 + 56 = 136. For 21 to 30, use (20 + a) × k = 20k + ak, so 27 × 6 = 120 + 42 = 162. Say each table aloud once a day for a week; then test yourself with random pairs in Calculation Practice.
Tables 1 to 10
Tables 11 to 20
Tables 21 to 30
The usual slips are 7 × 8 = 56 (not 54), 12 × 12 = 144, 13 × 7 = 91 (people say 81), 17 × 7 = 119, 19 × 7 = 133 and 23 × 4 = 92. Circle the ones you hesitate on and drill only those.
2. Squares 1 to 100
A square is a number times itself: 13² = 169. Squares 1 to 30 should be instant; 31 to 100 you can rebuild in a few seconds with the tricks below, but reading this table daily makes them instant too.
Squares are landmarks on the number line. When you see 729 or 841 you should immediately recognise 27² and 29², the same way you recognise a famous building.
Near 50: (50 + a)² = 2500 + 100a + a², so 53² = 2500 + 300 + 9 = 2809. Near 100: (100 − a)² = 10000 − 200a + a², so 96² = 10000 − 800 + 16 = 9216. Ending in 5: n5² = n(n+1) then 25, so 75² = 7 × 8 = 56, then 25 = 5625.
Patterns that help you check an answer
- A perfect square can only end in 0, 1, 4, 5, 6 or 9. If a number ends in 2, 3, 7 or 8 it is not a perfect square.
- Squares ending in 0 always end in an even number of zeros (100, 400, 2500).
- The last two digits of n² and (50 − n)² and (50 + n)² are the same: 21² = 441, 29² = 841, 71² = 5041, 79² = 6241 all end in 41.
- Difference of consecutive squares: (n + 1)² − n² = 2n + 1, so 41² = 40² + 81 = 1600 + 81 = 1681.
3. Cubes 1 to 50
A cube is a number multiplied by itself three times: 12³ = 12 × 12 × 12 = 1,728. Cubes 1 to 20 come up constantly in mensuration (volumes) and in cube-root questions.
If a square is a floor tile pattern, a cube is the whole box of tiles stacked up: side × side × side.
The last digit of n³ depends only on the last digit of n: 0→0, 1→1, 2→8, 3→7, 4→4, 5→5, 6→6, 7→3, 8→2, 9→9. So 2↔8 and 3↔7 swap; 0, 1, 4, 5, 6, 9 stay the same. This is how you find cube roots in seconds (section 6).
Useful identities
- (a + b)³ = a³ + b³ + 3ab(a + b), so 21³ = 8000 + 1 + 3 × 20 × 1 × 21 = 9261.
- a³ + b³ = (a + b)(a² − ab + b²) and a³ − b³ = (a − b)(a² + ab + b²).
- If a + b + c = 0, then a³ + b³ + c³ = 3abc.
4. Powers of Small Numbers & Unit-Digit Cycles
Powers show up in compound interest, growth, number system and remainder questions. You rarely need a huge power in full, but you must know the small ones and how the last digit repeats.
The last digit of a power goes round in a loop, like the hands of a clock. Once you know the loop length, you can jump straight to the 100th turn.
| Base | Powers |
|---|---|
| 2 | 21 = 2 · 22 = 4 · 23 = 8 · 24 = 16 · 25 = 32 · 26 = 64 · 27 = 128 · 28 = 256 · 29 = 512 · 210 = 1,024 · 211 = 2,048 · 212 = 4,096 · 213 = 8,192 · 214 = 16,384 · 215 = 32,768 · 216 = 65,536 · 217 = 131,072 · 218 = 262,144 · 219 = 524,288 · 220 = 1,048,576 |
| 3 | 31 = 3 · 32 = 9 · 33 = 27 · 34 = 81 · 35 = 243 · 36 = 729 · 37 = 2,187 · 38 = 6,561 · 39 = 19,683 · 310 = 59,049 · 311 = 177,147 · 312 = 531,441 |
| 4 | 41 = 4 · 42 = 16 · 43 = 64 · 44 = 256 · 45 = 1,024 · 46 = 4,096 · 47 = 16,384 · 48 = 65,536 · 49 = 262,144 · 410 = 1,048,576 |
| 5 | 51 = 5 · 52 = 25 · 53 = 125 · 54 = 625 · 55 = 3,125 · 56 = 15,625 · 57 = 78,125 · 58 = 390,625 · 59 = 1,953,125 · 510 = 9,765,625 |
| 6 | 61 = 6 · 62 = 36 · 63 = 216 · 64 = 1,296 · 65 = 7,776 · 66 = 46,656 · 67 = 279,936 · 68 = 1,679,616 |
| 7 | 71 = 7 · 72 = 49 · 73 = 343 · 74 = 2,401 · 75 = 16,807 · 76 = 117,649 · 77 = 823,543 · 78 = 5,764,801 |
| 8 | 81 = 8 · 82 = 64 · 83 = 512 · 84 = 4,096 · 85 = 32,768 · 86 = 262,144 · 87 = 2,097,152 |
| 9 | 91 = 9 · 92 = 81 · 93 = 729 · 94 = 6,561 · 95 = 59,049 · 96 = 531,441 · 97 = 4,782,969 |
| 11 | 111 = 11 · 112 = 121 · 113 = 1,331 · 114 = 14,641 · 115 = 161,051 · 116 = 1,771,561 |
| 12 | 121 = 12 · 122 = 144 · 123 = 1,728 · 124 = 20,736 · 125 = 248,832 |
| 13 | 131 = 13 · 132 = 169 · 133 = 2,197 · 134 = 28,561 · 135 = 371,293 |
| 15 | 151 = 15 · 152 = 225 · 153 = 3,375 · 154 = 50,625 |
Unit-digit cycles
| Last digit | Cycle of last digits | Cycle length |
|---|---|---|
| 0 | 0 | 1 |
| 1 | 1 | 1 |
| 2 | 2 → 4 → 8 → 6 | 4 |
| 3 | 3 → 9 → 7 → 1 | 4 |
| 4 | 4 → 6 | 2 |
| 5 | 5 | 1 |
| 6 | 6 | 1 |
| 7 | 7 → 9 → 3 → 1 | 4 |
| 8 | 8 → 4 → 2 → 6 | 4 |
| 9 | 9 → 1 | 2 |
Divide the exponent by the cycle length and use the remainder (a remainder of 0 means the last item in the cycle). Example: 799: cycle 7, 9, 3, 1 (length 4); 99 ÷ 4 leaves 3, so the 3rd item: 3. Check: 3.
For bases ending in 0, 1, 5 or 6 the last digit never changes. Do not divide by 4 for them.
5. Square Roots 1 to 50
The square root of a number is the value that, multiplied by itself, gives the number: √196 = 14. Perfect squares have whole-number roots; the others are decimals you should know to two or three places for estimation.
Finding a square root is walking the squares table backwards: you know the answer, you are looking for the question.
√2 = 1.414, √3 = 1.732, √5 = 2.236, √6 = 2.449, √7 = 2.646, √10 = 3.162. Almost every other root can be built from these: √12 = 2√3 = 3.464, √50 = 5√2 = 7.071.
Square root of a big perfect square in 10 seconds
- Look at the last digit to get the possible last digits of the root: 1→1 or 9, 4→2 or 8, 9→3 or 7, 6→4 or 6, 5→5, 0→0.
- Drop the last two digits and find the largest square below what is left. That gives the tens digit.
- Pick between the two candidates by comparing with (tens digit)(5)²: if the number is bigger, take the larger candidate.
Example: √5329. Ends in 9 → root ends in 3 or 7. Drop "29": 53, and 7² = 49 < 53 < 64, so the tens digit is 7. 75² = 5625 > 5329, so take the smaller: 73. Check: 73² = 5329.
6. Cube Roots
The cube root is the number that, cubed, gives the value: ∛2744 = 14. For perfect cubes up to six digits there is a two-step trick that needs only the cubes of 1 to 9.
Like reading a car number plate: the last digit tells you one thing, the front tells you the other, and together they identify it.
1) The last digit of the root comes from the last digit of the cube: 1→1, 8→2, 7→3, 4→4, 5→5, 6→6, 3→7, 2→8, 9→9, 0→0. 2) Drop the last three digits; the front of the root is the largest number whose cube is at most what remains. Example: ∛103823: ends in 3 → root ends in 7. Drop 823: 103, and 4³ = 64 ≤ 103 < 125 = 5³, so front = 4. Root = 47. Check: 47³ = {comma(47**3)}.
| Perfect cube | Cube root | Perfect cube | Cube root |
|---|---|---|---|
| 1 | 1 | 17,576 | 26 |
| 8 | 2 | 19,683 | 27 |
| 27 | 3 | 21,952 | 28 |
| 64 | 4 | 24,389 | 29 |
| 125 | 5 | 27,000 | 30 |
| 216 | 6 | 29,791 | 31 |
| 343 | 7 | 32,768 | 32 |
| 512 | 8 | 35,937 | 33 |
| 729 | 9 | 39,304 | 34 |
| 1,000 | 10 | 42,875 | 35 |
| 1,331 | 11 | 46,656 | 36 |
| 1,728 | 12 | 50,653 | 37 |
| 2,197 | 13 | 54,872 | 38 |
| 2,744 | 14 | 59,319 | 39 |
| 3,375 | 15 | 64,000 | 40 |
| 4,096 | 16 | 68,921 | 41 |
| 4,913 | 17 | 74,088 | 42 |
| 5,832 | 18 | 79,507 | 43 |
| 6,859 | 19 | 85,184 | 44 |
| 8,000 | 20 | 91,125 | 45 |
| 9,261 | 21 | 97,336 | 46 |
| 10,648 | 22 | 103,823 | 47 |
| 12,167 | 23 | 110,592 | 48 |
| 13,824 | 24 | 117,649 | 49 |
| 15,625 | 25 | 125,000 | 50 |
Handy decimal cube roots
| Number | Cube root |
|---|---|
| 2 | 1.260 |
| 3 | 1.442 |
| 4 | 1.587 |
| 5 | 1.710 |
| 6 | 1.817 |
| 7 | 1.913 |
| 9 | 2.080 |
| 10 | 2.154 |
7. Fraction ↔ Percentage ↔ Decimal (1/1 to 1/30)
Most percentage, profit and ratio questions become one-line mental sums once you can switch between a fraction and its percentage without dividing. 12.5% is just 1/8; 16 2/3% is 1/6.
It's the same amount in three languages. A good translator doesn't look words up; they just know that 1/8 and 12.5% mean the same thing.
x% of y = y% of x, so 16% of 25 = 25% of 16 = 4. And to take 37.5% of anything, take 3/8 of it: 37.5% of 640 = 3 × 80 = 240.
| Fraction | Percentage | Decimal | Fraction | Percentage | Decimal |
|---|---|---|---|---|---|
| 1/1 | 100% | 1 | 1/16 | 6 1/4% | 0.0625 |
| 1/2 | 50% | 0.5 | 1/17 | 5 15/17% | 0.0588… |
| 1/3 | 33 1/3% | 0.3333… | 1/18 | 5 5/9% | 0.0556… |
| 1/4 | 25% | 0.25 | 1/19 | 5 5/19% | 0.0526… |
| 1/5 | 20% | 0.2 | 1/20 | 5% | 0.05 |
| 1/6 | 16 2/3% | 0.1667… | 1/21 | 4 16/21% | 0.0476… |
| 1/7 | 14 2/7% | 0.1429… | 1/22 | 4 6/11% | 0.0455… |
| 1/8 | 12 1/2% | 0.125 | 1/23 | 4 8/23% | 0.0435… |
| 1/9 | 11 1/9% | 0.1111… | 1/24 | 4 1/6% | 0.0417… |
| 1/10 | 10% | 0.1 | 1/25 | 4% | 0.04 |
| 1/11 | 9 1/11% | 0.0909… | 1/26 | 3 11/13% | 0.0385… |
| 1/12 | 8 1/3% | 0.0833… | 1/27 | 3 19/27% | 0.037… |
| 1/13 | 7 9/13% | 0.0769… | 1/28 | 3 4/7% | 0.0357… |
| 1/14 | 7 1/7% | 0.0714… | 1/29 | 3 13/29% | 0.0345… |
| 1/15 | 6 2/3% | 0.0667… | 1/30 | 3 1/3% | 0.0333… |
"…" means the decimal keeps going (it is rounded to 4 places).
8. Multiples of Common Fractions
Once 1/8 = 12.5% is automatic, 3/8, 5/8 and 7/8 are just multiples: 37.5%, 62.5%, 87.5%. The denominators below are the ones that appear most in tests.
If one slice of pizza is 12.5% of the pizza, three slices are 37.5%. You never have to cut a new pizza.
| Denominator | Fractions in lowest terms and their percentages |
|---|---|
| /3 | 1/3 = 33 1/3% · 2/3 = 66 2/3% |
| /6 | 1/6 = 16 2/3% · 5/6 = 83 1/3% |
| /7 | 1/7 = 14 2/7% · 2/7 = 28 4/7% · 3/7 = 42 6/7% · 4/7 = 57 1/7% · 5/7 = 71 3/7% · 6/7 = 85 5/7% |
| /8 | 1/8 = 12 1/2% · 3/8 = 37 1/2% · 5/8 = 62 1/2% · 7/8 = 87 1/2% |
| /9 | 1/9 = 11 1/9% · 2/9 = 22 2/9% · 4/9 = 44 4/9% · 5/9 = 55 5/9% · 7/9 = 77 7/9% · 8/9 = 88 8/9% |
| /11 | 1/11 = 9 1/11% · 2/11 = 18 2/11% · 3/11 = 27 3/11% · 4/11 = 36 4/11% · 5/11 = 45 5/11% · 6/11 = 54 6/11% · 7/11 = 63 7/11% · 8/11 = 72 8/11% · 9/11 = 81 9/11% · 10/11 = 90 10/11% |
| /12 | 1/12 = 8 1/3% · 5/12 = 41 2/3% · 7/12 = 58 1/3% · 11/12 = 91 2/3% |
| /16 | 1/16 = 6 1/4% · 3/16 = 18 3/4% · 5/16 = 31 1/4% · 7/16 = 43 3/4% · 9/16 = 56 1/4% · 11/16 = 68 3/4% · 13/16 = 81 1/4% · 15/16 = 93 3/4% |
Two changes of a% and b% give a net change of a + b + ab/100. +20% then −20% = 20 − 20 − 400/100 = −4%, not zero.
The same fraction goes up and down differently. If A is 25% (1/4) more than B, then B is 1/5 = 20% less than A, because the base changed.
9. Factorials 0! to 20!
n! (n factorial) is 1 × 2 × 3 × … × n. It counts the ways to arrange n different things in a row. 0! is defined as 1.
5 people can line up in 5! = 120 ways: 5 choices for the first spot, 4 for the next, and so on.
| n! | Value | n! | Value | n! | Value |
|---|---|---|---|---|---|
| 0! | 1 | 7! | 5,040 | 14! | 87,178,291,200 |
| 1! | 1 | 8! | 40,320 | 15! | 1,307,674,368,000 |
| 2! | 2 | 9! | 362,880 | 16! | 20,922,789,888,000 |
| 3! | 6 | 10! | 3,628,800 | 17! | 355,687,428,096,000 |
| 4! | 24 | 11! | 39,916,800 | 18! | 6,402,373,705,728,000 |
| 5! | 120 | 12! | 479,001,600 | 19! | 121,645,100,408,832,000 |
| 6! | 720 | 13! | 6,227,020,800 | 20! | 2,432,902,008,176,640,000 |
Cancel before you multiply: 10!/8! = 10 × 9 = 90. Number of trailing zeros in n! = ⌊n/5⌋ + ⌊n/25⌋ + ⌊n/125⌋ + …, so 100! ends in 20 + 4 = 24 zeros.
Factorials grow very fast. 13! is already more than 6 billion, so in P&C questions always cancel first.
10. nCr and Pascal's Triangle (n up to 12)
nCr is the number of ways to choose r things from n when order does not matter. Row n of Pascal's triangle lists nC0, nC1, …, nCn; each number is the sum of the two above it.
Choosing 2 friends out of 5 for a trip: the order you invite them doesn't matter, so it's 5C2 = 10 ways.
nCr = nC(n − r), so 12C10 = 12C2 = 66. nC0 + nC1 + … + nCn = 2ⁿ: the rows add up to 1, 2, 4, 8, 16, …
11. Prime Numbers up to 200
A prime has exactly two factors: 1 and itself. There are 25 primes below 100 and 46 below 200. 1 is not prime, and 2 is the only even prime.
Primes are the atoms of numbers: every whole number is built by multiplying primes together in exactly one way.
Check divisibility only by primes up to its square root. For 191: √191 ≈ 13.8, so test 2, 3, 5, 7, 11, 13. None divide it, so 191 is prime.
91 = 7 × 13, 119 = 7 × 17, 133 = 7 × 19, 143 = 11 × 13, 161 = 7 × 23, 187 = 11 × 17 and 169 = 13² look prime but are not.
12. Log Values (base 10)
log x answers: 10 to what power gives x? Tests usually give you log 2 and log 3 and expect you to build the rest.
If log 2 and log 3 are two Lego bricks, log 6, log 8, log 12 and log 18 are all built by snapping them together.
| Value | log₁₀ | Value | log₁₀ | Value | log₁₀ |
|---|---|---|---|---|---|
| log 2 | 0.3010 | log 5 | 0.6990 | log 8 | 0.9031 |
| log 3 | 0.4771 | log 6 | 0.7782 | log 9 | 0.9542 |
| log 4 | 0.6021 | log 7 | 0.8451 | log 10 | 1.0000 |
log 4 = 2 log 2, log 5 = 1 − log 2, log 6 = log 2 + log 3, log 8 = 3 log 2, log 9 = 2 log 3. Number of digits in 250 = ⌊50 × 0.3010⌋ + 1 = 16. Check: 250 = 1,125,899,906,842,624.
13. Speed Tricks with Worked Examples
These tricks replace long multiplication with two or three small mental steps. Learn one a day and practise it on ten random numbers.
Taking a shortcut through the park instead of walking around the block: same place, half the time.
Multiply by 11
64 × 11
- Write the two digits apart: 6 _ 4.
- Put their sum in the middle: 6 + 4 = 10.
- It's two digits, so carry the 1 to the 6: 7 0 4.
Answer: 704
87 × 11
- 8 _ 7, middle = 8 + 7 = 15.
- Carry 1: 9 5 7.
Answer: 957
Multiply by 5, 25 and 125
× 5 = × 10 ÷ 2, × 25 = × 100 ÷ 4, × 125 = × 1000 ÷ 8.
736 × 5
- 736 × 10 = 7360.
- ÷ 2 = 3680.
Answer: 3680
48 × 25
- 48 × 100 = 4800.
- ÷ 4 = 1200.
Answer: 1200
Numbers near a base (100, 1000)
98 × 97
- Both are below 100: 98 is −2, 97 is −3.
- Cross-subtract: 98 − 3 = 95 (or 97 − 2). That's the left part.
- Multiply the differences: (−2) × (−3) = 06 (two digits for base 100).
- Join: 95 | 06.
Answer: 9506
998 × 997
- Differences from 1000: −2 and −3.
- 998 − 3 = 995.
- (−2)(−3) = 006 (three digits for base 1000).
- Join: 995 | 006.
Answer: 995,006
104 × 107
- Differences from 100: +4 and +7.
- 104 + 7 = 111.
- 4 × 7 = 28.
- Join: 111 | 28.
Answer: 11,128
Numbers equally spaced around a round number
47 × 53
- Both are 3 away from 50.
- Use (a − b)(a + b) = a² − b²: 50² − 3² = 2500 − 9.
Answer: 2491
Squaring
85²
- Ends in 5: take 8 × 9 = 72.
- Write 25 after it.
Answer: 7225
58²
- 58 = 50 + 8: 2500 + 100 × 8 = 3300.
- Add 8² = 64.
Answer: 3364
97²
- 97 = 100 − 3: 10000 − 600 = 9400.
- Add 3² = 9.
Answer: 9409
Division and percentages
What is 12.5% of 848?
- 12.5% = 1/8.
- 848 ÷ 8 = 106.
Answer: 106
What is 18% of 350?
- 10% = 35, 1% = 3.5.
- 18% = 35 + 8 × 3.5 = 35 + 28.
Answer: 63
Is 7,392 divisible by 8?
- Only the last three digits matter for 8: 392.
- 392 ÷ 8 = 49, a whole number.
Answer: Yes
Tricks are only faster once they're automatic. In the first week you'll be slower with them than without; that's normal. Use Calculation Practice with 2-digit × 1-digit and 2-digit × 2-digit to build speed.
14. 4-Week Memorisation Plan
Memorising works best in short daily sessions with self-testing, not long reading sessions. Fifteen minutes a day beats two hours on Sunday.
Like going to the gym: small, regular sessions build the muscle; one huge session just makes you sore.
| Week | Learn | Daily drill (15–20 min) | Target by the end of the week |
|---|---|---|---|
| 1 | Tables 1–20, squares 1–30 | Say 2 tables aloud; write squares 1–30 from memory; 20 random 2-digit × 1-digit in Calculation Practice | Tables 1–20 without hesitation; 2-digit × 1-digit in under 4 s |
| 2 | Tables 21–30, cubes 1–20, fractions 1/2 to 1/20 | Write out 2 tables; flash-card 10 fractions ↔ %; 20 mixed + − × questions | Every fraction ↔ % in under 2 s |
| 3 | Squares 31–100 with the tricks, square and cube roots, powers of 2 and 3 | Square 10 random numbers using the near-50/near-100 rule; 5 cube roots with the two-step trick | Any square up to 100 in under 5 s |
| 4 | Factorials, primes, log values, all speed tricks | Mixed 30-question sessions with 20 s per question; review your slowest ones | 80% of mixed questions inside the time limit |
After the four weeks, one 10-minute Calculation Practice session a day keeps everything sharp.