The most common technique for divisions is only suitable when dividing by small numbers, or when using pen and paper. Mental calculation of more advanced divisions requires a different method.
For example, when using the standard “long division” technique for mental calculation of 1829 ÷ 7.6543, we would need to calculate multiples of 7.6543, such as 7.6543 × 3 = 22.9629, and subtract each multiple from a remainder stored in our working memory.
Given all humans have a limited working memory, this becomes unmanageably difficult as the size of the calculations increases.
Luckily, there is an alternative method, based on cross-multiplication, which places only a moderate load on our working memory, and therefore allows quick division by numbers of arbitrary size, to arbitrary accuracy. You could use this method to calculate 123 ÷ π to a hundred decimal places if you really wanted.
I (re)discovered this method when preparing in 2015 for the Memoriad competition, and in this article I’ll show you how to do it yourself.
Note: when an estimate is sufficient, rather than many digits of accuracy, then estimation techniques are more useful.
Cross Division Method
To solve 1829 ÷ 7.6543 by cross division, we first take the first 1-2 digits—in this case the 76—which we will be using for the division. If your mental calculation is strong, you should use 2 digits. If that is very difficult for you, under some circumstances, you can use just 1 digit (7).
The digits after the 76—in this case, the 543—will be used the cross-division step.
Scale
765.43 < 1829 < 7654.3, so the answer will be between 100 and 1000, and the first digit will be in the 100s place.
First digit
To obtain each new digit of the answer, we follow the same steps—dividing the current number by 76, calculating the remainder and then subtracting some additional terms to account for the 543.
- Because 76 × 2 < 182 < 76 × 3, the first digit is 2 [answer is 200-something]
- Remainder is 182 – (76 × 2) = 30
- Multiply by 10, and add on the next digit from 1829, which is 9, to give 309
- Cross division: subtract 309 – 5 × 2 = 299
[In the next step I’ll explain why we subtracted the 5 × 2]
Next digit—with explanation
We just repeat the same steps for every new digit:
- Because 76 × 3 < 299 < 76 × 4, the next digit is 3 [answer is 230-something]
- Remainder is 299 – (76 × 3) = 71
- Multiply by 10, and add on the next digit from 1829.000…, which is 0, to give 710
- Cross division: subtract 710 – 5 × 3 – 4 × 2 = 687
Here is how to find the subtractions in the cross-division step:
- Start with the most significant digit that we are not dividing by—in this case it’s the 5 from the 543, and multiply that with the newest digit of the answer—which in this case is the 3 from the 230-something.
- Next, continue through the digits of the divisor, moving towards the least significant digits (the 4 then the 3) paired with the digits of the answer (so far just the 2), moving towards the older digits. So for this example we just have 4 × 2, but later we could have more subtractions.
- Stop when there are either no more digits in the answer, or the remaining digits in the divisor are all 0. In this case, there is one more non-zero digit in the divisor (the 3) but no earlier digits in the answer, so we already stop.
The cross division will take most practice as you begin, but is fairly mechanical.
Next digit—example with overflow issue
Continuing as before:
- Because 76 × 9 < 687, the next digit is 9 [answer is 239.something]
- Remainder is 687 – (76 × 9) = 3
- Multiply by 10, and add on the next digit from 1829.000…, which is 0, to give 30
- Cross division: subtract 30 – 5 × 9 – …
Here we have a problem because the answer is negative! When this happens, it means that we should have taken a smaller multiple of 76, and allowed an “improper” remainder, larger than 76. This is rare, but happens when the remainder (3) is very small, so with some experience you can spot these situations quickly! Let’s try it:
- Try using 8 for the next digit [answer is 238.something]
- Remainder is 687 – (76 × 8) = 79
- Multiply by 10, and add on the next digit from 1829.000…, which is 0, to give 790
- Cross division: subtract 790 – 5 × 8 – 4 × 3 – 3 × 2 = 732
Next digit
Continuing as before:
- Because 76 × 9 < 732, the next digit is 9 [answer is 238.9…]
- Remainder is 732 – (76 × 9) = 48
- Multiply by 10, and add on the next digit from 1829.000…, which is 0, to give 480
- Cross division: subtract 480 – 5 × 9 – 4 × 8 – 3 × 3 = 394
Next digit
- Because 76 × 5 < 394 < 76 × 6, the next digit is 5 [answer is 238.95…]
- Remainder is 394 – (76 × 5) = 14
- Multiply by 10, and add on the next digit from 1829.000…, which is 0, to give 140
- Cross division: subtract 140 – 5 × 5 – 4 × 9 – 3 × 8 = 55
Next digit
- Because 55 < 76 × 1, the next digit is 0 [answer is 238.950…]
- Remainder is just 55
- Multiply by 10, and add on the next digit from 1829.000…, which is 0, to give 550
- Cross division: subtract 550 – 5 × 0 – 4 × 5 – 3 × 9 = 503
Next digit
- Because 76 × 6 < 503 < 76 × 7, the next digit is 6 [answer is 238.9506…]
- Remainder is 503 – (76 × 6) = 47
- Multiply by 10, and add on the next digit from 1829.000…, which is 0, to give 470
- Cross division: subtract 470 – 5 × 6 – 4 × 0 – 3 × 5 = 425
And so on until you have the accuracy you required.
Visual Example
One of the our readers has submitted a visual illustration of the method above, with the same example. Credit to Elke Kuge—fellow Mental Calculation Coach at the JMCWC, and Hectoc expert!
Hints & Tips
Single-digit divisors
For beginners at mental math, you might prefer to take a single-digit divisor (e.g. 76.543 rather than 76.543) to make the remainder step easier. However, this means that the overflow issue occurs much more regularly!
In this case, 76 is a good approximation to 76.543, so the overflow issue is rare. But 70 is much further from 76.543, so you will have problems very often.
Therefore I recommend using 2-digit divisors as much as possible.
Calculating remainders
At first glance, solving e.g. 503 – (76 × 6) = 47 seems very challenging, as 76 × 6 = 456 is difficult, and then performing 503 – 456 without seeing either number is also difficult.
However, an easier way to think about these is that 76 × 6 = 75 × 6 + another 6, so:
503 – (76 × 6) = (503 – 450) – 6 = 53 – 6 = 47
Separating these remainder calculations in such ways makes it much easier for mental calculation.
Top performers
The fastest people in the world can calculate ten digits of accuracy in less than one minute!
Training
To begin, I suggest writing your own questions, dividing by 4 or 5 digits, but choosing an easy first two digits. For example, 45.123 is easy to divide by because 45 is a convenient number for making multiples of, and the digits 123 are small, so you probably avoid overflow issues automatically.
To get in touch with me (Daniel Timms) about mental calculation training, coaching, or anything on this site, you can contact me here.