I have discussed a similar topic with Kinma but I’ll post here to perhaps broaden the discussion. I find that one of the most common mental calculations I need to perform is where I need to divide by 7.25, 4.2, 72, 58. I realise for 7.25 I could use 7 but I want a little more accuracy, the answer only needs to be accurate to 1 decimal place maybe 2. What is important though is that I produce an answer quickly, so am happy to sacrifice some accuracy for speed. I guess I’m looking for a way to improve estimation. If you need to divide 87 by 7.25 you could adjust it down to 85 by 7 but you’d lose a little accuracy so am looking for something a little better.
The other calculation I need to perform often is converting fractions to percentages eg 7/16, I do use reciprocals and multiply and that works ok for numbers under 20, sometimes I get stuck on things like 9/37. Again extreme accuracy is not vital but accuracy to 1 or 2 decimal places would be good.
Division by 7.25 might look difficult until you … wait for it… just do it.
An example:
\frac{19}{7.25}
We could morph this into
\frac{19 *4}{29} = \frac{76}{29} and use the systems described here:
But what if… we actually just divided by 7.25?
This is what we will do here.
Here … it … goes:
Step 1: 7.25 into 19 goes 2 times. Subtract twice 7.25 = 14.5 from 19, leaving a remainder of 4.5. (Who said the remainder needs to be an integer?)
Answer so far: 2R4.5
Step 2: Shift decimal point: 4.5 => 45
Step 3: 7.25 into 45 goes 6 times. 6x7.25 = 42+1.5 = 43.5. Remainder is 1.5
Answer so far: 2.6R1.5
Step 4: 7.25 into 15 goes 2 times. 2X7.25 = 14.5. Remainder is 0.5.
Answer so far: 2.62 R 0.5
etc
This is how I would do it with writing down subtotals.
However; if you want to do this completely in the mind, no paper, no pen, I would use a slight different approach. One that is easier on the brain. Read: more, but smaller steps. Here is what I do:
Step 1: 7.25 into 19 goes 2 times. 2 parts. Focus on 7 first. Subtract twice 7 = 14 from 19, leaving a remainder of 5. Second part. Twice a quarter makes a half. Subtract from 5 = 4.5.
Answer so far: 2R4.5
Step 2: Shift decimal point: 4.5 => 45
Step 3: 7.25 into 45 goes 6 times. 6x7 = 42, leaving a remainder of 3. 6 times a quarter = one and a half. Subtract from 3, leaving a remainder of 1.5.
Answer so far: 2.6R1.5
Step 4: 7.25 into 15 goes 2 times. 2x7 = 14. Remainder is 1. 2 times a quarter makes a half. subtract from 1, leaves a half:
Answer so far: 2.62 R 0.5
etc.
If you want to visualize this, try to see 8 baskets. 7 normal ones and a stranger, smaller one.
7 baskets hold full size apples and the 8th one only accepts quarter apples.
Now do the division.
The only thing that matters in division is how many apples are there per basket. How you divided all apples (evenly) over all the baskets is irrelevant.
In the example of \frac{19}{7.25}, it is ok to first load the first 7 baskets with 2 apples each (14), then look at how many are still left to do (5), then put 2 quarter apples in the last basket, leaving 4.5 apples left to be divided.
By splitting this process in 2 parts it is easier on the brain.
Let’s take division by 58 now to show another example of this 2 step process.
Shall we try: \frac{180}{58}?
58 into 180 goes three times.
Subtract 150 from 180 = 30, then subtract 3x8=24 from 30 to get 6.
Answer so far: 3R6
58 into 60 goes one time.
Subtract 58 from 60 to get 2. No need to do the 2 step process here, right?
Answer so far: 3.1R2
58 into 20 does not go
Answer so far: 3.10R20
58 into 200 goes three times.
Subtract 150 from 200 to get 50. Subtract 24 from 50 = 26
Answer so far: 3.103R26
58 into 260 goes about four times.
Subtract 200 from 260 to get 60. Subtract 32 from 60 = 28
Answer so far: 3.1034R28
58 into 280 goes about four times.
Subtract 200 from 280 to get 80. Subtract 32 from 80 = 48
Answer so far: 3.10344R48
58 into 480 goes about 8 times.
Subtract 400 from 480 to get 80. Subtract 64 from 80 = 16.
16 is less than half of 58, so round down.
Answer in 7 digit precision: 3.103448
One important reason I use this 2 step process is that the first step makes the remainder a lot smaller. In general you lose the first digit. See the last example; after each step the remainder went from 3 digits (of which the last one is usually zero) to 2 digits.
In the example of division by 58, there is no need to learn the table of 58 (would be good if you did, 58, 116, 174, etc, but anyway). Just use 60 for the guess and work from there.
Thank you so muck Kinma, both of the approaches you mentioned work great! I certainly find them a lot quicker than the previous methods I’ve tried particularly the “Vedic” type approach which involves juggling and carrying multiple numbers at at time. With the 7.25 division, I find the direct division method faster ie dealing with decimal remainders. I don’t think I’ve ever come across this system anywhere but it works well. You are very skilled at developing algorithms that simplify calculations. I see a similar theme in your algorithms, using complements, avoiding the carry but I’ve never thought how that can be used in division, again thank you so much!