As is already answered, I would also use either 1000-9 or 1000 - 10 + 1
Raja’s method is solid if you know the answer is an integer.
My method is similar at first, but this also works for non integer answers:
Like Raja I would start wth (12 * 10)^2 = 14400 as a ball park figure.
The difference of 14400 - 15129 is about 5%.
5% / 2 = 2.5%
12 plus 2.5% is 12.3.
So I revise (12 * 10)^2 = 14400 to 123^2 = 15129.
123^2, I calculate using cross multiplication with 12 and 3.
The 12 is actually 120, but I position the numbers in my head automatically: 12^2 | 2 * 12 * 3 | 3^2
I round 991 up to 1000 and remember that for every 1000 I subtract, I have to add 9.
Start by subtracting 26*1000. Remainder is 757. Add 26*9 = 234.
757+234 = 991. The remainder is exactly one more time 991. So correct the 26 to 27.
Answer: 27
I count the number of 3s. 9 in total. I first round it to 333 (million).
18*18 = 324, which is the closest integer that squared is lower than 333.
First rough answer is 18,000.
The difference between 324 and 333 is 9.
9/2 = 4.5.
4.5/324 is about 1.4%.
I add 1.4% to 18,000 to get 18,180 + 72 = 18,252.
Then I realise that my answer can be improved upon.
9 is actually 9.333333, so we need a little more than 4.5: 4.6666665
And my 333 is actually 333.333333.
So we get: 4.6666665 / 333.333333.
In my mind 4.6666665 is 4 2/3 = 14/3 and 333.333333 = 1/3 (disregarding the decimal point).
Dividing by 1/3 = multiplying by 3.
14/3, multiplied by 3 = 14.
Putting the decimal point back and 14 becomes 1.4%.
My original answer of 18,000 plus 1.4% stays.
So my answer is 18,252.
A calculator gives 18,257.412…
We are 5 off, which is close enough for me.
If you want to get an exact answer with all the decimals, check out my post about the duplex method:
Tbh; too much work for me.
If my answer is exact in the first 4 decimals, this is fine for me and my purposes.