How fast can you do 22 * 7 in your head? I get a lot of mileage out of simply approximating pi=3, and optionally adding in a factor of 10% later. For your example, I know 7^2=49 immediately, which I round to 50, 50 * 3=150 which is fairly close to 153.938. (Use more digits of pi!) I can quickly improve my estimate by adding 4.9. (Edit: It's also fairly trivial to get even better estimates quickly from here, but at this point it's probably faster to grab a cell phone / [favorite language] repl. From 154.9, subtract the additional 3 gained from using 50 instead of 49, now 151.9, add 4.9/2~=2.4, now 154.3, subtract 4.9 * 10% ~= .5, 153.8.)

22 times a single digit is pretty easy, double the digit, shift the decimal, add the doubled number - but it's not trivial. As i said, a contrived example.

The point is, you yourself stick to integer arithmetic, then try to fix it up with a 10% modifier at the end. People have been using pi for a long time. Easy access to calculators is, what, about 50 years now? I'll happily agree that rational representation is a historical artifact. But i still believe the vast majority of people doing arithmetic pre 1960 with floating point numbers did it like you and i do. They would put off the decimal representation as long as possible.

> optionally adding in a factor of 10% later

You're better off not adding anything. If you could add 5% instead, then you'd be much better still.

Ideally you add 14%. Why do you say 5% (or nothing) is better than 10%?

He's taking the % of the wrong number: the number you get after multiplying by three, not the original. I do what you do: 3, +10% + 5% (which is easy after getting the 10%).

It is easier to add 5% first, then triple, skipping the middle step.

Not for me- I get 10% to get the 5% so I have it anyway.

However, you could work from the 3 you've already used as the tripling in your model, then take 10% of that then 1/2 of that to get there. I've just never done it that way (these are things I just do without thinking too deeply about).