My first job out of engineering college was at a Tire company . One night, a spindle needed to be replaced in a machine on the line to restart it . I used my deep knowledge of mechanical engineering and figured I needed a spindle with 28.8 CM diameter . Everyone around me had a good laugh at my expense . Turns out spindles come rounded in 5CM intervals . Also I need not have done any calculations – there was a simple chart posted on the machine that indicated what size was required 🙂
There is some irony in this story . I was very familiar with the theory of approximations in Mathematics while in college . Just that I was not smart enough to use the principle in real life scenarios .
My generation didn’t get to use a calculator in school – so by and large most people in my peer group can do mental math just fine . This unfortunately can’t be taken for granted any more and it pains me to watch massive confusion from cashiers and customers alike when I am waiting in line at a grocery or something . it’s not as if kids became any less smart – the educational system somehow derailed along the way .
Even amongst people who learned how to do mental math – approximating answers is not something that comes naturally when it comes to multiplication and division . This becomes a challenge both in personal life ( eg: what would I need to spend if I need new carpet for my house ) and at work ( eg: what percentage of customers might churn annually at current trajectory ) .
Say for example you need to find what is 17% of 58976 . The correct answer is 10025.92 for which most people will need a calculator to get to this precise level . But most decisions don’t need this level or precision – a close enough approximation is just fine .
One quick mental math is to round up and find 20% of 60000 . Which is 12000 – but since we have rounded up both sides , we should know how much to reduce from this to get a close enough approximation .
One possible next step is to find 15% of 60000 – which is still rounding up 58976 but rounding down 17% . That gives us 9000 .
Since 17 is closer to 15 than 20 – we can assume the final answer is closer to 9000 than to 12000 . The range between the two is still a big 3000 and we need to get somewhat closer for this approximation to be useful . How close though ? 17 is about one third the way from 15 and two thirds the way from 20 . So what does that tell us – it should be about one third the way between the answers for 15% and 20% which is 1000 (out of the range of 3000 between the two extremes we found) . Consequently the answer is 9000 + 1000 = 10000 !
The math is rarely the limiting factor in our ability to approximate . Worst case you can always go find a calculator and get the math work .
The limiting factor often is in how well we define the problem and it’s context . For example – if the math problem above was about paying taxes , IRS wouldn’t take it kindly if you approximated the math and paid less 🙂
Breaking down the problem into peace parts and making sure there is minimal overlap is where we usually struggle .
Let’s say we are trying to budget for new carpet. It’s not enough to approximate square feet and multiply by a price you find online – there are other issues at play like labor and transportation , cost of removal , sales tax etc . Then there is the issue of knowing how close your budget needs to be to what you will actually spend which is very specific to your situation . As a young immigrant couple of decades ago – I needed to know within 5% . Now I only need to know within say 20-% .
At work, this takes a whole another level of complexity . At its simplest level – we tend to over engineer for the short term , and completely under engineer for the long term . I have erred on this myself countless times . As numbers get bigger and the time frame increases – it gets really hard to make good approximations .
For example – a day has 24 hours , which is 86400 seconds . That means a week has 604800 seconds . So how many years will it take to spend 1 million seconds ? How about 1 billion seconds ? You will probably be surprised at how close you will approximate million versus billion .
It’s Sunday evening now and I am staring at a massive spreadsheet figuring out plans for 2023 – and making judgments on where I absolutely need bottoms up calculations with high accuracy and where a directionally correct top down approximation is perfectly fine .
The good thing is that my two wing men are standing by to help in whatever way they can 🙂