Ensuring Accurate Timecard Totals
The riddle of the missing dollar
By Jeff Krueger
Redcort Software
Time clock software programs are designed to accurately record employee hours over a given payroll period and display these records clearly and automatically. What can be more disconcerting to a new user than to see an occasional timecard that appears to have a one or two minute difference between daily hours reported as worked and the total hours worked for the period?
The answer to this common question is much like the old ‘Missing Dollar’ riddle. As the riddle goes, three friends check into a motel for the night and the clerk tells them the bill is $30. Each pays the clerk $10 and goes to their room. A few minutes later, the clerk realizes he has made an error and overcharged the trio by $5. He asks the bellhop to return $5 to the 3 friends who had just checked in. The bellhop reasons that the three friends would have a tough time dividing $5 evenly among them; so he decides to tell them the clerk made a mistake of only $3, giving a dollar back to each of the friends while pocketing the leftover $2. Now, each of the three friends gets a dollar back, thus they each paid $9 for the room which is a total of $27 for the night. Adding this to the $2 the bellhop pocketed accounts for $29 dollars. So, where did the other dollar go?
The reason this popular riddle works so well is the numerical information is presented in such a way that it looks obvious that a dollar is mysteriously unaccounted for. The reality is that the 3 men paid $27 ($9 each) for a room the hotel only charged them $25 for. The two extra dollars went in the bellman’s dishonest pocket!
Similarly, when new users discover ‘missing minutes’ they are often surprised (and skeptical) to be told the minutes only appear to be missing. While many are tempted to say, ‘Oh well, there must be a rounding error’ that is not the source of the difference either. Very much like the Missing Dollar riddle, the solution to the missing minutes is that the numerical information is presented in such a way that it only appears like some time is missing. Unlike the riddle, Virtual TimeClock is not performing intentional deception; it’s a side effect of how numbers are displayed in decimal format.
I can see you’re not easily convinced so let’s use my favorite example:
Consider an employee who works a 20-minute shift three times a day. How long does the employee work each day? “That’s easy” you say confidently, “20 minutes times 3 equals 60 minutes or one hour per day”.
Now, let’s look at his timecard report:
Start Stop Hours Worked
08:00 AM 08:20 AM .33
12:00 PM 12:20 PM .33
05:00 PM 05:20 PM .33
------
Total Hours 1.0
Everyone knows that .33 + .33 + .33 = .99. So where’s the missing minute? The answer is in the presentation. Well, presentation and a little fact you learned in grade school but have long since forgotten. You see, 20 minutes in decimal form isn't really .33 at all. In realty 20/60 = .3333333 to infinity. In grade school we all agreed that putting a line over the last 3 meant ‘and on to infinity’. This seemed all terribly academic at the time so we forgot all this and just use .33 to represent a third of something when in fact we all know it’s really a bit more than .33.
So then when we all agree that if we carried the 3s out to infinity, adding .3333333 (to infinity) three times gives us .9999999999 to infinity. In other words, we get an answer really, really close to 1 but not exactly 1. This seems ridiculous to me as anyone knows that a third and a third and a third equals exactly one. Why can one half an hour equal exactly .5 and yet a third of an hour have no accurate representation in the decimal world? I'm not a mathematician so I can’t give you the scholarly reason why our simple division is so complicated. I do think these smart guys who invented math made up infinity so there’s a place we cannot go to find the answer. The important point here is that as soon as we learned division in grade school, we learned that this was the way life worked with decimal numbers.
So then, coming back to the mundane world of timesheets, this means that even if we used hugely long decimals in our timecard reports, we’d still never be able to use decimal math to exactly say 20 minutes times 3 = 60 minutes. So, in the real world of ‘close enough’ we show 20 minutes simply as .33 hours and never give it a thought until 3 twenty minute shifts fail to add up to 1 whole hour!
The good news is that the Virtual TimeClock software records every time and does all its internal calculations using minutes rather than decimals. Times are always totaled for the exact number of minutes the employee worked. When Virtual TimeClock calculates a weekly or period timecard for you in decimal format, it sums the total minutes worked, divides by 60, and then shows you the decimal approximation with two digits of precision.
Using time and minutes for totaling rather than decimal math guarantees that Virtual TimeClock timesheets accurately report every minute an employee works!
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