How To Do Math in Your Head

Recently I was on the subway, and noticed an ad for a nicotine patch. It was trying to convince people to quit smoking, by hinting at how much money they were sending buying cigarettes. The text was in a cartoon "thought balloon" to give the impression that this was what the passenger under it was thinking It ran, "One pack a day is $2.50, times 7 days in a week in $17.50. Times 52 weeks is ... Wow!"

As I read that, it occurred to me that the advertiser was counting on the readers not being able to multiple 2.5 times 7 times 52 in their heads. And, I guess, most people who saw that ad would also assume that doing that calculation was beyond what they could in their heads, while sitting in a subway.

Naturally, I felt obliged to do the calculation in my head. And now, I feel obliged to teach it to you. In fact, I'll show you not one, but two different ways to find that answers, and along the way, show the tricks (skills) used to handle every problem like that.

The first and most important rule for doing math in your head is "Look for the Tens". Anything that you can do to reduce the problem to one of dealing with 10s will help you along the way.

Which brings us to the first problem with solving this problem that way it was begun in the ad itself. They started with the simple calculation 2.50 X 365, and confused the issue by converting it to 2.50 X 7 X 52, and then tried to "simplify" it by making it 17.50 x 52. Not really much simpler, right?

Now, I think most people will recognize that 4 X 2.50 = 10, and so, if we could rewrite this as 2.50 X 4 X (something) finding the answer would be simple. All we'd have to do is figure out what that "something" is. So, let us begin. We'll created our very on week -- one that's only four days long. We'll call it a ShortWeek. In each ShortWeek, we save $10, so now all we have to do is figure out how many ShortWeeks there are in a regular year.

OK, Now let's think about a regular month. It's about 4 (regular) weeks long, with a few extra days left over. Those extra days are annoying, so we'll just ignore them (for now), and declare 4 weeks a ShortMonth. Now, since 4 x 7 is the same as 7 X 4, we've now shown that there are 7 ShortWeeks in one ShortMonth. And since each ShortWeek is worth $10, then a ShortMonth is worth $70. Of course, at this point you could just multiple 70 X 12, and get a value that's close to the number we are looking for, but I know you wouldn't be satisfied with that. We now need to find out how many ShortMonths there are in a Real year.

OK, we know that there are 52 weeks in a real year, and 4 weeks in our ShortMonth. So, what other place in our day-to-day lives do the numbers 52 and 4 turn up? How about a deck of cards. There are 52 in a deck, and 4 suits of cards --- And it's important to note that, obviously, every rank is represented in each suit. This is important because it tells us that 52 is evenly divisible by 4, which means that a year can be evenly divided into ShortMonths. But, evenly into how many ShortMonths?

The serious card players probably already know by now, but for the rest, it's the number of ranks in a suit of cards. Here's a hint -- there are ten cards counting from Ace to Ten, and then we add the Jack, Queen and King. Some might call that 13, but --- what was the first rule? ("Look for the Tens"). So, it better if we think of it as 10 + 3.

So, what have we got now? We save $2.50 per day, there are 4 days in a ShortWeek, 7 ShortWeeks in a ShortMonth, and 10 + 3 ShortMonths in a real year. Patch it together, and we get:

2.50 X 4 X 7 X (10 + 3)

We know that a ShortWeek is worth $10, so we'll just pull that off, and work with the rest:

7 X (10 + 3)

Which is the same as:

7 X 10 + 7 X 3

If you can't do that in your head, you really must have been a sleep in the third grade. (It's 70 + 21 or 91)

So, we now know that there are 91 ShortWeeks in a year, and for each one, our theoretically future ex-smoker save $10, for a total of $910.00.

Well, not exactly that, since if you recall, if your birthday is on a Tuesday one year, it'll be on a Wednesday the next (ignoring Leap Years), which means that a year is really 52 weeks plus one day, so we have to add that in, for a final Grand Total of

$912.50

But, say you don't trust those calculations and want to double check.... OK, let's try it a different way. Again, "Look for the Tens"... We want to know how much is $2.50 X 365 -- So, we already know that four days is worth $10--- What about 40 days? Clearly $100. And 400 days? $1000.

Since 400 days is just a bit more that what we are looking for (365), so we'll start there. Now we've got to subtract something from that. If we subtract 40, we're now at 360. Ok, now let's add 4 and then add one. Translated into dollars, that's $1000 - $100 + $10 + $2.50

And again we've reached the grand total of $912.50