Wednesday, December 09, 2009

Will do math for chocolate

I don’t usually pay too much attention to the conversations in my office that are in German. My office mates both have many people coming in to talk to them and are both German, so speak in German with their German visitors. This particular conversation caught my attention primarily because it repeatedly used the work ‘Schokolade” one of the few German words I have no trouble with. Also there were four people involved. Eventually one of my office mates said, “Ask Dan, he is very interested in chocolate!”

So the conversation was about a simple demography puzzle, and my boss had offered a chocolate bar to whoever solved it first.
The problem is this: 100% of a population were alive at the beginning of a year. 60% were alive at the end of the year. Assuming that the mortality rate is constant throughout the year, what percentage of individuals were alive at the exact middle of the year?

Note that a constant mortality rate does not mean that the same absolute number of individuals die each day, but rather than the same proportion of those starting the day alive end the day dead.
Motivated by chocolate, and using one simple bit of algebra, I secured the Intense Orange Lindt dark chocolate bar.

So I can’t hang a chocolate bar in the comments section, but I can promise kudos if you can tell me what portion of individuals were alive after six months, and how you got that answer.


gml said...

After 6 months, 77.46% were alive.

1.00 X
---- = -
X .6

The population decreases by the same proportion in each 6 month period. It is like a problem in radioactive half-lives. GML

gml said...

The right half of the equation did not show up. I will put it it words, instead: 1.00 is to the unknown amount at 6 months as that amount (X) is to 0.60. GML

jte said...

Start with 100. Then


That is, (100) multiplied by (x percent mortality rate) multiplied by (365 days) = 60

So: 100x = 60/365

So: x = (60/365)/100 = 0.00164384 = 0.164384%

Apply the rate to successive days, so that the population on day 2 = population day 1 minus (pop on day 1 multiplied by 0.00164384). For day 3, replace day 1 above with day 2, and so on.

Half the year is day 182.5. The population on days 182 and 183 are 74.2 and 74.1, respectively.

Only problem is, when I run this for a full 365 days, I don't end up with 60, I end up with 55. So something in there is wrong. Drats. That's okay, I don't like fruit flavors in my chocolate anyway.

Dan Levitis said...

Quite right. Or to put it in demographer talk, survivorships multiply. This means that if x proportion of individuals survive through the first half of the year, and x proportion of those survive through the second half of the year, then x*x proportion are alive at the end of the year. If x*x=.6 then x=.6^.5 (x is the square root of .6) which is .7746.

jte said...

Ah, the difference between discrete and continuous numbers. Or just knowing how to do it the right way. Never was my strong suit.