6.3 More short investigations
Here are two short investigations involving large numbers for you to try. Please do not turn to the comments on these exercises until you have made some notes and had a go yourself.
Exercise 13: Where did I come from?
Family trees can be fascinating things. On the principle that everyone has two biological parents, extending a family tree back in time shows that everyone has an increasing number of ancestors the further back you go. So it seems that every generation of your direct ancestors must have contained twice the number of people as the generation which followed it: you have two parents, four grandparents, eight great-grandparents, 16 great-great-grandparents and so on.
If you went back, say 30 generations, how many ancestors should each person have? And where did they all come from.
Answer
Using the pattern established in the question gives a pattern similar to that in the previous exercise.
Generations back | 1 | 2 | 3 | 4 | 5 | ... | 30 |
---|---|---|---|---|---|---|---|
Number of ancestors | 2 | 4=2^{2} | 8=2^{3} | 16=2^{4} | 32=2^{5} | ... | 2^{30} |
Suppose one generation is approximately 25 years. Then 30 generations is about 750 years.
And 2^{30} is over 10^{9}, or one thousand million. It does not seem possible that you had that many ancestors in the thirteenth century, since the population of the entire world was probably only about four hundred million at that time.
There must be something wrong with the assumptions in the question. What is wrong is the implicit assumption that no one appears more than once in each generation in a family tree. For example, it is quite possible for one of your mother's grandmothers to be the same person as one of your father's grandmothers, and the further back you go the more likely this sort of double counting becomes.
Exercise 14: A colossal cash cache
A newspaper report in 1994 revealed that the former Philippines ruler, Ferdinand Marcos, had stashed away around 1240 tons of gold during his 21-year dictatorship. At that time it was reputedly sitting in an airport warehouse in Switzerland. A question that may come to mind on reading the article is to see just how much 1240 tons of gold is worth: thousands of pounds, or millions or billions? Spend a few minutes deciding what further information you would require to answer this question before reading on.
There is an issue here about where such information might come from, because not everyone has a comprehensive reference library on their doorstep. Firstly, what is the price of gold? The financial page of the same newspaper from which the article was taken stated that gold was selling at $391.25 per ounce. You will also need to know the rate of conversion between ounces and tons. A dictionary or an encyclopaedia could supply the information that there are 16 ounces in one pound and 2240 pounds in 1 ton.
Finally, the rate of conversion between dollars and pounds sterling (again from the financial page of the newspaper) was given as 1.497 dollars to the pound sterling.
Use the information provided above to make an estimate of Ferdinand Marcos’ purported fortune in 1994.
Answer
Using the figures given in the exercise:
This was worth
1240 × 2240 × 16 × 391.25 dollars
or
1240 × 2240 × 16 × 391.25 ÷ 1.497 pounds sterling.
The calculator produced an answer of £1.16 × 10^{10} (to 3 s.f.), which is nearly £12 billion.
(In fact, you will find that the price of gold is usually given per troy ounce, and 1 troy ounce=1.01 ordinary (or avoirdupois) ounces. Using this, the calculation above comes to a little less: £1.15 × 10^{10}).