HINTS for   Problem Set #1

__________________________________________

 

It might be a good idea to write out the first 2 problems "long-hand" just to make sure you understand what's going on.  However, it quickly becomes tedious, and you need to turn this doubling time stuff into formulae to solve real-world problems quickly (spreadsheets is another way to do it, but they can be tedious too). 

 

Think about the grains on a chessboard example from the video.  How can we calculate the number of grains to put on any square? To go from one to the next, you just double the number.  In the example, the first square had 1 grain but in problem 1 we started with 2 cockroaches.  You can start with any number.  Let's just call the number we start with N0.  In each step, we double this, or take it times 2. 

 

Step 0                     #= N0                                                                              # = N0* 20

Step 1                     #= (N0) * 2                                            # = N0* 21

Step 2                     #= (N0 * 2) *2                                     # = N0* 22

Step 3                     #= (N0 * 2 *2) *2                             # = N0* 23

. . .

so after n steps, we should have...

Step n                                                                                             # = N0* 2n

 

You have to know 2 of 3 things (#, N0, n) to solve for the 3rd.  If the unknown is up in the exponent, you have to use logs (addition and subraction are opposites, multiplication and division are opposites; the opposite of an exponent is a log).  To use a log, remember the rule...

a=bx   --->   log(a) = x log(b)      if a and b are just numbers, their logs are just numbers

 

Doubling time is easy to grasp (and the rule of 70 is very useful), but you often want to look at the growth in more detail or for a particular time.  To get to the bottom of this, think of how compound interest works.  Let's call the interest rate I.  It's usually given as a per cent, but 2 percent is 2 per 100 or 2/100 or .02.  The growth each period is the amount you start with times the interest rate (as a fraction or decimal number). The trick here is that the new number is equal to what you started with plus the growth (interest earned), which will work out to the initial investment times (1+I) multiplied by itself n times...

 

Step 0                     #= N0                                                                             

Step 1                     #= (N0)+(N0) * I                                                                    # = N0* (1+I)1

Step 2                     #= (N0+N0* I) + (N0+ N0 * I)*I                                # = N0* (1+I)2

. . .

so after n steps, we should have...

Step n                                                                                                                                 # = N0* (1+I)n

 

Notice that if the interest rate is 100%, 1+I=2 and this is just # = N0* 2n, the equation we had before!