Intermath | Workshop Support

Spreading Rumors
Problem Statement
A rumor starts by someone telling the rumor to two people on May 1. Each of those two people are responsible for telling the rumor to two others on the next day (May 2).At this time seven people know the rumor. On May 3, the four people who heard the rumor on May 2 must each tell two more people. It is important for 8000 people to hear the rumor by May 14. Is this outcome likely to occur if the pattern for spreading the rumor continues as started?

On what day will approximately one-half of the 8000 have heard the rumor? On what day will 256 new people be told the rumor? If the rumor process continues until May 20,how many new people will hear the rumor on that day?

If the rumor process continues for n days, how many new people will be told the rumor on the nth day? What will be the total number of people who know the rumor on that day?

Problem setup

Decide how much the number of people who know the rumor will increase as each day goes by until at least 8000 people have heard the rumor.  Can this happen before the 20th day?  Each day each person who knows the rumor will tell two more.  On what day will about 4000 people know the rumor?  What is the formula for determining the number of people who know the rumor on a given day?


Plans to Solve/Investigate the Problem

I will start by making a diagram to show the first person telling 2 more, those 2 each telling 2 more, and so on.  Put the date beside each progression, beginning with May 1.  The diagram quickly becomes a pattern where each amount is doubling day by day. 


Investigation/Exploration of the Problem

May 1 = 1

May 2 = 2

May 3 = 4

May 4 = 8

May 5 = 16 until May 14 = 9,192, over the 8,000 person mark by the deadline of May 14.

May 13 = 4096, over the 4,000 mark (the 13th day).

256 people will have heard the rumor by May 9, day 9.

By May 20 588,288 people will have heard the rumor.

The pattern here is the amount of people on a certain day equals 2 times 2 to the (day - 1) power.  For example, day 20 = 2 * 2 to the 19th power.  The number of new people would be the number of people on the (nth -1) day * 2.


Extensions of the Problem

Outside of math, use this to emphasize character ed - not to spread rumors about each other!