Double, Double,….
Surely if you’ve searched for this post using “mathematics” as a topic, you’ve heard the story about doubling a grain of rice every day for a month. With students, I would substitute money which is a lot more engaging to an eighth grader for whom rice is something that Chinese food sits upon. In case you need a refresher, if Mom offered you the choice of giving you an allowance of $100 a week for a month or, instead, a penny a day doubled for the month, which would be the better deal? Students think, “Hmmm, $400 sounds pretty good to me!” until I show them the simple Spreadsheet that amounts to a neat
sum of over $10 million in a 30-day month!
| Day | You Get | Total to Date |
| 1 | $0.01 | $0.01 |
| 2 | $0.02 | $0.03 |
| 3 | $0.04 | $0.07 |
| 4 | $0.08 | $0.15 |
| 5 | $0.16 | $0.31 |
| 6 | $0.32 | $0.63 |
| 7 | $0.64 | $1.27 |
| 8 | $1.28 | $2.55 |
| 9 | $2.56 | $5.11 |
| 10 | $5.12 | $10.23 |
| 11 | $10.24 | $20.47 |
| 12 | $20.48 | $40.95 |
| 13 | $40.96 | $81.91 |
| 14 | $81.92 | $163.83 |
| 15 | $163.84 | $327.67 |
| 16 | $327.68 | $655.35 |
| 17 | $655.36 | $1,310.71 |
| 18 | $1,310.72 | $2,621.43 |
| 19 | $2,621.44 | $5,242.87 |
| 20 | $5,242.88 | $10,485.75 |
| 21 | $10,485.76 | $20,971.51 |
| 22 | $20,971.52 | $41,943.03 |
| 23 | $41,943.04 | $83,886.07 |
| 24 | $83,886.08 | $167,772.15 |
| 25 | $167,772.16 | $335,544.31 |
| 26 | $335,544.32 | $671,088.63 |
| 27 | $671,088.64 | $1,342,177.27 |
| 28 | $1,342,177.28 | $2,684,354.55 |
| 29 | $2,684,354.56 | $5,368,709.11 |
| 30 | $5,368,709.12 | $10,737,418.23 |
I let some time pass before I pose the next question: A sheet of newspaper is .003 inch thick. I place one sheet on the floor. I then double the stack by placing another on top (I now have two sheets.) I double a second time, placing 2 sheets on top of the 2 already there, for a total of 4 sheets. After I do this 10 times, I have a stack about 3 inches high. How high is the stack after doubling 50 times?
Many students will think, “Hey, this is a simple proportion problem. Let’s see, 10:3 as 50:x” and they arrive at the answer “15 inches!” Other guesses include 3 feet, 5 feet, maybe 10 feet. The answer surprises most, but when you see it in a calculator, it is all clear. Scroll down for the Ah Ha moment:
This is a lesson in exponential expansion.
First, determine how many sheets of newspaper there will be: 2^50
Secondly, multiply that by the thickness (.003”) of each.
Then use dimensional analysis to convert to miles and the answer is:
Solution: (2^50) (.003”) (1 ft) (1 mile)/(12″) (5280′)= 53,309,654 miles
or, more than half the distance from the earth to the sun!
