Department of Mathematics, Statistics
and Computer Science
Wim Ruitenburg's Spring 2010 MATH 1300-101
Money from chapter 10
Despite all the money talk, chapter 10 is really about exponential growth, with
examples from money management.
From the chapter we can learn:
- Increasing an amount like 200 by 7 percent is essentially the same as
multiplying 200 by (1 + 7/100) or by 107/100 or by 1.07, and the result equals
214.
Decreasing an amount like 200 by 7 percent is essentially the same as
multiplying 200 by (1 - 7/100) or by 93/100 or by 0.93, and the result equals
186.
- Increasing an amount like 300 by 5 percent and then reducing the new
amount by 5 percent is NOT neutral.
For first we multiply 300 by 1.05 to get 315, and then we multiply 315 by 0.95
to get as final result 299.25.
- Compound interest is very powerful.
When we start with just 400 (dollars, say), and increase the amount by 3
percent each year for 5 years, the new amount equals 400 * 1.03 * 1.03 * 1.03 *
1.03 * 1.03, also written 400 * (1.03 ^ 5), which equals approximately
463.71.
However, if we have 400 at 3 percent each year for 200 years, the new
amount equals 400 * (1.03 ^ 200), which equals approximately 147742.33.
- Earning interest of 24 percent for one year is not the same as earning
interest of 2 percent a month for one year.
Interest of 24 percent for one year means multiplying your original amount by
only 1.24.
However, earning compound interest at 2 percent a month for one year means
multiplying you original amount by 1.02 ^ 12, which equals approximately 1.268.
We spend a lot of time on talking about geometric sequences.
For example, compute the sum S of 10 + 20 + 40 + 80 + 160 + 320 +640.
This sequence has start value 10 and (growth) ratio 2.
We write down the sequence for 2S, which equals 20 + 40 + 80 + 160 + 320 + 640
+ 1280.
Subtraction and canceling equal terms gives S - 2S = 10 - 1280.
So (1-2)S = 10 - 1280, thus S = (10 - 1280) / (1-2) = -1270/(-1) = 1270.
As another example we compute the sum T of 4 + 4 * 1.5 + 4 * (1.5)^2 + 4 *
(1.5)^3 + 4 * (1.5)^4.
This sequence has start value 4 and (growth) ratio 1.5.
We write down the sequence for 1.5S, which equals 4 * 1.5 + 4 * (1.5)^2 + 4 *
(1.5)^3 + 4 * (1.5)^4 + 4 * (1.5)^5.
Subtraction and canceling equal terms gives T - 1.5T = 4 - 4 * (1.5)^5.
So (1-1.5)T = 4 - 4 * (1.5)^5 = 4 * (1 - (1.5)^5).
So T = 4(1 - (1.5)^5) / (1-1.5) = 4 * ((1.5)^5 - 1) / (1.5-1) = 4 * (7.59375 -
1) / 0.5 = 8 * 6.59375 = 52.75.
Last updated: February 2010
Comments and suggestions: Email wimr@mscs.mu.edu