   SEARCH HOME Math Central Quandaries & Queries  Question from Kenneth: Hello: How is the average determined for the following amount if it earns compound interest? Here are my calculations, but I'm not certain that they are correct: Initial amount invested $500.00:$500.00 @ 5% = $525.00.$525.00 @ -2% = $514.50.$14.50/$500.00 = 2.9% divided by 2 equals an average of 1.45%. If this average, 1.45%, is correct, how can I use 5% and -2% to determine the same average? Is it possible? I thank you for your reply. Kenneth,$500.00 @ 5% yields

$500.00 + 0.05 ×$500.00 = (1 + 0.5) × $500.00 = 1.05 ×$500 = $525.00$525.00 @ -2% yields

$525.00 - 0.02 ×$525.00 = (1 - 0.02) × $525.00 = 0.98 ×$525.00 = $514.50 Hence 1.05 × 0.98 ×$500.00 = $514.50 or 1.029 ×$500.00 = $514.50. Thus there is a 2.9% increase over the two time periods or an average of 1.45% per time period. Hence the 2.9% comes from (1 + 0.05) × (1 - 0.02). I hope this helps, Penny Kenneth wrote back Hello Penny: In your reply, you indicated that$500.00 + 0.05 x $500.00 = (1 + 0.05) x$500.00.

How does $500.00 + 0.05 x$500.00 equal (1 + 0.05) x $500.00? In other words, how does$500.00 + 0.05 = (1 + 0.05)?

In the expression $500.00 + 0.05 x$500.00 you need to do the multiplication first. Since 0.05 x $500.00 =$25.00 the expression is

$500.00 + 0.05 x$500.00 = $500.00 +$25.00 = $525.00 Rather than calculate$500.00 + 0.05 x $500.00 this way I wanted to use the distributive law. I'm going to illustrate with samller numbers. 3 × 2 + 4 × 2 = 6 + 8 = 14 But this can also be written 3 × 2 + 4 × 2 =(3 + 4) × 2 = 7 × 2 = 14. In the expression 3 × 2 + 4 × 2 there is a factor of 2 which is common to both terms 3 × 2 and 4 × 2. This allows me to factor out the 2 and write 3 × 2 + 4 × 2 as (3 + 4) × 2. In my earlier expression$500.00 + 0.05 x $500.00 think of the first$500.00 as 1 × %500.00 then

$500.00 + 0.05 x$500.00

is

1 × $500.00 + 0.05 x$500.00

and $500.00 is a common factor so 1 ×$500.00 + 0.05 x $500.00 = (1 + 0.05) ×$500.00

Penny     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.