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