# Math Help - need help please!

Machine A can be bought for $15,000, but machine B, the automated model, will cost$40,000 to buy and install. The automated machine will reduce annual cost by $5743.50 over machine A. Either machine will have a service life of seven years. The rate of return on the extra investment required by buy machine B is 2. Originally Posted by JAMESANANE Machine A can be bought for$15,000, but machine B, the automated model, will cost $40,000 to buy and install. The automated machine will reduce annual cost by$5743.50 over machine A. Either machine will have a service life of seven years. The rate of return on the extra investment required by buy machine B is
Someone with more business saavy might approach this differently, but here goes:

The $40,000 purchase will more than pay for itself in 7 years considering the annual reduction in operating costs (7 X$5,743.50 = $40,204.50). The return on investment (ROI) given just that information would be $\frac{40204.50-40000}{40000}=.005 = .5\%$ If you figured from just the extra investment made ($40,000-$15,000=$25,000), then the ROI would be

$\frac{40204.50-25000}{40000}=.38=38\%$

3. 1. The rate of return on the extra investment is merely the extra profit you get from it each year divided by the additional initial investment. So that's 5743/25,000 = 23%

2. The formula for compounding interest is:

$
FV = PV*(1+i)^n
$

So here you have
$
7500 = 5000*(1+0.05)^n
$

or
$
1.5 = 1.05^n
$

Now solve for n. The way to do that is using logarithms. Take the log of both sides:

$
log(1.5) = log(1.05^n)
$

Recall from highschool that $log(a^b) = b \cdot log(a)$, so you have

$
log(1.5) = n \cdot log(1.05)$
, and hence
$
n = \frac {log(1.5)} {log(1.05)} = 8.3\ \text{years.}
$