# The value of an industrial machine has a decay factor of 0.75 per year. After six years, the machine is worth 7500. What was the original value of the machine?

Jul 21, 2016

I found: $30 , 720 , 000$ BUT I used a cumulative effect impacting $0.75$ upon the reduced values of each year.

#### Explanation:

Call the original value $x$;
after the first year the value will become:
$x - 0.75 x$
after the second year:
$\left(x - 0.75 x\right) - 0.75 \left(x - 0.75 x\right) = \left(x - 0.75 x\right) \left(1 - 0.75\right)$
after the third year:
$\left[\left(x - 0.75 x\right) \left(1 - 0.75\right)\right] - 0.75 \left[\left(x - 0.75 x\right) \left(1 - 0.75\right)\right] = \left(x - 0.75 x\right) \left(1 - 0.75\right) \left(1 - 0.75\right) = \left(x - 0.75 x\right) {\left(1 - 0.75\right)}^{2}$
so basically we have that after the nth year:

$\left(x - 0.75 x\right) {\left(1 - 0.75\right)}^{n - 1}$

in 6 years:
$\left(x - 0.75 x\right) {\left(1 - 0.75\right)}^{5} = 7500$
$0.25 x = \frac{7500}{1 - 0.75} ^ 5$
so:
$x = 30 , 720 , 000$