A brand new music system is worth $18,000 in 2001. If the music system's value decreases by 6% every year, what will its value be in 2003?

2 Answers
Feb 8, 2018

15904.8

Explanation:

You can use the formula:

y=i*(1+-x)^t

"i is the initial value."

"x is the change. Use negative for decay, and positive for growth." (+-)

"t is time."

Plug in:

18000*(1-0.06)^2,
(2 because 2003-2001 is 2).

Solve to get 15904.8.

Have a nice day!

Feb 8, 2018

The value of the music system in 2003 is "$15,904.80".

Explanation:

The function we will use is

A = P(1+r)^t

where

A= the final value of the music system
P= the initial value of the music system
r= the rate of growth per year
t= the number of years

(Since we're dealing with decay instead of growth, r will be negative.)

To solve for A, we plug in the other values we know:

A=18000(1-0.06)^2
color(white)(A)=18000(0.94)^2
color(white)(A)=18000(0.8836)
color(white)(A)=15904.8

So the depreciated value of the music system is "$15,904.80".

Bonus:

Why does this formula work? Let's think of how much the music system would be worth after 1 year (call this A_1). This is 6% less than its initial value of $18,000 (a.k.a. 94% of $18,000):

A_1=0.94 xx 18000

The value of the music system after a 2nd year (A_2) is then 94% of its value after one year -- that is, 94% of A_1:

A_2=0.94 xx A_1
color(white)(A_2)=0.94 xx 0.94 xx 18000
color(white)(A_2)=(0.94)^ 2 xx 18000

It's not hard to see that, if we extend this to a 3rd year, the value for A_3 will be (0.94)^3xx18000, and if we generalize it to t years, the depreciated value is (0.94)^txx18000. Since 0.94 was 1-0.06 (a.k.a. 1 minus the rate of decay) and 18000 was our initial value P, this gets us back to the form of the equation we started with:

A=(1-r)^txxP