Step by step
The question
Question: A car is worth $20000 and loses 15% of its value each year.
Write the model.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Find the start
t is the time in years.
We call the value V.
At the start no time has passed: t = 0.
Then the value is $20000.
That is the start.
2 Find the factor
Each year, 15% of the value is lost.
What is left is 100% take away 15%: 85%.
Percent means out of 100.
So 85% is 85/100: 0.85.
So the factor is 0.85: each year, we multiply by 0.85.
3 Make a small table
A small table: t = 0, 1, 2.
Each step, we multiply by 0.85: 20000 times 0.85 is 17000.
Times 0.85 again is 14450.
After 2 years, we multiplied by 0.85 two times.
Two times is the power 2: 0.85 squared.
After t years, we multiply t times.
So the power is t.
4 Write the model
Start times the factor, to the power t.
So the model is V = 20000 times 0.85 to the power t.
Careful!
Not 0.15.
That is the part that is lost.
We multiply by the part that is left: 0.85.
The right way: V = 20000 × 0.85t.
5 Check
Let's check the first year the story way.
15% of 20000 is 0.15 times 20000: 3000.
It loses 3000: 20000 minus 3000 is 17000.
The model at t = 1 gives 20000 times 0.85: 17000 too.
It works!
6 Pick the choice
Our answer is V = 20000 × 0.85t.
That is choice B.
7 Why not the others
A is V = 20000(0.15)ᵗ.
Wrong: 0.15 is the 15% lost: the car keeps 85%.
C is V = 20000(1.15)ᵗ.
Wrong: 1.15 is 15% more each year: the car loses value.
D is V = 20000 − 0.15t.
Wrong: takes away 0.15 dollars a year, not 15%.
Let's look back.
First: find the start.
Then: find the factor.
Then: make a small table.
Then: write the model.
Then: check.
Then: pick the choice.
Last: why not the others.