Step by step
The question
Question: A town has 3000 people and grows 4% 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 number of people P.
At the start no time has passed: t = 0.
Then there are 3000 people.
That is the start.
2 Find the factor
Each year, all the people stay: that is 100%.
And 4% more is added.
So it becomes 104% of what it was.
Percent means out of 100.
So 104% is 104/100: 1.04.
So the factor is 1.04: each year, we multiply by 1.04.
3 Make a small table
A small table: t = 0, 1, 2.
Each step, we multiply by 1.04: 3000 times 1.04 is 3120.
Times 1.04 again is 3244.8.
After 2 years, we multiplied by 1.04 two times.
Two times is the power 2: 1.04 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 P = 3000 times 1.04 to the power t.
Careful!
Not 0.04.
Times 0.04 would leave only a small part.
We keep all of it and add 4%: times 1.04.
The right way: P = 3000 × 1.04t.
5 Check
Let's check the first year the story way.
4% of 3000 is 0.04 times 3000: 120.
It gets 120: 3000 plus 120 is 3120.
The model at t = 1 gives 3000 times 1.04: 3120 too.
It works!
6 Pick the choice
Our answer is P = 3000 × 1.04t.
That is choice C.
7 Why not the others
A is P = 3000(0.04)ᵗ.
Wrong: 0.04 is the 4% alone: the town would shrink.
B is P = 3000(0.96)ᵗ.
Wrong: 0.96 = 100% − 4%: that is losing 4% a year.
D is P = 3000(1.4)ᵗ.
Wrong: 1.4 means 40% more, not 4%.
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.