$1000 grows 5% every 2 years. Write the model. Step by Step

Step by step

The question

Question: $1000 grows 5% every 2 years.

Write the model.

1 Find the start

t is the time in years.

We call the amount A.

A = amount, t = years

At the start no time has passed: t = 0.

Then the amount is $1000.

That is the start.

start, t = 0: A = $1000

2 Find the factor

Every 2 years, all of the amount stays: that is 100%.

And 5% more is added.

So it becomes 105% of what it was.

100% + 5% = 105%

Percent means out of 100.

So 105% is 105/100: 1.05.

105% = 105100 = 1.05

So the factor is 1.05: every 2 years, we multiply by 1.05.

factor: × 1.05 every 2 years

3 Make a small table

A small table: t = 0, 2, 4: one step is 2 years.

Each step, we multiply by 1.05: 1000 times 1.05 is 1050.

Times 1.05 again is 1102.5.

After two steps, 4 years, we multiplied by 1.05 two times.

Two times is the power 2: 1.05 squared.

t = 4: 1000 × 1.05 × 1.05 = 1000 × 1.05²

4 Count the steps

One step takes 2 years.

At t = 2: 2/2 is 1 step.

t = 2: 22 = 1 step

At t = 4: 4/2 is 2 steps.

Like the table.

t = 4: 42 = 2 steps

So in t years there are t/2 steps.

We divide the time by 2, the length of one step.

That is the power.

t: t2 steps → × 1.05^t2

5 Write the model

Start times the factor, to the power t/2.

So the model is A = 1000 times 1.05 to the power t/2.

A = 1000 × 1.05^t2

Careful!

Not 0.05.

Times 0.05 would leave only a small part.

We keep all of it and add 5%: times 1.05.

Careful: not A = 1000 × 0.05^t2 ✗

6 Check

Let's check the first 2 years the story way.

5% of 1000 is 0.05 times 1000: 50.

5% of 1000 = 0.05 × 1000 = 50

It gets 50: 1000 plus 50 is 1050.

1000 + 50 = 1050

The model at t = 2 gives 1000 times 1.05: 1050 too.

It works!

model, t = 2: 1000 × 1.05²² = 1050

Let's look back.

First: find the start.

Then: find the factor.

Then: make a small table.

Then: count the steps.

Then: write the model.

Last: check.

Answer