A house is worth $250000 and its value grows 6% each year. Write the… Step by Step

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The question

Question: A house is worth $250000 and its value grows 6% each year.

Write the model.

1 Find the start

t is the time in years.

We call the value V.

V = value, t = years

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

Then the value is $250000.

That is the start.

start, t = 0: V = $250000

2 Find the factor

Each year, all of the value stays: that is 100%.

And 6% more is added.

So it becomes 106% of what it was.

100% + 6% = 106%

Percent means out of 100.

So 106% is 106/100: 1.06.

106% = 106100 = 1.06

So the factor is 1.06: each year, we multiply by 1.06.

factor: × 1.06 each year

3 Make a small table

A small table: t = 0, 1, 2.

Each step, we multiply by 1.06: 250000 times 1.06 is 265000.

Times 1.06 again is 280900.

After 2 years, we multiplied by 1.06 two times.

Two times is the power 2: 1.06 squared.

t = 2: 250000 × 1.06 × 1.06 = 250000 × 1.06²

After t years, we multiply t times.

So the power is t.

after t years: × 1.06, t times = × 1.06^t

4 Write the model

Start times the factor, to the power t.

So the model is V = 250000 times 1.06 to the power t.

V = 250000 × 1.06^t

Careful!

Not 0.06.

Times 0.06 would leave only a small part.

We keep all of it and add 6%: times 1.06.

Careful: not V = 250000 × 0.06^t ✗

5 Check

Let's check the first year the story way.

6% of 250000 is 0.06 times 250000: 15000.

6% of 250000 = 0.06 × 250000 = 15000

It gets 15000: 250000 plus 15000 is 265000.

250000 + 15000 = 265000

The model at t = 1 gives 250000 times 1.06: 265000 too.

It works!

model, t = 1: 250000 × 1.06¹ = 265000

Let's look back.

First: find the start.

Then: find the factor.

Then: make a small table.

Then: write the model.

Last: check.

Answer