P = 2500(1.04)^t is the number of people in a town t years from now… Easy

The question

Question: P = 2500 × (1.04)ᵗ is the number of people in a town t years from now.

What does 1.04 mean?

We write the equation big, so we can label it.

1 What the letters mean

P is the number of people.

t is the number of years.

It is the small raised number, the power: how many times we multiply by 1.04.

2 Make a small table

To see what each number does, we try a few years: 0, 1 and 2.

0 is where it starts.

t = 0: no years have passed, so we multiply by 1.04 zero times.

That is why 1.04 to the power 0 is 1.

2500 times 1 is 2500.

t = 1: 1.04 to the power 1 is just 1.04.

2500 times 1.04 is 2600.

t = 2: 1.04 squared is 1.04 times 1.04.

2500 times 1.04 is 2600.

Times 1.04 again: 2600 times 1.04 is 2704.

Look at the values of P: 2500, 2600, 2704.

Each time t goes up 1, P is multiplied by 1.04.

The same number every time.

3 Find the numbers in the table

Now we write the equation again, and label its numbers.

2500 is P at t = 0: the first number in the table.

So 2500 is the number of people at the start.

1.04 is what we multiply by, once for each year.

That is the power t: it counts the years.

4 Write it as a percent

A percent is out of 100: one whole is 100%.

So we multiply by 100.

1.04 times 100 is 104.

So 1.04 is 104%.

104% is all of it, 100%, plus 4% more.

Careful!

104% is the new amount: the 100% it had, plus 4%.

The growth is only the part above 100%: 4%.

5 Say it in words

So 1.04 means: the number of people goes up by 4% each year.

6 Check

Let's check the 4%.

4% is 4/100, that is 0.04.

4% of 2500 is 0.04 times 2500: 100.

2500 plus 100 is 2600: the same as the table at t = 1.

It works!

Let's look back.

First: what the letters mean.

Then: make a small table.

Then: find the numbers in the table.

Then: write it as a percent.

Then: say it in words.

Last: check.

▶ Watch it solved

V = 20000(0.85)^t is the value, in dollars, of a car t years after… Medium

The question

Question: V = 20000 × (0.85)ᵗ is the value, in dollars, of a car t years after it is bought.

What does 20000 mean?

We write the equation big, so we can label it.

1 What the letters mean

V is the value of a car, in dollars.

t is the number of years.

It is the small raised number, the power: how many times we multiply by 0.85.

2 Make a small table

To see what each number does, we try a few years: 0, 1 and 2.

0 is where it starts.

t = 0: no years have passed, so we multiply by 0.85 zero times.

That is why 0.85 to the power 0 is 1.

20000 times 1 is 20000.

t = 1: 0.85 to the power 1 is just 0.85.

20000 times 0.85 is 17000.

t = 2: 0.85 squared is 0.85 times 0.85.

20000 times 0.85 is 17000.

Times 0.85 again: 17000 times 0.85 is 14450.

Look at the values of V: 20000, 17000, 14450.

Each time t goes up 1, V is multiplied by 0.85.

The same number every time.

3 Find the numbers in the table

Now we write the equation again, and label its numbers.

0.85 is what we multiply by, once for each year.

That is the power t: it counts the years.

20000 is V at t = 0: the first number in the table.

So 20000 is the value of a car at the start.

Careful!

20000 is not added each year.

It is the amount at t = 0, before any multiplying.

4 Say it in words

So 20000 means: the value of a car is 20000 dollars at t = 0.

That is the start.

5 Check

Let's check by going back one year, from t = 1 to t = 0.

To get to t = 1, we multiplied by 0.85.

To go back, we divide by 0.85.

17000 divided by 0.85 is 20000: the start.

It works!

Let's look back.

First: what the letters mean.

Then: make a small table.

Then: find the numbers in the table.

Then: say it in words.

Last: check.

▶ Watch it solved

P = 10000(1.025)^t is the number of people in a city t years from… Harder

The question

Question: P = 10000 × (1.025)ᵗ is the number of people in a city t years from now.

What does 1.025 mean?

We write the equation big, so we can label it.

1 What the letters mean

P is the number of people.

t is the number of years.

It is the small raised number, the power: how many times we multiply by 1.025.

2 Make a small table

To see what each number does, we try a few years: 0, 1 and 2.

0 is where it starts.

t = 0: no years have passed, so we multiply by 1.025 zero times.

That is why 1.025 to the power 0 is 1.

10000 times 1 is 10000.

t = 1: 1.025 to the power 1 is just 1.025.

10000 times 1.025 is 10250.

t = 2: 1.025 squared is 1.025 times 1.025.

10000 times 1.025 is 10250.

Times 1.025 again: 10250 times 1.025 is 10506.25.

Look at the values of P: 10000, 10250, 10506.25.

Each time t goes up 1, P is multiplied by 1.025.

The same number every time.

3 Find the numbers in the table

Now we write the equation again, and label its numbers.

10000 is P at t = 0: the first number in the table.

So 10000 is the number of people at the start.

1.025 is what we multiply by, once for each year.

That is the power t: it counts the years.

4 Write it as a percent

A percent is out of 100: one whole is 100%.

So we multiply by 100.

1.025 times 100 is 102.5.

So 1.025 is 102.5%.

102.5% is all of it, 100%, plus 2.5% more.

Careful!

102.5% is the new amount: the 100% it had, plus 2.5%.

The growth is only the part above 100%: 2.5%.

5 Say it in words

So 1.025 means: the number of people goes up by 2.5% each year.

6 Check

Let's check the 2.5%.

2.5% is 2.5/100, that is 0.025.

2.5% of 10000 is 0.025 times 10000: 250.

10000 plus 250 is 10250: the same as the table at t = 1.

It works!

Let's look back.

First: what the letters mean.

Then: make a small table.

Then: find the numbers in the table.

Then: write it as a percent.

Then: say it in words.

Last: check.

▶ Watch it solved