A village has 1200 people and its population decreases 5% each year… Step by Step

Step by step

The question

Question: A village has 1200 people and its population decreases 5% each year.

Write the model.

1 Find the start

t is the time in years.

We call the number of people P.

P = people, t = years

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

Then there are 1200 people.

That is the start.

start, t = 0: P = 1200

2 Find the factor

Each year, 5% of the people is lost.

What is left is 100% take away 5%: 95%.

100% − 5% = 95%

Percent means out of 100.

So 95% is 95/100: 0.95.

95% = 95100 = 0.95

So the factor is 0.95: each year, we multiply by 0.95.

factor: × 0.95 each year

3 Make a small table

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

Each step, we multiply by 0.95: 1200 times 0.95 is 1140.

Times 0.95 again is 1083.

After 2 years, we multiplied by 0.95 two times.

Two times is the power 2: 0.95 squared.

t = 2: 1200 × 0.95 × 0.95 = 1200 × 0.95²

After t years, we multiply t times.

So the power is t.

after t years: × 0.95, t times = × 0.95^t

4 Write the model

Start times the factor, to the power t.

So the model is P = 1200 times 0.95 to the power t.

P = 1200 × 0.95^t

Careful!

Not 0.05.

That is the part that is lost.

We multiply by the part that is left: 0.95.

Careful: not P = 1200 × 0.05^t ✗

5 Check

Let's check the first year the story way.

5% of 1200 is 0.05 times 1200: 60.

5% of 1200 = 0.05 × 1200 = 60

It loses 60: 1200 minus 60 is 1140.

1200 − 60 = 1140

The model at t = 1 gives 1200 times 0.95: 1140 too.

It works!

model, t = 1: 1200 × 0.95¹ = 1140

Let's look back.

First: find the start.

Then: find the factor.

Then: make a small table.

Then: write the model.

Last: check.

Answer