Step by step
The question
Question: A colony has 500 bacteria and grows 8% each hour.
How many bacteria after 6 hours?
1 Find the start and the factor
The start is 500.
Each hour it keeps all of it, 100%, and gets 8% more.
So it becomes 108% of what it was.
108% is 108/100: 1.08.
That is the factor.
start: a = 500
100% + 8% = 108% = 1.08
factor: b = 1.08
2 Write the rule
x is the number of hours.
Each hour we multiply by 1.08 again.
So y = 500 × 1.08ˣ.
y = 500 × 1.08x
Careful!
The 8% is taken of the NEW amount every hour, not always of 500.
Careful: not 500 + 6 × 40 = 740 ✗
3 Put in x = 6
After 6 hours, x = 6.
y = 500 × 1.086
1.08 to the power 6 is 1.586874.
1.086 = 1.586874
So we multiply 500 by 1.586874.
= 500 × 1.586874
That is 793.437161, about 793.
≈ 793
4 Check
Let's check by going back: 793.437161 divided by 1.08 to the power 6 gives the start, 500.
It works!
793.437161 ÷ 1.086 = 500
Let's look back.
First: find the start and the factor.
Then: write the rule.
Then: put in x = 6.
Last: check.
Answer
≈ 793