# Population drop word problem -- help with the Algebra please

Gold Member
Every ten minutes, a tenth of the bacteria are KILLED.
At 10 minutes: a tenth of 10^6 bacteria are killed. This leaves 100,000 remaining.
At 20 minutes : a tenth of the 100,000 bacteria are killed. This leaves 10,000 remaining.
At 30 minutes: a tenth of the 10,000 are killed. This leaves 1000 remaining.

This is the process that is happening.

Now I am asked at which period will 70% of the bacteria be killed, or 30% of the bacteria remain?

The initial population of the bacteria is 10^6. I want to find 30% of 10^6. Mathematically, I want 0.3(10^6). This is one side of the equation, where I list what I am asked for.

For the other side, I express the process of decay that is happening, which I listed in words previously.
I know that I am given an initial population of 10^6, and every ten minutes, the current population drops by a tenth, or (1/10).
To express this process mathematically, I have the given population dropping by a tenth every ten minutes, or 10^6(1/10)(1/10)(1/10)..... or 10^6(1/10)^x.

So combining both of these, I have ##0.3\left(10^6\right)=10^6\left(\frac{1}{10}\right)^x##

This is my train of thought here, looking at it as bacteria killed. The LHS represents 30% of the initial bacteria population, which means 70% was killed. The RHS represents the decay process. Could you be so kind as to highlight in bold where I make a faulty statement?

Gold Member
Was posting post #26 while you responded. Let me give it a read.

• symbolipoint
Gold Member
I still think it's ##0.3\left(10^6\right)=10^6\left(0.9\right)^x## The solution to this is somewhere between 11 and 12 and the answer in the back of the book say 110-120 minutes so I'm inclined to say that this is the correct formula.
The formula makes sense to me.
LHS expresses 30% of the initial population which means 70% was killed.
RHS expresses the initial population losing 10% x amount of times.

symbolipoint
Homework Helper
Education Advisor
Gold Member
I still think it's ##0.3\left(10^6\right)=10^6\left(0.9\right)^x## The s......
The formula makes sense to me.
LHS expresses 30% of the initial population which means 70% was killed.
RHS expresses the initial population losing 10% x amount of times.
Good! Most of that was what I was trying to help you to understand.

Note that the simplification for the factor 10^6 on both sides will allow the equation to become
0.3=1*(0.9)^x.
More simply, 0.3=0.9^x.

• opus
Gold Member
Jeez for such a basic problem that one had me in knots. Thanks for all the help!!

FactChecker
Science Advisor
Gold Member
Every ten minutes, a tenth of the bacteria are KILLED.
At 10 minutes: a tenth of 10^6 bacteria are killed. This leaves 100,000 remaining.
No. If a tenth are killed, that is 100,000 killed and 1,000,000-100,000 = 900,000 remaining.
At 20 minutes : a tenth of the 100,000 bacteria are killed. This leaves 10,000 remaining.
A tenth of 900,000 are killed. That is 90,000 killed and 900,000 - 90,000 = 810,000 remaining.
At 30 minutes: a tenth of 810,000 are killed. That is 81,000 killed and 810,000 - 81,000 = 729,000 remaining.
...
So the approach of calculating the killed at each step requires that you keep subtracting the killed from the remaining living at each time step.
It's easier to directly calculate the living by multiplying the remaining living at each step by 0.9. Then you don't have to keep doing a subtraction:
10^6 * 0.9 = 900,000
900,000 * 0.9 = 810,000
810,000 * 0.9 = 729,000
...
After ##n*10## minutes, there are ##10^6*0.9^n## remaining.

Last edited:
• opus
symbolipoint
Homework Helper
Education Advisor
Gold Member
No. If a tenth are killed, that is 100,000 killed and 1,000,000-100,000 = 900,000 remaining.A tenth of 900,000 are killed. That is 90,000 killed and 900,000 - 90,000 = 810,000 remaining.
At 30 minutes: a tenth of 810,000 are killed. That is 81,000 killed and 810,000 - 81,000 = 729,000 remaining.
...
So the approach of calculating the killed at each step requires that you keep subtracting the killed from the remaining living at each time step.
It's easier to directly calculate the living by multiplying the remaining living at each step by 0.9. Then you don't have to keep doing a subtraction:
10^6 * 0.9 = 900,000
900,000 * 0.9 = 810,000
810,000 * 0.9 = 729,000
...
After ##n*10## minutes, there are ##10^6*0.9^n## remaining.
From his comment of post #28 and #30, opus may finally understand.

• opus
Gold Member
No. If a tenth are killed, that is 100,000 killed and 1,000,000-100,000 = 900,000 remaining.A tenth of 900,000 are killed. That is 90,000 killed and 900,000 - 90,000 = 810,000 remaining.
At 30 minutes: a tenth of 810,000 are killed. That is 81,000 killed and 810,000 - 81,000 = 729,000 remaining.
...
So the approach of calculating the killed at each step requires that you keep subtracting the killed from the remaining living at each time step.
It's easier to directly calculate the living by multiplying the remaining living at each step by 0.9. Then you don't have to keep doing a subtraction:
10^6 * 0.9 = 900,000
900,000 * 0.9 = 810,000
810,000 * 0.9 = 729,000
...
After ##n*10## minutes, there are ##10^6*0.9^n## remaining.

I really need to work on my reading comprehension! Thanks for pointing that out to me.

symbolipoint
Homework Helper
Education Advisor
Gold Member
I really need to work on my reading comprehension! Thanks for pointing that out to me.
Most of Reading Comprehension for Mathematics is very, very literal.

• opus