# How to scale back up

So far we've figured out how to make a dilution, which we can then plate and count. But by definition, we're counting only a fraction of what was originally there. How do we account for that? By scaling up.

"Scaling up" means starting from a sample and figuring out how many were in the original brew, or stock, or whatever was originally there.

For example, we can start with a cup of wimpy coffee and figure out how much caffeine was in the original brew. All we need to know is what the overall dilution factor was. In the case of wimpy coffee, it was 1/50, or 0.02.

So, when we count the caffeine molecules in a cup of wimpy coffee, we know we got 1/50th of what was in a cup of the original brew -- or in other words, there was 50 times as much in a cup of the original. This method is called multiplying by the inverse (of the dilution factor).

• If the dilution factor is in the form of a fraction,
"flip" the fraction
(i.e., 1/50 becomes multiply by 50/1).
• If the dilution factor is in decimal form,
multiply by 1 over the decimal
(i.e., 0.02 becomes multiply by 1/0.02).

Finally, notice that I'm telling you the total number of caffeine molecules ONE CUP of the original brew. If I don't know the actual amount of original brew, I also won't know how much caffeine there was total.

## Back to the lab

Here are some problems, ranging from easy to a bit hard...

### I did a series of dilutions with an overall dilution factor of 1/20,000, and then plated a grew a 1mL sample. After 1 day, I counted 27 CFUs on the petri dish. How many CFUs would there be per 1mL of the original stock?

(To make this problem interactive, turn on javascript!)

• I need a hint ... :
inverse of 1/20,000 is 20,000
• ...another hint ... :
27*20,000 = ...

### I did a series of dilutions, with dilution factors of 0.1, 0.1, and 0.01. At the end I plated and grew a 1mL sample, and counted 48 CFUs. How many CFUs would there be per 1mL of the original stock?

(To make this problem interactive, turn on javascript!)

• I need a hint ... :
What's the overall dilution factor?
• ...another hint ... :
overall dilution factor = 0.1*0.1*0.01 = 0.0001
• ...another hint ... :
Multiply by the inverse...
• ...another hint ... :
overall dilution factor = 0.1*0.1*0.01 = 0.0001

### If the final 1mL sample had 152 colonies, what was the original concentration?

(To make this problem interactive, turn on javascript!)

• I need a hint ... :
The dilution factors were 1/10, 1/100, and 1/50
• ...another hint ... :
The overall dilution was 1/50,000