Expert: Paul Klarreich - 2/16/2007

Question
I have 1 liter of water and I need to make a 50ug/ml salt solution with this liter. If I already have a 100 ml of a 10 mg/ml salt solution, how much would I add from this salt solution to the 1 liter of water?

Answer
Questioner:   Dana
Category:  Advanced Math

Subject:  Volume Conversion
Question:  I have 1 liter of water and I want to make a 50ug/ml salt solution with this liter. If I already have a 100 ml of a 10 mg/ml salt solution, how much would I add from this salt solution to the 1 liter of water?
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Hi, Dana,

I assume that  ug/ml means micrograms per milliliter.  

This is a basic mixture problem, although in ninth grade algebra you always knew the final volume -- here you do not.  However, one of the principles still applies -- all the units must match.

WARNING: THE FOLLOWING DISCUSSION MAY CONTAIN FRACTIONS AND OTHER MATERIAL INAPPROPRIATE FOR CERTAIN COMPUTING SYSTEMS.  VIEW IT IN A FIXED-SIZE FONT, SUCH AS COURIER.

Let  x = the number of ml of  10 mg (milligrams)/ml to be added. [I'll call this the CONCENTRATE.  10 mg/ml is really strong, salty stuff, right?]

However, its concentration will have to be written as 10000 ug/ml, to make the units match.

Stuff          Amount used    Concentration  Amount of Salt
-------------+-------------+---------------+---------------
Water        |   1000 ml   |   0 ug/ml     |      0 ug  
-------------+-------------+---------------+---------------
Concentrate  |      x ml   |  10000 ug/ml  |  10000x ug
-------------+-------------+---------------+---------------
Final sol.   | 1000 + x ml |   50 mg/ml    |  50(1000+x)
-------------+-------------+---------------+---------------

Now you have your equation:  The total of the amounts of salt added equals the final amount of salt:

0 + 10000x = 50(1000 + x)

10000x = 50000 + 50x

9950x = 50000

x = 50000/9950, a little more than 5 milliliters of 'concentrate'.