Often, scientists and lab technicians express the concentration of a diluted solution in terms of a ratio to the original -- a 1:10 ratio, for example, meaning that the final solution has been diluted tenfold. Don't let this frighten you; it's just a different form of a simple equation. You, too, can calculate ratios between solutions. Here's how to set about solving these kinds of problems.
Determine what information you have and what you need to find. You might have a solution of known starting concentration and be asked to dilute it by some set ratio -- 1:10, for example. Or you might have the concentration of two solutions and need to determine the ratio between them.
If you have a ratio, convert it into a fraction. 1:10 becomes 1/10, for example, while 1:5 becomes 1/5. Multiply this ratio by the original concentration to determine concentration of the final solution. If the original solution has 0.1 mole per liter and the ratio is 1:5, for example, the final concentration is (1/5)(0.1) = 0.02 moles per liter.
Use the fraction to determine how much of the original solution should be added to a given volume when diluting.
Let's say, for example, that you have a 1 molar solution and need to do a 1:5 dilution to prepare a 40 mL solution. Once you convert the ratio to a fraction (1/5) and multiply it by the final volume, you have the following:
(1/5)(40 mL) = 8 mL
meaning you need 8 mL of the original 1 molar solution for this dilution.
If you need to find the ratio of concentration between two solutions, just turn it into a fraction by placing the original solution in the denominator and the dilute solution in the numerator.
Example: You have a 5 molar solution and a diluted 0.1 molar solution. What is the ratio between these two?
Answer: (0.1 molar) / (5 molar) is the fractional form.
Next, multiply or divide both numerator and denominator of the fraction by the smallest number that will convert them to a whole-number ratio. The whole goal here is to get rid of any decimal places in numerator or denominator.
Example: (0.1 / 5) can be multiplied by 10/10. Since any number over itself is just another form of 1, you are merely multiplying by 1, so this is mathematically acceptable.
(10/10)(0.1 / 5) = 1/50
If the fraction had been 10 / 500, on the other hand, you could have divided both numerator and denominator by 10 -- essentially dividing by 10 over 10 -- to reduce to 1 / 50.
Change the fraction back into a ratio.
Example: 1/50 converts back to 1 : 50.