To find the total amount of nicotine in a given volume of juice, multiply the STRENGTH of the juice (in mg/ml) by the AMOUNT there is (in ml). This will give you the TOTAL amount of nicotine (in mg) that is in the juice.
Now that you know how many mg of nicotine you have, and you (presumably) have a target strength in mind, divide the TOTAL amount of nicotine (in mg) by the TARGET STRENGTH (in mg/ml). This will give you the FINAL amount of juice you need.
Finally, subtract the initial AMOUNT of juice you started with from the FINAL amount that you need, to find out how much VG you need to add.
Example: Say you have 10ml of 18mg/ml juice, which you're trying to dilute down to 6mg/ml. There are 10*18=180 mg of nicotine in your juice. This will make 180/6=30 ml at 6mg. You would need to add 30-10=20 ml of VG to your initial 10ml.
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Alternatively, if you're aiming for a target amount (say, the empty 10ml bottle you mentioned), then you would do it like this:
Multiply the TARGET AMOUNT (10ml) by the TARGET STRENGTH (6mg/ml) to find the TOTAL amount of nicotine you need in the bottle (in this case, 60mg). Divide this total (60) by the STRENGTH of your starting liquid (18) to find out how much of the starting liquid you need, to obtain this total (60/18=3.333....). Again, you subtract this starting amount from the target amount, to find out how much you need to add. So in this case, you would put about 3.3ml of the 18mg juice in the bottle, and then add 6.7ml of VG for a total of 10ml.
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Don't forget to check your work. Example:
3.3ml of juice, at 18mg/ml, contains 3.3*18=59.4mg of nicotine.
59.4mg nic, in 10ml of total liquid, has a concentration of 5.94 mg/ml
(it's not exactly six mg/ml, because we had to round the amount to 3.3ml instead of exactly 3 and 1/3, or 3.33333333333... ml)
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Another way to look at it is, if your target strength (6mg/ml) is 1/3 of the starting strength, then you need to cut it by 2/3. Just add "1 part" juice and "2 parts" VG. This method works best when the strengths in question are factors of each other, like 18 and 6 (18 = 3*6) If you were aiming for a target of say 5 or 10 mg/ml, this method would be a lot trickier since 5 and 10 are not factors of 18.
