I need help solving a problem.

Question

Bob is getting ready for a date.  He frequently uses Blue Patchouli cologne because 30% of it is real bits of Patchouli, but it's too strong for this date.  He decides to water it down; he guesses 10% Patchouli would be about right.  How much water (0% Patchouli) should Bob add to the .5 ounces of Patchouli (30% Patchouli) in order to create a cologne that is 10% Patchouli?

Answer 1

First, figure out the proportions of Patchouli in the cologne.  If it's 30% Patchouli, then in 0.5 oz of cologne there is 0.15 oz Patchouli. 

 

The cologne is made up of Patchouli and "other" (water?), so Patchouli / cologne = 30%.  

 

30% =  P  / 0.5 oz cologne

 

P = 0.15 oz

 

Check:

0.15/0.5 = 0.3 = 30%

 

Ok, now, we want to add water to this so that we end up with 10% Patchouli. In other words, we're going to change volume of the cologne but NOT the volume of Patchouli (since we're only adding water).

 

0.15 / 0.5 + x        = 10% or 0.10

                               where x is the amount of water added

 

 

Cross multiply to figure out what x is:

 

0.15 = (0.5 + x) (0.1)

 

0.15 = 0.05 + 0.1x

 

0.10 = 0.1x

 

1 = x

 

 

Therefore, we're going to add 1 oz of water.

 

 

Check:

 

0.15    / (0.5 + 1)

 

0.15 / 1.5 = 0.1 = 10%

 

Remember, we're not adding any Patchouli, so there is still going to be only 0.15 oz in the cologne.  We add 1 oz of water, so the total volume is 1.5 oz, and then the proportion of Patchouli to the total is 0.1, or 10%