Expert: Paul Klarreich - 11/22/2010

Question
I'm having a lot of trouble figuring out why the numbers go where they do, and when to add the x or multiply. Example: How many liters of sports drink which is 70% water should be added to a 6 liter drink which is 90% water to get a drink which is 85% water? (Example Problem from my book I don't understand.)  

Answer
Questioner:   Megan
Country:  United States
Category:  Advanced Math
Private:  No
 
Subject:  Precalculus: Mixture Problems
Question:  I'm having a lot of trouble figuring out why the numbers go where they do, and when to add the x or multiply. Example: How many liters of sports drink which is 70% water should be added to a 6 liter drink which is 90% water to get a drink which is 85% water? (Example Problem from my book I don't understand.)
......................................
Hi, Megan,

There are a whole class of problems that work the same way:

Rate-time-distance.
Mixture problems.
Percent solution problems.

-- and a few others.  Set up a little table.  (And PLEASE don't ever call it a chart -- makes me throw up.)

OK, enough of that stuff.

Set it up like this:

(BTW, if the solution is 70% water, what is the other stuff?  Ok, I won't ask.)

Kind of stuff        Number            Ltrs of
                   of ltrs. * Pct  =  Water
----------------------------------------------
70% solution        |       |        |       |
----------------------------------------------
90% solution        |       |        |       |
----------------------------------------------
85% mixture         |       |        |       |
----------------------------------------------

Now fill in information from the problem, deciding that

x = number of liters of 70% solution to add.

Kind of stuff        Number            Ltrs of
                   of ltrs. * Pct  =  Water
----------------------------------------------
70% solution        |   x   |  .7    |       |
----------------------------------------------
90% solution        |   6   |  .9    |       |
----------------------------------------------
85% mixture         |       |  .85   |       |
----------------------------------------------


Fill in more, such as the fact that the ltrs of water are the result of multiplication:


Kind of stuff        Number            Ltrs of
                   of ltrs. * Pct  =  Water
----------------------------------------------
70% solution        |   x   |  .7    |  .7 x |
----------------------------------------------
90% solution        |   6   |  .9    | 5.4   |
----------------------------------------------
85% mixture         |       |  .85   |       |
----------------------------------------------

and that the numbers of liters must add up:

Kind of stuff        Number            Ltrs of
                   of ltrs. * Pct  =  Water
----------------------------------------------
70% solution        |   x   |  .7    |  .7 x |
----------------------------------------------
90% solution        |   6   |  .9    | 5.4   |
----------------------------------------------
85% mixture         |  x+6  |  .85   |  ???? |
----------------------------------------------

Now the BIG THING:  The ???? is the product across: .85(x+6).

But the ???? is also the sum:  .7x + 5.4

Got the clue, now?  Those are equal:

.7x + 5.4 = .85(x + 6)

You will solve that equation.  (Hint: clear the decimals first -- multiply all terms by 100.)

I will leave the rest to you.  My fingers are tired.