Question is in the description. I have an exam tomorrow and this is the only confusing question. Please help.

Question

The Doctor and Clara are ?ghting several Cybermen. The Doctor knows that he needs 10 liters of a solution with exactly 50% acid to dissolve the metal exoskeletons of the Cybermen. Clara only has 4 liters of a 75% acid solution, but she has a very large container of 25% acid solution. How many liters of the 50% solution can Clara make? How much of each solution should she mix together?

Answer 1

Remember: calm, confident, focused students actually score higher on tests than when they are nervous and distracted – even when they know roughly the same amount of information.

 Read and re-word the problem until you are confident that you understand it.  How’s this:

  How many liters (we need 10) of 50% acid solution can be made from 4 liters of 75% acid solution and unlimited 25% acid solution?  What is the mix?   (note: when I grew up, we solved problems that used Kryptonite).

 

There are a couple very important points about Mixture (Solutions) problems:

 

   1.  The two amounts must add up to the total amount.  For example, If Clara adds 10 liters to the 4 liters that she already has, she will get 14 liters of mixture.

 

  2.  In word (story) problems, the word “of” usually means multiply.  So, when Clara has 10 liters of 50% acid solution, she has 10 liters with 50% of 10 litersthat is acid.  Clara can make x liters of 50% solution (that is, 50% of x liters) by adding y liters of the 25% solution ( that is, <what goes here?> ).

 

Now, you remember that 50% may be written as 0.50 or 50/100 (that is, 1/2) as needed.

 

O.K., now a picture of three containers is very helpful (but it won’t look pretty):

 

 

                              +                           =

              a%                       b%                           c%

        x amount            y amount           x+y amount

 

 

Multiply the percent “of” times the amounts:

           (a%)(x)         +        (b%)(y)         =      (c%)(x+y)

 

Problems can change by having you solve for various parts of that formula, but it’s always the same formula.

 

 

Let’s try it !   (while Clara still has time)

 

    Let x = number of liters of 25% solution added

 

        has                          adds              produces

     (75%)(4)    +       (25%)(x)     =    (50%)(x+4)

 

Let’s use fractions for % and solve for x (now, the math is easy):

      (3/4)4  + (1/4)x  =  (1/2)(x + 4)

          3       +  x/4     =  x/2   + 2                (distribute)

         12      +   x      =   2x    +  8             (multiply everything by 4 to get rid of fractions)

               4    =     x                      (subtract x from both sides; subtract 8 from both sides)

 

So, by adding 4 liters of the 25% solution to the 4 liters of the 75% solution that she already has, Clara has made 8 liters of the needed 50% solution.

 

Clara is as frantic as a person about to take an algebra test !!   She needs 1 more liter of 75% solution to add to an available 1 liter of 25% solution to get 2 more liters of the needed 50% solution (that would make the 10 that the Doc needs).

 

The Doctor realizes that Clara was not the best person to hire for this job, because the Cybermen … (sorry, that's ’here stories like this often end).