Tony has two concentrations of antifreeze in his garage 50% and 100%. In most climates, a 50% concentration is optimal, but Tony lives in Minnesota and wants a 65% concentration of antifreeze in his car. The cooling system holds 13 quarts. How many quarts of each type of antifreeze should he use to fill his system with a 65% concentration?

Question

radar · Answer

Somehow I am missing the boat.  I keep coming up with an 85% solution rather than 65%.  Please review my work and find the error.

Let X = number of quarts of the 50% concentrate needed.
Let Y = number of quarts of the 100% solution (pure)
X + Y = 13
Any thing wrong so far?????

radar · Answer

The amount of antifreeze for a 65% solution in a 13 quart radiator would be:
(.65) x 13 = 8.45 quarts or 8.45 quarts of pure antifreeze (100%)
Does this make sense?  By the way I feel sorry for Tony in Minnesota.

radar · Answer

Using the variable assignments (labels) in the first post then we can state this equation.
.5X + Y =8.45 qts (pure antifreeze).

radar · Answer

We now have two equations, two unknowns thus it can be solved.
Taking the equation X + Y = 13 State in terms of X getting X= 13-Y substitute this in our other equation.
.5(13-Y) + Y = 8.45 this becomes
6.5 - .5Y + Y = 8.45
.5Y = 8.45 - 6.5
.5Y =1.95
Y = 3.9 qts of the 100% solution       substuting  in the X + Y = 13 we solve for X
X=13-3.9 =9.1 quarts of the 50% solution.  So far looking correct,  but we now need to verify.

radar · Answer

X+Y = 13............9.1 + 3.9 = 13    yup
(.5)(9.1) + 3.9 =8.45.......4.55+3.9 =8.45     yup but is the solution 65%   what do you think?

radar · Answer

It checks out o.k.     Those are the correct answers.

anonymous · Answer

Thanks