SOLUTION: How many gallons of a 80% antifreeze solution must be mixed with 70 gallons of 30% antifreeze to get a mixture that is 70% antifreeze?

Algebra.Com
Question 1163316: How many gallons of a 80% antifreeze solution must be mixed with 70 gallons of
30% antifreeze to get a mixture that is 70% antifreeze?
Found 2 solutions by Edwin McCravy, greenestamps:
Answer by Edwin McCravy(20055)   (Show Source): You can put this solution on YOUR website!
80% antifreeze solution is 80% alcohol and 20% other stuff.
30% antifreeze solution is 30% alcohol and 70% other stuff.
70% antifreeze solution is 70% alcohol and 30% other stuff.
        
how many| percent | how many  |
gallons | alcohol |  gallons  |
  of    |  as a   |    of     |
solution| decimal |  alcohol  |
--------|---------|-----------|
   x    |   0.8   | 0.8x      |
  70    |   0.3   | 0.3(70)   |  
--------|---------|-----------| 
  x+70  |   0.7   | 0.7(x+70) |

The equation comes from the last column

0.8x + 0.3(70) = 0.7(x+70)

Solve that.  The answer is more than 200 and less than 300 gallons.

That's a big answer, but we should expect that because 30% is pretty 
weak compared to 70%, so it should take a lot.

Edwin
Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


You should understand the formal algebraic solution method shown by the other tutor; it is a good example of problem solving using algebra.

However, if a formal algebraic solution is not required, you can get to the answer to any two-part mixture problem like this quickly and easily using logical reasoning and simple mental arithmetic.

As noted in the response from the other tutor, it is going to take a lot of the 80% antifreeze to mix with the 30% antifreeze to get 70% antifreeze, because 70% is much closer to 80% than it is to 30%.

Putting numbers to that general concept, the ratio of the amounts of the two ingredients is exactly dependent on where the 70% target lies between the 30% and 80% of the two ingredients.

Simple mental calculation shows that 70% is 4/5 of the way from 30% to 80%. (From 30 to 80 is a difference of 50; from 30 to 70 is a difference of 40; 40/50 = 4/5.)

That means 4/5 of the mixture must be the 80% antifreeze.

So the 70 gallons of 30% antifreeze is 1/5 of the mixture; that means the 4/5 of the mixture that is the 80% antifreeze is 4*70 = 280 gallons.

ANSWER: 280 gallons of the 80% antifreeze are needed.

CHECK:
.30(70)+.80(280) = 21+224 = 245
.70(70+280) = .70(350) = 245
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