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?
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-> 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?
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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):
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
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.
