Question 1155684: Find how many quarts of 6% butterfat milk and 3% butterfat milk should be mixed to yield 45 quarts of 4% butterfat milk. Found 4 solutions by Boreal, Theo, josgarithmetic, greenestamps:Answer by Boreal(15235) (Show Source):
You can put this solution on YOUR website! x is amount of 6%
45-x is amount of 3%
.06x+.03(45-x)=45*0.04=1.80 qts "pure)
.06x+1.35-0.03x=1.8
0.03x=0.45
x=15 qts of 6% (0.9 pure)
45-x=30 qts 3% (0.9 pure)
You can put this solution on YOUR website! x = number of quarts of 6% butterfat milk.
y = number of quarts of 3% butterfat milk.
you have 2 equations that need to be solved simultaneously.
they are:
x + y = 45
.06x + .03y = .04 * 45
simplify the second equation to get .06x + .03y = 1.8 quarts of butterfat.
solve for y in the first equation to get y = 45 - x
replace y in the second equation to get .06 * x + .03 * (45-x) = 1.8
simplify to get .06x + .03*45 - .03x = 1.8
simplify further to get .06x + 1.35 - .03x = 1.8
subtract 1.35 from both sides of the equation to get .06x - .03x = 1.8 - 1.35
combine like terms to get .03x = .45
solve for x to get x = 45 / .03 = 15
since x + y = 45, then y = 30
you have:
x = 15 and y = 30
.06 * x + .03 * y becomes .06 * 15 + .03 * 30 = 1.8 quarts of butterfat
your solution is that 15 quarts of 6% butterfat and 30 quarts of 3% butterfat must be mixed together to get 45 quarts of 4% butterfat.
4% cuts the lowest and the highest concentration into three parts; and is one of those parts above the 3%, or one-third the way from 3 to 6. Two-thirds of the 45 quarts should be of the 3% butterfat, and one-third of the 45 quarts should be of the 6% butterfat.
30 quarts of the 3%
15 quarts of the 6%
--------------Another Method----------------
v, how much of 6%
45-v, how much of 3%
-----to account for just the butterfat by percentages
- -------Same as
The method shown by tutor @josgarithmetic is the fastest and easiest way to solve a two-part mixture problem like this.
If the numbers in the problem are "nice", a solution by this method can be obtained in a small fraction of the time required for a formal algebraic solution.
In this problem, the numbers ARE nice; being good at mental arithmetic, I solved the problem within a couple of seconds after reading it.
I find her explanation of the method difficult to understand, so here I am showing two different ways to understand the method.
(1) 4% is "twice as close" to 3% as it is to 6%; therefore the mixture should use twice as much 3% as it does 6%.
...or...
(2) 4% is 1/3 of the way from 3% to 6%; therefore 1/3 of the mixture should be the 6%.
Either way of thinking leads quickly to the answer of 30 quarts of the 3% butterfat milk and 15 quarts of the 6% butterfat milk.
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