Question 609155: find how many quarts of 6% butterfat milk and 3% butterfat milk should be mixed to yield 75 quarts of 5% butter milk?
Answer by math-vortex(648)   (Show Source):
You can put this solution on YOUR website! Hi, there--
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To solve this problem we will create a system of equations modeling the information in the problem.
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Let x be the number of quarts of 6% butterfat milk.
Let y be the number of quarts of 3% butterfat milk.
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6% butterfat means that six-hundredths of each quart is butterfat. Since we want to know the amount of butter fat in x quarts, we multiply, 0.06*x or just .06x.
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Using similar reasoning, the amount of butterfat in the 3% butterfat milk would be .03y.
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We want to end up with 75 quarts of 5% butterfat milk. Five-hundredths of each of these quarts is butterfat and we have 75 quarts so we multiply again. There are (.05)(75)=3.75 quarts of butterfat in the final 75 quarts.
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We can use this information to write an equation:
[the butterfat in the 6% milk] + [the butterfat in the 3% milk] = [butterfat in the final product]
0.06x + 0.03y = 3.75
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We can write a second equation about the total volume of the milk:
[the amount of 6% milk] + [the amount of 3% milk] = [amount of 5% milk]
x + y = 75
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Now we have two equations with two unknowns. We can solve using substitution. Set the second equation equal to y.
x + y = 75
y = 75 - x
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Substitute 75 - x for y in the first equation.
0.06x + 0.03y = 3.75
0.06x + 0.03(75 - x) = 3.75
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Simplify and solve for x.
0.06x + 2.25 - 0.03x = 3.75
0.03x +2.25 = 3.75
0.03x = 1.50
x = 50
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Solve for y using the second equation.
x + y = 75
50 + y = 75
y = 25
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There are 50 quarts of 6% butterfat milk and 25 quarts of 3% butterfat milk.
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We can check our work by making use the amount of butterfat is correct:
.06(50) = 3 quarts of butterfat from the 6% solution.
.03(25) = 0.75 quarts of butterfat from the 3% solution.
Together, we have 3.75 quarts of butterfat.
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Check!
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Hope this helps. Feel free to email me via gmail if you have questions about the solution.
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Ms.Figgy
math.in.the.vortex@gmail.com
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