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A dairy needs 291 gallons of milk containing 6% butterfat. How many gallons each of milk containing 7% butterfat and milk containing 4% butterfat must be used to obtain the desired 291 gallons?

User Max Droid
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2 Answers

9 votes
9 votes

Final answer:

To obtain the desired 291 gallons of milk containing 6% butterfat, a system of equations can be used to find the amounts of milk with different butterfat percentages that need to be used.

Step-by-step explanation:

To find out how many gallons of milk each of different butterfat percentages must be used to obtain the desired 291 gallons of milk containing 6% butterfat, we can use a system of equations.

Let's assume x gallons of milk containing 7% butterfat will be used, and y gallons of milk containing 4% butterfat will be used.

The total gallons of milk can be expressed as: x + y = 291

The total butterfat in the milk can be expressed as: 0.07x + 0.04y = 0.06(291)

Solving this system of equations will give us the values of x and y, which represent the gallons of each type of milk that need to be used to obtain the desired milk.

Method 1: Substitution

Solve the first equation for y: y = 291 - x

Substitute this expression for y in the second equation: 0.07x + 0.04(291 - x) = 0.06 * 291

Simplify and solve for x: 0.03x = 9.66, x = 322 gallons

Substitute x back into the first equation to find y: 322 + y = 291, y = -31 gallons (This negative value doesn't make sense in the context of the problem, so we need to revise our approach.)

Method 2: Elimination

Multiply the first equation by 4: 4x + 4y = 1164

Multiply the second equation by 7: 0.49x + 0.28y = 165.7

Subtract the second equation from the first equation: 3.72x = 998.3

Solve for x: x = 268 gallons

Substitute x back into the first equation to find y: 268 + y = 291, y = 23 gallons

Therefore, the dairy needs 268 gallons of milk containing 7% butterfat and 23 gallons of milk containing 4% butterfat to obtain the desired 291 gallons with 6% butterfat.

User Greg Whitfield
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18 votes
18 votes

Answer:

Step-by-step explanation:

Each gallon of 7% milk contains 0.07 gallon of butterfat.

Each gallon of 4% milk contains 0.04 gallon of butterfat.

291 gallons of 6% milk contain 291×0.06 = 17.46 gallons of butterfat.

Let x be the number of gallons of 7% milk used. 291-x is the number of gallons of 4% milk used.

0.07x + 0.04(291-x) = 17.46

0.03x + 11.64 = 17.46

x = 200

Use 200 gallons of 7% and 91 gallons of 4%.

User HelloWindowsPhone
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