Problem

Liquid mixture problems. How many quarts of a $10…

05:51

Problem 41 Hard Difficulty

To make low-fat cottage cheese, milk
containing 4 $\%$ butterfat is mixed with milk containing $1 \%$
butterfat to obtain 15 gallons of a mixture containing $2 \%$
butterfat. How many gallons of each milk must be used?

Answer

$4 \%: 5$ gal; $1 \%: 10$ gal

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Algebra

Elementary and Intermediate Algebra

Chapter 2

Equations, Inequalities, and Problem Solving

Section 6

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Video Transcript

problem. 41 reads to make low fat cottage cheese milk containing 4% butterfat is mixed with milk containing 1% butterfat to obtain 15 gallons would make sure containing 2% butterfat. How many gallons of each milk must be used? So we know that the low fat milk is the 2%. So as a decimal, that's gonna be 0.2 And the high fat milk is 4% which has a decimal will be 0.4 and the little fat milk is 1%. So as a decimal, that 0.1 we want to know how much of the high fat and the little fat milk it would take to make 15 gallons of the low fat. So 15 is gonna go in the amount column. And if X represents the amount of the high fat milk, that means that 15 minus X will represent the amount for the little fat milk. And then to find the amount of butterfat, we're gonna have Smalls by the amount times the strength. So for high fat, that 0.4 x for a little fat, that 0.1 times 15 minus X for low fat. That's 15 times 0.2 which is 0.3. So now that we have this information, we're going to make an equation. So the amount of butterfat and the high fat and the low fat or the little fat milk is going to have to equal the amount of butterfat and the low fat milk. So our equations gonna be 0.4 x plus 0.1 times 15 minus x is equal to 0.3. So we're gonna have to multiply both sides of the equation by 100 to get rid of the decimals on both sides. So 100 times airplanes, therefore X, is for X, and then we have 100 times their 1000.1 which is one then we have one times 15 minus x, and then we have 0.3 times 100 which is 30. We have to distribute the one to the 15 and then the ones the negative X So we have four x plus one times 15 is 15 and then one times negative. X is negative. One X is equal to 30. Now we're gonna combine like terms. We have four X minus one X for X minus one X is three X then we have three X plus 15 is equal to 30. We want to move our constant, which is 15 to the other side, the equation. So to do that, we're gonna have to subtract both sides by 15. 15 minus 15 is zero. So that leaves us with three. Eggs is equal to 30 minus 15 which is 15. And lastly, we want to isolate the variable of X. So we're gonna have to divide both sides by three. So three divided by three is one which leaves us with X is equal to and then 15 divided by three is five. Our unit of measurement is going to be gallons. So we have five gallons of the high fat milk, which is the 4% milk. And then to figure out how many gallons of the little fat milk that we have, we're gonna have to subtract 15 minus X, so we have 15 minus five is equal to 10. So that means we have 10 gallons of the little fat milk of the 1% butter fat milk. So that is our solution to this problem.

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