In the "Peanuts" cartoon shown, solve the problem that is...

In the "Peanuts" cartoon shown, solve the problem that is sending Peppermint Patty into an agitated state. How much cream and how much milk, to the nearest hundredth of a gallon, must be mixed together
185 of cream that contains $12.5 \%$ butterfat?
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Key Concepts

Mixture Problems

Mixture problems involve combining two or more substances with different properties to attain a final mixture that meets specific criteria. This typically involves determining the proper amounts of each component such that the total quantity and the overall concentration or characteristic of the mixture reach a desired value.

Percentage and Concentration

Understanding percentage and concentration is essential when dealing with mixtures. This concept involves expressing a component’s quantity as a percentage of the total amount, which helps in formulating the relationships between different substances in the mixture model.

Weighted Average

The weighted average is a method used to compute the overall concentration of a mixture when different components contribute unequally. By multiplying each component's value by its proportion in the mixture and then summing these values, one can determine the average that reflects the contribution of each part.

System of Linear Equations

Mixture problems are often solved by setting up a system of linear equations that represent both the total quantity and the overall concentration of the mixture. This algebraic approach allows for the simultaneous solution of unknown quantities by relating the individual amounts and their respective percentages or concentrations.

Transcript

00:01 In this problem based off of a peanuts comic, we are dealing with some amount of cream, c, and some amount of milk, m.

00:07 This is going to be in gallons.

00:09 Now, we know that the total amount in a mixture between cream and milk should be 50 gallons.

00:14 So c plus m should equal 50.

00:17 Additionally, we want a certain fat content in our final product.

00:21 That is, we want it to be 12 .5 % fat in 50 gallons.

00:25 So the amount of fat will be 12 .5 % in decimal form, that is, 0 .1 .1 .2%.

00:30 125 times 50 gallons.

00:32 I convert that percent to a decimal by moving the decimal point over two spots to the left.

00:37 We know that cream is 25 percent fat and we know that milk is 3 .5 percent fat.

00:46 Thus we can have an equation something like this.

00:49 0 .25 times c plus 0 .035 times m equals 0 .125 times 50.

00:55 This is just determining the total fat content in the cream plus total fat content in milk equals total fat content in the mixture.

01:04 Now we've got two equations, two variables, so we can solve.

01:07 Let's get started.

01:08 The first thing that i'm going to do here is multiply out this 0 .125 times 50, which, when multiplied, will give us 6 .25.

01:18 Now, i'm going to do something that might look a bit strange here, but i'm going to multiply this top equation by negative 0 .25 on both sides...

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