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 thousandth of a gallon, must be mixed together to obtain 50 gallons of cream that contains 12.5$\%$ butterfat?

Key Concepts

Percent Concentration

Percent concentration refers to the proportion of a particular component within a mixture, typically expressed as a percentage. This concept is crucial when setting up equations for mixture problems, as it helps translate the desired composition into a mathematical relationship, often by converting percentages to decimals for calculation.

Mixture Problems

Mixture problems involve combining different substances with varying properties, such as concentration, to form a mixture that meets a specific criterion. They require establishing relationships, typically through algebraic expressions, between the amounts and properties (like percentage composition) of each component and the final desired mixture.

Systems of Equations

Systems of equations are used to solve problems with multiple unknown quantities by establishing equations that capture the relationships between these quantities. In mixture problems, one equation often accounts for the total volume while another represents the concentration or content of a specific component, allowing for the simultaneous determination of each unknown quantity.

Weighted Average Concentration

Weighted average concentration is a method used to calculate the overall concentration of a mixture when combining components with different concentrations. It involves considering the proportional contribution of each component’s concentration relative to its amount in the overall mixture to ensure the final mixture meets the targeted concentration.

Transcript

00:01 In the peanuts cartoon shown the problem that is sending peppermint paid into agitated state.

00:07 How much cream and how much milk to the nearest thousandth of gallons must be mixed together to obtain 50 gallons that contains 4 .5 butter fat.

00:24 So we have got milk and cream that is supposed to be mixed to make 50 gallons that have 12 .5 butter.

00:30 5 butter fat.

00:32 So let's just let first.

00:35 So let cream equals to eggs and also let milk equal to y.

00:48 So it's clear that if you eat cream and milk you should get 50 gallons.

00:57 And the next one is the fat, the butter fat.

01:02 So in cream the cream contains 0 .25x.

01:14 It contains 0 .25 % of butterfat.

01:18 And the milk contains 3 .5 % of butter fat.

01:27 And all in all, the 50 gallons should have about 12 and 1 half percent of butter fat.

01:37 Percent of butterfed.

01:41 So 12 and a half percent of 50 is 6 .25.