A grocery store offers a cheese sampler that includes a...

A grocery store offers a cheese sampler that includes a pepper cheddar cheese that- costs $\$ 16$ per kilogram and Pennsylvania Jack that costs $\$ 12$ per kilogram. How many kilograms of each were used to make a 5 -kilogram mixture that costs $\$ 13.20$ per kilogram?

Key Concepts

Linear Equations

Linear equations are fundamental in representing relationships in mixture problems. They allow us to express the balance between different components and their contributions, and can be solved using methods like substitution or elimination to find the precise amounts needed.

Mixture Problems

Mixture problems involve combining substances that have different qualities, such as cost or concentration, to achieve a final product with a desired overall property. They require setting up relationships between the quantities of the different components and the characteristics of the final mixture.

Weighted Average

A weighted average is used to calculate the overall attribute (such as cost per kilogram) of a mixture by taking into account the proportion and individual contribution of each component. It ensures that components with larger quantities contribute more significantly to the final average.

System of Equations

In mixture problems, a system of equations is often set up to represent the constraints of the situation, such as the total weight of the mixture and the total cost. Solving these equations simultaneously allows one to determine the unknown quantities of each component.

Transcript

00:01 Hello, welcome to this lesson.

00:03 In this lesson, we have a grocery store that offer pepper, that offer cheese sample that includes pepper, sugar, cheese.

00:12 All right, so the cost of that is $16 per kilogram.

00:17 And that of pennsylvania jack also cost $12 per kilogram.

00:23 All right, so here we are looking at the number of kilograms of each that were used to make five kilogram mixture.

00:31 So let's the number of pepper, cheese, sugar, that number of kilograms used to be x and the number of kilograms used for the pennsylvania jack to be y.

00:47 All right, so we're given that the total mixture weighs five kilograms.

00:54 So we have the two actually summing up to five kilograms.

01:03 And also given that the mixture, which was five kilograms, cost 13 .2 kilograms per, it cost 13 .2 per kilogram, all right? so this is our first equation.

01:23 The second equation, we have the quantity, all right, of the pepper sugar that was actually x times the cost of that, which was 16, then plus the quantity of the pennsylvania jack that was y.

01:45 Times the cost of that, that was 12, so 12y.

01:48 This is equals to 5, which was the weight of the mixture, then times how much each of the kilogram cost.

02:07 So this is 16x plus 12y equals to 66.

02:14 And now if we divide through by 2, we have 8x plus 6y equals to 33.

02:22 So this is our second equation...

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