A grocer needs to mix raisins at 2.00 per pound with...

A grocer needs to mix raisins at $\$ 2.00$ per pound with granola at $\$ 3.25$ per pound to obtain 10 pounds of a mixture that costs $\$ 2.50$ per pound. How many pounds of raisins and how many pound of granola must be used?

Key Concepts

Mixture Problems

Mixture problems involve combining two or more substances with different properties or costs to achieve a desired mixture. These problems typically require setting up equations based on the amounts and characteristics of each component so that the final mixture meets specific criteria, such as a target cost or concentration.

Weighted Average

The weighted average concept is used to determine the overall cost or concentration of a mixture when the components have different values. In these problems, each ingredient's contribution is weighted by its proportion in the final mixture, and the sum of these contributions equals the specified average value for the entire mixture.

System of Equations

A system of equations is a set of equations with multiple unknowns that need to be solved simultaneously. In mixture problems, one equation usually represents the total quantity of the mixture while another represents the weighted cost or concentration, allowing for a complete solution for the unknown quantities.

Substitution Method

The substitution method is a technique for solving a system of equations where one equation is solved for one variable, and that expression is substituted into the other equation. This helps to reduce the system to a single equation with one unknown, making it easier to find the solution step by step.

Transcript

00:01 Okay, problem 60, a grossness to mix raisins at $2 per pound with granola at $3 .25 per pound to obtain 10 pounds of mixture that costs $2 .50 per pound.

00:18 How many pounds of raisins and how many pounds of granola must be used? so to grapple with this one we need to do the model.

00:29 So the model is we start by letter.

00:34 So let the pounds of raisins be x.

00:43 Let the pounds of raisins equals to x and the pounds of granola equals to y.

01:08 So the first of this model, the first equation is always straightforward.

01:16 If you add the pounds of granola, i mean the pounds of raisins, plus the pounds of granola, you obtain 75 pounds.

01:33 You obtain 75 pounds, right? i'm not mistaken, okay, you obtain 10 pounds of mixture.

01:58 So it's 10 pounds, not 75, i wonder where i got it, maybe it's from the previous question.

02:04 So 10 pounds.

02:06 Then the second one is modeled around the aspect of cost.

02:12 Now you include costs into it.

02:15 So how much does the raise cost? raise cost two dollars each so it's two so the total cost for raises it's two times the number or the amount or the quantity of raisins plus a unit cost per canola is 325 3 .25 of why this will also give you a total cost of the 10 times 250.

03:03 So 10 times 250 is 25.

03:07 So this is equation one, this is equation two.

03:11 We are going to use substitution, so we use equation three that we are going to model or create or derive from equation one.

03:22 So it will be 10 minus x.

03:26 And then this is equation three...

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