Bree sells a dried-fruit mixture for 5.80 per pound and...

Bree sells a dried-fruit mixture for $\$ 5.80$ per pound and Hawaiian macadamia nuts for $\$ 14.75$ per pound. She wants to blend the two to get a $15-$ bb mixture that she will sell for $\$ 9.38$ per pound. How much of each should she use?

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Key Concepts

Weighted Average

The weighted average concept is used when combining items with different attributes (such as prices) in such a way that the overall average of the mixture is known. In these problems, each component contributes to the overall average based on its proportion in the mixture.

Mixture Problem

A mixture problem involves combining two or more substances with known characteristics to achieve a desired overall property. The key is to set up equations based on the total amount and the total value (or concentration) of the substances in the mixture.

System of Linear Equations

Mixture problems often require setting up a system of linear equations. One equation typically represents the total quantity of the mixture, and another represents the combined cost (or other property) of the mixture. Solving this system yields the amounts of each component used.

Substitution Method

The substitution method is a common technique for solving systems of equations. It involves solving one of the equations for one variable and substituting that expression into the other equation. This reduces the system to a single equation that can be solved for the unknown variable.

Transcript

00:01 In order to solve this problem, what we first have to do is designate some variables.

00:05 So i decided to use x to represent our dried fruit mixture and y to represent our macadamia nut.

00:12 And so we know that x plus y is what we will get to give us a 15 pound bag of our fruit and nut mixture.

00:26 We're also given how much each of these costs.

00:28 So $5 .80, 5 .8x added to $14 .75, $14 .75 for y, will give us our $9 .38 per pound for a 15 pound bags.

00:57 We'll multiply this by 15.

01:05 Is going to give us 140 .7.

01:11 And so that is how much our 15 pound bag will cost in total.

01:17 So now that we have our two equations, we can go ahead and write this as a matrix.

01:23 To remind you, when there is no variable in front of our x or our y or any of our variables, we go ahead and put a number one.

01:33 So this will be 1, 1, 115, on our top row.

01:43 In our second row will be 5 .8, 14 .75, and 140 .7.

02:06 So what we're going to do to get the first column of row 2 to be a 0 is we're going to multiply our top row by negative 5 .8.

02:17 And doing this will allow for this place to cancel out.

02:24 That will become a 0.

02:26 So when we multiply our top row by negative 5 .8, our matrix will look like this...

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