We are making trail mix using peanuts and chocolate candies.
The cost of the peanuts is 2.65 per pound.
The chocolate costs 5.15 per pound.
We want 17 of trail mix that is worth 3.47 per pound.


Start with a six-cell table:


**Ingrediant 1**

**Ingrediant 2**

**Mixture**

**Price per lb**

Cell1

Cell2

Cell3

**Weight**

Cell4

Cell5

Cell6


There are two different methods for solving mixture problems: One Variable (chapter 2), and Two Variables (chapter 4). This question requires student to understand the ONE VARIABLE method.
Use 𝑥 as your variable.



Peanuts

Chocolate

Mixture

Price Per lb

Weight

𝑥




Use the table above to create a single equation (using x) that could be used to answer this problem: 


Solve the problem. Round to one decimal place.
Pounds of peanuts = 
Pounds of chocolate = 


Q

Let's set up the problem using the given information and create a table to organize the data.

### Table
|            | **Ingredient 1** | **Ingredient 2** | **Mixture**      |
|------------|------------------|------------------|------------------|
| **Type**   | Peanuts          | Chocolate        | Trail Mix        |
| **Price/lb** | \$2.65           | \$5.15           | \$3.47           |
| **Weight** | $ x $           | $ 17 - x $     | 17 lbs           |

Now, we'll create a single equation using $ x $ to represent the pounds of peanuts.

#### Mixture Cost Calculation

1. **Cost of peanuts**: $ 2.65x $
2. **Cost of chocolate**: $ 5.15(17 - x) $
3. **Total cost of the mixture**: $ 3.47 \times 17 $

Since the total cost of the mixture is the sum of the costs of peanuts and chocolate, we can set up the equation:

\[
2.65x + 5.15(17 - x) = 3.47 \times 17
\]

Let's solve the equation:

\[
2.65x + 5.15(17 - x) = 3.47 \times 17
\]

\[
2.65x + 5.15(17 - x) = 58.99
\]

\[
2.65x + 87.55 - 5.15x = 58.99
\]

\[
-2.5x + 87.55 = 58.99
\]

\[
-2.5x = 58.99 - 87.55
\]

\[
-2.5x = -28.56
\]

\[
x = \frac{-28.56}{-2.5}
\]

\[
x = 11.4
\]

Thus, the pounds of peanuts, $ x $, is $ 11.4 $. The pounds of chocolate, $ 17 - x $, is:

\[
17 - 11.4 = 5.6
\]

**Pounds of peanuts** = **11.4**

**Pounds of chocolate** = **5.6**

Would you like further details or have any questions? Here are some related questions to consider:

1. What would be the total cost if we increased the amount of chocolate by 2 lbs?
2. How would the cost per pound of the mixture change if the price of peanuts increased?
3. How would you solve the problem using the Two Variables method?
4. Can we use a different type of candy with a different price per pound and still achieve the same cost per pound for the mixture?
5. How would the solution change if we only had 15 lbs of the mixture?
6. What is the significance of the variable $ x $ in this context?
7. How would the problem change if the mixture had to be worth $4.00 per pound instead of $3.47?
8. What other scenarios can we use this one-variable method for?

**Tip:** When working on mixture problems, always make sure to balance the cost per pound by taking into account both the quantity and the price of each component.

Learn how to solve a trail mix mixture problem involving peanuts and chocolate candies. Calculate the optimal quantities of peanuts and chocolate to achieve a specified cost per pound. Suitable for high school students studying mixture problems and linear equations.

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