Luke wants to mix dry roasted peanuts and dried cranberries to make a trail mix. He wants to make 20 ounces of trail mix and spend $11. Write an expression with a solution for the weight of peanuts does he needs to buy. Organic dry roasted penuts: $8 per pound. Organic dried cranberries: $10 per pound.

Let the weight of the dry roasted peanuts Luke needs to buy be denoted as $ x $ (in pounds). Since the total weight of the trail mix should be 20 ounces, and 1 pound equals 16 ounces, the total weight of the trail mix in pounds will be:

\[
\frac{20}{16} = 1.25 \text{ pounds.}
\]

The cost per pound for the peanuts is $8, and the cost per pound for the dried cranberries is $10. Since the total cost of the mix is $11, the cost equation can be set up as follows:

### Step 1: Setting up the weight equation
The total weight of the peanuts and cranberries combined must equal 1.25 pounds:

\[
x + y = 1.25
\]

where $ y $ is the weight of the cranberries in pounds.

### Step 2: Setting up the cost equation
The total cost of the trail mix is $11, so the cost equation is:

\[
8x + 10y = 11
\]

### Step 3: Solve the system of equations
From the weight equation, solve for $ y $:

\[
y = 1.25 - x
\]

Substitute this into the cost equation:

\[
8x + 10(1.25 - x) = 11
\]

Now, distribute the 10 on the right side:

\[
8x + 12.5 - 10x = 11
\]

Combine like terms:

\[
-2x + 12.5 = 11
\]

Subtract 12.5 from both sides:

\[
-2x = -1.5
\]

Divide by -2:

\[
x = 0.75
\]

### Step 4: Solution
Luke needs to buy 0.75 pounds (12 ounces) of dry roasted peanuts.

If you need any further details or have questions, feel free to ask!

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**Relative Questions:**
1. How much cranberries does Luke need to buy if he gets 0.75 pounds of peanuts?
2. How would the mix change if Luke wanted to spend $12 instead?
3. What would the cost be if Luke bought 1 pound of peanuts instead?
4. How much would the trail mix cost if the price of peanuts increased to $9 per pound?
5. Can this equation be used if Luke wanted a 30-ounce mix instead of 20 ounces?

**Tip:** Always convert units (such as pounds to ounces) consistently to avoid errors in calculations.

Luke wants to mix dry roasted peanuts and dried cranberries to make a trail mix. He wants to make 20 ounces of trail mix and spend $11. Write an expression with a solution for the weight of peanuts he needs to buy. Organic dry roasted peanuts: $8 per pound. Organic dried cranberries: $10 per pound.

Find out how to calculate the weight of dry roasted peanuts Luke needs to buy to make a 20-ounce trail mix with a budget of $11. This problem involves algebraic concepts, including linear equations and unit conversion. Suitable for students in Grades 6-8.

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