A can contains `1/4` pound of cashews.
The can is then filled with a mixture that has equal weights of cashews , pecans , and walnuts . If the final weight is 1 pound , what fraction of the final nut mixture is cashews ?

Options

A. `1/4`
B. `1/3`
C. `1/2`
D. `2/3`

Answer

To solve the problem, we need to find the fraction of cashews in the final nut mixture after adding pecans and walnuts. Let's break it down step by step.

### Step 1: Identify the initial weight of cashews
The can contains \( \frac{1}{4} \) pound of cashews.

### Step 2: Determine the total weight of the final mixture
The final weight of the mixture is given as 1 pound.

### Step 3: Calculate the weight of the mixture added
To find the weight of the mixture added (pecans and walnuts), we subtract the initial weight of cashews from the final weight:
\[ \text{Weight of mixture added} = \text{Final weight} - \text{Initial weight of cashews} = 1 - \frac{1}{4} = \frac{3}{4} \text{ pounds} \]

### Step 4: Set up the equation for equal weights of cashews, pecans, and walnuts
Let \( x \) be the weight of each of the nuts (cashews, pecans, and walnuts) in the mixture added. Since they are in equal weights, we have:
\[ \text{Weight of cashews added} + \text{Weight of pecans} + \text{Weight of walnuts} = 3x \]
This total weight must equal the weight of the mixture added:
\[ 3x = \frac{3}{4} \]

### Step 5: Solve for \( x \)
To find \( x \), we divide both sides by 3:
\[ x = \frac{3}{4} \div 3 = \frac{3}{4} \times \frac{1}{3} = \frac{1}{4} \text{ pounds} \]
So, the weight of each type of nut (cashews, pecans, walnuts) added is \( \frac{1}{4} \) pound.

### Step 6: Calculate the total weight of cashews in the final mixture
The total weight of cashews in the final mixture is the initial weight plus the weight added:
\[ \text{Total weight of cashews} = \text{Initial weight of cashews} + \text{Weight of cashews added} = \frac{1}{4} + \frac{1}{4} = \frac{1}{2} \text{ pounds} \]

### Step 7: Find the fraction of cashews in the final mixture
To find the fraction of cashews in the final mixture, we divide the total weight of cashews by the total weight of the mixture:
\[ \text{Fraction of cashews} = \frac{\text{Total weight of cashews}}{\text{Final weight of mixture}} = \frac{\frac{1}{2}}{1} = \frac{1}{2} \]

Thus, the fraction of the final nut mixture that is cashews is \( \frac{1}{2} \).

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