Ten pounds of mixed nuts contain 50 percent peanuts . How many pounds of peanuts must be added so that the final mixture has 60 percent peanuts ?

Options

A. 2.5
B. 5
C. 6
D. 10

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Determine the amount of peanuts in the original mixture.
The original mixture weighs 10 pounds and contains 50% peanuts. To find the amount of peanuts, we can use the formula:

\[ \text{Amount of peanuts} = \text{Percentage of peanuts} \times \text{Total weight of mixture} \]

Substituting the values:

\[ \text{Amount of peanuts} = 0.50 \times 10 = 5 \text{ pounds} \]

### Step 2: Set up the equation for the new mixture.
Let \( x \) be the amount of peanuts we need to add. After adding \( x \) pounds of peanuts, the new amount of peanuts will be:

\[ 5 + x \text{ pounds} \]

The total weight of the mixture after adding \( x \) pounds will be:

\[ 10 + x \text{ pounds} \]

### Step 3: Set up the equation for the percentage of peanuts in the new mixture.
We want the new mixture to contain 60% peanuts. Therefore, we can set up the equation:

\[ \frac{5 + x}{10 + x} = 0.60 \]

### Step 4: Solve the equation.
To eliminate the fraction, we can cross-multiply:

\[ 5 + x = 0.60(10 + x) \]

Expanding the right side:

\[ 5 + x = 6 + 0.60x \]

Now, we will rearrange the equation to isolate \( x \):

\[ 5 + x - 0.60x = 6 \]

This simplifies to:

\[ 5 + 0.40x = 6 \]

Subtract 5 from both sides:

\[ 0.40x = 1 \]

Now, divide both sides by 0.40:

\[ x = \frac{1}{0.40} = 2.5 \]

### Step 5: Conclusion
We need to add **2.5 pounds** of peanuts to the mixture to make the final mixture contain 60% peanuts.

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