the royal physician failed to produce a cure for the prince's laziness but the queen said she would still let the physician live if he could find a cure for her husband's snoring instead. Now the physician is nervously combing a 10 ounce substance containing 6% cottonseed with another substance containing 12% cottonseed to (hopefullly) create a snore- suppressing substance with 8% cottonseed. How many ounces of the 12% substance must he use?

Answer 1

Let \( x \) be the amount (in ounces) of the 12% cottonseed substance that the physician will use. Therefore, the amount of the 6% cottonseed substance used will be \( 10 - x \) ounces, since the total amount of the mixture should equal 10 ounces.

Next, we can set up an equation based on the concentration of cottonseed in the final mixture.

The amount of cottonseed in the 6% substance: \[ 0.06(10 - x) \]

The amount of cottonseed in the 12% substance: \[ 0.12x \]

The total amount of cottonseed in the mixture should be: \[ 0.08 \times 10 = 0.8 \text{ ounces} \]

Combining the amounts of cottonseed from both substances gives us the equation: \[ 0.06(10 - x) + 0.12x = 0.8 \]

Now, let's simplify and solve this equation:

1. Distribute \(0.06\) in the left-hand side: \[ 0.6 - 0.06x + 0.12x = 0.8 \]

2. Combine like terms: \[ 0.6 + 0.06x = 0.8 \]

3. Subtract \(0.6\) from both sides: \[ 0.06x = 0.8 - 0.6 \] \[ 0.06x = 0.2 \]

4. Divide by \(0.06\): \[ x = \frac{0.2}{0.06} \approx 3.33 \]

So, the physician should use approximately \(3.33\) ounces of the 12% cottonseed substance.

Now, to find the amount of the 6% substance: \[ 10 - x = 10 - 3.33 \approx 6.67 \] Thus, he will use about \(6.67\) ounces of the 6% cottonseed substance.

To summarize, the physician must use approximately: \[ \boxed{3.33} \text{ ounces} \] of the 12% cottonseed substance.