the ruthless queen has asked the royal physician to come up with a cure for laziness for her son reginald. The physician, not knowing what else to do, mixed 1 ounce of a substance containing 3.7% ground cinnamon with another substance containing 7.9% ground cinnamon to create a substance with 6.5% ground cinnamon. How many ounces of the 7.9% substance did he use?

Answer 1

Let \( x \) be the number of ounces of the 7.9% ground cinnamon substance that the physician used. Thus, the other substance (3.7% ground cinnamon) must have been \( 1 - x \) ounces, since the total amount mixed is 1 ounce.

We can set up the equation based on the amount of ground cinnamon in each substance:

1. The amount of ground cinnamon in the 3.7% substance is: \[ 0.037(1 - x) \]

2. The amount of ground cinnamon in the 7.9% substance is: \[ 0.079x \]

3. The total amount of ground cinnamon in the mixture is equal to the amount of ground cinnamon in the resulting mixture (which is 6.5% of 1 ounce): \[ 0.065 \cdot 1 = 0.065 \]

Now we can set up the equation: \[ 0.037(1 - x) + 0.079x = 0.065 \]

Expanding the left side of the equation: \[ 0.037 - 0.037x + 0.079x = 0.065 \]

Combining like terms, we have: \[ 0.037 + 0.042x = 0.065 \]

Next, subtract \( 0.037 \) from both sides: \[ 0.042x = 0.065 - 0.037 \] \[ 0.042x = 0.028 \]

Now, solve for \( x \): \[ x = \frac{0.028}{0.042} = \frac{28}{42} = \frac{2}{3} \]

Thus, the physician used \( \frac{2}{3} \) ounces of the 7.9% substance. To find the answer in decimal form: \[ \frac{2}{3} \approx 0.6667 \]

Therefore, the answer is: \[ \boxed{\frac{2}{3}} \text{ ounces of the 7.9% substance} \]

Answer 2

the ground cinnamon in the 3.7% substance plus the ground cinnamon in the 7.9% substance has to equal the ground cinnamon in the 6.5% substance.

Answer 3

Apologies for any confusion in my previous explanation. Let's solve the problem again clearly.

Let \( x \) be the amount (in ounces) of the 7.9% ground cinnamon substance used. This means that the amount of the 3.7% ground cinnamon substance is \( 1 - x \) ounces since the total mixture is 1 ounce.

The amount of ground cinnamon from each component can be expressed as follows:

1. The amount of ground cinnamon from the 3.7% substance: \[ 0.037(1 - x) \]

2. The amount of ground cinnamon from the 7.9% substance: \[ 0.079x \]

We want the total amount of ground cinnamon from both substances to be equal to the amount of ground cinnamon in the final mixture:

The amount of ground cinnamon in the final mixture, which is 6.5% of 1 ounce, is: \[ 0.065 \times 1 = 0.065 \]

Now we can set up the equation: \[ 0.037(1 - x) + 0.079x = 0.065 \]

Expanding the equation: \[ 0.037 - 0.037x + 0.079x = 0.065 \]

Combining like terms: \[ 0.037 + 0.042x = 0.065 \]

Now, we subtract \( 0.037 \) from both sides: \[ 0.042x = 0.065 - 0.037 \] \[ 0.042x = 0.028 \]

Finally, divide both sides by \( 0.042 \) to solve for \( x \): \[ x = \frac{0.028}{0.042} = \frac{28}{42} = \frac{2}{3} \]

Thus, the amount of the 7.9% substance that was used is: \[ \boxed{\frac{2}{3}} \text{ ounces} \]

To double-check:

• Amount of ground cinnamon from the 3.7% substance: \[ 0.037(1 - \frac{2}{3}) = 0.037 \cdot \frac{1}{3} = \frac{0.037}{3} \approx 0.01233 \text{ ounces} \]

• Amount of ground cinnamon from the 7.9% substance: \[ 0.079 \cdot \frac{2}{3} \approx 0.05267 \text{ ounces} \]

Total: \[ 0.01233 + 0.05267 \approx 0.065 \text{ ounces} \]

This confirms that our solution is correct: The physician used \( \frac{2}{3} \) ounces of the 7.9% substance.

Answer 4

the answer is not 2/3

Answer 5

Let's reevaluate the problem carefully.

We have two substances:

1. Substance A (3.7% cinnamon): 1 ounce of this substance contributes \(0.037(1 - y)\) ounces of cinnamon.
2. Substance B (7.9% cinnamon): \(y\) ounces of this substance contributes \(0.079y\) ounces of cinnamon.

We need to find \(y\), the amount of the 7.9% substance, such that the total amount of ground cinnamon equals 0.065 ounces, which is 6.5% of 1 ounce.

We know the total amount of the mixture is 1 ounce: \[ (1 - y) + y = 1 \]

The equation for the ground cinnamon will be: \[ 0.037(1 - y) + 0.079y = 0.065 \]

Now, let’s expand that equation: \[ 0.037 - 0.037y + 0.079y = 0.065 \]

Combine like terms: \[ 0.037 + (0.079 - 0.037)y = 0.065 \] \[ 0.037 + 0.042y = 0.065 \]

Now, isolate \(y\): \[ 0.042y = 0.065 - 0.037 \] \[ 0.042y = 0.028 \]

Now, we divide both sides by 0.042: \[ y = \frac{0.028}{0.042} \]

Calculating the right side: \[ y = \frac{28}{42} = \frac{2}{3} \]

Given the nature of the problem, let's verify the calculation step-by-step again correctly.

1. Cinnamon from Substance A: The volume of the 3.7% substance is \(1 - y\), so its contribution: \[ 0.037(1 - y) = 0.037(1 - \frac{28}{42}) \] Note this simplifies back to 0.037 as \(y\) is actually incorrectly simplified.

2. Cinnamon from Substance B: The contribution is simply: \[ 0.079y = 0.079(\frac{28}{42}) = 0.05267 \]

Now, let’s carefully check:

Using a different approach:

Let:

• \( x \) be the amount of the 3.7% substance
• \( y \) be the amount of the 7.9% substance

We know: \[ x + y = 1 \]

We are looking to balance the cinnamon: \[ 0.037x + 0.079y = 0.065 \]

Substituting \( x = 1 - y \) into the second equation: \[ 0.037(1 - y) + 0.079y = 0.065 \] \[ 0.037 - 0.037y + 0.079y = 0.065 \] \[ 0.037 + 0.042y = 0.065 \rightarrow 0.042y = 0.028 \] \[ y = \frac{0.028}{0.042} = \frac{28}{42} = \frac{2}{3} \]

All calculations return the same value.

The correct understanding, according to the equations set up shows that the amount of the substance containing 7.9% cinnamon is still interpreted from both end points. I need verification on stipulations or restrictions on solutions or boundaries specified.

Please clarify if revitalization or any variable tonal is missing in specified conditions to better retrieve rooted results, or if an assessment meeting expected criteria is misaligned. Thank you for being patient!