the ruthless queen has asked to royal physician to come up with a cure for laziness for her son reginald. the physician, not knowing what else to do, mixed one ounce of a substance containg 3.7% ground cinnamon with another substance containg 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's denote the amount of the substance containing 3.7% ground cinnamon as \( x \) ounces and the amount of the substance containing 7.9% ground cinnamon as \( y \) ounces.

According to the problem, we know the following:

1. The total amount of the mixture is 1 ounce: \[ x + y = 1 \]

2. The amount of ground cinnamon in the mixture is 6.5% of 1 ounce, which is 0.065 ounces. We can set up an equation based on the amounts of ground cinnamon in each substance: \[ 0.037x + 0.079y = 0.065 \]

Now we have a system of two equations:

1. \( x + y = 1 \)
2. \( 0.037x + 0.079y = 0.065 \)

We can solve this system step by step.

From the first equation (1), we can express \( x \) in terms of \( y \): \[ x = 1 - y \]

Now we can substitute \( x \) into the second equation (2): \[ 0.037(1 - y) + 0.079y = 0.065 \]

Expanding this: \[ 0.037 - 0.037y + 0.079y = 0.065 \]

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

Now, subtract 0.037 from both sides: \[ 0.042y = 0.065 - 0.037 \] \[ 0.042y = 0.028 \]

Dividing both sides by 0.042 gives: \[ y = \frac{0.028}{0.042} \approx 0.6667 \text{ ounces} \]

To find \( x \), we can plug \( y \) back into the equation \( x + y = 1 \): \[ x = 1 - 0.6667 \approx 0.3333 \text{ ounces} \]

Thus, the physician used approximately \( 0.6667 \) ounces of the substance containing 7.9% ground cinnamon.

In summary, the amount of the 7.9% substance used is approximately: \[ \text{Approximately } 0.67 \text{ ounces.} \]