An herbalist is mixing a 9-kilogram batch of a medicinal blend. The blend, which costs $25.05 per kilogram to make, is made up of ground ginseng at $33.85 per kilogram and golden chia at $16.25 per kilogram. How many kilograms of each should be used to make the 9-kilogram batch?

Question

Write your answers as whole numbers or as decimals rounded to the nearest tenth.
kilograms of ground ginseng
kilograms of golden chia

Answer 1

To solve this problem, we can set up a system of equations based on the information provided.

Let:

• \( x \) = the kilograms of ground ginseng
• \( y \) = the kilograms of golden chia

We have the following equations:

1. The total weight of the blend: \[ x + y = 9 \]

2. The total cost of the blend: The cost per kilogram for the blend is $25.05. Therefore, for 9 kilograms, the total cost is: \[ 9 \times 25.05 = 225.45 \] The costs for ginseng and chia can be represented as: \[ 33.85x + 16.25y = 225.45 \]

Now, we have the system of equations: \[ \begin{align*}

1. & \quad x + y = 9 \
2. & \quad 33.85x + 16.25y = 225.45 \end{align*} \]

We can solve these equations step-by-step. From the first equation, we can express \( y \) in terms of \( x \): \[ y = 9 - x \]

Now we substitute \( y \) into the second equation: \[ 33.85x + 16.25(9 - x) = 225.45 \]

Expanding this, we get: \[ 33.85x + 146.25 - 16.25x = 225.45 \]

Combining like terms: \[ (33.85 - 16.25)x + 146.25 = 225.45 \] \[ 17.60x + 146.25 = 225.45 \]

Next, we subtract 146.25 from both sides: \[ 17.60x = 225.45 - 146.25 \] \[ 17.60x = 79.20 \]

Now we can solve for \( x \): \[ x = \frac{79.20}{17.60} = 4.5 \]

Using the first equation to find \( y \): \[ y = 9 - 4.5 = 4.5 \]

Thus, the solution is:

• Kilograms of ground ginseng: 4.5 kg
• Kilograms of golden chia: 4.5 kg

Therefore, the final answers are:

• 4.5 kilograms of ground ginseng
• 4.5 kilograms of golden chia