Question

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?
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**