Question

An herbalist is making a medicinal blend of herbs. The blend includes ground ginseng, which costs $33.85 per kilogram, and golden chia, which costs $16.25 per kilogram. Overall, the blend costs $25.05 per kilogram. If the herbalist already has 2 kilograms of ginseng prepared, how many kilograms of golden chia should she add?
Write your answer as a whole number or as a decimal rounded to the nearest tenth.
kilograms

Answer 1

Let \( x \) be the kilograms of golden chia to be added. The herbalist has 2 kilograms of ginseng, and we need to find \( x \) so that the total cost of the blend is $25.05 per kilogram.

The total cost of the ginseng is:
\[
2 \, \text{kg} \times 33.85 \, \text{USD/kg} = 67.70 \, \text{USD}
\]

The cost of \( x \) kilograms of golden chia is:
\[
x \, \text{kg} \times 16.25 \, \text{USD/kg} = 16.25x \, \text{USD}
\]

The total amount of herbs (ginseng and golden chia) is:
\[
2 + x \, \text{kg}
\]

The total cost of the blend is:
\[
67.70 + 16.25x \, \text{USD}
\]

The average cost per kilogram of the blend is given by:
\[
\frac{67.70 + 16.25x}{2 + x}
\]

We set this equal to the target cost of $25.05 per kilogram:
\[
\frac{67.70 + 16.25x}{2 + x} = 25.05
\]

Now we can cross-multiply to eliminate the fraction:
\[
67.70 + 16.25x = 25.05(2 + x)
\]

Expanding the right side:
\[
67.70 + 16.25x = 50.10 + 25.05x
\]

Now, we will isolate \( x \):
\[
67.70 - 50.10 = 25.05x - 16.25x
\]

This simplifies to:
\[
17.60 = 8.80x
\]

Dividing both sides by 8.80 gives:
\[
x = \frac{17.60}{8.80} = 2
\]

Thus, the herbalist should add \(\boxed{2}\) kilograms of golden chia.