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

1 answer

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.