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?

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

GPT-4o mini · 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.