Key Concept: Resource Depletion Analysis
c) 200 units
[Solution Description]
Let the initial resource be $R$. Each generation consumes an equal fraction of the remaining resource. After three generations, the remaining resource is 125 units.
Let each generation consume $x$ units. After the first generation: $R - x$.
After the second generation: $(R - x) - x = R - 2x$.
After the third generation: $(R - 2x) - x = R - 3x$.
Given that the final remaining resource is 125 units: $R - 3x = 125$.
To find the relationship between $R$ and $x$, assume each generation consumes half of the remaining resource:
First generation: $x = \frac{R}{2}$.
Second generation: $x = \frac{R - x}{2} = \frac{R - \frac{R}{2}}{2} = \frac{R}{4}$.
Third generation: $x = \frac{R - \frac{R}{2} - \frac{R}{4}}{2} = \frac{R}{8}$.
Substituting into the equation $R - 3x = 125$: $R - 3 \times \frac{R}{8} = 125$.
Simplifying: $R - \frac{3R}{8} = 125$ $\Rightarrow$ $\frac{5R}{8} = 125$.
Solving for $R$: $R = 125 \times \frac{8}{5} = 200$ units.
Your Answer is correct.
c) 200 units
[Solution Description]
Let the initial resource be $R$. Each generation consumes an equal fraction of the remaining resource. After three generations, the remaining resource is 125 units.
Let each generation consume $x$ units. After the first generation: $R - x$.
After the second generation: $(R - x) - x = R - 2x$.
After the third generation: $(R - 2x) - x = R - 3x$.
Given that the final remaining resource is 125 units: $R - 3x = 125$.
To find the relationship between $R$ and $x$, assume each generation consumes half of the remaining resource:
First generation: $x = \frac{R}{2}$.
Second generation: $x = \frac{R - x}{2} = \frac{R - \frac{R}{2}}{2} = \frac{R}{4}$.
Third generation: $x = \frac{R - \frac{R}{2} - \frac{R}{4}}{2} = \frac{R}{8}$.
Substituting into the equation $R - 3x = 125$: $R - 3 \times \frac{R}{8} = 125$.
Simplifying: $R - \frac{3R}{8} = 125$ $\Rightarrow$ $\frac{5R}{8} = 125$.
Solving for $R$: $R = 125 \times \frac{8}{5} = 200$ units.