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Linear Fitness Decrease Model: Finding Function and Value at Generation 60

Published on March 8, 2026

Word-Problem

This problem requires finding the equation of a line given two points (generation, fitness) to model a linear decrease in population fitness, and then using that function to predict the fitness at 60 generations.

Question

Answer

The problem states that the mean fitness ( ) of a population decreases linearly each generation ( ). This means the relationship between and can be modeled by a linear equation:
where is the slope (rate of change of fitness per generation) and is the -intercept (the fitness at generation ).
We are given two data points:
  1. After 25 generations, the fitness was 0.96. This gives the point .
  2. After 100 generations, the fitness was 0.88. This gives the point .
Step 1: Calculate the slope ( ).
The slope is calculated as the change in fitness divided by the change in generations:
To express this as a decimal or a simpler fraction:
As a decimal:
We will use the fraction form for exactness in the next step.
Step 2: Calculate the -intercept ( ).
We use the point-slope form or substitute one point into the linear equation . Let’s use :
Since :
Now, solve for :
Convert to a fraction with a denominator of 3:
As a decimal approximation for :
Step 3: Create the function describing fitness as a function of generation, .
Alternatively, using the decimal values calculated from the fraction for the slope:
Let’s verify the intercept using :
To get a common denominator of 75:
This matches the given data.
The function is:
Step 4: Calculate the fitness after 60 generations ( ).
Substitute into the function:
Simplify the first term:
Now calculate the value of :
Alternatively, using fractions for precision:
Multiply numerator and denominator by 10:
Divide by 2:
Convert the fraction to a decimal:
To get the exact decimal representation:
Since , .
Wait, let’s stick to the result from the arithmetic:
If we use :
The fitness after 60 generations is , or approximately .
Answer:
The function describing fitness as a function of generation is:
The fitness after 60 generations is:
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