Given:
In the United States between 1850 and 1880, the number of farmers continued to increase, but at a rate lower than that of the general population.
Prove that following statement directly contradicts the information presented above.
The proportion of farmers in the general population increased from 68 percent in 1850 to 72 percent in 1880.
There was the answer:
Let $x_0$ be the number of farmers in 1850 Let $x_1$ be the number of farmers in 1850
Let $y_0$ be the number total population in 1850 Let $y_1$ be the number total population in 1850
The first statement says
$x_1>x_0$ and $\frac {x_1}{x_0} < \frac{y_1}{y_0}$
The contradictory statement says:
$\frac {x_0}{y_0} = 0.68$ and $\frac {x_1}{y_1} = 0.72$ or $\frac{x_0}{y_0} < \frac{x_1}{y_1}$
So the two are not compatible.
This was the objection to the answer:
How do you know the rate of farmers increasing isn't the derivative with respect to time? and therefore $\frac{x_1-x_0}{30}<\frac{y_1-y_0}{30}$?
The answer isn't solid because it can't explain why the problem statement didn't refer to "rate" as the change in farmers with respect to time, and the growth of the population with respect to time.
If the problem did refer to the derivative in physics, then the "following statement" doesn't contradict the statement and therefore the answer is false?
Correct. We can provide a very simple counterexample which is consistent with both statements:
$$x_0=34$$ $$y_0=50$$ $$x_1=72$$ $$y_1=100$$