Given:
Original question:Alternative Proof to "Prove that it cannot be proven that "The United States had more fallow acreage than planted acreage"
My Question:
A ten year comparison between the United States and the Soviet Union in terms of crop yields per acre revealed that when only planted acreage is compared, Soviet yields were equal to 68 percent of United States yields. When total agricultural acreage (planted acreage plus fallow acreage) is compared, however, Soviet yield was 114 percent of US yield. From the information above, generate and show how to generate a counter example to the statement "the United States had more fallow acreage than planted acreage."
The answer to the linked question by Hagen von Eitzen:
The information given is consistent with the following values:
- US planted acres $=34$
- US fallow acres $=23$
- US total crop harvest $=100$
- SU planted acres $=34$
- SU fallow acres $=0$
- SU total crop harvest $=68$
To check: $$\frac{\text{SU crop yield per planted acre}}{\text{US crop yield per planted acre}} =\frac{\frac{68}{34}}{\frac{100}{34}}=\frac{68}{100}=68\,\%,$$ $$\frac{\text{SU crop yield per total acre}}{\text{US crop yield per total acre}} =\frac{\frac{68}{34}}{\frac{100}{57}}=\frac{57}{50}=114\,\%.$$ Someone gave the answer:
As this consistent set of values has less fallow than planted US acres, no proof of the contrary is possible.
What's the intuition needed to come up with this counterexample?
Did the person just guess this answer and arrive at some plausible set of numbers or is there some tricks to quickly think of this counterexample?
Stackexchange should somehow fix the ">" and "<"
Well 68% of planted acres means, if the US has the same fallow acreage as planted, SU acres ratio will be 34% taking a rough 194% markup to equal US acres total. This leads to roughly 235% of the fallow acreage as planted acreage for the SU to get to 114% of US acres. Now the proof is in that these numbers fit the relevant data technically about yeilds. But, fails to have the US have more fallow than planted acreage. So, it shows there are sets of numbers that defy a proof of the fallow to planted acreage claim. It counters the claim. Therefore, is a counterexample. A statement with a counterexample can not be proven.