Given:
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, show that it cannot be proven that The United States had more fallow acreage than planted acreage.
Reference:Prove that it cannot be proven that "The United States had more fallow acreage than planted acreage"
Proof:
Let $p_u,f_u$ be the planted/fallow acreage in the U.S. and $p_s,f_s$ be the planted/fallow acreage in the Soviet Union. Also, let $y_u$ be the yield per planted acre in the U.S. and $y_s$ be the yield per planted acre in the Soviet Union. The information given says that
$$\frac{y_s}{y_u} = 0.68 \; \iff \; y_s = 0.68y_u \tag{1}\label{eq1}$$
The total yield in the U.S. is $y_u p_u$, so the yield per all of the acres would be
$$y_{au} = \frac{y_u p_u}{p_u + f_u} \tag{2}\label{eq2}$$
Similarly, for the Soviet Union, it's yield per all of the acres would be
$$y_{as} = \frac{y_s p_s}{p_s + f_s} \tag{3}\label{eq3}$$
By cross-multiplying and combining the terms for $f_s$ and $p_s$, you get
\begin{align} y_{as}(p_s + f_s) & = y_s p_s \\ y_{as}p_s + y_{as}f_s & > = y_s p_s \\ y_{as}f_s & = y_s p_s - y_{as}p_s \\ f_s & = \frac{p_s(y_s - y_{as})}{y_{as}} \tag{4}\label{eq4} \end{align}
It's also given that
$$\frac{y_{as}}{y_{au}} = 1.14 \; \iff \; y_{as} = 1.14y_{au} > \tag{5}\label{eq5}$$
This is the only information provided, so if $2$ sets of values are found which are consistent with the above equations but with one showing that $f_u \gt p_u$ and the other showing that $f_u \lt p_u$, then this would answer what's requested.
Let's set $y_u = 100$. Then from \eqref{eq1}, you get $y_s = 68$. Next, let $p_u = 10,000,000$ and $f_u = 11,000,000$. Substituting these into \eqref{eq2} gives $y_{au} = 47.619\ldots$. From \eqref{eq5}, this gives $y_{as} = 54.285\ldots$. From \eqref{eq4}, you get
$$f_s = \frac{p_s(68 - 54.285\ldots)}{54.285\ldots} > \tag{6}\label{eq6}$$
Note you can plug any value of $p_s$ you want to get a specific value of $f_s$, e.g., if $p_s = 10,000,000$, then $f_s = > 2,526,315.789\ldots$.
Next, consider $f_u = 9,000,000$. Then \eqref{eq2} gives $y_{au} = > 52.631\ldots$. From \eqref{eq5}, this gives $y_{as} = 60$. From \eqref{eq4}, you get
$$f_s = \frac{p_s(68 - 60)}{60} \tag{7}\label{eq7}$$
If you use $p_s = 10,000,000$ again, then $f_s = 1,333,333.333\ldots$.
All of these values are consistent with the equations relating the only information which was provided, but with one set showing more fallow acreage than planted acreage in the U.S. (i.e., $f_u = > 11,000,000 \gt p_u = 10,000,000$) and the other one showing the opposite (i.e., $f_u = 9,000,000 \lt p_u = 10,000,000$).
A main reason why you can't prove which of the fallow and planted acreage in the U.S. is greater is because there are $6$ unknowns of $p_u,f_u,p_s,f_s,y_u$ and $y_s$, but only $4$ equations relating them. As these equations are consistent with each other, it's an under-determined system of equations, with $2$ degrees of freedom in this case (in general, you would have more than $2$ if any of the equations are linearly dependent). Also, note the question numeric value restrictions are for comparing values between the U.S. and the Soviet Union, meaning there are fewer constraints among values within the U.S. (and the Soviet Union as well).
The proof above is too complicated to understand, or maybe it's more important to see a big picture than rather know how to prove this. Is there a more concise, shorter way, simpler way to prove? I don't know, with pictures, WLOG?
The information given is consistent with the following values:
To check: $$\frac{\text{SU crop yield per planted acre}}{\text{US crop yield per planted acre}} =\frac{\frac{68}{34}}{\frac{100}{34}}=\frac{68}{100}=68\,\%,$$ $$\frac{\text{SU crop yield per total acre}}{\text{US crop yield per total acre}} =\frac{\frac{68}{34}}{\frac{100}{57}}=\frac{57}{50}=114\,\%.$$
As this consistent set of values has less fallow than planted US acres, no proof of the contrary is possible.