GRE exam word problem: show that a higher percentage of total agricultural acreage was fallow in United States than in the Soviet Union.

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, show that a higher percentage of total agricultural acreage was fallow in United States than in the Soviet Union.

Proof:

Let $USP$ for US planted and $SUP$ for Soviet Union planted. The same for $USF$ and $SUF$. We know that $SUP/USP= 0.68.$

How does the sentence

When total agricultural acreage (planted acreage plus fallow acreage) is compared, however, Soviet yield was 114 percent of US yield.

imply that

$(SUF+SUP)/(USF+USP)= 1.14$? (that's the next step of the proof which is the part I'm stuck)

Answer 1

Let the US total output be $C_1$, its planted acreage be $P_1$ and its fallow acreage be $F_1$. Let the Soviet total output be $C_2$, its planted acreage be $P_2$ and its fallow acreage be $F_2$. The yields per planted acre are ${C_1\over P_1}$ and ${C_2\over P_2}$ respectively, so we know $${C_2P_1\over C_1P_2}=.68.\tag{1}$$ The yields per total acre are ${C_1\over P_1+F_1}$ and ${C_2\over P_2+F_2}$ respectively, so we know $${C_2(P_1+F_1)\over C_1(P_2+F_2)}=1.14.\tag{2}$$

From $(1)$ we have $${C_2\over C_1}=.68{P_2\over P_1}$$ Substituting into $(2)$ gives $$ {P_2(P_1+F_1)\over P_1(P_2+F_2)} = {1.14\over.68}>1$$ so that $$ {P_2\over P_2+F_2}>{P_1\over P_1+F_1}$$ This says that a greater fraction of arable land is planted in the Soviet Union than the United states, which is to say that a larger fraction lies fallow.

I can't see any way to make sense of the proof you indicated.

Answer 2

If, in a country, there are $x$ fallow acres for every planted acre, yield per planted acre is $1+x$ times the yield per total acre.

Thus the ratio of yields per planted acre between the Soviet Union (S) and the U.S. (U), $.68$, is $1+x_S\over1+x_U$ times the ratio of yields per total acre, $1.14$. Therefore ${1+x_S\over1+x_U}< 1$, whence $x_S<x_U$. This means that in the U.S., there are more fallow acres per planted acre than there are in the Soviet Union, so the percentage of arable land left fallow is higher in the U. S.