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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.

Was the explanation for the 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.

How do you prove

"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."?

1 Answers1

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Let's go through this one step at a time:

Say there are $n$ planted acres. For every planted acre, there are $x$ fallow acres, so there are $nx$ fallow acres. The number of total acres is therefore $nx + n$, or $(x+1)n$.

Now let $y$ be the total yield. To get the yield per something, we need to divide by that something. So the yield per planted acre is $\frac{y}{n}$, and the yield per total acre is $\frac{y}{(x+1)n}$.

Now $1+x$ times the yield per total acre $\frac{y}{(x+1)n}$ is equal to $\frac{(x+1)y}{(x+1)n}$, or $\frac{y}{n}$. This is equal to the yield per planted acre, thus the statement is proven.

Magma
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  • What is the interpretation of $1+x$? What does that represent? Ratio of land mass? –  Jul 18 '19 at 00:11
  • @Germania As shown above, $(x+1)n$ is the total # of acres and $n$ is the # of planted acres. Thus, $x + 1 = \frac{(x+1)n}{n}$ is the ratio of the total # of acres to the # of planted acres. This is essentially what you're asking to prove. – John Omielan Jul 18 '19 at 02:12