Angela and Barry share a piece of land. The ratio of the area of Angela's portion to the area of Barry's portion is $3: 2$. They each grow corn and peas on their piece of land. The entire piece of land is covered by corn and peas in the ratio $7: 3$. On Angela's portion of the land, the ratio of corn to peas is $4: 1$. What is the ratio of corn to peas for Barry's portion?
(A) $11: 9$
(B) $2: 3$
(C) $3: 2$
(D) $3: 7$
(E) $1: 4$
Solution 1

Suppose that Angela and Barry share 100 hectares of land. (We may assume any convenient total area, since this is a multiple choice problem.)

Since the ratio of the area of Angela's land to the area of Barry's land is $3: 2$, then Angela has $\frac{3}{5}$ of the 100 hectares, or 60 hectares. Barry has the remaining 40 hectares.

Since the land is covered by corn and peas in the ratio $7: 3$, then $\frac{7}{10}$ of the 100 hectares (that is, 70 hectares) is covered with corn and the remaining 30 hectares with peas.

On Angela's land, the ratio is $4: 1$ so $\frac{4}{5}$ of her 60 hectares, or 48 hectares, is covered with corn and the remaining 12 hectares with peas.

Since there are 70 hectares of corn in total, then Barry has $70-48=22$ hectares of corn.

Since there are 30 hectares of peas in total, then Barry has $30-12=18$ hectares of peas.

Therefore, the ratio of corn to peas on Barry's land is $22: 18=11: 9$.

## Solution 2

Suppose that the total combined area of land is $x$.

Since the ratio of the area of Angela's land to the area of Barry's land is $3: 2$, then Angela has $\frac{3}{5}$ of the land, or $\frac{3}{5} x$, while Barry has the remaining $\frac{2}{5} x$.

Since the land is covered by corn and peas in the ratio $7: 3$, then $\frac{7}{10} x$ is covered with corn and the remaining $\frac{3}{10} x$ with peas.

On Angela's land, the ratio is $4: 1$ so $\frac{4}{5}$ of her $\frac{3}{5} x$, or $\frac{4}{5}\left(\frac{3}{5} x\right)=\frac{12}{25} x$, is covered with corn and the remaining $\frac{3}{5} x-\frac{12}{25} x=\frac{3}{25} x$ with peas.

Since the area of corn is $\frac{7}{10} x$ in total, then Barry's area of corn is $\frac{7}{10} x-\frac{12}{25} x=\frac{35}{50} x-\frac{24}{50} x=\frac{11}{50} x$. Since the area of peas is $\frac{3}{10} x$ in total, then Barry's area of peas is $\frac{3}{10} x-\frac{3}{25} x=\frac{15}{50} x-\frac{6}{50} x=\frac{9}{50} x$. Therefore, the ratio of corn to peas on Barry's land is $\frac{11}{50} x: \frac{9}{50} x=11: 9$.

ANswer: (A)