## 15. The Emir and His Oil

My emirate encompasses a vast desert, in the middle of which I have built my palace. It also includes territorial waters along the coastline, which I value greatly. The territorial waters are quite extensive (covering an area that reaches a third of the desert's area), and part of my oil field is located there. Knowing that the area of the desert with oil deposits is three times the area of the marine territory without oil and that one-seventh of my territory without oil is in the sea, could you determine exactly what fraction of my oil field is in the sea?
15. Let \( S \) be the total area of the emirate. The area of its marine part is \( S / 4 \).

Let \( x \) be the fraction of the total area that

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is occupied by the sea without oil-bearing areas. Then the part of the sea with oil fields has an area of \( (1 / 4 - x) S \). (But the area of the sea without oil is \( x S \).) The area of the desert rich in oil is \( 3 x S \), and the area of the desert without oil is \( 6 x S \). The total area of the emirate is

\[
S / 4 - x S + x S + 3 x S + 6 x S = S
\]

From this, \( x = 1 / 12 \). Therefore, the marine territory containing oil fields constitutes

\[
\frac{\left(\frac{1}{4} - \frac{1}{12}\right) S}{\frac{3}{12} S + \left(\frac{1}{4} - \frac{1}{12}\right) S} = \frac{\frac{1}{6}}{\frac{1}{4} + \frac{1}{6}} = \frac{2}{5}
\]

of the total area of the emirate.