Three partners shared the profit in a business in the ratio 5 : 7 : 8. They had partnered for 14 months, 8 months and 7 months respectively. What was the ratio of their investments? 

$$\begin{gathered} \mathrm{Let} \mathrm{their} \mathrm{investments} \mathrm{be} \mathrm{Rs}. \mathrm{x} \mathrm{for} 14 \mathrm{months}, \mathrm{Rs}. \mathrm{y} \mathrm{for} 8 \mathrm{months} \\ \mathrm{and} \mathrm{Rs}. \mathrm{z} \mathrm{for} 7 \mathrm{months} \mathrm{respectively}. \end{gathered}$$

$$\begin{gathered} 14x:8y:7z=5:7:8 \\ \text{ Now, }\frac{14\mathrm{x}}{8\mathrm{y}}=\frac{5}{7}\iff 98\mathrm{x}=40\mathrm{y}\iff \mathrm{y}=\frac{49}{20}\mathrm{x} \\ \text{ And, }\frac{14\mathrm{x}}{7\mathrm{z}}=\frac{5}{8}\iff 112\mathrm{x}=35\mathrm{z}\iff \mathrm{z}=\frac{112}{35}\mathrm{x}=\frac{16}{5}\mathrm{x} \end{gathered}$$

$\therefore x:y:z=x:\frac{49}{20}x:\frac{16}{5}x=20:49:64.$