A boat covers 32 km upstream and 36 km downstream in 7 hours. Also, it covers 40 km upstream and 48 km downstream in 9 hours. The speed of the boat in still water and that of the stream is

Let the speed of the boat in still water be x km/hr and the speed of the stream be y km/hr. 
Then
Speed upstream $=(\mathrm{x}-\mathrm{y})$ km/hr
$\text{Speed downstream }=(\mathrm{x}+\mathrm{y})\mathrm{km}/\mathrm{hr}$
Now, Time taken to cover 32 km upstream $=\frac{32}{\mathrm{x}-\mathrm{y}}\mathrm{hrs}$
$\text{ Time taken to cover }36\mathrm{km}\text{ downstream }=\frac{36}{\mathrm{x}+\mathrm{y}}\mathrm{hrs}$
But, total time of journey is 7 hours.
$\therefore \frac{32}{\mathrm{x}-\mathrm{y}}+\frac{36}{\mathrm{x}+\mathrm{y}}=7$….(i)
Time taken to cover 40 km upstream $=\frac{40}{\mathrm{x}-\mathrm{y}}$
$\text{ Time taken to cover }48\mathrm{km}\text{ downstream }=\frac{48}{\mathrm{x}+\mathrm{y}}$
In this case, total time of journey is given to be 9 hours.
$\therefore \frac{40}{\mathrm{x}-\mathrm{y}}+\frac{48}{\mathrm{x}+\mathrm{y}}=9$….(ii)
$\text{ Putting }\frac{1}{\mathrm{x}-\mathrm{y}}=\mathrm{u}\text{ and }\frac{1}{\mathrm{x}+\mathrm{y}}=\mathrm{v}\text{ in equations (i) and (ii), we get }$
$32\mathrm{u}+36\mathrm{v}=7\Rightarrow 32\mathrm{u}+36\mathrm{v}-7=0$….(iii)
$40\mathrm{u}+48\mathrm{v}=9\Rightarrow 40\mathrm{u}+48\mathrm{v}-9=0$…..(iv)
Solving these equations by cross – multiplication, we get
$\begin{array}{l}\frac{\mathrm{u}}{36\times (-9)-48\times (-7)}=\frac{-\mathrm{v}}{32\times (-9)-40\times (-7)}=\frac{1}{32\times 48-40\times 36}\\ \Rightarrow \frac{\mathrm{u}}{-324+336}=\frac{-\mathrm{v}}{-288+280}=\frac{1}{1536-1440}\\ \Rightarrow \frac{\mathrm{u}}{12}=\frac{\mathrm{v}}{8}=\frac{1}{96}\\ \Rightarrow \mathrm{u}=\frac{12}{96}\text{ and }\mathrm{v}=\frac{8}{96}\\ \Rightarrow \mathrm{u}=\frac{1}{8}\text{ and }\mathrm{v}=\frac{1}{12}\end{array}$
$\text{ Now, }\mathrm{u}=\frac{1}{8}\Rightarrow \frac{1}{\mathrm{x}-\mathrm{v}}=\frac{1}{8}\Rightarrow \mathrm{x}-\mathrm{y}=8$…..(v)
$\text{ And, }\mathrm{v}=\frac{1}{12}\Rightarrow \frac{1}{\mathrm{x}+\mathrm{y}}=\frac{1}{12}\Rightarrow \mathrm{x}+\mathrm{y}=12$…..(vi)
Solving equations (v) and (vi), we get x = 10 and y = 2
Hence, Speed of the boat in still water = 10 km/hr.