SSC CGL 26 Oct 2014 Shift 2 Paper

Speed of boat in still water = x kmph
Speed of current = y kmph
∴ Rate upstream = (x – y) kmph
Rate downstream = (x + y) kmph
Case I,
$\frac{24}{\mathit{x}-\mathit{y}}$ + $\frac{28}{\mathit{x}+\mathit{y}}$ = 6
⇒ $\frac{24\left(\mathit{x}+\mathit{y}\right)+28\left(\mathit{x}-\mathit{y}\right)}{\left(\mathit{x}-\mathit{y}\right)\left(\mathit{x}+\mathit{y}\right)}$ = 6
⇒ 52 x – 4 y = 6 (${\mathit{x}}^2\mathit{–}{\mathit{y}}^2$)
⇒ 26 x – 2 y = 3 (${\mathit{x}}^2\mathit{–}{\mathit{y}}^2$) ... (i)
Case II,
$\frac{30}{\mathit{x}-\mathit{y}}+\frac{21}{\mathit{x}+\mathit{y}}$ = $\frac{13}{2}$
⇒ $\frac{30\left(\mathit{x}+\mathit{y}\right)+21\left(\mathit{x}-\mathit{y}\right)}{\left(\mathit{x}+\mathit{y}\right)\left(\mathit{x}-\mathit{y}\right)}$ = $\frac{13}{2}$
⇒ $\frac{30\mathit{x}+30\mathit{y}+21\mathit{x}-21\mathit{y}}{{\mathit{x}}^2-{\mathit{y}}^2}$ = $\frac{13}{2}$
⇒ 51x + 9y = $\frac{13}{2}\left({\mathit{x}}^2-{\mathit{y}}^2\right)$ ..... (ii)
On dividing equation (i) by (ii),
$\frac{36\mathit{x}-2\mathit{y}}{51\mathit{x}+9\mathit{y}}$ = $\frac{\frac{3}{13}}{2}$ = $\frac{6}{13}$
⇒ 338 x – 26 y = 306 x + 54 y
⇒ 338 x – 306 x = 26 y + 54 y
⇒ 32 x = 80 y
⇒ 2 x = 5 y
⇒ y = $\frac{2\mathit{x}}{5}$ ... (iii)
From equation (i),
26 x - $\frac{2\times 2\mathit{x}}{5}$ = 3 $\left({\mathit{x}}^2-\frac{4{\mathit{x}}^2}{25}\right)$
⇒ $\frac{130\mathit{x}-4\mathit{x}}{5}$ = $3\left(\frac{25{\mathit{x}}^2-4{\mathit{x}}^2}{25}\right)$
⇒ $\frac{126\mathit{x}}{5}$ = $\frac{3\times 21{\mathit{x}}^2}{25}$
⇒ 3 × 21 = 126 × 5
⇒ x = $\frac{126\times 5}{3\times 21}$ = 10 kmph