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प्रश्न
Ashish goes to his friends house which is 12 km away from his house. He covers half of the distance at a speed of $$x$$ km per hour and the remaining at $$(x + 2)$$ km per hour. If he takes 2 hrs 30 min. to cover the whole distance, find the value of $$x$$.
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उत्तर
Total distance $$= 12\text{ km}$$.
Half of the distance $$= 6\text{ km}$$.
Speed for first half $$= x\text{ km/hr}$$, time taken $$= \frac{6}{x}\text{ hrs}$$.
Speed for second half $$= (x + 2)\text{ km/hr}$$, time taken $$= \frac{6}{x + 2}\text{ hrs}$$.
Total time taken $$= 2\text{ hrs } 30\text{ min} = 2\frac{1}{2}\text{ hrs} = \frac{5}{2}\text{ hrs}$$.
Therefore: $$\frac{6}{x} + \frac{6}{x + 2} = \frac{5}{2}$$
$$6\left(\frac{x + 2 + x}{x(x + 2)}\right) = \frac{5}{2}$$
$$6\left(\frac{2x + 2}{x^2 + 2x}\right) = \frac{5}{2}$$
$$12(2x + 2) = 5(x^2 + 2x)$$
$$24x + 24 = 5x^2 + 10x$$
$$5x^2 - 14x - 24 = 0$$
Factoring: $$5x^2 - 20x + 6x - 24 = 0$$
$$5x(x - 4) + 6(x - 4) = 0$$
$$(x - 4)(5x + 6) = 0$$
$$x = 4 \quad \text{or} \quad x = -\frac{6}{5}$$
Since speed cannot be negative, reject $$x = -\frac{6}{5}$$.
Hence, $$x = 4$$.
