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Abdul travelled 300 km by train and 200km by taxi, it took him 5 hours 30 minute.But if he travels 260 km by train and 240 km by taxi, he takes 6 mins longer.Find the speed of the train and that of the taxi.

Answer 1

ANSWER:

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Let the speed of the train be x km/hr and
let the speed of the taxi be y km/hr.
$time = \frac{distance}{speed} \\ \\ time \: taken \: to \: cover \: 300km \: by \: train = \frac{300}{x} hrs \\ \\ time \: taken \: to \: \: cover \: 200km \: by \: taxi = \frac{220}{y} hrs$

According to the first condition,

$\frac{300}{x} + \frac{200}{y} = 5 \frac{30}{60}............(1hr = 60min) \\ \\ therefore \\ \frac{300}{x} + \frac{200}{y} = \frac{11}{2}........1$

Similarly,

$time \: taken \: to \: cover \: 260km \: by \: train = \frac{260}{x} hrs \\ \\ time \: taken \: to \: cover \: 240km \: by \: taxi = \frac{240}{y} hrs$

According to the second condition,

$\frac{260}{x} + \frac{240}{y} = 5 \frac{36}{60} \\ \\ therefore \\ \frac{260}{x} + \frac{240}{y} = \frac{28}{5}.......2$

$substituting \: \frac{1}{x} = m \: and \: \frac{1}{y} = n \: in \: equations \: 1 \: and \: 2$

$300m + 200n = \frac{11}{2} \\ i.e. \: 600m + 400n = 11.........3$
Similarly,

$260m + 240n = \frac{28}{5} \\ i.e.1300m + 1200n = 28...........4$

Multiplying equation (3) by 3,

$1800m + 1200n = 33........5$

Subtracting equation (4) from (5)

$\: \: \: \: \: \: \: 1800m + 1200n = 33 \\ \: \: \: \: \: \: 1300m + 1200n = 28 \\ ( - ) \: \: \: \: \: \: \: \: \: \: \: \: ( - ) \: \: \: \: \: \: \: \: \: \: ( - ) \\ - - - - - - - - - - - - \\ \: \: \: \: \: \: 500m \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = 5$

$m = \frac{5}{500} \\ m = \frac{1}{100}$

Substituting values of m in equation (3) we get,

$600 \times \frac{1}{100} + 400n = 11 \\ \\ 6 + 400n = 11 \\ \\ 400n = 11 - 6 \\ \\ 400n = 5 \\ \\ n = \frac{5}{400} \\ \\ n = \frac{1}{80}$

Resubstituting the value of m and n,

$\frac{1}{x} = \frac{1}{100} and \frac{1}{y} = \frac{1}{80} \\ \\ therefore \\ \\ x = 100 \: and \: \\ y = 80$

Therefore,
the speed of train is 100km/hr and

the speed of taxi is 80km/hr.

Answer 2

Case 1:

Distance travelled by train = 300 km

Distance travelled by taxi = 200 km

Let the speed of train = x

And the speed of taxi = y

Time taken by train =
$\frac{distance}{speed} = \frac{300}{x}$

Time taken by Taxi =

$\frac{distance}{speed} = \frac{200}{y}$

Total Time Taken = 5 hours and 30 minutes = 5½ hours

$\frac{300}{x} + \frac{200}{y} = \frac{11}{2} \: \: \: = > eq.1$

Case 2 :

Distance covered by train = 260 km

Distance covered by taxi = 240 km

Again , Let the speed of train = x

And the speed of taxi = y

Time taken by train =

$\frac{260}{x}$

Time taken by taxi =

$\frac{240}{y}$

This time the time taken is 6 minutes more than the previous one :

Total time taken = 5 hours 36 sec

$5 \frac{36}{60} = \frac{28}{5} \: hours$

$\frac{260}{x} + \frac{240}{y} = \frac{28}{5} \: \: \: = > \: eq2$

On solving we get x = 100 km/sec and y = 80 km/hr