Explanation:

solution: Let the speed of train is x km/HR

and speed of taxi is y km/hr

Since,Travelling time is given in both the cases,calculate the time taken by train and by taxi and form linear equations

Case1: Abdul travelled 300km by trains and 200km by taxi, it took 5 hours 30 minutes.

Time taken by train to cover 300 km with the speed x km/hr

Time= Distance travelled/speed

= 300/x hr

Time taken by taki to travel 200 km,

= 200/y

\begin{gathered} \frac{300}{x} + \frac{200}{y} = \frac{11}{2} \: \: \: \: eq1\\ \end{gathered}

x

300

+

y

200

=

2

11

eq1

CASE2: Abdul travels 260km by train and 240 km by taxi,he takes 6 minutes longer..

Time taken by train to cover 260 km with the speed x km/hr

Time= Distance travelled/speed

= 260/x hr

Time taken by taki to travel 240 km,

= 240/y

\begin{gathered} \frac{260}{x} + \frac{240}{y} = 5 \: hr \: 36 \: min \\ \\ \frac{260}{x} + \frac{240}{y} = 5 \frac{36}{60} \\ \\ \frac{260}{x} + \frac{240}{y} = 5 \frac{3}{5} \\ \\ \frac{260}{x} + \frac{240}{y} = \frac{28}{5} \: \: \: \: \: eq2 \\ \\ \end{gathered}

x

260

+

y

240

=5hr36min

x

260

+

y

240

=5

60

36

x

260

+

y

240

=5

5

3

x

260

+

y

240

=

5

28

eq2

Eq1 and 2 can be converted to linear equation

\begin{gathered}let \\ \\ \frac{1}{x} = a \\ \\ \frac{1}{y} = b \\ \\ 300a + 200b = \frac{11}{2} \: \: \: \: ...eq3\\ \\ 65a + 60b = \frac{7}{5} \: \: \:... eq4 \\ \\ \end{gathered}

let

x

1

=a

y

1

=b

300a+200b=

2

11

...eq3

65a+60b=

5

7

...eq4

On solving these linear equations by elimination method,

Multiply eq3 by 60 and eq4 by 200 and subtract both

\begin{gathered}60[300a + 200b = \frac{11}{2}]\\\\18000a+12000b=330 \: \: \: \: ...EQ5\\ \\ 200[65a + 60b = \frac{7}{5}]\\\\ 13000a+12000b=280 \: \: \:... eq6\\ \\ \end{gathered}

60[300a+200b=

2

11

]

18000a+12000b=330...eq5

200[65a+60b=

5

7

]

13000a+12000b=280...eq6

Eq5-eq6

\begin{gathered}5000a=50\\\\a=\frac{50}{5000}\\\\a=\frac{1}{100}\\\\\end{gathered}

5000a=50

a=

5000

50

a=

100

1

put the value of a in eq 4

b= 1/80

Now

\begin{gathered}a = \frac{1}{100} \\ \\ b = \frac{1}{80} \\ \\ \end{gathered}

a=

100

1

b=

80

1

Thus,

\begin{gathered} \frac{1}{x} = a \\ \\ \frac{1}{x} = \frac{1}{100} \\ \\ x = 100 \\ \\ \frac{1}{y} = b \\ \\ \frac{1}{y} = \frac{1}{80} \\ \\ y = 80 \\ \\ \end{gathered}

x

1

=a

x

1

=

100

1

x=100

y

1

=b

y

1

=

80

1

y=80

Thus,the speed of train is 100km/hr and speed of taxi is 80 km/hr.

Hope it helps you.