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A train covers a distance at a uniform speed . If the train could have been \[{\text{10 kmph}}\] faster , it would have taken two hours less than the scheduled time. And if the train were slower by \[{\text{10 kmph}}\], it would have taken \[{\text{3}}\] hours more than the scheduled time. Find the distance covered by the train.

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Hint – Use the formula of speed and make equations according to the question .

Let the distance covered by the train \[ = d\;km\]
Speed \[ = s\;kmph\]
\[t{\text{ }}\]be the scheduled time
Distance = Speed X Time = constant in this case
\[
  st = (s + 10)(t - 2)\; = d{\text{ }}\;{\text{ }}\;{\text{ }}......\left( i \right) \\
  st = (s - 10)(t + 3) = d\;{\text{ }}\;{\text{ }}\;{\text{ }}\;.....\left( {ii} \right) \\
\]
Since the distance travelled is same in all the cases
Simplifying the \[\left( i \right)\],
\[
  st = st - 2s + 10t - 20 \\
   \Rightarrow 2s - 10t = - 20\;{\text{ }}\;{\text{ }}.....\left( {iii} \right) \\
\]
Simplifying the \[\left( {ii} \right)\] we get ,
\[
  st = st + 3s + 10t - 30\; \\
   \Rightarrow 3s - 10t = 30\;{\text{ }}\;............\left( {iv} \right) \\
\]
Multiplying \[\left( {iii} \right)\] by \[3\;\& {\text{ }}\left( {iv} \right)\] by \[\;2\], we get,
\[
  6s - 30t = - 60\;{\text{ }}\;{\text{ }}\;{\text{ }}\;{\text{ }}\;{\text{ }}\;{\text{ }}\;{\text{ }}..........\left( v \right) \\
  6s - 20t = 60\;{\text{ }}\;{\text{ }}\;{\text{ }}\;{\text{ }}\;{\text{ }}\;{\text{ }}\;{\text{ }}\;..........{\text{ }}\left( {vi} \right) \\
\]
Subtracting equation \[\left( {vi} \right)\] from \[\left( v \right)\], we get,
\[
   - 10t = - 120 \\
  t = 12hrs \\
\]
Putting the value of \[t{\text{ }}\] in equation \[\left( {iii} \right)\],
Then we get speed as,
\[s = 50\,kmph\,\]
And we know Distance = Speed x Time
So,
\[d = st = 50 \times 12 = 600\;km\;\]
Hence the distance is \[600\;km\].

Note – Whenever you struck with this type of problem always try to make equations of various variables according to question , then solve it and get the asked parameter. During solving this problem we kept in mind that the distance travelled is always same whatever the speed may be.
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