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A journey of 192 km from Mumbai to Pune takes two hours less by a fast train than by a slow train. If the average speed of the slow train is 16 km/hr less than that of the fast train, find the average speed of each train.

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Last updated date: 13th Jun 2024
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Answer
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Hint: We are required to find the average speed of the two trains based on the conditions provided to us. To solve this question, we need to know about the concept of speed, distance, and time. We will use the formula of the speed and based on that we will solve our given conditions.

Formula Used: We will use the formula \[{\rm{Speed = }}\dfrac{{{\rm{Distance}}}}{{{\rm{Time}}}}\] to solve our question .

Complete step-by-step answer:
We are given that the total distance between Mumbai and Pune is 192 km.
Now we are given two types of trains in the question. One is a fast train and another one is the slow train.
Let us assume that the speed of the fast train is \[x\].
Now we are given that the slower train has an average speed of 16 km/hr less than that of the fast train.
This means that the average speed of the slower train is,
The average speed of slower train \[ = x - 16\]
Now the time taken to cover the total distance by both the trains will be,
Using the formula of speed,
\[{\rm{Time = }}\dfrac{{{\rm{Distance}}}}{{{\rm{Speed}}}}\]
Time taken by fast train to cover a distance of 192 km with \[x\]km/hr speed \[ = \dfrac{{192}}{x}\]
Time taken by slow train to cover a distance of 192 km with \[x - 6\]km/hr speed \[ = \dfrac{{192}}{{x - 16}}\]
Now the question says that the fast train takes two hours less than the slow train. This means that the difference in the time taken by the slow train and the fast train will be 2 hours.
So, according to the question,
Time taken by slow train\[ - \]Time taken by fast train\[ = 2\]
On substituting the values in the above equation, we get
\[\dfrac{{192}}{{x - 16}} - \dfrac{{192}}{x} = 2\]
Taking LCM on the left hand side, we get
\[ \Rightarrow \dfrac{{192x - 192\left( {x - 16} \right)}}{{\left( {x - 16} \right)x}} = 2\]
On cross multiplication, we get
\[ \Rightarrow 192x - 192x + 3072 = 2\left\{ {\left( {x - 16} \right)x} \right\}\]
Simplifying the equation further, we get
\[\begin{array}{l} \Rightarrow 3072 = 2\left( {{x^2} - 16x} \right)\\ \Rightarrow 1536 = {x^2} - 16x\\ \Rightarrow {x^2} - 16x - 1536 = 0\end{array}\]
Factorising above equation, we get
\[\begin{array}{l} \Rightarrow {x^2} - 48x + 32x - 1536 = 0\\ \Rightarrow \left( {x - 48} \right)\left( {x + 32} \right) = 0\end{array}\]
We will now find the value of \[x\]
\[\begin{array}{l}x - 48 = 0\\ \Rightarrow x = 48\\x + 32 = 0\\ \Rightarrow x = - 32\end{array}\]
Since \[x\]is the speed and speed cannot be negative so we reject \[x = - 32\].
Thus, the average speed of our fast train is 48 km/hr.
The average speed of the slower train is \[ = x - 16 = 48 - 16 = 32{\rm{km/hr}}\]

Hence, the average speed of the slower train is 32 km/hr.

Note: We can check our answer by substituting the value of \[x = 48\] in the equation \[\dfrac{{192}}{{x - 16}} - \dfrac{{192}}{x} = 2\]. If the equation will be satisfied so our solution will be correct otherwise wrong.
On doing the substitution, we get
\[\begin{array}{l}\dfrac{{192}}{{x - 16}} - \dfrac{{192}}{x} = 2\\ \Rightarrow \dfrac{{192}}{{48 - 16}} - \dfrac{{192}}{{48}} = 2\end{array}\]
Simplifying the equation, we get
\[\begin{array}{l} \Rightarrow 6 - 4 = 2\\ \Rightarrow 2 = 2\end{array}\]
As the equation is satisfied so the solution is correct.