Certainly! Let's break down the problem step by step.
(i) Calculate the distance between the stations:
We are given that the train is running at a speed of 80 km/h and it takes 2 hours to travel between Delhi and Mathura.
To find the distance, we use the formula:
[tex]\[ \text{Distance} = \text{Speed} \times \text{Time} \][/tex]
Thus,
[tex]\[ \text{Distance} = 80 \, \text{km/h} \times 2 \, \text{hours} = 160 \, \text{km} \][/tex]
So, the distance between the stations is 160 km.
(ii) How long will it take to cover the same distance if its speed is decreased by 20 km/h?
If the speed of the train is decreased by 20 km/h, then the new speed of the train is:
[tex]\[ 80 \, \text{km/h} - 20 \, \text{km/h} = 60 \, \text{km/h} \][/tex]
Now, we need to calculate the time taken to cover the same distance (160 km) at the new speed.
Using the formula:
[tex]\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \][/tex]
Thus,
[tex]\[ \text{Time} = \frac{160 \, \text{km}}{60 \, \text{km/h}} \approx 2.67 \, \text{hours} \][/tex]
Therefore, it will take approximately 2.67 hours (or 2 hours and 40 minutes) to cover the same distance at the reduced speed.
(iii) An express train covers the distance in 1 h 15 min. Calculate the speed of the express train.
First, convert 1 hour and 15 minutes to hours. Since 15 minutes is equal to [tex]\( \frac{15}{60} = 0.25 \)[/tex] hours, we have:
[tex]\[ 1 \, \text{hour} + 0.25 \, \text{hours} = 1.25 \, \text{hours} \][/tex]
Using the formula:
[tex]\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \][/tex]
Thus,
[tex]\[ \text{Speed} = \frac{160 \, \text{km}}{1.25 \, \text{hours}} = 128 \, \text{km/h} \][/tex]
So, the speed of the express train is 128 km/h.
(iv) Calculate the ratio of their speeds.
We need to find the ratio of the initial train's speed to the express train's speed. The initial train's speed is 80 km/h and the express train's speed is 128 km/h.
The ratio of their speeds is:
[tex]\[ \frac{80 \, \text{km/h}}{128 \, \text{km/h}} = 0.625 \][/tex]
Therefore, the ratio of their speeds is 0.625.
To summarize:
(i) The distance between the stations is 160 km.
(ii) It will take approximately 2.67 hours to cover the same distance if the speed is decreased by 20 km/h.
(iii) The speed of the express train is 128 km/h.
(iv) The ratio of their speeds is 0.625.
