Calculating Speed Increase for Time Saving

This problem involves the relationship between distance, speed, and time. We are given the initial speed and time taken to cover a certain distance and are asked to find the increase in speed required to cover the same distance in less time.

Understanding the Problem

The core concept here is that for a fixed distance, speed and time are inversely proportional. If you increase the speed, the time taken to cover the same distance decreases.

The given information is:

• Initial Time ($T_1$): 7 hours
• Initial Speed ($S_1$): 50 km/h
• Time to be saved: 2 hours

We need to find the required speed increase ($\Delta S$).

Step-by-Step Solution

Step 1: Calculate the total distance

The distance covered is constant in both scenarios. We can calculate this distance using the initial speed and time.

The formula relating distance, speed, and time is:

\( \text{Distance} = \text{Speed} \times \text{Time} \)

Using the initial values:

\( D = S_1 \times T_1 \)

\( D = 50 \text{ km/h} \times 7 \text{ h} \)

\( D = 350 \text{ km} \)

So, the distance between the two towns is 350 km.

Step 2: Determine the new required time

The problem states that 2 hours of travelling time must be saved. The initial time was 7 hours.

New Time ($T_2$) = Initial Time - Time Saved

\( T_2 = T_1 - 2 \text{ hours} \)

\( T_2 = 7 \text{ hours} - 2 \text{ hours} \)

\( T_2 = 5 \text{ hours} \)

The distance of 350 km must now be covered in 5 hours.

Step 3: Calculate the new required speed

Now we use the same distance formula, but this time to find the new speed ($S_2$) required to cover the distance (D) in the new time ($T_2$).

\( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \)

\( S_2 = \frac{D}{T_2} \)

\( S_2 = \frac{350 \text{ km}}{5 \text{ hours}} \)

\( S_2 = 70 \text{ km/h} \)

To save 2 hours of travel time, the speed must be 70 km/h.

Step 4: Calculate the increase in speed

The question asks for the amount by which the speed should be *increased*. This is the difference between the new required speed and the initial speed.

Speed Increase ($\Delta S$) = New Speed ($S_2$) - Initial Speed ($S_1$)

\( \Delta S = S_2 - S_1 \)

\( \Delta S = 70 \text{ km/h} - 50 \text{ km/h} \)

\( \Delta S = 20 \text{ km/h} \)

The speed should be increased by 20 km/h.

Summary of Calculations

Thus, the speed must be increased by 20 km/h to save 2 hours of travel time.

Revision Table: Distance, Speed, and Time

Additional Information: Speed Calculation Tips

When solving problems involving distance, speed, and time, always ensure the units are consistent. For example, if speed is in km/h, time should be in hours and distance in km.

Here are a few points to remember:

• Distance is usually measured in units like kilometers (km) or meters (m).
• Speed is typically measured in units like kilometers per hour (km/h) or meters per second (m/s).
• Time is measured in units like hours (h), minutes (min), or seconds (s).
• Make sure to read the question carefully to understand what is being asked – sometimes it's the new speed, and other times it's the increase/decrease in speed.

This type of problem is common in quantitative aptitude sections of many exams.