Understanding the Speed, Distance, and Time Problem

This problem involves calculating the time saved when the speed of travel increases over a fixed distance. We are given the initial speed and the time taken to cover a certain distance. We first need to find the total distance covered. Then, using the new speed, we can find the new time taken to cover the same distance and finally calculate the difference in time, which is the time saved.

Calculating the Total Distance Covered

The relationship between speed, distance, and time is given by the formula:

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

We are given:

• Initial Speed: 62 km/hr
• Initial Time: \(3\frac{1}{2}\) hours

Let's convert the initial time to a decimal or improper fraction for easier calculation:

\( 3\frac{1}{2} \text{ hours} = 3.5 \text{ hours} \)

Now, we can calculate the distance:

\( \text{Distance} = 62 \text{ km/hr} \times 3.5 \text{ hours} \)

\( \text{Distance} = 62 \times \frac{7}{2} \text{ km} \)

\( \text{Distance} = 31 \times 7 \text{ km} \)

\( \text{Distance} = 217 \text{ km} \)

So, the total distance between the two places is 217 km.

Calculating the New Speed and New Time

The problem states that the speed is increased by 8 km/hr.

• Initial Speed: 62 km/hr
• Increase in Speed: 8 km/hr

New Speed = Initial Speed + Increase in Speed

New Speed = 62 km/hr + 8 km/hr

New Speed = 70 km/hr

Now we need to find the time taken to cover the same distance (217 km) at the new speed (70 km/hr). The formula for time is:

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

Using the distance and the new speed:

\( \text{New Time} = \frac{217 \text{ km}}{70 \text{ km/hr}} \)

\( \text{New Time} = \frac{217}{70} \text{ hours} \)

Let's simplify the fraction:

\( \frac{217}{70} = \frac{31 \times 7}{10 \times 7} = \frac{31}{10} = 3.1 \text{ hours} \)

So, the new time taken is 3.1 hours.

Calculating the Time Saved

The time saved is the difference between the initial time and the new time.

• Initial Time: 3.5 hours
• New Time: 3.1 hours

Time Saved = Initial Time - New Time

Time Saved = 3.5 hours - 3.1 hours

Time Saved = 0.4 hours

Converting Time Saved to Minutes

The question asks for the time saved in minutes. We know that 1 hour is equal to 60 minutes.

Time Saved in Minutes = Time Saved in Hours \(\times\) 60 minutes/hour

Time Saved in Minutes = 0.4 hours \(\times\) 60 minutes/hour

Time Saved in Minutes = \(\frac{4}{10} \times 60\) minutes

Time Saved in Minutes = \(4 \times 6\) minutes

Time Saved in Minutes = 24 minutes

Therefore, 24 minutes would be saved if the speed is increased by 8 km/hr.

Conclusion on Time Saved Calculation

By first calculating the distance, then the new time required with the increased speed, and finally finding the difference between the original and new travel times, we determined that 24 minutes would be saved.

Revision Table: Speed Distance Time Calculations

Review the key concepts used in this problem:

• The relationship \( \text{Distance} = \text{Speed} \times \text{Time} \) is fundamental.
• Units must be consistent (e.g., km and hours).
• Fractions and decimals can be used for time.
• Converting between hours and minutes is necessary for the final answer.

Additional Information: Understanding Time, Speed, and Distance

The concepts of time, speed, and distance are interconnected. If you know any two of these values, you can always find the third using the relevant formula derived from \( \text{Distance} = \text{Speed} \times \text{Time} \). Increasing speed while keeping the distance constant will always decrease the time taken. Conversely, decreasing speed will increase the time taken.

• Speed: The rate at which someone or something is able to move or operate (Distance per unit Time).
• Distance: The length of the space between two points.
• Time: The measured or measurable period during which an action, process, or condition exists or continues.
• Unit Conversion: It is crucial to pay attention to units. If speed is in km/hr, time should be in hours and distance in km. If the final answer is required in minutes, remember to convert hours to minutes by multiplying by 60.