Calculating Anil's Journey Start Time

This problem involves calculating the start time of a journey given the fractions of the journey covered at two different times. We need to use the information about the distance covered and the time elapsed to determine the speed or rate of travel, and then work backwards to find the start time.

Understanding the Journey Details

We are given two key pieces of information about Anil's journey:

• By 10 a.m., he covered \(\frac{1}{2}\) of his journey.
• By 1 a.m., he covered \(\frac{4}{5}\) of his journey. (Assuming '1 a.m.' here means 1 p.m., as the journey progresses past 10 a.m. and the options suggest a morning start time).

Determining the Distance Covered Between Two Points in Time

Let the total distance of the journey be \(D\).

• At 10 a.m., the distance covered is \(\frac{1}{2}D\).
• At 1 p.m., the distance covered is \(\frac{4}{5}D\).

The distance covered between 10 a.m. and 1 p.m. is the difference between the distance covered at 1 p.m. and the distance covered at 10 a.m.

Difference in distance \( = \frac{4}{5}D - \frac{1}{2}D \)

To subtract these fractions, we find a common denominator, which is 10.

\( \frac{4}{5}D = \frac{4 \times 2}{5 \times 2}D = \frac{8}{10}D \)

\( \frac{1}{2}D = \frac{1 \times 5}{2 \times 5}D = \frac{5}{10}D \)

Difference in distance \( = \frac{8}{10}D - \frac{5}{10}D = \frac{3}{10}D \)

So, Anil covered \(\frac{3}{10}\) of the total journey between 10 a.m. and 1 p.m.

Calculating the Time Taken for This Portion of the Journey

The time elapsed between 10 a.m. and 1 p.m. is:

• From 10 a.m. to 12 p.m. is 2 hours.
• From 12 p.m. to 1 p.m. is 1 hour.
• Total time = 2 hours + 1 hour = 3 hours.

Anil covered \(\frac{3}{10}\) of the journey in 3 hours.

Determining the Rate of Travel

Assuming Anil traveled at a constant speed, we can find the fraction of the journey covered per hour.

Rate \( = \frac{\text{Fraction of journey covered}}{\text{Time taken}} \)

Rate \( = \frac{\frac{3}{10}}{3 \text{ hours}} \)

Rate \( = \frac{3}{10} \times \frac{1}{3} \text{ per hour} \)

Rate \( = \frac{1}{10} \text{ of the journey per hour} \)

This means Anil covers \(\frac{1}{10}\) of the total journey every hour.

Calculating the Time Taken for the First Half of the Journey

We know that by 10 a.m., Anil had covered \(\frac{1}{2}\) of the journey.

The rate of travel is \(\frac{1}{10}\) of the journey per hour.

Time taken to cover \(\frac{1}{2}\) of the journey \( = \frac{\text{Fraction of journey}}{\text{Rate}} \)

Time taken \( = \frac{\frac{1}{2}}{\frac{1}{10}} \text{ hours} \)

Time taken \( = \frac{1}{2} \times \frac{10}{1} \text{ hours} \)

Time taken \( = \frac{10}{2} \text{ hours} \)

Time taken \( = 5 \text{ hours} \)

So, it took Anil 5 hours to cover the first half of his journey.

Finding the Journey Start Time

Anil finished covering the first half of his journey at 10 a.m., and it took him 5 hours to do so. To find the start time, we subtract the time taken from the end time.

Start Time = 10:00 a.m. - 5 hours

Starting from 10:00 a.m., subtracting 5 hours:

• 10 a.m. - 1 hour = 9 a.m.
• 9 a.m. - 1 hour = 8 a.m.
• 8 a.m. - 1 hour = 7 a.m.
• 7 a.m. - 1 hour = 6 a.m.
• 6 a.m. - 1 hour = 5 a.m.

Therefore, Anil started his journey at 5:00 a.m.

Verification

Let's check if this start time is consistent with the information given:

• Start time: 5:00 a.m.
• By 10:00 a.m., 5 hours have passed (10 a.m. - 5 a.m. = 5 hours). Since it takes 5 hours to cover half the journey, at 10 a.m., he has indeed covered \(\frac{1}{2}\) of the journey.
• By 1:00 p.m., 8 hours have passed (1 p.m. - 5 a.m. = 8 hours). The rate is \(\frac{1}{10}\) of the journey per hour. In 8 hours, he would cover \(8 \times \frac{1}{10} = \frac{8}{10} = \frac{4}{5}\) of the journey. This also matches the given information.

The calculations are consistent with the provided details, assuming '1 a.m.' in the second instance refers to 1 p.m.

Revision Table: Journey Time Calculation

Additional Information on Speed, Distance, and Time Problems

Problems involving speed, distance, and time often assume a constant speed unless stated otherwise. When fractions of a journey are involved, it's helpful to think of the total journey as a unit (like 1 or \(D\)) and express the covered distances as fractions of that unit. The relationship between speed, distance, and time is fundamental:

• Speed = Distance / Time
• Distance = Speed × Time
• Time = Distance / Speed

In problems like this, where speed is constant, the fraction of the journey covered is directly proportional to the time taken. This allows us to set up ratios or calculate a 'rate' (fraction of journey per unit time) as we did above.

For example, if it takes 3 hours to cover \(\frac{3}{10}\) of the journey, then it takes \(\frac{3 \text{ hours}}{3/10} = 3 \times \frac{10}{3} = 10\) hours to cover the entire journey (or \(1 \times D\)).

Once the total time for the journey is known (10 hours in this case), and the start time is 5 a.m., the journey would finish at 5 a.m. + 10 hours = 3 p.m.

Understanding how to manipulate fractions and relate them to time intervals is crucial for solving such quantitative aptitude problems.