Introduction

Understanding the time and distance aspects of a journey is crucial for effective travel planning. This article details the mathematical approach to determining the start time of a journey, providing a clear example and explanation to help readers grasp the concept.

The Scenario

A person set out on a journey in the morning. By 11 A.M., he had covered 3/8th of the journey, and by 4:30 P.M., he had completed 5/6th of it. We aim to find out at what time the journey began.

Step-by-Step Calculation

We begin by denoting the total duration of the journey as T hours. Let’s break down the problem into smaller, manageable steps to find the start time.

At 11 A.M.

The person has covered frac{3}{8} of the journey. This means he has been traveling for a time denoted as t_1. The distance covered can be represented as:

Distance frac{3}{8} times Total Distance

The time taken to cover frac{3}{8} of the journey is:

t_1 frac{3}{8} T

At 4:30 P.M.

By 4:30 P.M., the traveler has covered frac{5}{6} of the entire journey. The time traveled until this point is denoted as t_2. The distance covered is:

Distance frac{5}{6} times Total Distance

The time taken to cover frac{5}{6} of the journey is:

t_2 frac{5}{6} T

Time Interval Calculation

Calculate the time interval from 11 A.M. to 4:30 P.M.:

The time difference between 11 A.M. and 4:30 P.M. is:

4:30 P.M. - 11:00 A.M. 5 hours and 30 minutes 5.5 hours

The equation can be set up as:

t_2 - t_1 5.5

Substitute the expressions for t_1 and t_2 into the equation:

frac{5}{6} T - frac{3}{8} T 5.5

Finding a Common Denominator and Solving for T

The least common multiple of 6 and 8 is 24. Rewrite the fractions:

frac{5}{6} frac{20}{24} quad and quad frac{3}{8} frac{9}{24}

Substitute back into the equation:

frac{20}{24} T - frac{9}{24} T 5.5

Simplify:

frac{11}{24} T 5.5

Multiply both sides by 24:

11T 5.5 times 24

11T 132

T 12 ; text{hours}

Calculating the Start Time

If the total journey duration T is 12 hours, and the person had reached frac{5}{6} of the journey by 4:30 P.M., we can determine the start time as follows:

The time taken to cover frac{5}{6} of the journey is:

t_2 frac{5}{6} times 12 10 ; text{hours}

Therefore, if he completed 10 hours of travel by 4:30 P.M., he must have started:

4:30 P.M. - 10 ; text{hours} 6:30 A.M.

Thus, the person started his journey at 6:30 A.M.

Conclusion

Understanding the principles of journey duration and time calculation can help in making efficient travel plans. By following the step-by-step approach outlined in this article, you can accurately determine the start time of any journey.