Journey Distance Calculation

The problem asks us to find the total distance of a journey given the total time and the speeds for the first and second halves of the journey.

Problem Breakdown

We are given:

• Total time ($T$) = 11 hours
• Speed for the first half ($v_1$) = 25 km/h
• Speed for the second half ($v_2$) = 30 km/h
• The journey is divided into two equal halves by distance. Let the distance of each half be $d$.
• The total distance ($D$) is $2d$.

Calculating Time for Each Half

The time taken to cover a distance is calculated as Distance / Speed.

• Time for the first half ($t_1$) = $\frac{d}{v_1} = \frac{d}{25}$ hours.
• Time for the second half ($t_2$) = $\frac{d}{v_2} = \frac{d}{30}$ hours.

Finding the Distance

The total time is the sum of the times for both halves:

$T = t_1 + t_2$

Substitute the expressions for $t_1$ and $t_2$ and the value of $T$:

$11 = \frac{d}{25} + \frac{d}{30}$

To solve for $d$, first find a common denominator for 25 and 30, which is 150:

$11 = d \left( \frac{1}{25} + \frac{1}{30} \right)$

$11 = d \left( \frac{6}{150} + \frac{5}{150} \right)$

$11 = d \left( \frac{11}{150} \right)$

Now, isolate $d$:

$d = 11 \times \frac{150}{11}$

$d = 150$ km

This is the distance for one half of the journey. The total distance ($D$) is twice this value:

$D = 2d = 2 \times 150$ km

$D = 300$ km