Solving the Cab Speed Problem

This problem involves calculating the speed of a faster taxi given the distance, the speed of a slower taxi, and the difference in their travel times. We can use the relationship between speed, distance, and time to solve this.

Understanding the Given Information

We are given the following details:

• Distance between town C and town D (\(D\)) = 540 km
• Speed of the slower taxi (\(S_A\)) = 90 km/hr
• The slower taxi takes 1 hour more than the faster taxi.

Let \(S_B\) be the speed of the faster taxi in km/hr, \(T_A\) be the time taken by the slower taxi in hours, and \(T_B\) be the time taken by the faster taxi in hours.

From the problem, we know that \(T_A = T_B + 1\).

Applying the Speed, Distance, Time Formula

The basic formula relating speed, distance, and time is:

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

Rearranging this, we get the formula for time:

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

Calculating the Time Taken by the Slower Taxi

Using the formula, we can calculate the time taken by the slower taxi (\(T_A\)):

\(T_A = \frac{D}{S_A}\)

Substituting the given values:

\(T_A = \frac{540 \text{ km}}{90 \text{ km/hr}}\)

$T_A = 6 \text{ hours}

So, the slower taxi takes 6 hours to cover the 540 km distance.

Setting up the Equation for the Faster Taxi's Speed

We know that the slower taxi takes 1 hour more than the faster taxi. So, the time taken by the faster taxi (\(T_B\)) is:

\(T_B = T_A - 1\)

\(T_B = 6 \text{ hours} - 1 \text{ hour}\)

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

Now, we can express the time taken by the faster taxi using its speed \(S_B\) and the distance \(D\):

\(T_B = \frac{D}{S_B}\)

Substitute the known values into this equation:

\(5 \text{ hours} = \frac{540 \text{ km}}{S_B}\)

Solving for the Speed of the Faster Taxi

To find \(S_B\), we can rearrange the equation:

\(S_B = \frac{540 \text{ km}}{5 \text{ hours}}\)

Performing the division:

$S_B = 108 \text{ km/hr}

The speed of the faster taxi is 108 km/hr.

Analyzing the Options

Let's compare our calculated speed with the given options:

• Option 1: 117 km/hr
• Option 2: 126 km/hr
• Option 3: 108 km/hr
• Option 4: 99 km/hr

Our calculated speed of 108 km/hr matches Option 3.

Revision Table: Speed, Distance, Time Concepts

Here is a summary of the formulas used in speed, distance, and time problems:

Additional Information: Solving Related Problems

Problems involving speed, distance, and time often appear in competitive exams. Understanding the basic formulas is crucial. Other variations include:

• Problems involving relative speed (when objects are moving towards or away from each other).
• Problems involving trains passing poles, platforms, or other trains.
• Problems involving boats and streams (considering the speed of the current).
• Problems where one part of the journey is covered at one speed and another part at a different speed.

Practice with different types of problems helps in mastering this topic.