Understanding the Time and Distance Problem

This question involves a classic time and distance problem where a total journey is divided into parts, each covered at a different speed. We are given the total distance, total time, and the speed for each part of the journey. Our goal is to find the distance covered in one specific part, which is the distance travelled by bicycle.

Setting Up the Problem Variables

Let's define the variables we will use to solve this problem:

• Total distance travelled = 50 km
• Total time taken = 9 hours
• Speed while travelling on foot = 5 km/h
• Speed while travelling by bicycle = 10 km/h
• Let the distance travelled on foot be \(d_f\) km.
• Let the distance travelled by bicycle be \(d_b\) km.
• Let the time spent travelling on foot be \(t_f\) hours.
• Let the time spent travelling by bicycle be \(t_b\) hours.

Formulating Equations

We can create two equations based on the given information:

1. The total distance is the sum of the distances travelled on foot and by bicycle:

   \(d_f + d_b = 50\)    (Equation 1)

2. The total time is the sum of the time spent on foot and by bicycle:

   \(t_f + t_b = 9\)     (Equation 2)

We also know the relationship between distance, speed, and time: Time = Distance / Speed.

• Time spent on foot: \(t_f = \frac{d_f}{5}\)
• Time spent by bicycle: \(t_b = \frac{d_b}{10}\)

Substitute these expressions for \(t_f\) and \(t_b\) into Equation 2:

\(\frac{d_f}{5} + \frac{d_b}{10} = 9\)     (Equation 3)

Solving for the Distance Travelled by Bicycle

We have a system of two equations with two unknowns (\(d_f\) and \(d_b\)):

1) \(d_f + d_b = 50\)

2) \(\frac{d_f}{5} + \frac{d_b}{10} = 9\)

From Equation 1, we can express \(d_f\) in terms of \(d_b\):

\(d_f = 50 - d_b\)

Now substitute this expression for \(d_f\) into Equation 3:

\(\frac{50 - d_b}{5} + \frac{d_b}{10} = 9\)

To eliminate the denominators, multiply the entire equation by the least common multiple of 5 and 10, which is 10:

\(10 \times \left(\frac{50 - d_b}{5}\right) + 10 \times \left(\frac{d_b}{10}\right) = 10 \times 9\)

\(2(50 - d_b) + d_b = 90\)

Distribute the 2:

\(100 - 2d_b + d_b = 90\)

Combine the \(d_b\) terms:

\(100 - d_b = 90\)

Subtract 90 from both sides and add \(d_b\) to both sides:

\(100 - 90 = d_b\)

\(d_b = 10\)

So, the distance travelled by bicycle is 10 km.

Verification

Let's check if this answer makes sense. If \(d_b = 10\) km, then \(d_f = 50 - 10 = 40\) km.

Time taken on foot \(t_f = \frac{d_f}{5} = \frac{40}{5} = 8\) hours.

Time taken by bicycle \(t_b = \frac{d_b}{10} = \frac{10}{10} = 1\) hour.

Total time = \(t_f + t_b = 8 + 1 = 9\) hours. This matches the total time given in the question, confirming our calculation is correct.

The distance travelled on the bicycle is 10 km.

Revision Table: Time and Distance Concepts

Additional Information: Solving Mixed Travel Problems

Problems involving travel at different speeds for different parts of a journey can be solved using systems of equations. Key steps include:

• Identify the total distance and total time.
• Define variables for the distance and/or time for each part of the journey.
• Use the relationship Distance = Speed \(\times\) Time to form equations based on the total distance and total time.
• Solve the system of equations to find the unknown variables.
• Always check your answer by plugging the calculated values back into the original conditions.