To determine the percentage of the total distance travelled at a uniform speed of 10 kmph, we need to first find out the exact distance covered at that specific speed. This problem involves understanding concepts of distance, speed, and time for different parts of a journey.

A person travelled a total distance of 80 km in 6 hours. This journey was divided into two parts, each with a different uniform speed. The first part was at 10 kmph, and the remaining part was at 18 kmph.

Travelled Distance Problem Setup

Let's define the variables for the two parts of the journey:

• Let \(D_1\) be the distance (in km) travelled at a uniform speed of 10 kmph.
• Let \(D_2\) be the distance (in km) travelled at a uniform speed of 18 kmph.

We are given the following information:

• Total Distance Travelled: 80 km
• Total Time Taken: 6 hours
• Speed for First Part: 10 kmph
• Speed for Remaining Part: 18 kmph

From the given information, we can form two equations based on the total distance and total time.

Distance Equation

The sum of the distances of the two parts must equal the total distance travelled:

\[ D_1 + D_2 = 80 \quad \text{(Equation 1)} \]

Time Equation

The time taken for each part of the journey can be calculated using the formula: \(\text{Time} = \frac{\text{Distance}}{\text{Speed}}\). The sum of the times for both parts must equal the total time taken.

• Time taken for the first part (\(T_1\)) = \(\frac{D_1}{10}\) hours
• Time taken for the second part (\(T_2\)) = \(\frac{D_2}{18}\) hours

So, the total time equation is:

\[ \frac{D_1}{10} + \frac{D_2}{18} = 6 \quad \text{(Equation 2)} \]

Solving for Distances Travelled

Now, we need to solve these two simultaneous equations to find the value of \(D_1\).

From Equation 1, we can express \(D_2\) in terms of \(D_1\):

\[ D_2 = 80 - D_1 \]

Substitute this expression for \(D_2\) into Equation 2:

\[ \frac{D_1}{10} + \frac{80 - D_1}{18} = 6 \]

To eliminate the denominators, we find the Least Common Multiple (LCM) of 10 and 18. The LCM of 10 and 18 is 90.

Multiply the entire equation by 90:

\[ 90 \times \left( \frac{D_1}{10} \right) + 90 \times \left( \frac{80 - D_1}{18} \right) = 90 \times 6 \]

\[ 9D_1 + 5(80 - D_1) = 540 \]

Now, distribute the 5 into the parenthesis:

\[ 9D_1 + 400 - 5D_1 = 540 \]

Combine the \(D_1\) terms:

\[ (9D_1 - 5D_1) + 400 = 540 \]

\[ 4D_1 + 400 = 540 \]

Subtract 400 from both sides:

\[ 4D_1 = 540 - 400 \]

\[ 4D_1 = 140 \]

Divide by 4 to find \(D_1\):

\[ D_1 = \frac{140}{4} \]

\[ D_1 = 35 \text{ km} \]

So, the distance travelled at a uniform speed of 10 kmph is 35 km.

Percentage of Total Distance Travelled at 10 kmph

The question asks for the percentage of the total distance that was travelled at a uniform speed of 10 kmph.

To calculate the percentage, use the formula:

\[ \text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100\% \]

In this case:

• Part: Distance travelled at 10 kmph (\(D_1\)) = 35 km
• Whole: Total distance travelled = 80 km

Substitute the values into the formula:

\[ \text{Percentage} = \frac{35}{80} \times 100\% \]

First, perform the division:

\[ \frac{35}{80} = 0.4375 \]

Now, multiply by 100 to get the percentage:

\[ \text{Percentage} = 0.4375 \times 100\% = 43.75\% \]

Therefore, 43.75% of the total distance was travelled at a uniform speed of 10 kmph.

This detailed step-by-step calculation helps to clearly understand how the percentage of total distance travelled at a specific uniform speed is derived in a mixed speed travel problem.