Distance Percentage Calculation at 10 kmph

The problem asks for the percentage of the total distance travelled at a specific speed (10 kmph). We are given the total distance, total time, and two different speeds for two parts of the journey.

Calculating Travelled Distance Percentage

Let's define the variables:

• Total Distance ($D$) = 80 km
• Total Time ($T$) = 6 hours
• Speed for the first part ($v_1$) = 10 kmph
• Speed for the remaining part ($v_2$) = 18 kmph
• Let $d_1$ be the distance travelled at $v_1$, and $t_1$ be the time taken.
• Let $d_2$ be the distance travelled at $v_2$, and $t_2$ be the time taken.

Setting Up Equations

We know that the sum of distances equals the total distance and the sum of times equals the total time:

1. Distance equation: $d_1 + d_2 = D \implies d_1 + d_2 = 80$ km
2. Time equation: $t_1 + t_2 = T \implies t_1 + t_2 = 6$ hours

Using the formula distance = speed × time ($d = v \times t$), we can express $d_1$ and $d_2$:

• $d_1 = v_1 \times t_1 = 10 t_1$
• $d_2 = v_2 \times t_2 = 18 t_2$

Solving for Time ($t_1$)

Substitute the expressions for $d_1$ and $d_2$ into the distance equation:

$10 t_1 + 18 t_2 = 80$

From the time equation, we can express $t_2$ in terms of $t_1$: $t_2 = 6 - t_1$. Substitute this into the equation above:

$10 t_1 + 18 (6 - t_1) = 80$

Now, solve for $t_1$:

$10 t_1 + 108 - 18 t_1 = 80$

$-8 t_1 = 80 - 108$

$-8 t_1 = -28$

$t_1 = \frac{-28}{-8} = \frac{28}{8} = 3.5 \text{ hours}$

Calculating Distance ($d_1$)

Now that we have $t_1$, we can find the distance travelled at 10 kmph:

$d_1 = 10 \times t_1 = 10 \times 3.5 = 35 \text{ km}$

Calculating Percentage of Total Distance

Finally, calculate the percentage of the total distance travelled at 10 kmph:

$\text{Percentage} = \frac{d_1}{D} \times 100$

$\text{Percentage} = \frac{35 \text{ km}}{80 \text{ km}} \times 100$

$\text{Percentage} = \frac{35}{80} \times 100 = \frac{35}{8} \times 10 = \frac{350}{8} = 43.75\%$

The percentage of the total distance travelled at a uniform speed of 10 kmph is 43.75%.