A delivery van travels at a speed of 90 km/hr while driving on the highway and 30 km/hr while driving through city streets. If the vehicle's overall average speed for the entire journey is 50 km/hr, what fraction of the total distance was covered on the highway?

The correct answer is $\frac{3}{5}$.

Step-by-Step Explanation:

1. Define the Variables:
Let $d_h$ be the distance covered on the highway.
Let $d_c$ be the distance covered through city streets.
The total distance covered during the entire journey is:
$D=d_h+d_c$

2. Express Time in Terms of Distance and Speed:
Time taken on the highway:
$t_h=\frac{d_h}{90}$
Time taken through city streets:
$t_c=\frac{d_c}{30}$
Total time for the journey:
$T=t_h+t_c=\frac{d_h}{90}+\frac{d_c}{30}$

3. Apply the Overall Average Speed Formula:
Average speed is calculated as:
$\text{Average Speed}=\frac{\text{Total Distance}}{\text{Total Time}}$
Given that the overall average speed is 50 km/hr:
$50=\frac{d_h+d_c}{\frac{d_h}{90}+\frac{d_c}{30}}$

4. Solve for the Relationship Between Distances:
Cross-multiplying both sides gives:
$d_h+d_c=50\times \left(\frac{d_h}{90}+\frac{d_c}{30}\right)$
Expanding the terms:
$d_h+d_c=\frac{50}{90}{d}_h+\frac{50}{30}{d}_c$
$d_h+d_{\mathrm{c}}=\frac{5}{9}{d}_h+\frac{5}{3}{d}_{\mathrm{c}}$
Rearranging the equation to solve for $d_c$ in terms of $d_h$:
$d_h-\frac{5}{9}{d}_h=\frac{5}{3}{d}_c-d_{\mathrm{c}}$
$\frac{4}{9}{d}_h=\frac{2}{3}{d}_c$
Multiplying both sides by 9:
$4d_h=6d_c$
$d_c=\frac{4}{6}{d}_h=\frac{2}{3}{d}_h$

5. Determine the Fraction of Distance Covered on Highway:
Express the total distance $D$ in terms of $d_h$:
$D=d_h+\frac{2}{3}$
The fraction of the total distance covered on the highway is:
$\frac{d_h}{D}=\frac{d_h}{\frac{5}{3}{d}_h}=\frac{3}{5}$

Therefore, the fraction of the total distance covered on the highway is $\frac{3}{5}$.