Raj drove at $80 \mathrm{~km} / \mathrm{h}$ for the first $\frac{4}{9}$ of the journey from his house to a village. He completed the remaining $30 \mathrm{~km}$ of the journey in $30 \mathrm{~min}$ . How long did Raj take to complete the whole journey?

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Q. Raj drove at

80 \mathrm{~km} / \mathrm{h}

for the first

\frac{4}{9}

of the journey from his house to a village. He completed the remaining

30 \mathrm{~km}

of the journey in

30 \mathrm{~min}

. How long did Raj take to complete the whole journey?

Calculate Total Distance: First, let's find the total distance of the journey. Since Raj drove the first $(4)/(9)$ of the journey at $80$ km/h and the remaining $30$ km at a different speed, we need to find the full distance to calculate the time for the first part of the journey.
Find Speed for Remaining Journey: Raj completed the remaining $30\,\text{km}$ in $30$ minutes, which is $0.5$ hours. Since speed equals distance divided by time, we can find the speed for the remaining journey: Speed $=$ Distance $/$ Time, so Speed $= \frac{30\,\text{km}}{0.5\,\text{h}} = 60\,\text{km/h}$ .
Determine Total Distance: Now, we know that the remaining journey was at $60\,\text{km/h}$ and it was not the $\frac{4}{9}$ part, so it must be the $\frac{5}{9}$ part of the journey. If $30\,\text{km}$ is $\frac{5}{9}$ of the total distance, we can set up a proportion to find the total distance: $\frac{5}{9}$ of total distance = $30\,\text{km}$ .
Calculate Distance at $80$ km/h: To find the total distance, we solve for the total distance using the proportion: Total distance = $(30\,\text{km} \times 9) / 5 = 54\,\text{km}$ .
Calculate Time for First Part: Now that we have the total distance, we can find the distance Raj drove at $80\,\text{km/h}$ . This is the $\frac{4}{9}$ part of the journey, so $\frac{4}{9}$ of total distance = $\frac{4}{9} \times 54\,\text{km} = 24\,\text{km}$ .
Find Total Time: Next, we calculate the time Raj took to drive the first $24\,\text{km}$ at $80\,\text{km/h}$ . Time = Distance / Speed, so Time = $\frac{24\,\text{km}}{80\,\text{km/h}} = 0.3\,\text{hours}$ or $18\,\text{minutes}$ .
Find Total Time: Next, we calculate the time Raj took to drive the first $24\,\text{km}$ at $80\,\text{km/h}$ . Time = Distance / Speed, so Time = $24\,\text{km}$ / $80\,\text{km/h}$ = $0.3$ hours or $18$ minutes.Finally, we add the time Raj took for the first part of the journey ( $18$ minutes) to the time he took for the remaining part of the journey ( $30$ minutes) to get the total time for the whole journey. Total time = $18$ minutes + $30$ minutes = $80\,\text{km/h}$ $0$ minutes.