Rahul is driving from Gurgaon to Agra at a speed of $80 ext{ km/h}$. After covering $120 ext{ km}$, he reduces his speed by $12.5\%$ and completes the entire journey in $5.5$ hours. What would be the total time taken by him to cover the same distance if he had reduced his speed only after driving for $2$ hours?

Question

Rahul is driving from Gurgaon to Agra at a speed of $80 	ext{ km/h}$. After covering $120 	ext{ km}$, he reduces his speed by $12.5\%$ and completes the entire journey in $5.5$ hours. What would be the total time taken by him to cover the same distance if he had reduced his speed only after driving for $2$ hours?

Accepted Answer

**Solution:**

1. Let the total distance be $D$ km.

2. Rahul drives the first $120$ km at $80 	ext{ km/h}$.

3. After $120$ km, speed is reduced by $12.5\%$:
   - New speed = $80 - 12.5\% 	imes 80 = 80 - 10 = 70 	ext{ km/h}$

4. Total time taken is $5.5$ hours, so:
   $$\frac{120}{80} + \frac{D - 120}{70} = 5.5$$

5. Calculate time for first part:
   $$\frac{120}{80} = 1.5 	ext{ hours}$$

6. Substitute and solve for $D$:
   $$1.5 + \frac{D - 120}{70} = 5.5$$
   $$\frac{D - 120}{70} = 4$$
   $$D - 120 = 280$$
   $$D = 400 	ext{ km}$$

7. Now, if Rahul reduces speed after $2$ hours instead of after $120$ km:

- Distance covered in first $2$ hours at $80 	ext{ km/h}$:
     $$80 	imes 2 = 160 	ext{ km}$$

- Remaining distance:
     $$400 - 160 = 240 	ext{ km}$$

- Speed after reduction is $70 	ext{ km/h}$.

8. Time to cover remaining distance:
   $$\frac{240}{70} = \frac{24}{7} = 3 \frac{3}{7} 	ext{ hours}$$

9. Total time:
   $$2 + 3 \frac{3}{7} = 5 \frac{3}{7} 	ext{ hours}$$

**Answer:** $5 \frac{3}{7} 	ext{ hours}$

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