Calculating Journey Time with Changing Speed

The problem asks us to find the time taken for a car to travel a certain distance at a new speed, given the initial speed and time taken.

Understanding the Relationship

The core relationship is between distance, speed, and time: Distance = Speed × Time.

If the distance remains constant, speed and time are inversely proportional. This means if the speed decreases, the time taken increases proportionally.

Step-by-Step Solution

1. Calculate the Total Distance: First, find the distance of the journey using the initial speed and time.

   Initial Speed ($v_1$) = $50 \text{ km/hour}$

   Initial Time ($t_1$) = $8 \text{ hours}$

   Distance ($d$) = $v_1 \times t_1 = 50 \text{ km/h} \times 8 \text{ h} = 400 \text{ km}$.

2. Calculate the New Time: Now, use the same distance and the new speed to find the time required.

   New Speed ($v_2$) = $40 \text{ km/hour}$

   Distance ($d$) = $400 \text{ km}$

   New Time ($t_2$) = $\frac{d}{v_2} = \frac{400 \text{ km}}{40 \text{ km/h}} = 10 \text{ h}$.

Alternatively, using the inverse proportion:

$v_1 \times t_1 = v_2 \times t_2$

$50 \text{ km/h} \times 8 \text{ h} = 40 \text{ km/h} \times t_2$

$400 = 40 \times t_2$

$t_2 = \frac{400}{40} = 10 \text{ h}$.

Conclusion

It will take 10 hours to travel the same distance at a speed of 40 km/hour.