Calculating Speed with Distance and Time

This problem involves the fundamental relationship between speed, distance, and time. The formula connecting these three quantities is:

\[ \text{Distance} = \text{Speed} \times \text{Time} \]

From this formula, we can also derive expressions for speed and time:

\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]

\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]

In this question, Yug first covers a certain distance at a given speed and time. We need to calculate this distance first.

Step 1: Calculate the total distance covered

Yug's initial speed is given as 56 km/h, and the time taken is \(\frac{3}{2}\) hours.

Using the formula Distance = Speed \(\times\) Time:

\[ \text{Distance} = 56 \text{ km/h} \times \frac{3}{2} \text{ hours} \]

\[ \text{Distance} = 56 \times \frac{3}{2} \text{ km} \]

\[ \text{Distance} = \frac{56 \times 3}{2} \text{ km} \]

\[ \text{Distance} = 28 \times 3 \text{ km} \]

\[ \text{Distance} = 84 \text{ km} \]

So, the distance covered by Yug is 84 km.

Step 2: Calculate the new speed required

Now, Yug wants to cover the same distance (84 km) in a new time of 7 hours. We need to find the new speed required for this.

Using the formula Speed = \(\frac{\text{Distance}}{\text{Time}}\):

\[ \text{New Speed} = \frac{\text{Distance}}{\text{New Time}} \]

\[ \text{New Speed} = \frac{84 \text{ km}}{7 \text{ hours}} \]

\[ \text{New Speed} = \frac{84}{7} \text{ km/h} \]

\[ \text{New Speed} = 12 \text{ km/h} \]

Thus, Yug's new speed needs to be 12 km/h to cover the distance of 84 km in 7 hours.

Summary of Calculations

The required new speed is 12 km/h.

Revision Table: Speed, Distance, and Time Formulas

Additional Information: Inverse Relationship

For a fixed distance, speed and time are inversely proportional. This means if you want to cover the same distance in less time, you need to increase your speed. Conversely, if you take more time to cover the same distance, your speed must decrease.

In this problem, the time increases from \(\frac{3}{2}\) hours (1.5 hours) to 7 hours. Since the time increases, the speed must decrease. The initial speed was 56 km/h, and the new speed is 12 km/h, which is a decrease, confirming the inverse relationship principle.