(a) Let the distance travelled be 'd'. The speed of \(^{\prime} A^{\prime}, v_{A}=\frac{d}{20} m s^{-1}\) The speed of ' \(\mathrm{B}^{\prime}, \mathrm{v}_{\mathrm{B}}=\frac{\mathrm{d}}{22} \mathrm{~m} \mathrm{~s}^{-1}\). \(\Rightarrow \frac{\mathrm{v}_{\mathrm{A}}}{\mathrm{v}_{\mathrm{B}}}=\frac{\frac{\mathrm{d}}{20}}{\frac{\mathrm{d}}{22}}=\frac{22}{20}=\frac{11}{10}=11: 10\) (b) Let them run for 't's. Then, \(\mathrm{d}_{\mathrm{A}}=\mathrm{v}_{\mathrm{A}} \times \mathrm{t}=\frac{\mathrm{d}}{20} \times \mathrm{t}\) \(\mathrm{d}_{\mathrm{B}}=\mathrm{v}_{\mathrm{B}} \times \mathrm{t}=\frac{\mathrm{d}}{22} \times \mathrm{t}\) \(\Rightarrow \frac{\mathrm{d}_{\mathrm{A}}}{\mathrm{d}_{\mathrm{B}}}=\frac{\frac{\mathrm{d}}{20}(\mathrm{t})}{\frac{\mathrm{d}}{22}(\mathrm{t})}=\frac{22}{20}=\frac{11}{10}=11: 10\)

Understanding the relationship between speed and distance is critical in solving problems related to motion. Speed is defined as the rate at which an object covers a certain distance, and it is usually measured in units like meters per second (m/s) or kilometers per hour (km/h). In many problems, especially those involving comparative speeds, we are interested in the ratio of speeds between different moving objects.

For example, in the case of two runners, A and B, we calculate their individual speeds by dividing the distance each runs by the time they take. If Runner A covers a distance in 20 seconds and Runner B covers the same distance in 22 seconds, we can compare their speeds by forming a ratio. Even when the exact distance 'd' is not given, understanding that the ratio of speeds will remain constant regardless of the actual value of 'd' is crucial. This concept plays a significant role in solving various physics problems, most notably when considering the movement of objects over time.

Ratios are mathematical expressions used to compare quantities, showing how many times one number contains another. In solving ratios, it's important to understand that they represent a relationship of proportionality. This means that if the ratio of the speeds of two runners is 11:10, if one runs 11 meters in a certain period, the other runs 10 meters in the same period, regardless of how long that period is.

The ability to see patterns in ratios can help students solve more complex problems. By simplifying ratios, it becomes easier to understand and apply them in various contexts, such as determining speeds over different distances and times or comparing the output of two machines. It's not just about finding the answer; it's about comprehending the relationship that the ratio represents between two or more quantities.

Physics numericals for class 7 often involve basic principles of speed, distance, and time. It’s important to first conceptualize what is being asked in a problem, whether it is to find a distance, a speed, or a time duration. Once that is established, using formulas like distance = speed × time becomes second nature.

In our context, to further improve understanding, visualize simpler examples—like two trains or cars moving at different speeds—and using diagrams to represent their movement can be incredibly beneficial. This allows students to better grasp the abstract concepts. Additionally, always encourage double-checking answers by plugging them back into the given formulas to verify that they make sense within the problem’s context.