In a race, two athletes 'A' and 'B' reach the finishing point in 20 second and 22 second respectively. (a) What is the ratio of their speeds? (b) If both are allowed to run with their respective speeds for a given time, what is the ratio of the distance covered by them?

Understanding the concept of ratio and proportion is crucial when it comes to comparing quantities and their relationships with each other. A ratio is a way to compare two quantities by using division. In our exercise, we encounter the situation where two athletes, 'A' and 'B', have different times to complete the same distance. To find out the ratio of their speeds, we use their race completion times.

Proportion, on the other hand, tells us that two ratios are equivalent. It's like saying one relationship or comparison is just as valid as another. When we talk about speeds in particular, we are often looking to see if the relationship between speed and time for one athlete is proportional to that of another. The beauty of proportion lies in its ability to simplify complex relationships into something understandable. In practice, if we know that Athlete 'A' runs at a certain speed, and this speed forms a certain ratio with Athlete 'B', we can predict other related quantities, like distances covered in the same time frame, using this ratio.

Speed calculation is essential when we're dealing with movement. Speed tells us how quickly an object is moving and is usually expressed as a distance covered over a given time. It's calculated using the formula:

Speed = Distance / Time.

In our example with athletes 'A' and 'B', we know they both cover the same race distance, but in different times. The direct comparison of their times isn't enough to determine who's faster because we are more interested in their speeds. By using the speed formula in reverse, we can use their known completion times to calculate the ratio of their speeds, assuming the race distance is constant. This type of speed calculation relies on understanding that time is inversely related to speed—the less time you take, the greater your speed. By understanding this inverse relationship, we deduce that since Athlete 'A' finishes the race quicker than 'B', Athlete 'A' has a higher speed.

The distance-time relationship is an essential concept in physics and everyday life, telling us how the two quantities are interconnected. In simple terms, the longer an object moves, or the faster it moves, the more distance it will cover. In the context of our exercise, if we let the athletes run for the same duration, the one with a higher speed will cover more distance.

When Athlete 'A' and Athlete 'B' run the race, this relationship tells us that their speeds and the times taken to finish will determine the distances they've travelled. This relationship is seen when we calculate the ratio of distances covered: if both athletes run for the same amount of time and their speed ratio is known, we can conclude that the ratio of their distances covered will be the same as their speed ratio. This is crucial in understanding movement and predicting how far an object will go over time, based on its speed. For the athletes in question, knowing the ratio of their speeds gives us a direct way to know the ratio of distances they would cover in any equal time period.