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Rate problems are mathematical questions involving ratios of two different quantities measured in different units. These problems often involve finding out how much of one quantity corresponds to one unit of another quantity, such as speed (distance per unit of time) or density (mass per unit of volume). Solving a rate problem typically requires the understanding that you are dividing two quantities to determine the ratio per one unit.

For instance, in the exercise mentioned, we have a distance measure and a time measure. To solve the problem, we assess how much distance is covered in a standardized unit of time; here, that standardized unit is one hour. The concept effectively combines unit conversion with division and is critical in a variety of real-world contexts, such as calculating the speed of a vehicle, the flow of a substance, or the cost of goods per unit.

The 'distance over time' calculation is essential for determining the speed of an object or the rate at which it covers a certain distance. This calculation is universally important in fields such as physics, transportation, and everyday activities like planning a trip.

To perform this calculation, one needs to divide the total distance travelled by the total time taken to travel that distance. The result expresses the rate at which the distance changes with time. Remembering the order of these quantities is vital: distance always comes first, and time is the denominator. For example, if a cyclist travelled 30 miles in 1.5 hours, the calculation would be \(\frac{30\text{ miles}}{1.5\text{ hours}}\) to find the average speed.

Miles per hour (mph) is a unit of speed expressing the number of miles travelled in one hour. It is a standard unit of measurement for speed in the United States and the United Kingdom.

To calculate mph, we use the unit rate formula for distance over time as illustrated in the given exercise. After finding that a car travels 105 miles in 2 hours, we convert this to a unit rate by dividing the number of miles by the number of hours, resulting in a speed of 52.5 mph. This means if the car continued to travel at this constant speed, it would cover 52.5 miles in one hour.

The concept of mph is crucial for understanding and obeying speed limits, estimating travel time, and for various fields such as transportation planning and vehicle engineering.