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Welcome to our Math lesson on What Are Rates? Clarifying Misconceptions, this is the first lesson of our suite of math lessons covering the topic of Rates. Applications of Ratios and Rates in Practice, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.
One of the biggest challenges a learner has to face when dealing with rates is the similarity of rates and ratios. By definition, rate is a ratio-like structure involving two different types of quantities. Recall that a ratio is a comparison through the division of two like quantities.
A rate is usually expressed as a fraction; it is rare for the colon symbol to be used to express rates.
Rates are very common in physics. For example, velocity represents the rate of position change. More specifically, when the position of a moving object changes by 210 km in 3 hours, the rate of position change is 210 km / 3 h = 70 km/h.
At first sight, ratios and rates are similar. Indeed, some modern theories (from 2011 onwards) tend to include rates within the ratios category. This is because each ratio has an associated rate, which is equal to the value of the ratio when the second number is equal to 1. We call this special rate a unit rate. For example, when we say the ratio of boys to girls watching a football match is 35 boys to 7 girls, the unit rate is 35 / 7 = 5 boys per each girl watching the match. However, in this maths course we will consider ratios and rates as separate categories based on the distinction criterion mentioned above (ratios involve like quantities, while rates have different ones).
Recall the examples in the previous tutorial where the number of items produced in a certain period as a function of the number of workers was expressed as a ratio. In reality, those situations are closer to rates than ratios, as they involve different types of quantities (workers & items). The unit rate in those cases represent the number of items per worker produced in a day.
One of the distinctive features of rates is that in many cases they involve time. In other words, the quantities expressed as a rate where the number written in the denominator represents time are all rates. For example, in physics the rate of position change (Δx) represents the velocity v (v = Δx / t where t is the time elapsed); the rate of velocity change (Δv) gives the acceleration a (a = Δv / t); the volumetric rate of water flow R is given by R = ΔV / t, where ΔV is the volume of water flowing through the pipe, and so on.
It takes 25 seconds to fill a 5L bucket with water. What is the flow rate of water from the tap?
To calculate the rate of water flow in litres per second, we must divide the water volume (in litres) by the time elapsed (in seconds). Thus, we have
To have a clearer idea on the relationship of rates to ratios, let's consider the following example.
A cat eats 2 kg of food in 30 days. How many kilograms of food does this cat eat in 135 days?
We can use two different approaches to solve this exercise.
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