| Tutorial ID | Title | Tutorial | Video Tutorial | Revision Notes | Revision Questions | |
|---|---|---|---|---|---|---|
| 4.2 | Rates. Applications of Ratios and Rates in Practice |
In this Math tutorial, you will learn:
What do you understand by the term water flow rate?
What about interest rates in banking? Do you know what they mean?
Could you produce a list of numbers where the next number is obtained from the sum of the previous two?
Are you good at Physics? If yes, how do you define velocity and acceleration? What about power? Induced electromotive force?
In this tutorial, we will discuss rates - a very important concept that is often confused with ratios. Hence, a significant number of examples are provided to clarify the misconceptions that are associated with this term.
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.
It is possible to write two unit rates for each situation involving rates. Thus, if a library contains 1200 books for every 75 readers, we can obtain the following unit rates:
Number of books for each reader (N1). We have
Number of readers for each book (N2). We have
Since both of these unit rates belong to the same event, they are called associated unit rates.
Earlier we discussed rate examples where time was involved. In such situations, we want to calculate how the other quantity change over time. Therefore, the rates expressing these changes are obtained by dividing two differences: the difference in the given quantity and the difference in time, which in many cases is not evident because we often take the initial time as zero and therefore, the difference in time simply gives the final time.
In general, we have Y for the rate of a quantity change:
The price of an item in 2019 was $80 and in 2022 the same item was priced at $104. What is the yearly (annual) rate of the price increase?
Here, the quantity Y represents the price. Thus, using the general formula of rate of change
we obtain
The rate of a quantity change can be also negative. This means the final value of the quantity involved is smaller than the initial value because the quantity decreases. Look at the example below.
The price of petrol during lockdown caused by the COVID-19 pandemic dropped uniformly from $2.15/gallon to $1.73/gallon in two weeks due to the reduced use of cars. What was the rate of price change during this period?
Price reduction here represents the rate of price change. We have
Here, Y represents the price and tfinal - tinitial = 2 weeks = 14 days. Hence,
This result means the price dropped by a rate of $0.03 per day.
There are numerous applications of rate in practice. Here, we will briefly discuss some of them.
Interest rates are a very fundamental term in banking, where the banks offer extra money to people who deposit their capitals, based on the annual interest rates. These rates are usually expressed as a percentage, which is another form of rational numbers expression, as a percentage is nothing more than a fraction with a denominator of 100.
For example, if a bank offers annual interest rates of 3%, this means the amount of money deposited by a customer increases by 3% or by 3/100. Therefore, if a customer deposits $50,000 in this bank, he earns
after one year. Thus, after one year, the total in the bank is $50,000 + $1,500 = $51,500.
Earlier we explained some of the quantities that involve rates (especially rates of change) in physics. Some other quantities that are obtained as rates of something changing in physics include:
Rate of energy change, which represents the power delivered by a source of energy in a given time, i.e.
Rate of electric flux change in a coil, which represents the electromotive force induced in the coil during a given time, i.e.
Rate of momentum change, which represents the force acting on an object, i.e.
where Δp is the change in momentum of the object, and so on.
Heart rate is an important parameter of humans' health. A healthy person has a heart rate of 70 - 80 beats per minute.
The Annual Incidence Rate of an acute disease is calculated by the number of new cases of that disease during a particular year / the estimated or counted average population at risk, observed within that year. For example, if 50,000 new cases of an acute disease are detected during one year in a country where 10 million people are at risk of being affected by it, the Annual Incidence Rate for this disease is 50,000 / 10,000,000 = 0,005.
The Point Prevalence Rate of a condition is the number of cases of that condition at a particular point in time / population at risk at that point in time,
and so on.
Rates are very common in civil engineering, when they are used to express specific quantities of materials that are to be mixed with another material to form stronger or more flexible structures. For example, the cement humidity rate in concrete must be at certain values to avoid cracks, etc.
Besides the applications of ratio discussed in the previous tutorial, there is another important and special application of ratio in practice, known as the golden ratio. Thus, if we have two numbers a and b (where a is the biggest) which meet the condition
we say that a and b form a golden ratio, expressed by the letter φ.
The golden ratio is a constant number, whose value is
Since √5 is irrational, the digits after the decimal point continue on forever without repeating.
Let's use the properties of fractions to find a possible relationship between the numbers a and b involved in a golden ratio. Thus, we can write
We will explain in the algebra section that
Therefore, substituting in the previous equation, we obtain
Hence, the numbers forming a golden ratio must meet the condition "sum × difference = product".
A two-storey building forms a golden ratio when the ground floor (the highest from the two) is 5.2 m high. What is the height of the first floor?
We have a = 5.2 m. Since
we obtain for the height of the first floor:
Obviously, it is very difficult to guess which numbers meet the condition to be in a golden ratio. Most numbers qualified for this are not integers, which makes finding the values of a and b an impossible task. This issue was addressed by Fibonacci - an Italian mathematician, who discovered a special sequence of numbers, where the next number is obtained by the sum of the previous two. The first numbers of Fibonacci sequence is
For example, 1 + 2 = 3; 2 + 3 = 5; 3 + 5 = 8 and so on.
But, what is the relationship between the golden ratio and Fibonacci numbers? Well, the more the numbers in the Fibonacci sequence increase, the closer they go to the value of the golden ratio. Thus,
As you see, the values get closer and closer to 1.6180339887, which is the true value of the Fibonacci sequence.
Items that form a golden ratio between them produce a very fascinating view that has been used since antiquity in art and construction. This is because the building structures that form a golden ratio are stronger and more stable. We can find the golden ratio in a number of things, including:
A piece of furniture is 233 cm high. How wide must it be to form the nicest view? (Width must be shorter than height.) Use the Fibonacci series to find the missing dimension.
The nicest view is obtained when height and width form a golden ratio. From the Fibonacci series, we have 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,
Therefore, the width must be 144 cm as
This value is very close to that of the golden ratio, so we obtain the nicest view of the furniture by using the above dimensions.
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