Rates. Applications of Ratios and Rates in Practice

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4.2Rates. Applications of Ratios and Rates in Practice

In this Math tutorial, you will learn:

  • What are rates? Where do they differ from ratios?
  • What is a unit rate?
  • What are the associated unit rates? How many are there in a rate?
  • What is the rate of a quantity change? What is the common quantity in all rates of change?
  • What are some applications of rates in practice?
  • What is the golden ratio? How to find it?
  • What is the Fibonacci sequence? Why it is so important?
  • What are some uses of the golden ratio in practice?

Introduction

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.

What Are Rates? Clarifying Misconceptions

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.

Example 1

It takes 25 seconds to fill a 5L bucket with water. What is the flow rate of water from the tap?

Solution 1

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

Rate of water flow = Volume of water/Time elapsed
= 5 L/25 s
= 0.2 L/s

To have a clearer idea on the relationship of rates to ratios, let's consider the following example.

Example 2

A cat eats 2 kg of food in 30 days. How many kilograms of food does this cat eat in 135 days?

Solution 2

We can use two different approaches to solve this exercise.

  1. The ratio approach. In this case, we take the ratio of the number of days and then we multiply it by the amount of food the cat eats in 30 days. This is because we have the following ratio:
    30 days : 135 days = 2 kg : Total amount of food consumed
    Since the second quantity in the ratio is bigger, we can find 1:R, as explained in the previous guide. Thus,
    1/R = 135 d/30 d = 4.5
    Thus,
    Total amount of food consumed/2 kg = 4.5
    Total amount of food consumed = 2 kg × 4.5
    = 9 kg
    Therefore, the cat consumes 9 kg food in the given period.
  2. The rate approach. In this approach, we can calculate the unit rate first, i.e. the daily food consumption of the cat and then, multiply it by the total number of days. We can write:
    Unit rate = Amount of food consumed/Number of days
    = 2kg/30 days
    = 1/15 kg/day
    Then, we multiply this unit rate to the total number of days involved. Thus,
    Total amount of food consumed = Unit rate × Total number of days
    = 1/15 kg/day × 135 days
    = 135/15 kg
    = 9 kg

Two Associated Unit Rates

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

N1 = Number of books/Number of readers
= 1200 books/75 readers
= 16 books/reader

Number of readers for each book (N2). We have

N2 = Number of readers/Number of books
= 75 readers/1200 books
= 0.0625 reader/book

Since both of these unit rates belong to the same event, they are called associated unit rates.

Rate of a Quantity Change

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:

Rate of Y change = Yfinal - Yinitial/tfinal - tinitial
= Y2 - Y1/t2 - t1

Example 3

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?

Solution 3

Here, the quantity Y represents the price. Thus, using the general formula of rate of change

Rate of Y change = Yfinal - Yinitial/tfinal - tinitial

we obtain

Annual rate of price change = Final price - Initial price/Final year - Initial year
= $104 - $80/2022 - 2019
= $24/3 years
= $8/year

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.

Example 4

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?

Solution 4

Price reduction here represents the rate of price change. We have

Rate of Y change = Yfinal - Yinitial/tfinal - tinitial

Here, Y represents the price and tfinal - tinitial = 2 weeks = 14 days. Hence,

Rate of price change = $1.73 - $2.15/14 days
= -$0.03/day

This result means the price dropped by a rate of $0.03 per day.

Applications of Rates in Practice

There are numerous applications of rate in practice. Here, we will briefly discuss some of them.

a. In banking

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

3/100 × $50,000 = $1,500

after one year. Thus, after one year, the total in the bank is $50,000 + $1,500 = $51,500.

b. In Physics

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.

P = ∆E/∆t

Rate of electric flux change in a coil, which represents the electromotive force induced in the coil during a given time, i.e.

εi = ∆Φ/Δt

Rate of momentum change, which represents the force acting on an object, i.e.

F = Δp/Δt

where Δp is the change in momentum of the object, and so on.

c. In medicine

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.

d. In civil engineering

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.

Applications of Ratios in Practice - The Golden Ratio and Fibonacci Numbers

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

a + b/a = a/b = φ

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

φ = 1 + √5/2 = 1.6180339887

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

a + b/a = a/b
a/a + b/a = a/b
1 + b/a = a/b
1 = a/b - b/a
1 = a2 - b2/a × b
a2 - b2 = a × b

We will explain in the algebra section that

a2 - b2 = (a - b) × (a + b)

Therefore, substituting in the previous equation, we obtain

(a + b) × (a - b) = a × b

Hence, the numbers forming a golden ratio must meet the condition "sum × difference = product".

Example 5

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?

Solution 5

We have a = 5.2 m. Since

a + b/a = a/b ≈ 1.618

we obtain for the height of the first floor:

5.2/b ≈ 1.618
b = 5.2/1.618
≈ 3.21 m

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

Fibonacci sequence = {1,2,3,5,8,13,21,34,55,89}

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,

1 + 2/2 = 3/2 = 1.5
2 + 3/3 = 5/3 = 1.666
3 + 5/5 = 8/5 = 1.6
5 + 8/8 = 13/8 = 1.625
8 + 13/13 = 21/13 = 1.6153
13 + 21/21 = 34/21 = 1.6190
21 + 34/34 = 55/34 = 1.6176
etc.

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:

  • In plants: You can find the golden ratio in the spiral arrangement of leaves (called a phyllotaxis) on some plants, or in the golden spiral pattern of pinecones, cauliflower, pineapples, and the arrangement of seeds in sunflowers.
  • In art: In many cases, artists have been inspired by the aesthetics of the golden ratio and incorporated it into their works.
  • In architecture: The Parthenon in Greece incorporates the golden ratio in many of its design elements. In the twentieth century, Swiss architect Le Corbusier used the golden ratio in his Modulor system for the scale of architectural proportion. The United Nations Secretariat Building in New York City was designed using the golden ratio: the size and shape of the windows, columns, and some sections of the building are based on the golden ratio.

Example 6

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.

Solution 6

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

H + W/H = 233 cm + 144 cm/233 cm
= 377 cm/233 cm
= 1.61802

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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