Proportion
Overview
Proportion is an equation used to compare the magnitudes of quantities. It can be defined as an equation that presents equality between two ratios. In other words, if the ratio between two characteristics of an object is equal to the ratio between the same two characteristics of another object, the two objects are considered to be in proportion. These characteristics could be anything that can be measured (such as size, quantity, dimension, etc.). For example, consider two rectangles, the first having length and width equal to 8 in. (20 cm) and 4 in. (10 cm) respectively, and the other having a length of 6 in. (15 cm) and width of 3 in. (7.5 cm). These two rectangles are in proportion as the ratio of the length and width of each rectangle is equal.
Although proportion is a concept mainly used in design, it is widely applied to other aspects of daily life as well. One of the most common examples is grocery shopping, where proportion is frequently used to compare prices of items with different sizes. In addition, proportion finds uses in numerous other fields, including architecture, art, maps, astronomy, business, imaging, technology, and even cooking.
Fundamental Mathematical Concepts and Terms
As stated earlier, proportion is indicated by the equality between two ratios. Mathematically, it can be expressed in two ways—a/b = c/d or a:b = c:d. The outer terms of the equation are known as extremes, while the inner (or middle) terms are known as means. For example, in the above equation "a" and "d" are extremes, whereas "b" and "c" are means.
SOLVING RATIOS WITH CROSS PRODUCTS
One way to test equality is by simply calculating the values of the ratios. However, a more commonly used method involves the use of cross products. Cross products can be calculated by multiplying the outer terms (or extremes) and then the inner terms (means). If both values are equal, the ratios are in proportion.
Consider the ratios 2/5 and 3/7.5. In this case, the cross product of the extremes is 2 × 7.5 = 15, while the cross product of means is 5 × 3 = 15. Hence, the ratios are in proportion. Note that simple division here would have been far more complex and time consuming, as compared to calculating cross products. This is one of the reasons for the popularity of the cross product method.
The cross product method also has another significant benefit. Real life applications use the concept of proportion mainly to compare two things or objects. In many cases, there may be a missing term in the proportion. For example, a grocery store owner charges __BODY__.50 for 1 lb. (0.4 kg) of beef roast. He wants to set the price of a 3 lb. (1.2 kg) roast, such that it is in proportion with the price of the 1 lb. (0.4 kg) roast. This can be easily done by writing the proportion equation, and then using cross product to determine the price.
The equation can be written as—1.50/1 = x/3, where "x" is the price of the 3 lb. roast. By calculating the cross products of the means and extremes, the value of "x" comes out to be 4.50. In other words, the 3 lb. (1.2 kg) roast should be priced at $4.50 for it to be in proportion. Simply put, you can calculate a missing term from a ratio if this ratio is in proportion to another known ratio. This underlying concept of proportion is extremely useful in real-life applications.
DIRECT PROPORTION
If change in one component causes a change of equal magnitude (size, percentage) in another component, the two components are said to be in direct proportion. Another way of expressing this is by stating that the first component is directly proportional to the second component. In a nutshell, direct proportion is a concept that pertains to the change in the values of two (or more) components that are already in proportion.
For example, imagine the price of a candy bar is __BODY__.50. The number of candy bars is always proportional to the total price of the bars—the ratio of the number of candy bars to the total price always remains same. One bar costs __BODY__.50, two bars cost __BODY__.00, four cost $2.00, eight bars cost $4.00, and so on. Put simply, a change in the number of bars causes a change in the total price. Moreover, the magnitude of the change is also the same. In other words, the change in the number of bars as well as the price can be represented by a common factor. The number of bars keeps doubling (or 1 × 2 = 2, 2 × 2 = 4, 4 × 2 = 8). Similarly, the price also doubles (__BODY__.50 × 2 = __BODY__.00, __BODY__.00 × 2 = $2.00, $2.00 × 2 = $4.00). Hence, the number of candy bars is directly proportional to the total price of the bars. Also the change is represented by a common factor (two in this case).
Mathematically, direct proportion is indicated as a α b (a is directly proportional to b). The main advantage of direct proportions is that they can be expressed in the form of an equation. For example, the relationship between the total number of bars and the total price, in the above case, can be shown as:
Total number of candy bars = k × Total Price, where k is the common factor.
The common factor is known as the proportionality constant. This equation may be used to easily calculate the total price if the number of candy bars is known, and vice versa. All direct proportion relationships can be expressed by such equations. Consequently, they are used extensively in various real-life activities and applications.
INVERSE PROPORTION
Like direct proportion, inverse proportion also pertains to the change in two (or more) components. However, in the case of inverse proportion, an incremental change in one component causes a decrement in the other component. In other words, if the magnitude of one component increases, the value of the other component decreases, and vice versa.
Consider, for example, a car traveling from one place to another. If the car has a constant speed (and assuming it does not stop anywhere), the more it travels, the less the remaining distance to the target destination. Hence, in this case, as the total travel time increases, the distance to the destination decreases—travel time is inversely proportional to distance remaining.
Similar to direct proportion, the change can be represented by a factor. However, the factors that represent change for both components are multiplicative inverses of each other. In simple terms, if the value of one component changes by a factor of three, the change in the value of the other component will be 1/3. Consequently, inverse proportion is also known as reciprocal proportion, and is mathematically indicated as a α 1/b (or travel time α 1/distance remaining, for the above example).
An inverse proportion relationship can also be expressed in the form of an equation. For instance, the two components (travel time and distance remaining) in the above example can be shown as:
Travel time × k/distance remaining, where k is the proportionality constant.
A Brief History of Discovery and Development
Throughout history, proportion has been used extensively in numerous areas. The Greek mathematician Pythagoras (580 B.C.–500 B.C.) who is most well known for the Pythagorean theorem, developed the Theory of
Proportion to relate music with mathematics. He established musical scales that were based on the concept of proportion.
Subsequently, evidence of proportion can be seen in many works of art and architecture, especially in ancient Greece and Rome. Some of the most popular paintings by renowned artists such as Michelangelo (1475–1564), Raphael (1483–1520), and Leonardo da Vinci (1452–1519) were based on proportion. The concept of proportion is vital to art and architecture as it describes the size, location, or amount of one element to another within the entire work (e.g., Vitruvian Man by Leonardo da Vinci). The proportion of various parts of the body in this painting is very similar to the proportion seen in an average human body.
Similarly, much like modern architecture, ancient structures and buildings also incorporated proportion. The ancient Egyptians used it in the construction of the pyramids. The Parthenon in Athens, Greece, is another structure where proportion, along with ratio and scale is used extensively to create a "harmony" among various elements.
Interestingly, Isaac Newton's (1643–1727) second law of motion states that the acceleration of an object in motion is directly proportional to the force applied on it—a classic equation indicating direct proportion between two properties, acceleration and force.
Historians and mathematicians also believe that the great musicians Mozart (1756–1791) and Beethoven (1770–1827) used proportion to compose music. Proportional scaling allows the composition of harmonic, pleasant-sounding, music—a concept initially put forward by Pythagoras.
Subsequently, by the nineteenth century proportion was applied to numerous applications including those in business and sciences.
Real-life Applications
ARCHITECTURE
Architecture uses mathematical concepts such as proportions and ratio extensively. Since ancient times, architects and designers have been building various parts of a structure in proportion to attain visual appeal, unity, stability, and order. These principles hold true even today. Proportion is employed in a number of ways in architecture. Most popular buildings and structures—ancient as well as modern, are based on what is commonly known as the divine proportion or golden proportion.
The divine proportion consists of two or more ratios that are equal to phi (or 1.618). In other words, if the ratio (also known as divine ratios) of various parts of a building (or a structure) is equal to the number 1.618, then the proportion of these various parts is known as the divine proportion. Throughout the world, monuments, famous buildings, and other structures have been created using the divine proportion. This includes the pyramids of Giza, the Parthenon in Greece, the Colosseum in Rome, numerous cathedrals including St. Peter's Cathedral in the Vatican, the Taj Mahal in India, the Pentagon in the United States of America, and many more.
As stated earlier, proportions are used on various elements (or parts) of the entire structure. For example, the front elevation of the Parthenon is built to the divine proportions: its width is 1.618 times its height. Besides divine proportion, basic principles of proportion are also used. For example, the Pentagon is made up of five internal (or concentric) pentagons. Each of these internal pentagons is in proportion to the outer pentagon.
The concept of proportion is used widely in modern architecture as well. Apartment buildings, or houses within the same community may have different sizes of apartments (or houses). However, they are typically in proportion to each other. Sports stadiums also incorporate proportion: the distance between the bases in a baseball field is always proportional to the length (or width) of the field. Similarly, the width of a goal post in a soccer field is proportional to the width of the entire field.
In addition, architects design miniature models before building the actual structure. These models, known as scale models, serve as detailed representations of the final structure. These scale models are much smaller in size, but are in proportion to the final structure. For example, if the scale model of a house is a hundred times smaller than the actual house, every room (or part) of the model would also be a hundred times smaller than the corresponding room (or part) in the actual house—all parts of the model are in proportion with the actual house. Similarly, different parts within the model are also in proportion. If the actual house should be built such that there are two rooms—one room twice the size of the other, the model would also depict two rooms, where the size of one room is twice that of the other. Simply put, the ratio of the sizes of the two rooms is equal in both cases.
The main advantage of a model is that it allows the architect to visualize a structure before it is built. Also, once the model is created, using proportion, various measurements of the final structure can be easily determined and constructed accordingly.
ART, SCULPTURE, AND DESIGN
Like architecture, painting and sculpting also relies on the concept of proportion. Some of the great painters and sculptors, for centuries, have used mathematical models of proportion to attain visual appeal and symmetry (balance) in their work. Portraits and paintings depicting natural scenery are, more often than not, in proportion with the real thing. For a portrait of a person, a good painter would ensure that the measurements of body parts in the painting are in proportion to the actual measurements of the person. This can be seen in most of the ancient as well as modern day portraits.
In addition, different elements within the same painting are also in proportion. In a painting of natural scenery
depicting a house, trees, fences, and mountains, the size of each of these is not similar. A house in the painting would be bigger than the size of the fence (unless they are supposed to be at different locations far away). In other words, depending on their location, the sizes are always in proportion—similar to what we see in the real world.
The same holds true for sculptures as well. Like a painting, the sculpture of a person may be bigger (or smaller) in size than the person. However, in most cases, the measurements are in proportion. The advantage of proportion for creating sculptures is evident when the difference in size of the actual object and that of the sculpture is large. Mount Rushmore, in South Dakota, is a classic example. The design and development of the famous memorial to the four presidents—George Washington, Thomas Jefferson, Abraham Lincoln, and Theodore Roosevelt, is based on a number of mathematical concepts, such as ratio, proportion, and scale.
Prior to sculpting the faces of the presidents on the mountain itself, the designer of the memorial, Gutzon Borglum, developed a smaller model. The size and measurements of the memorial on the mountain are in proportion to the model. Carving the faces on the mountain directly would have been an extremely difficult task for the designer and his team. However, a smaller but proportional model greatly simplified the process. Many technical aspects such as distance between the faces, size of each face, measurements within a particular face, could be easily calculated in the model. Once all measurements were recorded, the designer used the proportion equation to calculate the actual measurements of the memorial (in order for it to look exactly like the model itself).
The principles of interior design also rely on proportion. Furniture, for example, is designed so that its parts are proportionate to each other. This is critical in achieving stability and balance. Furniture that is out-of-proportion is not considered visually appealing. The parts of a chair—the arms, legs, seat, and back—are in proportion to each other, and the chair as a whole.
MEDICINE
Medicines are very essential to treat many illnesses and diseases. Medicines are also used during surgeries and medical diagnosis. They often contain more than two ingredients or compositions that are essential to have desired effect. The proportion of this composition becomes very important. In other words, every medicine contains a specific proportion of its ingredients.
Prescriptions, as well as over the counter drugs, require the mixture of various chemicals, and other additional constituents, to be in certain proportions. For example, over the counter medicines for pain relief often contain aspirin, a required drug to relieve pain, along with other drugs. The proportion of each constituent present in medicine is important as they are meant to treat a certain type of disease, illness, or ailment.
Changing the proportion of the constituents can have different effects. Several common ingredients are used to treat different types of illnesses. The reason for this is that medicines, when prepared using different proportions of the same drugs (or ingredients), act differently, and hence are meant for different diseases.
Proportion is also used frequently by doctors and nurses, while preparing dosages for patients. Patients may require dosages of drugs that vary in quantity and strength. For example, some times a patient may need a dosage that contains 200 mg of a drug that comes as 100 mg diluted in 1 ml of fluid. The technical specifications associated with dosage measurement are beyond the scope of this article. However, for our purpose, the above dosage can be thought of containing a drug in specific quantity (200 mg), having specific strength (100 mg diluted in 1 ml of fluid). The quantity of a drug is proportional to its strength. Using this relation, health care professionals can calculate the quantity of the drug to be administered for a particular strength.
MAPS
Maps may represent a large geographical area and can be of various types depending on the features they emphasize. The area represented by a map can vary from a small room to the entire universe.
There exists a relationship between a specific distance on the map and its actual distance. This relationship is defined by the mathematical concept of scale (or map scale). However, it is important to note that the map scale is based on proportion. In simple terms, the size of the map and the size of the area it shows are always in proportion.
Consider a map that depicts an area that is a hundred times larger than the size of the map. In this case, the relationship between the map and the actual area can be shown as the map scale (a ratio in this case) 1:100—one unit of measurement (cm, inch, feet, etc.) on the map is equal to hundred units in the actual area. The ratio between any part of the map to its actual size remains the constant (1:100). Therefore, every part of the map is in proportion to its actual size. For example, if the actual distance between two points is 100 inches, then the distance between the same two points on the map would be 1 inch. Similarly, if the actual distance between two points is 500 inches, the distance between these two points on the map is 5 inches—the distance between any two points on the map is proportional to the actual distance between them.
Maps can be categorized into two types—the large scale map, and the small scale map. The large scale map shows a smaller area but in greater detail, whereas a small scale map shows a larger area in less detail. The map scale for these maps would differ; however, the maps are always in proportion to the actual size. A city map would be an example of a large scale map as compared to a world map (small scale).
ERGONOMICS
Ergonomics is a science that studies technology and how well it suits the human body. Ergonomics involves understanding basic body parts, their functions and abilities to operate equipments, machinery, products, and other technological devices. Ergonomics is commonly used while designing cars, among other things. Ergonomic car designs are based on the principles of proportion.
Consider, for example, a car seat for drivers. Its height from the surface, inclination, and movements patterns are all designed in proportion to the human body. The size of the seat has to be in proportion with the size of an average human driver. In addition, you do not expect a person to have a giant steering wheel in front of him/her—the size of the wheel (the diameter of the wheel) has to be in proportion to the size of the hand grip, shoulder width, and distance between the wheel and person driving the car.
Ergonomics is used extensively in many areas as well. This includes design of kitchen and appliances, design of home and office furniture, bathroom appliances, electronics, computer systems, airplane and train interiors, and much more. Every ergonomically designed object is proportional to the size of the human body (or a part of it).
For example, a bed is usually designed in proportion to the human body. The length of a bed is proportional to the average height of a person. Many beds in Europe are around seven feet (2 m), whereas those in Asia are around six feet (2 m) long. This also influences other design standards such as height of the bed from the floor, and width of the bed.
ENGINEERING DESIGN
Engineers apply the principle of proportion in many ways including when designing automobiles, airplanes, and trains. Representative two-dimensional models (similar to scale models discussed earlier) are designed before finalizing and manufacturing a car, plane, or train. These are detailed models depicting each and every characteristic. The automobile is then built such that its size and other measurements are directly proportional to the model. In other words, a relation based on proportion is established between the model and the actual object.
The main benefit of creating models for automobiles (as well as airplanes and trains) is to easily study design issues. For example, after calculating the measurements of a seat in the car model, using proportion, the actual size of the seat can be calculated. This will enable the designer to analyze whether the size of the seat is appropriate for a person.
As the dimensions and size of the car are proportional to the model, any change in the model would affect the car. Besides, parts of the model (or car) are also proportional to the model (or car) as a whole. Put simply, if for example, the size of the leg room is changed, the change in the total size of the car can be calculated. if leg room needs to be increased, and at the same time the size of the car must remain constant, the designer would have to reduce the size of some other part of the car.
Once a model with ideal measurements is created, manufacturing the final object becomes a lot easier.
MUSICAL INSTRUMENTS
Since ancient times, mathematicians have always established relationships between principles of mathematics and music. Pythagoras was the first people known to study and apply concepts of proportion and scale to music. These principles are also valid for most musical instruments.
It is widely believed that instruments designed using specific proportions produce superior music. This can be seen in both ancient as well as modern day instruments. For example, to achieve better quality of music, the distance between strings on a guitar (or a violin) is proportional to its entire width. In fact, proportion is used for designing every part of the instrument. Similarly, for a piano to function properly, all its parts have to be in proportion to one another.
CHEMISTRY
Chemicals are often a mixture of a variety of substances. These substances are present in certain ratios. For example, the chemical composition of ammonia is NH3. Here, the amount of nitrogen (N) is directly proportional to the amount of hydrogen (H)—the ratio of nitrogen atoms to hydrogen atoms is 1:3. In other words, if the number of nitrogen atoms increases by one, the number of hydrogen atoms have to be increased by three. Similarly, if two nitrogen atoms are added, six hydrogen atoms must also be added to continue for the substance to be ammonia. The ratio between nitrogen and hydrogen is always maintained.
Setting up equations as proportions is one of the most effective ways of solving a number of problems in chemistry. For example, to prepare chemical solutions, the chemicals are usually dissolved in water or alcohol. The quantity of chemical present in the solution is known as the strength of the solution. In simple terms, a 70% solution would contain 70% of chemical and 30% of alcohol (or water). While preparing the solution of a specific concentration, the amount of chemical is always proportional to the amount of alcohol (or water). This relationship is especially useful while preparing solutions in different quantities but the same concentration.
A 50 mL (four tablespoons) of chemical solution contains 20 mL (a little more than one tablespoon) of alcohol. If the amount of chemical solution has to be increased to 80 mL (a little more than five tablespoons), what would be the amount of alcohol present in this solution? This can be calculated by setting up a proportionality equation as shown below:
20 mL alcohol / 50 mL solution = x mL alcohol/80 mL solution, where x is unknown amount of alcohol. The quantity of alcohol should be 32 mL (two tablespoons) for an 80 mL solution.
Such equations are used widely by doctors, scientists, and students.
DIETS
Dieticians and fitness experts often apply mathematical approaches to developing "balanced" diets. They indicate that every meal should have proteins, carbohydrates, and fats in a certain proportion to each other (and the entire meal). This relationship helps greatly in calculating the amount of proteins, carbs, and fats for different meal portions.
For example, a particular meal amounts to 400 calories—160 calories from proteins, 160 calories from carbohydrates, and 80 calories from fat. If another meal is equivalent to 600 calories, the amount of proteins, carbs, and fats would increase to 240 calories, 240 calories, and 120 calories respectively. Note that the amount of proteins, carbs, and fats is in proportion.
Most food items list the amount (in grams) of protein, carbohydrate, and fat content. For instance, 100 grams (3.5 oz) of ice-cream may contain 20 grams (0.7 oz) of fat. The amount of fat in 50 grams (1.7 oz) of the same ice-cream would be 10 grams (0.3 oz), and so on—fat content is proportional to the total quantity. Food items are always available in specific quantities. Put simply, by applying proportion equations, the content of proteins, carbohydrates, and fats can easily estimated for different quantities.
The same concept is also applied to cooking. While preparing a food item, the ingredients are in proportion to each other (and to the total quantity of the food item).
STOCK MARKET
Mathematical concepts such as proportion and ratio have a lot of business applications. One such example is in the stock market. There are factors that contribute to the share value of a company. However, more often than not, a company's share value fluctuates based on profit it makes. Besides, the value also depends on the number of buyers of the company shares. Simply put, the value of a share is proportional to a combination of factors, including the profit and number of buyers.
Most companies divide a percentage of profits amongst all its shareholders (people who own the company's shares). The amount given per share is known as dividend. Higher the number of shares a person owns, higher the dividend. Another way to look at this is that the total dividend is proportional to the number of shares owned.
PROPORTION IN NATURE
The number Phi is an unusual number with astounding mathematical properties. As explained earlier, the golden section, a principle on which ancient Greek architecture was based, is derived from a ratio that further results in the number phi. Phi appears in proportions of the human body as well as the proportions of various other animals. The renaissance artists referred to the golden section as the divine proportion and used it for achieving balance in arts. The divine proportion principle is found in abundance in nature. The spirals of a sea shell, the galaxy, the body of a dolphin, the structure of a butterfly, a peacock feather, the patterns of flowers and plants, the rings of Saturn, all follow the divine proportion principle.
The average human face is also an example of divine proportion. The head forms the golden rectangle with eyes exactly at the center. The mouth and nose are each placed at golden sections of the distance between the eyes and the bottom of the chin. Assume that the eyes are represented by A, nose by B, mouth by C and chin by D. The ratio of line AC to line AD is the same as ratio of line BC to line AC. This means that the ratio of distance between eyes and mouth to the distance between eyes and chin is in proportion with the ratio of distance between nose and mouth and eyes and mouth. Some scientists who study psychological reactions to faces assert that concepts of beauty may be related to facial symmetry and proportion.
Interestingly, the average human face, when viewed from side also reflects the divine proportion principle. Even the dimensions of human teeth are based on this principle. Some dentists are even considering the knowledge of this principle to enhance their aesthetic dentistry skills. The human hand is also an example of the divine proportion.
