Ratio
Overview
A ratio defines the numerical relationship between two comparable quantities. Examining the ratios between two or more values often provides valuable insight into the patterns and behaviors of numbers.
Ratios exist naturally throughout the universe. The ratio of the size of one planet to another nearby planet can affect the orbits of both planets. The ratio of owls to mice plays a big role in the survival of both species. The ratio of height to trunk width limits the growth of trees. Humans have used ratios in almost all of our creations throughout history. The physical stability of a building depends on several ratios—involving height, width, angles, and the strength of materials that must be carefully analyzed to ensure the safety of the people inside. The accurate mixing of chemicals that allows us to create stronger materials is also reliant on ratios that define how much of each substance is needed with respect to the other materials. People around the world use ratios on a daily basis to organize time and finances.
Fundamental Mathematical Concepts and Terms
A ratio between two numbers X and Y is usually expressed in one of three ways:
- X/Y (much like a fraction)
- X:Y
- "X to Y"
Each of these expressions represents the ratio of X to Y.
For example, if there are 12 cars for every three trucks, then the ratio of cars to trucks can be written as 12/3, as 12:3, or as "12 to 3." Given this information about cars and trucks, it is also true that the ratio of trucks to cars is 3/12, 3:12, or "3 to 12".
All of these expressions for the ratio of cars to trucks (or trucks to cars) state exactly the same thing: for every 12 cars, there are three trucks. Suppose that people in a certain neighborhood always keep their cars in their garages, but leave their trucks out in the driveway. If three trucks are visible in the neighborhood, then there are 12 cars in the neighborhood, even though they are hidden in garages.
The foundation of the idea of a ratio is that whatever happens to one of the numbers also happens to the other. Suppose that six trucks can be seen in driveways around the neighborhood. This means that there are 24 cars hidden in garages. The number of trucks was doubled (multiplied by 2) so the number of cars must have doubled as well. Division of ratios works in the same way. If there was only one truck in the entire neighborhood, then there would be only four cars. Here, the number of trucks and cars are both divided by two to arrive at the ratio 1:4. In fact, this is the simplest form of the ratio of trucks to cars. In a case such as this, the ratio can be simplified so that one of the values is one, which is a good illustration of how ratios work: no matter how many trucks are in the neighborhood, the number of cars is four times as large. Not all ratios can be simplified this neatly—2:3 for example. In cases like this, a decimal can be used as 2:3 simplifies to 1:1.5. In any case, it is easiest to understand the relationship between the two values when the ratio is simplified.
Ratios can be multiplied together to discover new ratios. For instance, if there are two cars for every truck, and three trucks at every house, then there are six cars at every house. That is, 2:1 multiplied by 3:1 is equal to 6:1. Perhaps money provides a better illustration of this concept. There are four quarters to every dollar and five nickels to every quarter; so there are 20 nickels to every dollar. This can be verified by multiplying the five pennies in each nickel by 20 (the number of nickels in a dollar) to get 100 pennies to every dollar.
Although often expressed as a quotient (one number divided by another, such as 2/3), ratios are not the same thing as fractions. For example, if Otis has two dogs and four cats, then the ratio of dogs to cats in his house is two to four, which simplifies to 1:2 or 1/2. This indicates how many dogs there are compared to cats (there are half as many dogs as cats). However, the fraction of animals in Otis' house that are dogs is two out of the total number of animals or 2/6, which simplifies to 1/3. This means that one third of all of his animals are dogs. Be careful to understand how fractions are related to ratios when using the quotient style of notation. To avoid confusion, this text most often uses the X:Y style of notation for ratios.
A Brief History of Discovery and Development
The term ratio stems from an early sixteenth century Latin word meaning reason or computation. However, the mathematical concept of ratios helped people understand the universe around them long before that.
For example, the relationship between a circle's diameter (the length of any line connecting one side of the circle to the other through the center of the circle) and circumference (the length of the boundary of the circle) was approximated for thousands of years before the Greek mathematician Archimedes discovered a way to define the relationship exactly. This ratio can be used to determine the circumference of a circle if its diameter is known, and vice versa. The circumference of any circle is equal to the diameter multiplied by this ratio, commonly represented by the Greek letter pi, and approximately equal to 3.14159265.
Ancient Egyptians approximated pi (though they did not call it pi) as 3.1605. The Old Testament of the Judeo-Christian Bible contains a reference to an approximation of 3:1 for the ratio of a circle's radius to the circumference of a circle. Although ancient Babylonians generally agreed with this approximation throughout most of their history, a stone tablet believed to have been created by Babylonians sometime between 1900 and 1680 B.C. referred to a slightly more accurate approximation of 3.125 for pi.
Early approximations of pi were dependent on approximations of the circumference of circles. It is believed that most approximations of circumference were found using methods similar to those used by Archimedes. First a circle was placed inside of the smallest hexagon (a polygon with six sides) that it could fit into. The length of the perimeter of the hexagon was calculated by measuring one side and multiplying this value by six. Next, the perimeter of largest hexagon that could fit inside the circle was calculated. Because the smaller hexagon just barely fits into the circle, and the circle just barely fits into the larger hexagon, the circumference of the circle is somewhere between the lengths of the perimeters of the two hexagons. To arrive at a better approximation, the number of sides of the two surrounding polygons was increased. As more sides were added, the two polygons fit the circle more snugly and the perimeters became closer and closer to the circumference of the circle. Archimedes used these approximations as clues that eventually led him to find a way to define the ratio of diameter to circumference exactly.
Another important ratio studied throughout history is the Golden Ratio, also known as the Golden Mean, the Divine Section, the Golden Section, the Golden Cut, the Divine Proportion, and many other names. The main reason that this ratio has so many names is that it has been discovered at different times by civilizations that use different languages and, most importantly, different numbering systems. The Golden Ratio is approximately 1.6180339887498948482 to 1 (how the Golden Ratio is calculated is beyond the scope of this text). The Golden Ratio is usually denoted by the Greek letter phi (φ).
The Golden Ratio can be found throughout nature—from the patterns found in leaves, pinecones, and seashells, to the reproductive patterns of certain animal species. It is also argued that the Golden Ratio provided a basis for the architecture of the ancient Egyptians (including the designs of pyramids and tombs), Greeks (the Parthenon), and Romans. Some ancient Egyptian hieroglyphics show signs of the Golden Ratio as well. Leonardo da Vinci, Mozart, and Beethoven purposely incorporated this ratio into their works. The seemingly endless applications of the Golden Ratio provide brilliant illustrations of the fascinating relationships between numbers.
Real-life Applications
LENGTH OF A TRIP
Ratios can be used to estimate length. For an example let us assume that Tom needs to drive from New York to Miami for a business convention on Saturday evening. He has never driven that far and wants to figure out about how long it will take, so he buys a map of the United States. He notices two bars labeled Scale in the corner of the map. The longer of the two bars represents 100 miles, and the shorter bar represents 100 kilometers. He uses his ruler and finds that the 100-mile bar is one inch long; so the ratio of inches to miles on Tom's map is one to 100. Using the other side of his ruler, he finds that the 100 kilometer bar is one centimeter long; so the ratio of centimeters to kilometers is also one to 100.
Tom is more comfortable thinking in terms of miles, so he chooses to approximate the length his trip based on the inch to mile ratio of 1:100. All he needs to do is find out how many inches separate New York and Miami on the map. Tom lays his ruler on the map, with the beginning of the ruler (representing zero in inches) at New York. The shortest driving route is not a straight line, so he must approximate how long, in inches, his route is on the map. Starting from New York, he measures one inch in the direction of the route that he will take, and marks the spot on the map with a pencil. Then he moves the beginning of the ruler to the mark he just made and measures another inch, following his intended route as accurately as possible. Continuing in this way, he makes 13 marks. The last mark is a little past Miami on the map, so he figures that the route is a little less than 13 inches long. He can't be late to his convention, so he decides to use 13 inches as the base of his calculations. As he found before, the ratio of inches to miles represented on the map is 1:100.
Tom then wants to figure out how many miles are represented by 13 inches, so he must multiply the ratio through by 13 to get a ratio of 13:1,300. This ratio indicates that 13 inches on the map represents 1,300 miles in the real world. So Tom's trip will be about 1,300 miles in distance (length).
Tom now needs to utilize another ratio to help him decide when to leave New York. Without exceeding the speed limit, he can drive about 500 miles in a day. So his mile to day ratio is 500:1. This means that he can drive 500 miles in a single day, 1,000 miles in two days, 1,500 miles in three days, and so on. He needs to go a total of 1,300 miles, so he cannot make it in two days. He can make it easily in three days. He may be a little early but he will not be rushed. He decides that if he leaves on Thursday morning, he will get to the convention with time to spare.
COST OF GAS
In the previous example, Tom calculated 1,300 miles as a slight overestimate for the length of his trip from New York to Miami. He now wants to calculate how much money he will need for gas so that he can plan the budget for his trip. His car gets an average of 25 miles per gallon, which is a mile to gallon ratio of 25:1. Tom uses this ratio to calculate how many gallons of gas his car will need to go 1,300 miles. 1,300 miles is 52 times as long as 25 miles, which means that Tom must multiply both sides of the ratio by 52. In this way, he calculates the mile to gallon ratio 1,300:52. To go 1,300 miles, Tom's car will need 52 gallons of gas.
Next, Tom looks on the Internet and discovers that the average cost of gas along his route is two dollars per gallon. So the ratio of dollars to gallons of gas is 2:1. To find out how much 52 gallons of gas will cost, Tom multiplies both sides of the ratio by 52 to get a dollars to gallon of gas ratio of 104:52, meaning that Tom needs $104 to buy 52 gallons of gas for his car. After working this figure into his budget, he finds that he has plenty of money for his trip to Miami.
GENETIC TRAITS
In 1866, Austrian monk and geneticist Gregor Johann Mendel (1822–1937), published his results from an extensive series of experiments that investigated how characteristics are passed to offspring. One such experiment involved the cross-pollination (transferring the pollen of one plant to another) of two different varieties of pea plants, a green wrinkly pea plant and a yellow rounded pea plant. In this experiment, Mendel discovered that the ratio of yellow rounded offspring to green wrinkly offspring was 3:1, meaning that the cross-pollination process produced three yellow rounded pea
plants for every green wrinkly pea plant. This suggested that the yellow characteristic is three times as likely to appear in the offspring as the green characteristic, and the round characteristic is three times as likely to appear as the wrinkly characteristic. This dominance of yellow and rounded characteristics led Mendel to believe that there are two different types of genetic traits: dominant traits and recessive traits. In these two types of pea plants, the yellow color and round shape dominate the green color and wrinkly shape.
Mendel used this idea of dominance to explain why the ratio of dominant traits to recessive traits is always 3:1. For instance, if the dominant yellow color trait is represented by Y, and the recessive green color trait is represented by g, then the four possible combinations of these traits are YY, Yg, gY, and gg (where each parent plant provides either a Y or a g). A dominant trait only needs to appear once in the combination in order for the dominant characteristic to appear in the offspring. Y appears in three of the combinations, and the only combination that results in a green pea plant is gg. Therefore, the ratio of offspring that show a dominant characteristic to offspring that show a recessive characteristic is three to one, or 3:1. This is true for the shape trait as well. Keep in mind that Mendel's actual experiments and results were more complicated than described here.
Mendel's experiments and conclusions explained a phenomenon that had confused people for thousands of years, that traits can appear after skipping generations. For example, cross-pollination of two yellow pea plants can result in a green offspring as long as at least one parent had a Yg or gY gene.
The importance of Mendel's results was not truly recognized until the beginning of the twentieth century when multiple researchers independently rediscovered Mendel's conclusions in their own experiments. Since then, Mendel's findings have been the foundation of many genetic studies and practices, including the creation of new flowers and new species of pet fish, and enhancements in farm production that improve the quality of produce found in most grocery stores.
STUDENT-TEACHER RATIO
The student-teacher ratio compares the number of students to the number of teachers at a given school. For example, if a school has 44 teachers and 968 students, the student-teacher ratio at this school is 968:44, which simplifies to 22:1. This can be interpreted in a few different ways. A mother might see it as an indication of how much attention her child will receive in school (e.g., her child will share each teacher with 22 other students). This ratio enables a teacher to predict how many students he or she will teach and how many papers she will be grading (about 22 students per class multiplied by the number of classes he or she teaches). A school official may see it as an indication of how many teachers need to be hired in the following year. A prospective college student will usually take the student-teacher ratio into consideration when choosing a school for higher studies.
MUSIC
Ratios can be found in every facet of music. Rhythm, the speed and pattern of beats, has been the foundation of music dating back to ancient drum circles. Rhythm determines how many beats there are in a measure (the standard unit in the arrangement of song). For example, if there are four beats in a measure, then the ratio of beats to measures is 4:1. This means that the musician considers four beats—possibly counted by tapping a foot—to be a single standard unit in the arrangement of the song. Different ratios of beats to measures affect different types of music. For example, a typical waltz has a ratio of 3:1.
The relationship between the pitch of two musical notes is also a ratio. Whether created by your vocal chords, a guitar, or a finger moving around the rim of a wine glass, the sound of a note is determined by the frequency (speed) of the vibration causing it. As you move from left to right on the keys of a piano, the difference from one note to the next is determined by the ratio between the frequencies of the two notes: the ratio between the frequencies of two subsequent notes is always the same. Harmony (whether or not two or more notes sound good when played together) is determined by the ratios of their frequencies as well.
Ratios in music allow songwriters and musicians to communicate the intended shape and feel of a song. The many ratios in a composition define how the various sounds relate to each other in time and, whether consciously noticed or not, give the music both structure and beauty.
AUTOMOBILE PERFORMANCE
The safe and efficient operation of any automobile is dependent on many ratios. Oils and fluids must be present in certain ratios to keep the engine and brakes operating properly. The relationships between the size, weight, and position of various parts ensure that a car or truck can make turns while traveling at reasonable speeds and can stop quickly when necessary. Two ratios found in automobiles are compression ratios and gear ratios.
The compression ratio is used to predict how efficiently an engine will perform. In general, a higher compression ratio indicates better engine performance. High compression ratios are often associated with requirements of more expensive fuel and frequent engine maintenance. The determination of an engine's compression ratio involves the relationship between the sizes of the parts of the engine that cause combustion (the small explosions that provide an engine with power).
The speed and power of an automobile depend partially on the ratio between the sizes of gears that cause the wheels to turn. A larger gear turns slower because it has more teeth and takes longer to complete a full revolution. If a gear that is powered by the engine is attached to a smaller gear, the smaller gear will turn more quickly than the large gear. This increases the speed of revolution without increasing the need for power. Given certain gear ratios for an automobile, a specialist can determine how many revolutions per minute (RPM) are required to go a certain speed, or how many tons can be pulled without overexerting the engine. A typical car has multiple sets of gears intended to perform different actions. The first gear has a high gear ratio in order to provide the car with enough power to get the car started. In higher gears, the gear ratio is increased in order to enable faster speeds. Also, a car would eventually get stuck without an additional gear set that caused the car to move in reverse.
SPORTS
Ratios are often used to assess the performance of an athlete or athletic team. The relationship between two or more statistics often proves a better indication of performance than a single statistic alone.
As an example, a point guard's contribution to a basketball team is partly measured by his assist-to-turnover ratio. This ratio is determined by comparing the number of assists (passes that lead to an immediate basket) to the number of turnovers (anything that causes the ball to be lost to the other team). Suppose Gary has had 53 assists this year, and has turned the ball over to the other team 44 times. Gary's assist-to-turnover ratio is 53:44.
Gary's talents could be judge based on turnovers alone. If Gary had more turnovers than anyone else this season, sports analysts might think that he is the worst point guard because he gives the ball up more often than anyone else. But what if he also happened to have the most assists? Would the analysts still think so poorly of him? The converse is true as well: if Gary's talents were judged based only on the number of assists that he has without taking into account the fact that he turns the ball over quite often, the analysts would not have a very accurate picture of how Gary actually performs on the court.
AGE OF EARTH
In 1905, New Zealand/English physicist Ernest Rutherford (1871–1937) announced a discovery that would forever change the approximations of the age of Earth. He suggested that the age of rocks could be computed by analyzing one of two ratios: the ratio of uranium to lead or the ratio of uranium to helium. These ratios can be used to determine how long radioactive materials have been decaying, and in turn, to determine how long ago rocks were formed. Prior to this discovery, the process of radioactive decay was poorly understood, and guesses at the age of Earth were just that: guesses. Since Rutherford's discoveries, new tools and methods have been derived to improve estimations of Earth's age. For example, calculations in the dating process, including values for decay rates, have been repeatedly improved upon. As of 2005, the best estimation for the age of Earth is in the neighborhood of 4.5 billion years.
HEALTHY LIVING
A person's height-to-weight ratio is the relationship between how tall that person is and how much that person weighs. If a person is six feet tall and weighs 180 pounds, then his height-to-weight ratio is six feet to 180 pounds, or 1 foot per 30 pounds. This ratio can be seen as an indication of how healthy a person is. There are, of course, many other important considerations—including body type, bone thickness, and muscle density—that help determine an individual's optimal weight. All of these factors can be put into terms of ratios.
COOKING
Whenever a chef follows a recipe, he uses ratios to determine how much of each ingredient to stir in. Suppose a chef is cooking his favorite soup for a large dinner party. He has a recipe that tells him how much of everything is required for making enough of the soup to serve 20 people, but there will be 140 people at the dinner party. The ratio of people served by his recipe to the actual number of people that he needs to serve is 20:140, which simplifies to 1:7 (by dividing both sides by 20). This tells the chef that he needs to buy seven times the amount of ingredients suggested by the recipe in order to make enough soup for the dinner party.
The chef can also use ratios to determine how much of one ingredient will be needed based on the required amount of another ingredient. For instance, the chef knows that the ratio of sugar to butter in this recipe is 1:3. This means that the amount of sugar needed to make any amount of this recipe is a third of the amount of butter needed. The chef has already calculated that he needs six cups of butter to make the soup for 140 people. With no further calculations, he knows that he needs two cups of sugar to make this amount of soup.
CLEANING WATER
Chlorine is the main chemical that is used to clean both drinking water and the water in swimming pools. The biggest difference between the processes for cleaning drinking water and swimming water is the concentration of the chemicals, the ratio of the amount of chemicals to the amount of water. This ratio is much lower in drinking water than in swimming water. That is, the water you drink has a smaller amount of chemicals in it than the water in most swimming pools. The concentration of chemicals in drinking water must be precisely monitored in order to ensure that enough chemicals are present to kill bacteria, but not enough to be harmful when swallowed by humans. Water in a swimming pool must contain a higher concentration of chemicals because the water is constantly in contact with contaminants from swimmers and the air above. The fact that a swimming pool is open to the air also allows the chemicals to evaporate, so new chemicals must be added periodically. These ratios between water and chemicals are essential for the different uses of water. Water from a swimming pool is not safe to drink in large quantities; and swimming in water with the concentration of chemicals found in drinking water would quickly result in the growth of algae and bacteria in the pool.
Potential Applications
STEM CELL RESEARCH
Stem cells are special cells in the human body that have the ability to become any type of human cell. This single type of cell can create skin and muscle tissue, bones and bone marrow, and organs such as the liver and lungs. This characteristic has made stem cells the main focus of regenerative medicine, a field of research involving the recreation of cells in the human body. The regeneration of cells may be the solution to many problems that have been unsolvable in the past. Potential uses of cell regeneration include regaining skin and muscle tissue lost in physical accidents; allowing someone bound to a wheelchair to walk; and curing diseases such as Parkinson's, cancer, Alzheimer's, and diabetes. Unfortunately, it may be many years before stem cells are regularly used in routine medical procedures.
Scientists have much to learn about manipulating stems cells to create a desired part of the body. Ratios play a big role in many stem cell research projects. For example, the ratio of blood cells in a donor to blood cells in the recipient may be an important factor in the success of stem cell transplants.
OPTIMIZING LIVESTOCK PRODUCTION
In nature, the sex ratio (ratio of males to females) of many species remains close to 1:1 (often referred to as 50:50 or fifty-fifty), meaning that about half of the population is male and about half is female. In species with males that can mate with multiple females, this may not seem a very efficient ratio. Nevertheless, the sex ratio remains approximately 1:1.
Farmers have for millenia artificially kept the ratio of female cattle (cows) to male cattle (bulls) high, because a single male cow can fertilize multiple female cows. Suppose a single male cow can regularly fertilize up to 20 cows. If a dairy farm with 100 cows had 50 males and 50 females, then only half of their cows would be producing milk, and the male cows would not be performing to their capacity. But if there were 5 males and 95 females, then the farm would have more cows producing milk, and the males would be able to do their job at a rate closer to their limit.
DETERMINING THE ORIGIN OF THE MOON
In 2003, German scientists compared the ratios of two elements present in rocks from the Earth, Moon, Mars, and various meteorites to arrive at a better approximation of how and when the moon was formed. The two elements compared were niobium (a metal commonly found in alloy steels) to tantalum (an acid-resistant metal commonly found in dental and surgical instruments).
Most astronomers have long subscribed to the theory that the moon was formed when a celestial body (roughly half the size of Earth) crashed into Earth causing a large mixture of rocky debris from both bodies to fly into space, some of which lumped together to form the moon, while the rest dropped back to Earth. The amount of the Moon that is made up of material from the body that struck Earth has long been passionately debated; as has the amount made up of material from the Earth itself. The percentage of the Moon that is made up of material from the impacting body, for example, was approximated at as low as 1% by some scientists, and as high as 90% by others. By comparing the ratios of niobium and tantalum, the German team of scientists was able to determine that the amount of the moon that is composed of material from the body that struck Earth is somewhere between 35% and 65%. The rest of the Moon is composed of material from Earth.
The approximate age of the moon, another value that scientists have had a hard time agreeing about, was also refined during these studies. Calculations based on the ratios of niobium and tantalum suggest that the Moon must have been created at about the same time as Earth: about 4.5 billion years ago. As scientists continue to study the moon, the approximations of its composition and age will become increasingly accurate.
