Ratios

Ratio is another important concept in the development of numbers. It emerges from multiplicative thinking and captures the essence of the relation between 2 or more changing quantities. Ratios are numbers through which we make sense of the change that we see every area of life.

A ratio in simple terms is a quotient when one number is divided by another. But depending on the context, the interpretation of a ratio, can provide many insights. Though it is written using 2 numbers, it needs to be seen and interpreted as one.

Ratio & Greek Math

Ratio is one of the oldest ideas in mathematics.

Greeks did not use numerical measuring units to denote lengths of line segments or angles. They called them as linear & angular magnitudes. A magnitude was an idea used by Greeks to represent "measured quantities" like length, volume, area etc. Possibly the number system had not evolved to an extent that these measures can be expressed using numbers as we do today.

They compared linear magnitudes additively or multiplicatively. Additively they compared 2 magnitudes as more/ less/ equal to each other.

They also compared magnitudes through ratios. But it may be surprising to students that originally ratios were not thought of as relations between numbers. They were relations between magnitudes.

Along with this developed the idea that any two magnitudes can be compared using ratios which use only integer values. Hence a rational number was defined as involving only integer values.

It was a shock to the Greeks to discover irrational numbers!

Since all measurements are irrational numbers, ratios started as relations between irrational numbers.

So lengths were compared by "mentally" placing them alongside each other. Greeks used the idea of ratios of lengths and developed their geometrical ideas. One of the famous theorems they formulated was that if the sides of two triangles were in the same ratio, the two triangles were similar to each other. Even the Pythagoras Theorem was not defined in terms of squares of numbers. It was defined in terms of areas of squares constructed on the sides of a right angled triangle.

In our daily life, we use many numbers, which, if we analyse, are actually ratios. The famous number π, is actually a ratio between the circumference of any circle and its diameter! The unit price of many things we buy, like vegetables & provisions are also ratios, of the total cost that we pay to the total quantity that we receive. The marks that we receive in a test is a ratio of the marks that we received to the total marks for the test. When we say that 45% of the students of a school are boys, we are referring to a ratio.

Scale & Rate

Ratios can be divided, for convenience, into 2 ideas – Scale & Rate.

When the ratio is taken of two quantities, both of which are measured with the same units, we call the ratio as Scale and the quotient as “scale factor”. A scale factor has no units. It is just a number. For example, a map of a country could be drawn on a sheet of paper, where 1cm represents 100 kms. Hence, we can say that the scale of the map is 1cm to 10,000,000 cms or the scale factor is 1:10,000,000.

Another way a ratio can be used is to combine 2 quantities in a composed unit. If 3 kgs of oranges cost Rs 120, we can take the ratio of both and say that the price of oranges is Rs 40/kg. Here the ratio is used to indicate a “rate”. If the distance of 120 kms is covered in 3 hours, we can take the ratio and say the speed of the car is 40 kms/hour.

Rate is an idea we use very frequently in daily life. It has also come to be known by specific terms like price, speed etc. So by convention, the term ratio has come to denote just scale. In fact the term scale is rarely used, except in certain situations like maps & construction drawings.

Interpretation of Ratios

In a ratio, we are not interested in the absolute values of the parameters compared. We are interested in the way they change, relative to each other. It helps us to isolate parameters that we are interested in and study their relation and thus get more information about the relation. Hence the meaning of a ratio depends on the context.

We will study the role played by ratios in our life, in some detail, in the next chapter.

Writing Ratios

Ratios are written like this - Price of pineapples: Price of Oranges :: 4 : 3.

It can also be written as Price of Oranges: Price of Pineapples :: 3 : 4.

It is read as “Price of Oranges To Price of Pineapples Is as 3 Is To 4.”

So the order in which the 2 quantities are written and the order in which the ratio appears should be the same.

Ratios can be between Irrational Numbers

While dealing with ratios for daily life processes, we normally keep the terms of the ratio as rational numbers.

But it is likely that ratios encountered in physical & other natural processes can be irrational. This is because most measurements are likely to be irrational numbers.

Π and φ are examples.Π is the ratio of the circumference of any circle to its diameter. φ is a ratio which frequently occurs in biological structures.