We looked at some basic concepts in our earlier published material.

**Ratio**

If are two quantities of the same kind and in the same units such that then the quotient is called the ratio between .

Points to remember:

- Ratio has no units and can be written as .
- In the ratio , we call as the first term or antecedent and as the second term or consequent. The second term of the ratio cannot be zero.
- In a ratio , cannot be zero. Similarly, In a ratio , cannot be zero.
- If both terms of the ratio are multiplied by or divided by the same number, the ratio does not change. The same is not true if we were to add or subtract the same number from terms of the ratio.
- A ratio must always be represented in the lowest terms. If the H.C.F of both the terms is 1, then we can say that the ratio of both the terms is the lowest.
- Also and are equal only if . What that means is that the order of the terms of the ratio is important.

Increase or decrease in ratio

If the quantity increases or decreases in the ratio , then the new resulting quantity would be times the original quantity. Let us say that the original quantity was , then the new quantity

Commensurable and incommensurable quantities:

If the ratio between any two quantities of the same units can be expressed in the ratios of integers, then the quantities are said to be commensurable or else they are incommensurable quantities. So for example, are commensurable quantities while are inconsummerable quantities.

Composition of Ratios

Compound Ratio: When two or more ratios are multiplied term wise, the ratio thus obtained is called compound ratio. Example:

For ratios and , the compound ratio is .

Similarly if there were three ratios, , and , then the compound ratio would be .

Duplicate Ratio: It is the compound ratio of two equal ratios.

Duplicate ratio of

= Compound ratio of and

= =

Triplicate Ratio: It is the compound ratio of three equal ratios.

Triplicate ratio of

= Compound ratio of , and

= =

Sub-duplicate Ratio: For any ratio , its sub-duplicate ratio is

Sub-triplicate Ratio: For any ratio , its sub-triplicate ratio is

Reciprocal Ratio: For any ratio , where , its reciprocal ratio

**Proportion**

Four non zero quantities, , are said to be in proportion if

In the above case, is the first term, is the second term, is the third and is the fourth term.

and are called the extremes and and are called means (middle terms).

When

In quantities and should be of the same units and and should be of the same units. Example

are in continued proportion if . Here is the mean proportion between . And is the third proportion between .

Important Properties of Proportions

If then the following propertied hold

By Invertendo:

By Alternendo:

By Componendo:

By Dividendo:

By Componendo and Dividendo: