## Introduction to Ratio and Proportion

Welcome to Curious Toons Blog. In this article, we will discuss ** Ratio and Proportion in Mathematics **at the level of Competition.

**Ratio and Proportion in Mathematics or Science** are explained generally based on fractions. When a fraction is represented in the form of a : b, then it is a ratio and a proportion states that two ratios are equal.

The ratio and proportion are the two important concepts. It is the foundation to understand the various concepts in mathematics as well as in science.

### Ratio

The ratio of two quantities *a* and *b* in the same units is the fraction * a/b* and we write it as

**A ratio is an ordered pair of numbers a and b, written a/b where b does not**

*a*:*b*.**equal 0**. Here, a and b are any two integers.

In the ratio** a : b**, we call

*as the first term or*

**a***and b, the second term or*

**antecedent**

*consequent*.*For Example the ratio 5 : 9 represent 5 / 9 with antecedent = 5, consequent = 9*

**Key Points to Remember: **

- The ratio can only exist between the quantities of the same kind.
- While comparing two things, the units should be similar.
- There should be significant order of terms.
- The comparison of two ratios can be performed, if the ratios are equivalent like the fractions.

#### Rule

The multiplication or division of each term of a ratio by the same non-zero number does not affect the ratio. *For Example, 4 : 5 = 8 : 10 = 12 : 15. Also, 4 : 6 = 2 : 3.*

#### Types of Ratios

**Comparison of Ratios: **We can say that *(a : b) > (c : d) <=> (a / b) > (c / d)*

**Compounded Ratio:** The compounded ratio of the ratios: (*a* : *b*), (*c* : *d*), (*e* : *f*) is (*ace* : *bdf*).

**Duplicate Ratios:**

- Duplicate ratio of (
*a*:*b*) is (*a*^{2}:*b*^{2}). - Sub-duplicate ratio of (
*a*:*b*) is (√*a*: √*b*). - Triplicate ratio of (
*a*:*b*) is (*a*^{3}:*b*^{3}). - Sub-triplicate ratio of (
*a*:*b*) is (*a*^{1/3}:*b*^{1/3}).

### Proportion

Proportion is an equation that defines that the two given ratios are equivalent to each other. In other words, the proportion states the equality of the two fractions or the ratios. In proportion, if two sets of given numbers are increasing or decreasing in the same ratio, then the ratios are said to be directly proportional to each other.

*“The equality of two ratios is called proportion.”*

If *a* : *b* = *c* : *d*, we write *a* : *b* **::** *c* : *d* and we say that *a, b, c, d* are in proportion. Here *a* and *d* are called** extremes,** while

*b*and

*c*are called

*mean terms*.One important property is – Product of means = Product of extremes.

Thus, *a* : *b* **::** *c* : *d* <=> (*b* x *c*) = (*a* x *d*).

#### Types of Proportions

**Fourth Proportional: **If *a* : *b* = *c* : *d*, then * d* is called the fourth proportional to

*a, b, c*.

**Third Proportional: ***a* : *b* = *c* : *d*, then * c* is called the third proportion to

*a*and

*b*.

**Mean Proportional: **Mean proportional between *a* and *b* is √*ab*.

#### Important Properties of Proportion

The following are the important properties of proportion:

- Addendo – If a : b = c : d, then a + c : b + d
- Subtrahendo – If a : b = c : d, then a – c : b – d
- Dividendo – If a : b = c : d, then a – b : b = c – d : d
- Componendo – If a : b = c : d, then a + b : b = c+d : d
- Alternendo – If a : b = c : d, then a : c = b: d
- Invertendo – If a : b = c : d, then b : a = d : c
- Componendo and dividendo – If a : b = c : d, then a + b : a – b = c + d : c – d

**Variations:**

We say that *x* is directly proportional to *y*, if *x* = *ky* for some constant *k* and we write, *x* ∞ *y*.

We say that *x* is inversely proportional to *y*, if *xy* = k for some constant *k and* we write, *x* ∞ 1/*y*.

In summary, we find that we can understand ratio and proportion very easily after reading. If you have any queries please don’t hesitate to write to us, we are here for to share knowledge and help each other.

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