**Properties of Ratio and Proportion :**

In this section, we will learn the properties of ratio and proportion.

The properties explained below will be much useful to solve problems on ratio and proportion.

1. Ratio exists only between quantities of the same kind.

2. Quantities to be compared (by division) must be in the same units.

3. The order of the terms in a ratio is important.

4. Both terms of a ratio can be multiplied or divided by the same (non–zero) number. Usually a ratio is expressed in lowest terms (or simplest form).

5. The order of the terms in a ratio is important.

6. To compare two ratios, convert them into equivalent like fractions.

7. If a quantity increases or decreases in the ratio a : b, then new quantity is

The fraction by which the original quantity is multiplied to get a new quantity is called the factor multiplying ratio.

8. One ratio is the inverse of another, if their product is 1. Thus a : b is the inverse of b : a and vice–versa.

9. A ratio a : b is said to be of greater inequality if a > b and of less inequality if a < b.

10. The ratio compounded of the two ratios a : b and c : d is ac : bd.

11. A ratio compounded of itself is called its duplicate ratio.

Thus a^{2} : b^{2} is the duplicate ratio of a : b. Similarly, the triplicate ratio of a : b is a^{3} : b^{3}.

12. The sub–duplicate ratio of a : b is √a : √b and the sub triplicate ratio of a : b is ^{3}√a : ^{3}√b.

13 If the ratio of two similar quantities can be expressed as a ratio of two integers, the quantities are said to be commensurable; otherwise, they are said to be in-commensurable.

√3 : √2 cannot be expressed as the ratio of two integers and therefore, √3 and √2 are in-commensurable quantities.

14. Continued Ratio is the relation (or compassion) between the magnitudes of three or more quantities of the same kind. The continued ratio of three similar quantities a, b, c is written as a: b: c.

1. An equality of two ratios is called a proportion.

Four quantities a, b, c, d are said to be in proportion,

if a : b = c : d (also written as a : b :: c : d).

That is, a / b = c / d

2. The quantities a, b, c, d are called terms of the proportion; a, b, c and d are called its first, second, third and fourth terms respectively.

First and fourth terms are called extremes (or extreme terms). Second and third terms are called means (or middle terms).

3. In a proportion,

product of extremes = product of means.

If a : b = c : d are in proportion, then,

ad = bc

This is called cross product rule.

4. Three quantities a, b, c of the same kind (in same units) are said to be in continuous proportion.

Now, we can write a, b, c in proportion as given below.

a : b = b : c

Using cross product rule, we have b² = ac

5. If a, b, c are in continuous proportion, then the middle term b is called the mean proportional between a and c, a is the first proportional and c is the third proportional.

Thus, if b is mean proportional between a and c,

then b^{2} = ac or b = √ac

6. In a proportion a : b = c : d, all the four quantities need not be of the same type. The first two quantities should be of the same kind and last two quantities should be of the same kind.

7. Invertendo :

If a : b = c : d, then b : a = d : c

8. Alternendo :

If a : b = c : d, then a : c = b : d

9. Componendo :

If a : b = c : d, then (a+b) : b = (c+d) : d

10. Dividendo :

If a : b = c : d, then (a-b) : b = (c-d) : d

11. Componendo and Dividendo :

If a : b = c : d, then (a+b) : (a-b) = (c+d) : (c-d)

12. Addendo :

If a : b = c : d = e : f =.........., then each of these ratios is equal

(a + c + e + ........) : (b + d + f + ........)

13. Subtrahendo :

If a : b = c : d = e : f =.........., then each of these ratios is equal

(a - c - e - ........) : (b - d - f - ........)

After having gone through the stuff given above, we hope that the students would have understood the properties of ratio and proportion.

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