## Tips: have a cup of tea ! i know you are going to love this .

## RATIO

A Ratio is the fraction or comparison of two or more quantities of the same type by division , i.e. if a and b are two quantities of same kind then the fraction a/b is called the ratio a to b . Here , a is called the first term (or antecedent) and b is called the second term (or consequent ) .

e.g. The ratio of 7m to 9m or 7 : 9.

### RULES FOR RATIO

RULE 1:

If both terms a and b of a ratio are multiplied or divided by the same quantity then ratio remains unchanged i.e. a / b = na / nb => a : b is same as na : na.

and , a / b = (a / n) / (b / n)

RULE 2:

Let a and b are two non zero numbers then some conditions are given below,

(a) If a > b , then a : b is called a ratio of greater inequality.

(b) If a

Lets see some properties for you , hehe i was kidding , lets see some properties of proportions,

(c) If a = b , then a : b is called a ratio of equality .

### COMPOSITION OF RATIOS

(1) Compounded ratio : When tow or more ratios are multiplied together , they are said to be compounded ratio.

so, if a / b and c / d are ratios then ac / bd is their compounded ratio.

Similarly , if a / b, c / d, e / f and g / b are all ratios , then their compounded ratio is aceg : bdfh .

e.g. Compound ratio of 2 / 3 , 4 / 7 and 1 / 3 = (2*4*1) / (3*7*3) = 8 / 63 .

(2) Duplicate ratio : When a ratio is compounded with itself, the resulting ratio is called the duplicate ratio . So , a² : b² is duplicate ratio of a : b .

(3) Triplicate ratio : If a ratio is compounded three times with itself then resulting ratio is called triplicate ratio . So , a³ : b³ is the triplicate ratio of a : b .

(4) Sub duplicate ratio : If a square root is applied on a ratio then resulting ratio is called sub duplicate ratio. If a : b is ratio then the sub duplicate ratio is √a : √b .

(5) Sub triplicate ratio : If cube root is applied on a ratio then resulting ratio is called sub triplicate ratio is ∛a : ∛b . e.g. Sub triplicate ratio of 64 : 27 is 4 : 3.

(6) Reciprocal ratio : If a : b is a ratio then 1 / a : 1 / b is its reciprocal ratio i.e. b : a .e.g. reciprocal ratio of 3 : 7 is 1 / 3 : 1 / 7 i.e. 7 : 3 .

**You study quiet faster than i thought you would.....**

## PROPORTION

An equality of two ratios is called a proportion . If a , b, c and d are four quantities of same kind then a : b :: c : d is called the proportion which means a : b = c : d .

If the proportion are in the form of a / b = c / d = e / f = k (say)

then we can write a = bk, c = bk, c= bk and e = fk

The quantities a , b , c and d are called the terms of the proportion a , b ,c and d are the first , second , third and fourth terms respectively . First and fourth terms are called extreme terms and second and third terms are called means or middle terms.

If quantities a , b , c and d are in proportion then a / b = c / d => ad = bc

i.e. product of extreme terms = product of middle terms . This is also called cross product rule .

### TYPES OF PROPORTION

(a) Continued proportion :

The non zero quantities of same kind , a, b, c, d, e, f ,..... are said to be in continued proportion , if a / b = b / c = c / d = e / f = ......

(b) Mean proportional :

If a , b and c are in continued proportion then b is called the mean proportional of a and c .

i.e. a / b = b / c => b² = √ac

(c) Third proportional : If a : b :: b : c , then c is called the 3rd proportional to a and b . Now, c will be calculated as below :

a : b :: b : c => a : b = b : c => b² = ac .

Lets see some properties for you , hehe i was kidding , lets see some properties of proportions,

### PROPERTIES OF PROPORTION

If four non zero quantities a , b , c and d are in proportion , then some properties of proportion are given below :

(a) Invertendo : If a : b :: c : d then b : a :: d : c i . e. a / b = c / d => a / c = b / a

(b) Alternendo : If a : b :: c : d then a : c :: b : d i.e. a / b = c / d => a / c = b / a

(c) Componendo : If a : b :: c : d then ( a + b ) : b :: (c + d) : d

(d) Dividendo : If a : b :: c : d then a : (a-b) :: c : (c - d)

(e) Componendo and Dividendo : If a : b :: c : d , then (a + b ) : (a - b) :: (c + d ) : (c - d)

**Well Done !**

**Now my dear friend , get a pen and a notebook to practice these.**

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