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`x/(b+c) = y/(c+a) = z/(a + b) = (y+z-x)/(c+a+a+b-b-c)` [appliying addendo process ] <br> ` = (y + z - x)/(2a)` <br> ` :. ` each ratio ` = (y + z - x)/(2a) ` ……………. (1) <br> Again , ` x/(b +c) = y/(c +a) = z/(a+b) = (z+x-y)/(a + b + b+c-c-a)` [applying addendo process ] <br> ` = (z = x - y)/(2b)` <br> ` :. ` each ratio ` = (z + x - y)/(2b)` ............... (2) <br> Also, ` x/(b +c) = y/(c + a) = z/(a + b) = (x + y - z)/( b + c + c + a - a - b)` [applying addendo process] <br> ` = (x + y - z)/(2c)` <br> ` :. ` each ratio ` = (x + y -z)/(2c) ` ................... (3) <br> Now, from (1), (2) and (3) we get , ` (y + z - x)/(2 a) = ( z + x - y)/(2b) = (x + y - z)/(2c) ` <br> or, ` (y + z - x)/a = (z + x - y)/b = ( + y - z)/c ` [multiplying by 2] <br> ` :. a/(y + z-x) = b/(z + x - y) = c/(x + y - z)` **Comparison by difference**

**Comparison by division**

**RATIO: The ratio of two quantities of the same kind and in the units is a fraction that shows how many times the one quantity is of the other.**

**RATIO IN THE SIMPLEST FORM: A ratio `a : b` is said to be in the simplest form if its antecedent a and consequent b have no common factor other than 1.**

**Equivalent ratios**

**PROPORTION: An equality of two ratios is called a proportion.**

**CONTINUED PROPORTION: Three numbers `a, b ,c` are said to be in continued proportion if `a ,b ,b ,c` are in proportion.**

**MEAN PROPORTIONAL: If `a b c` are in continued proportion then b is called the mean proportional between `a and c`.**