Given :  The weight of three heaps of gold are in the ratio  5:6:7.

To Find :  By what fractions of themselves must the  first two be increased so that the ratio of the weights  MAY be changed to 7:6:5

Solution:

The weight of three heaps of gold are in the ratio  5:6:7

Let say weight  are 5G  , 6G  , 7G  

Let say  A added to 5G   & B added to 6G

5G +A  : 6G + B  : 7 G  

TOTAL Weight = 5G + A + 6G + B + 7G = 18G + A + B

now ratio  7 : 6 : 5

5G +A   = (7/18)  (18G + A + B)

=> 90G + 18A  = 126G + 7A + 7B

=> 11A = 36G  + 7B

6G + B  = (6/18)(18G + A + B)

=> 6G + B  = (1/3)(18G + A + B)

=> 18G + 3B  = 18G + A  + B

=> A = 2B

11A = 36G  + 7B  

=> 11(2B) = 36G + 7B

=> 15B = 36G

=> 5B = 12G

=> B = 12G/5  

=> B = ( 2/5)(6G)

B = 12G/5  

=> 2B = 24G/5

=> A  = 24G/5

=> A =  24G * 5 / 5 * 5

=> A =  (24/25) 5G

1st weight to be increased by (24/25) ratio

2nd weight to be increased by  ( 2/5)  ratio

 Example   for verification :

 25x      30x    35x   ( 5 : 6 : 7)

 24x      12x      0    (25x  increased by 24/25 &  30x   increased by 2/5)

 49x      42x    35x

 7          6         5

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