Crown

diamond and ruby = 5:6 and ruby and emerald is 8:3

5:6 nd 8:3 solve it by ratio proportion method that is 5×8:6×8:6×3=40:48:18=20:24:9 add them that is 53.

Obviously, in the two ratios we can see the existence of rubies. Therefore, to make it easy for comparison, we's think of transforming the two in a way that the ratio of rubies can be the same. The lowest common multiple of 6 and 8 are 24, so 5:6 should be 20:24 (multiplied by 3) and likewise 8:3 should be 24:9 ( x3 too). Combining the two, we a ratio 20:24:9 respectively of diamonds, rubies and emeralds. Moreover, the ratio 20:24:9 cannot be divided by any interger, the smallest total should be 20+24+9 = 53. I just try to rewrite it in a way for some of you to better understand!

let rubies be r , diamonds will then be 5r/6 and emeralds are 3r/8 . Adding them up , you end up with 53r/24 and this has to be an integer . So, the minimum number of gems have to be 53 that is when r=24