The jewels in a crown consist of diamonds, rubies, and emeralds. If the ratio of diamonds to rubies is 5/6 and the ratio of rubies to emeralds is 8/3, what is the least # of jewels that could be in the tiara?
Let "d" be the number of diamonds, "r" the number of rubies, and "e" the number of emeralds.
Saying "the ratio of diamonds to rubies is 5/6" means that d/r= 5/6 or d= (5/6)r.
Saying "the ratio of rubies to emeralds is 8/3" means that r/e= 8/3 or r= (8/3)e.
In order that d be a whole number, r must be a multiple of 6. In order that r be a whole number, e must be a multiple of 3. We can pick up a factor of 2 by multiplying by 8 but to get the remaining factor of 3 in 6= 2(3), e/3 must be a multiple of 3 which means that e itself must be a multiple of 3(3)= 9.
The smallest possible value of e, then, is 9. Now you can find both r and d and then the total number of jewels.
In order to keep proportions, we can only multiply by positive integers the given ratios.
So the problems boils down to find the lowest common multiple (LCM) between 6 and 8, being rubies involved in both ratios.
The rest follows accordingly as in a domino game.
So the LCM between 6 and 8 is 24. And the original proportions must be multiplied accordingly (by 4 and by 3) because the rubies must be now 24 (so from 6 to 24 is multiplying by 4, and from 8 to 24 is multiplying by 3)
so there must be 24 rubies, then 20 diamonds (5 x 4) and 9 emeralds (3 x3)