all unicorns on rainbow island are either green or yellow. every time the clock strikes midnight the unicorns change their color. Yesterday there were 6 green for every 5 yellow ones. Today there are 3 green one and 4 yellow ones. What is the smallest possible number of unicorns living on rainbow island

77 unicorns, because yesterday the ratio was 6:5 and today it is 3:4. If you add the numbers for both ratios together, you get 11 and 7. Finding the LCM of these two gives 77 (because 7 and 11 are both prime).

You find the LCM of the sums because when you find values based on ratios, you divide value by the sum of the ratio.

e. g. I have 100 fruits. The ratio of apples to oranges is 2:3. How many apples are there?

To solve this you'd add 2 and 3 (5), divide 100 by 5 (20), multiply that by 2 (40) because that's the ratio of apples to oranges.

In this case, you do the reverse. So If I have 77 unicorns, then yesterday there were (77/11 = 7, 7*6 = 42, 7*5 = 35) 42 green unicorns and 35 yellow. The sum of 42 and 35 is 77. If you do this process with the other ratio you'll get similar results as well.