Minimum number of balls: Problem 1 (simpler)

Question

In a bin, if the ratio of blue balls to green balls is 6:8 and the ratio of green balls to red balls is 8:14, what is the minimum number of balls in the bin that are not green?

Answer 1

Since the number of green balls in both the ratios is the same number (here 8), you can combine the two ratios as follows: Blue:Green:Red :: 6:8:14. Is the answer just 6+14 (Blue+Red) = 20?

That would be too many. To see why, we first reduce the ratio to its lowest terms.

Since 2 is a factor of 6, 8, and 14, we divide all the number by 2 to and get: 3:4:7. We note that in terms of ratios, 3:4:7 is the same as 6:8:14.

Since the only common factor of 3, 4, and 7 is 1, we cannot reduce it further.

So the fewest number of non-green balls is 3+7 = 10.