when a fraction is divided by 5/9 and 4/15 respectively, the quotients obtained are whole numbers. What is the smallest possible value of this fraction?

Question

5/9 / 4/15 = x

Answer 1

Let "x/y" be the fraction you want to find. When you divide this with other fractions (5/9 and 4/15), you are multiplying it by their inverses (9/5 and 15/4). It looks like this:

(x/y) / (5/9) becomes (x/y)*(9/5)
(x/y) / (4/15) becomes (x/y)*(15/4)

To end up with whole numbers as the quotient, X needs to be the multiple of the new denominators (5 and 4) so that they cancel out after multiplying. Use the Least Common Multiple, which is 20. So for now the "x/y" fraction looks like 20/y.

(20/y)*(9/5) = 36/y
(20/y)*(15/4) = 75/y

Now your quotients will be whole numbers if y = 1. BUT we want Y to be as big as possible in order to make the value of "x/y" as small as possible. So we have to find the Greatest Common Factor for 36 and 75 (if you don't now how to do this, please reply me). The GCF is 3, which we plug into Y. Let's check:

36/3 = 12, and
75/3 = 25. Yup, whole numbers, and no more common factors.

Now you have your "x/y" fraction: 20/3. The equation you wrote in your description is actually a different problem than the question. Let me know if you need any clarification for what your question is trying to ask.