The answer is one and a half.

Though this looks tricky to discover, it is easy to find by finding a common denominator. This is done by multiplying the bottom numbers of each fraction, in this case 3 and 2, to get 6. There are two sixths in a third (or 2/6), and three sixths in a half (3/6). These forms, a third and a half, are the simplest expressions of fractions. Thus, to find the ratio of a third to a half, and given the sixths that we have, we must find out what fraction of six sixths, or one, there are in each size of cup. There are 2 lots of 3/6s in a whole cup, and 3 lots of 2/6s in a whole cup, when adding together 1/3 cups and 1/2 cups, so the ratio is 3:2. This works out in its simplest form as 1.5:1 and so, for every one 1/2 cup, there are 1.5 1/3 cups.

Mathematics specialises in tools which can be utilised to work out such equations, in its branch called pure maths. Applied mathematics is more useful for statistics and probabilities, and in use for computational mathematics which uses binary forms of numbers to encode new programs.

Though this looks tricky to discover, it is easy to find by finding a common denominator. This is done by multiplying the bottom numbers of each fraction, in this case 3 and 2, to get 6. There are two sixths in a third (or 2/6), and three sixths in a half (3/6). These forms, a third and a half, are the simplest expressions of fractions. Thus, to find the ratio of a third to a half, and given the sixths that we have, we must find out what fraction of six sixths, or one, there are in each size of cup. There are 2 lots of 3/6s in a whole cup, and 3 lots of 2/6s in a whole cup, when adding together 1/3 cups and 1/2 cups, so the ratio is 3:2. This works out in its simplest form as 1.5:1 and so, for every one 1/2 cup, there are 1.5 1/3 cups.

Mathematics specialises in tools which can be utilised to work out such equations, in its branch called pure maths. Applied mathematics is more useful for statistics and probabilities, and in use for computational mathematics which uses binary forms of numbers to encode new programs.