ana is 5 times as old as beth and 2 2/3 times as old as cathy. What fraction of cathy's age is Beth's?

Beth's age = Ana's age / 5

Cathy's = Ana's age / ( 2 2/3 ) = Ana's age / ( 8 / 3 )

1/ ( 8 / 3 ) / ( 1 / 5 ) 0 = ( 3 / 8 ) * 5 = 15 / 8

To find the fraction of Cathy's age that Beth is, we first need to determine the ages of all three individuals.

Let's assume Beth's age as x.

According to the problem, Ana is 5 times as old as Beth. Therefore, Ana's age would be 5x.

Additionally, the problem states that Ana is 2 and 2/3 times as old as Cathy. To find Cathy's age, we can set up the equation:

2 and 2/3 * Cathy's age = Ana's age

Since Ana's age is 5x, we can rewrite the equation as:

8/3 * Cathy's age = 5x

To isolate Cathy's age, we divide both sides of the equation by 8/3:

Cathy's age = (5x) / (8/3)
Cathy's age = (5x) * (3/8)
Cathy's age = 15x/8

Now that we know Cathy's age in terms of x, we can find the fraction of Cathy's age that Beth is by dividing Beth's age by Cathy's age:

Fraction = Beth's age / Cathy's age
Fraction = x / (15x/8)

To simplify this fraction, we can multiply the numerator and denominator by the reciprocal of the fraction in the denominator:

Fraction = x / (15x/8) * (8/15x)
Fraction = (x * 8) / (15x * 1)
Fraction = 8/15

Thus, the fraction of Cathy's age that Beth is can be expressed as 8/15.