Ratios cannot be expressed as whole-number integers. These numbers are known as rational numbers and are a superset above integers, whole numbers and natural numbers. The mathematical manipulation of ratios is commonly first presented in pre-algebra studies. The division of one ratio by another creates what is known as a complex fraction. Complex fractions are evaluated using standard rules of algebra. In this manipulation, the division operation is changed, and the complex fraction broken into two smaller fractions.

Create a fraction that has a numerator equal to the ratio being divided and the denominator equal to the ratio it is being divided by. For example, (3/5) / (1/3) represents 3/5 divided by 1/3.

Invert the denominator and change the division symbol to a multiplication symbol. Continuing the example, (3/5) / (1/3) = (3/5) * (3/1).

Multiply the numerators and denominators. For example, (3/5) * (3/1) = 9/5.

Simplify the fraction as much as possible.