The FOIL method is the standard procedure for multiplying binomials -- expressions that contain two terms such as "x + 3" or "4a - b." Binomials may have fractions either as constants (free numbers) or as coefficients (numbers that are multiplied by variables). When using the FOIL method with fractions as either coefficients, constants or both, you will need to remember the rules for multiplying and adding fractions.

### The FOIL Method

"FOIL" is an acronym for the steps involved in multiplying binomial factors. To find the product of two binomials (a + b) and (c + d), multiply the first terms (a and c), the outside terms (a and d), the inside terms (b and c) and the last terms (b and d), and add the products together (ac + ad + bc + bd). FOIL stands for First-Outside-Inside-Last, which represents the order of the products in the sum.

### Multiplying Fractions

When binomial factors have fractions either as coefficients or constants, the FOIL method will involve fraction multiplication. To find the product of two fractions, multiply their numerators to get the numerator of the product and multiply their denominators to get the denominator of the product. For example, the product of 2/3 and 4/5 is 8/15. When multiplying fractions by whole numbers, rewrite the whole number as a fraction with a denominator of 1.

### Combining Fractions

It is necessary to combine like terms after the FOIL method if the product contains like terms. For example, the product (x + 4/3)(x +1/2) is x^2 + (1/2)x + (4/3)x + 2/9 contains two like terms -- (1/2)x and (4/3)x. To combine like terms containing fractions, the fractions must have a common denominator. The common denominator of (1/2) and (4/3) is 6, so the expression can be rewritten as (3/6)x + (8/6)x. Combine fractions with a common denominator by adding the numerators and keeping the denominator the same: (3/6)x + (8/6)x = (9/6)x.

### Reducing Fractions

The final step of the FOIL method with fractions is reducing the fractions in the product. A fraction is written in simplest form when its numerator and denominator have no common factors other than 1. For example, the fraction 6/9 is not in simplest form because 6 and 9 have a common factor of 3. To reduce fractions to simplest form, divide both the numerator and denominator by their common factor. Divide 6 and 9 by 3 to get 2/3, which is the fraction's simplest form.