The distributive property works with fractions in the same way it works with whole-number coefficients. The distributive property states that a*(x + y) = a*x + a*y, where "a" is any coefficient, such as 2 or 1/2. A fraction used as a coefficient to a group of terms inside parentheses can be distributed to each individual term and turns that term into a fraction.
The distributive property is used in algebra because it makes it easier to isolate a particular variable. When mathematicians distribute the coefficient, they multiply it to each term inside of the parentheses and then separate those terms. When coefficients inside or outside of the parentheses are fractions, the only added step is to find like denominators when adding or subtracting fractions.
The following equation demonstrates how the distributive property works with fractions.
1/2*(2/3 + 2/5) = 1/2*(2/3) + 1/2*(2/5)
1/2*(16/15) = 1/3 + 1/5
8/15 = 8/15
Sometimes fractions can be canceled or turned into integers through distribution. For example:
8*(1/2*x - 1/4*y) = 8*1/2*x - 8*1/4*y = 8/2*x - 8/4*y = 4x - 2y
To cancel any coefficient, multiply it by its reciprocal. The reciprocal of 1/2 is 2/1 or the whole number 2.
To isolate y in the equation 4x - 2y = 0, a person could use distribution. Because the reciprocal of 2 is 1/2, both sides would be multiplied by 1/2.
1/2*(4x - 2y) = 1/2*0
1/2*(4x) - 1/2*(2y) = 2x - y = 0
2x = y