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If a recipe calls for half a cup of flour and half a cup of sugar, you could say you need half a cup of the flour-sugar mix combined. In algebra, pulling out that shared fraction is called factoring.
A rational coefficient is any fraction or decimal multiplying a variable, such as 12\frac{1}{2} in 12x\frac{1}{2}x or 0.750.75 in 0.75y0.75y. Factoring simply runs the distributive property in reverse to pull that number outside parentheses.
What happens when you factor a fraction out of a whole number, like pulling 13\frac{1}{3} out of 22?

Factoring Out Unit Fractions

To factor out a fraction like 13\frac{1}{3}, divide every term inside the expression by 13\frac{1}{3}. Since dividing by a fraction means multiplying by its reciprocal (flip), dividing by 13\frac{1}{3} is the same as multiplying by 33.
Let's factor 13\frac{1}{3} out of the linear expression 13x−2\frac{1}{3}x - 2. Dividing 13x\frac{1}{3}x by 13\frac{1}{3} leaves xx, and dividing −2-2 by 13\frac{1}{3} gives −2×3=−6-2 \times 3 = -6, resulting in 13(x−6)\frac{1}{3}(x - 6).
How do we choose which fraction to pull out when the terms have different denominators, like 12x+14\frac{1}{2}x + \frac{1}{4}?

Finding a Common Fractional Factor

When terms have different fractions, factor out the fraction with the least common denominator so that the terms inside parentheses become whole integers. For 12x+14\frac{1}{2}x + \frac{1}{4}, the common denominator is 44, so we factor out 14\frac{1}{4}.
Dividing 12x\frac{1}{2}x by 14\frac{1}{4} gives 12×4=2x\frac{1}{2} \times 4 = 2x, and dividing 14\frac{1}{4} by 14\frac{1}{4} gives 11. The expression becomes 14(2x+1)\frac{1}{4}(2x + 1).