Product and quotient of fractions with letters

Calculating with letters: Fractions with letters

Product and quotient of fractions with letters

Product of two fractions

The product of two fractions is the fraction whose numerator is the product of the numerators and whose denominator is the product of the denominators. If possible, simplify your answer.

Example

\[\begin{aligned} \frac{x}{6y}\cdot \frac{2y^2}{z^2} &= \frac{x\cdot 2y^2}{6y\cdot z^2}\\[0.3cm] &= \frac{xy}{3z^2}\end{aligned}\]

Brackets In a product of fractions you use brackets when the numerator and/or denominator consist of several terms. \[\begin{aligned} \frac{x+1}{6y}\cdot \frac{2y^2}{z+1} &= \frac{(x+1)\cdot 2y^2}{6y\cdot (z+1)}&\blue{\text{product of numerators and denominators}}\\[0.3cm] &= \frac{(x+1)y}{3(z+1)}&\blue{\text{simplification}}\end{aligned}\]

Quotient of two fractions

Division by a fraction is the same as multiplication by the inverted fraction.

The inverted fraction is obtained by interchanging the numerator and the denominator.

Example

\[\begin{aligned} \frac{12x}{2x+1} \div \frac{4x^2}{2x-1} &= \frac{12x}{2x+1}\cdot \frac{2x-1}{4x^2}\\[0.25cm] &= \frac{3(2x-1)}{x(2x+1)}\end{aligned}\]