Algebra

NZ Level 6 (NZC) Level 1 (NCEA)

Multiply and divide algebraic fractions

Lesson

When it comes to working with algebraic fractions and applying the four operations, the process is exactly the same as when we worked with numeric fractions.

Let's have a look at a simple example of multiplying two numerical fractions.

Simplify $\frac{3}{4}\times\frac{5}{7}$34×57

$\frac{3}{4}\times\frac{2}{5}$34×25 | $=$= | $\frac{3\times5}{4\times7}$3×54×7 Multiplying numerators and denominators |

$=$= | $\frac{15}{28}$1528 Simplifying the numerator |

Since $\frac{15}{28}$1528 doesn't have any common factors between the numerator and denominator, that is the most simplified form of our answer.

Now let's apply the same process to multiplying algebraic fractions.

Simplify $\frac{y}{5}\times\frac{3}{m}$`y`5×3`m`

$\frac{y}{5}\times\frac{3}{m}$y5×3m |
$=$= | $\frac{y\times3}{5m}$y×35m Multiplying numerator and denominators |

$=$= | $\frac{3y}{5m}$3y5m Simplifying the numerator |

Again, since the numerator $3y$3`y` and the denominator $5m$5`m` don't have any common factors, $\frac{3y}{5m}$3`y`5`m` is the simplest form of our answer.

Simplify the expression:

$\frac{a}{7}\times\frac{a}{12}$`a`7×`a`12

Simplify the expression:

$\frac{8u}{3v}\times\frac{2v}{7u}$8`u`3`v`×2`v`7`u`

Again, the process for dividing is the same as when we divided numeric fractions.

Simplify $\frac{2}{3}\div\frac{3}{5}$23÷35

$\frac{2}{3}\div\frac{3}{5}$23÷35 | $=$= | $\frac{2}{3}\times\frac{5}{3}$23×53 | Dividing by a fraction is the same as multiplying by its reciprocal. So invert and multiply. |

$=$= | $\frac{2\times5}{3\times3}$2×53×3 | Multiply numerators and denominators respectively. | |

$=$= | $\frac{10}{9}$109 |

Since $\frac{10}{9}$109 doesn't have any common factors between the numerator and denominator, that is the most simplified form of our answer.

Now let's apply the same process to dividing algebraic fractions.

Simplify $\frac{m}{3}\div\frac{5}{x}$`m`3÷5`x`

$\frac{m}{3}\div\frac{5}{x}$m3÷5x |
$=$= | $\frac{m}{3}\times\frac{x}{5}$m3×x5 |
Dividing by a fraction is the same as multiplying by its reciprocal. So invert and multiply. |

$=$= | $\frac{m\times x}{3\times5}$m×x3×5 |
Multiply numerators and denominators respectively. | |

"=" |
$\frac{mx}{15}$ |

Again, since the numerator $mx$`m``x` and the denominator $15$15 don't have any common factors, $\frac{mx}{15}$`m``x`15 is the simplest form of our answer.

Simplify the expression:

$\frac{m}{8}\div\frac{3}{n}$`m`8÷3`n`

Simplify the following: $\frac{-2x}{11}\div\frac{7y}{5}$−2`x`11÷7`y`5

Simplify $\frac{-2x}{11}\div\frac{2x}{3}$−2`x`11÷2`x`3.

Form and solve linear equations and inequations, quadratic and simple exponential equations, and simultaneous equations with two unknowns

Generalise the properties of operations with rational numbers, including the properties of exponents

Apply algebraic procedures in solving problems