Algebra

NZ Level 6 (NZC) Level 1 (NCEA)

Mixed Operations with Algebraic Fractions

Lesson

We looked at how to add and subtract fractions in More and Less Parts.

The most important things to remember when adding and subtracting fractions (of any kind) are

- we need
*like denominators (bases)* - we need to keep our fractions
*equivalent*

Now we are going to build on this knowledge and look at how to add and subtract algebraic fractions.

$\frac{4m}{5}-\frac{2}{5}$4`m`5−25

**Think**: The first thing we need to do is check that the denominators are the same. In this case, both denominators are $5$5, so then we move on to the next step.

**Do**: Because our denominators are the same, we can write the numerators together as a single expression over the common denominator.

$\frac{4m-2}{5}$4`m`−25

**Reflect**: Because we cannot simplify the numerator $4m-2$4`m`−2 any further, this means this is a simplified as we can get.

Let's look at a very similar example, but where the denominators are not initially the same.

$\frac{2y}{3}+\frac{5}{6}$2`y`3+56

**Think**: We cannot add or subtract fractions unless the denominators are common. In this case we one denominator of $3$3, and the other with $6$6. We need to find a common denominator. Let's choose $6$6, as it is a common multiple of both $3$3 and $6$6.

**Do**: Change the first fraction to have a denominator of $6$6.

$\frac{2y}{3}$2y3 |
$=$= |
$\frac{2y\times2}{3\times2}$2 |

$=$= | $\frac{4y}{6}$4y6 |

So this means our expression now becomes:

$\frac{2y}{3}+\frac{5}{6}=\frac{4y}{6}+\frac{5}{6}$2`y`3+56=4`y`6+56

Success, now we can add the fractions as the denominators are common.

**Do**: Now write the numerators as a single expression above the common denominator.

$\frac{4y}{6}+\frac{5}{6}=\frac{4y+5}{6}$4`y`6+56=4`y`+56

**Reflect**: Is this simplified enough? As the terms $4y$4`y` and $5$5 are not like terms, yes, this is simplified as much as we can.

Our final example is where we have different denominators, and some simplification to perform at the final step.

$\frac{3x}{4}+\frac{3x}{2}$3`x`4+3`x`2

**Think**: Our first goal is to have common denominators. Looking at the denominators we have, $4$4 and $2$2, we can see that $4$4 is a common multiple. So use $4$4.

**Do**:

$\frac{3x}{4}+\frac{3x}{2}$3x4+3x2 |
$=$= | $\frac{3x}{4}+\frac{3x\times2}{2\times2}$3x4+3x×22×2 |

$=$= | $\frac{3x}{4}+\frac{6x}{4}$3x4+6x4 |

**Think**: Now we have a common denominator, we write the fraction as a single expression over the common denominator and then simplify where we can.

**Do**:

$\frac{3x}{4}+\frac{6x}{4}$3x4+6x4 |
$=$= | $\frac{3x+6x}{4}$3x+6x4 |

$=$= | $\frac{9x}{4}$9x4 |

**Reflect**: We collected the like terms of $3x$3`x` and $6x$6`x`. Are there any other common terms? Are they any common factors with the $9x$9`x` and $4$4? No, so this is a simplified as this answer gets.

Simplify the following:

$\frac{6x}{2}-\frac{7x}{2}$6`x`2−7`x`2

Simplify the following: $\frac{3x}{5}-\frac{x}{7}$3`x`5−`x`7

Simplify the expression $\frac{11x}{14}+\frac{7x}{21}$11`x`14+7`x`21.

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