Equivalent algebraic fractions Simplifying algebraic fractions Manipulating algebraic fractions Multiplying and dividing algebraic fractions Adding algebraic fractions Subtracting algebraic fractions

Algebraic fractions 3x 4x2 2a 3a + 2 and are examples of algebraic fractions. The rules that apply to numerical fractions also apply to algebraic fractions. For example, if we multiply or divide the numerator and the denominator of a fraction by the same number or term we produce an equivalent fraction. It is important to realize that, like numerical fractions, multiplying or dividing the numerator and the denominator of an algebraic fraction by the same number, term or expression does not change the value of the fraction. This fact is used both when simplifying algebraic fractions and when writing algebraic fractions over a common denominator to add or subtract them. For example, 3x 4x2 = 3 4x = 6 8x = 3y 4xy = 3(a + 2) 4x(a + 2)

Equivalent algebraic fractions It is important to realize that, like numerical fractions, multiplying or dividing the numerator and the denominator of an algebraic fraction by the same number, term or expression does not change the value of the fraction. This fact is used both when simplifying algebraic fractions and when writing algebraic fractions over a common denominator to add or subtract them.

Simplifying algebraic fractions We simplify or cancel algebraic fractions in the same way as numerical fractions, by dividing the numerator and the denominator by common factors. For example, Simplify 6ab 3ab2 2 6ab 3ab2 = 6 × a × b 3 × a × b × b = 2 b

Simplifying algebraic fractions Sometimes we need to factorize the numerator and the denominator before we can simplify an algebraic fraction. For example, Simplify 2a + a2 8 + 4a 2a + a2 8 + 4a = a (2 + a) 4(2 + a) = a 4

Simplifying algebraic fractions b2 – 36 is the difference between two squares. Simplify b2 – 36 3b – 18 b2 – 36 3b – 18 = (b + 6)(b – 6) 3(b – 6) b + 6 3 = Pupils should be encouraged to spot the difference between two squares whenever possible. If required, we can write this as 6 3 = b + b 3 + 2

Manipulating algebraic fractions Remember, a fraction written in the form a + b c can be written as b a + However, a fraction written in the form c a + b cannot be written as b a + Stress that if two terms are added or subtracted in the numerator of a fraction, we can split the fraction into two fractions written over a common denominator. The converse is also true. However, if two terms are added or subtracted in the denominator of a fraction, we cannot split the fraction into two. Verify these rules using the numerical example. For example, 1 + 2 3 = 2 1 + 3 1 + 2 = 2 1 + but

Multiplying and dividing algebraic fractions We can multiply and divide algebraic fractions using the same rules that we use for numerical fractions. In general, a b × = c d ac bd a b ÷ = c d × = ad bc and, Point out to pupils that in the example we could multiply out the brackets in the denominator. However, it is usually preferable to leave expressions in a factorized form. For example, 3p 4 × = 2 (1 – p) 3 6p 4(1 – p) = 3p 2(1 – p) 2

Multiplying and dividing algebraic fractions What is 2 3y – 6 ÷ 4 y – 2 ? This is the reciprocal of 4 y – 2 2 3y – 6 ÷ = 4 y – 2 2 3y – 6 × 4 y – 2 2 3(y – 2) × = 4 y – 2 2 1 6 =

Adding algebraic fractions We can add algebraic fractions using the same method that we use for numerical fractions. For example, What is 1 a + 2 b ? We need to write the fractions over a common denominator before we can add them. 1 a + 2 b = b ab + 2a = b + 2a ab If necessary review the method for adding numerical fractions. In general, + = a b c d ad + bc bd

Adding algebraic fractions 3 y What is + ? x 2 We need to write the fractions over a common denominator before we can add them. 3 x + y 2 = + 3 × 2 x × 2 y × x 2 × x + 6 2x xy = = 6 + xy 2x

Subtracting algebraic fractions We can also subtract algebraic fractions using the same method as we use for numerical fractions. For example, What is – ? p 3 q 2 We need to write the fractions over a common denominator before we can subtract them. – = p 3 q 2 – = 2p 6 3q 2p – 3q 6 In general, – = a b c d ad – bc bd

Subtracting algebraic fractions 2 + p 4 3 2q What is – ? = – 2 + p 4 3 2q – (2 + p) × 2q 4 × 2q 3 × 4 2q × 4 = – 2q(2 + p) 8q 12 = 2q(2 + p) – 12 8q 6 The denominators in this example share a common factor. That means that we will either have to cancel at the end of the calculation (as shown) or use a common denominator of 4q in the first step. 4 = q(2 + p) – 6 4q

Addition pyramid – algebraic fractions Start by revealing the three fractions on the bottom row of the wall. Add the fractions together to find the missing values in the bocks above. Each block is the sum of the two fractions below it. Ask pupils if it is true to say that the fraction in the top row is the sum of the three fractions in the bottom row. Conclude that if the three factions on the bottom row are a, b and c then the fraction on the top row is a + 2b + c. The activity can be varied by revealing one fraction in each row and using subtraction to find those that are missing.