Unit 10

9 marks · Normal

This question is about rational and irrational numbers and simplifying algebraic fractions.

(a)

State, with a reason, whether each of the following is rational or irrational:

(i) (ii) (iii)

[4 marks]

(b)

Simplify the following algebraic fraction as much as possible:

[3 marks]

(c)

Given that , write the answer to part (b) in its simplest form.

[2 marks]

Work this out on paper, then compare with the answer and worked solution below. With a free account you can submit written answers for marking.

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(a) (i) Irrational (reason: 18 is not a perfect square, so is not rational); (ii) Rational (reason: 0.727272... is recurring and can be written as ); (iii) Rational (reason: which is an integer) · must show classification with sound reasons

(b) for · must show fully factorised and simplified form

Common mistakes: Stating square roots of non-square numbers are rational; Not fully cancelling the algebraic fraction; Not substituting surd in simplest form

Show mark scheme and worked solution

Marking points

B1 (a) · 1 mark · Identifies (i) as irrational with a correct reason.

B2 (a) · 1 mark · Identifies (ii) as rational with reasoning relating to recurring decimals.

B3 (a) · 1 mark · Identifies (iii) as rational with explanation.

B4 (a) · 1 mark · Clear, mathematically sound reasons for each decision.

M1 (b) · 1 mark · Attempts to factorise numerator or cancel common terms.

A1 (b) · 2 marks · Correctly simplifies to with all restrictions stated.

M2 (c) · 1 mark · Attempts substitution of . (follow-through)

A2 (c) · 1 mark · Correct surd simplification: . (follow-through)

Worked solution

(a) (i) is irrational because 18 is not a perfect square, so its square root cannot be written as a fraction. (ii) 0.727272... is rational because it is a recurring decimal and can be written as . (iii) which is an integer, so it is rational.

(b) Factorise numerator: . Denominator is 6x. Cancel 3x top and bottom (for ):