Algebraic Fractions (Grade 9) — Worksheet & Memo

∑ DANEMATHICS

FREE CAPS MATHS RESOURCES

Grade 9 · Algebraic Expressions

Algebraic Fractions (Grade 9)

MARKING GUIDELINE

Marks

Duration

45 minutes

Questions

Name:

Class:

Date:

Mark

/ 28

Instructions and Information

Answer ALL the questions in this question paper.
Answer QUESTION 1 by circling the letter (A–D) in the answer grid at the end of that section.
Show ALL calculations clearly.
Show all units where applicable.
Number the answers correctly according to the numbering system used in this question paper.
A non-programmable calculator may be used, unless stated otherwise.
Write neatly and legibly.

Question 1

[10 MARKS]

Four options are given as possible answers to the following questions. Choose the correct answer and circle the letter (A–D) in the grid at the end of this section. If you want to change your choice, put a cross through the wrong letter and circle your new choice.

1.1
Simplify: 12a⁵b²4a³b
(1)
A)3a²b✓
B)8a²b
C)3a⁸b³
D)3a²b²
Answer: A — 12 ÷ 4 = 3, a⁵⁻³ = a² and b²⁻¹ = b.
- B — SUBTRACTED the coefficients instead of dividing
- C — ADDED the exponents instead of subtracting
- D — forgot to subtract the exponent of b
1.2
Simplify: x² − 9x + 3
(1)
A)x − 9
B)x − 3✓
C)x + 3
D)x² − 3
Answer: B — Factorise the top: (x + 3)(x − 3), then cancel (x + 3).
- A — cancelled the 3 on its own — only FACTORS cancel
- C — cancelled the wrong bracket
- D — cancelled only part of the numerator
1.3
2x + 3x = …
(1)
A)52x
B)5x✓
C)6x
D)5x²
Answer: B — The denominators are already the same, so add the numerators only.
- A — added the denominators too
- C — multiplied the numerators
- D — multiplied the denominators
1.4
12x + 13x = …
(1)
A)25x
B)56x✓
C)16x
D)26x
Answer: B — 3/6 + 2/6 = 5/6.
- A — added the tops and the bottoms
- C — MULTIPLIED the two fractions
- D — used a common denominator but added wrongly
1.5
Simplify: 6x3
(1)
A)2x✓
B)3x
C)18x
D)2
Answer: A — 6 ÷ 3 = 2, and the x is untouched.
- B — SUBTRACTED 3 from 6
- C — multiplied instead of dividing
- D — cancelled the x away as well
1.6
ab × cd = …
(1)
A)adbc
B)acbd✓
C)a + cb + d
D)abcd
Answer: B — Multiply the tops and multiply the bottoms.
- A — that is a/b ÷ c/d
- C — multiplication is not addition
- D — paired the letters wrongly
1.7
ab ÷ cd = …
(1)
A)acbd
B)adbc✓
C)a + db + c
D)bcad
Answer: B — Multiply by the RECIPROCAL of the second fraction.
- A — that is a/b × c/d
- C — division is not addition
- D — inverted the wrong fraction
1.8
For what value of x is 5x − 2 undefined?
(1)
A)5
B)2✓
C)0
D)−2
Answer: B — The denominator is 0 when x = 2, and division by 0 is undefined.
- A — that is the numerator
- C — x = 0 gives 5/(−2), which is fine
- D — x = −2 gives 5/(−4), which is fine
1.9
Simplify: 4x + 84
(1)
A)x + 8
B)x + 2✓
C)4x + 2
D)x
Answer: B — BOTH terms of the numerator are divided by 4.
- A — divided only the first term
- C — divided only the second term
- D — cancelled the 8 away entirely
1.10
x3 − x4 = …
(1)
A)x12✓
B)0
C)2x7
D)7x12
Answer: A — 4x/12 − 3x/12 = x/12.
- B — the two terms are not equal, so the answer is not 0
- C — subtracted the tops AND the bottoms
- D — ADDED the two fractions instead of subtracting

Answer grid — marking guideline

1.1	A	B	C	D
1.2	A	B	C	D
1.3	A	B	C	D
1.4	A	B	C	D
1.5	A	B	C	D
1.6	A	B	C	D
1.7	A	B	C	D
1.8	A	B	C	D
1.9	A	B	C	D
1.10	A	B	C	D

Question 2

[9 MARKS]

Simplify the following.

2.1
6x²y3xy³
(3)
6 ÷ 3 = 2 (1)
x² ÷ x = x, and y ÷ y³ = y⁻² (1)
= 2xy² (1)
2.2
x² + 3xx
(3)
Factorise the numerator first: x(x + 3)x (2)
= x + 3 (1)
2.3
2x3 × 94x²
(3)
= 18x12x² (2)
= 32x (1)

Question 3

[9 MARKS]

Simplify the following. Show ALL your steps.

3.1
x4 ÷ x2
(3)
Flip the second fraction: x4 × 2x (2)
= 12 (1)
3.2
x2 + x3
(3)
LCD = 6 (1)
= 3x6 + 2x6 (1)
= 5x6 (1)
3.3
Explain why you may NOT cancel the x in x² + 3xx before factorising.
(3)
You may only cancel FACTORS, never terms (1)
x² + 3x is a SUM, not a product (1)
Factorising first turns it into the product x(x + 3), and then the x may be cancelled (1)

TOTAL: 28 marks

This question paper consists of 3 questions.