(Also called “fractions whose denominator is a monomial”)

What Is a Monomial Denominator?

A monomial is an algebraic expression with only one term.

Examples: 3x, 5y², ab, 7 (just a number is also a monomial).

A monomial denominator means the bottom part (denominator) of a fraction is just one term (a monomial).

Example: ^5x⁄₃ has denominator 3 which is a monomial.

Example: ^{2a + 3}⁄_5b has denominator 5b, which is a monomial.

But ^{x + 1}⁄_{a + b} does not have a monomial denominator, because a + b is two terms.

So an algebraic fraction with a monomial denominator is a fraction like:
^{(some algebraic expression)}⁄_{(a single term)}.

Examples:

1. ^3x⁄₄


2. ^{2a + 5}⁄_b


3. ^{5y2 − 3y}⁄₇


4. ^xy⁄_5x

Why We Study It (Use / Purpose)

1. To simplify algebraic expressions involving fractions.


2. To perform addition, subtraction, multiplication, division of algebraic fractions (in simple cases).


3. To use in word problems where parts are expressed by fractions.


4. To prepare for more complex algebra (when denominator has multiple terms).


5. Because the curriculum places “Algebraic Fraction with monomial denominators” under JSS2 topics.

Things Students Should Be Able To Do

1. Recognize when an algebraic fraction has a monomial denominator.


2. Simplify such fractions (by canceling common factors).


3. Add or subtract algebraic fractions when denominators are the same (monomial).


4. Multiply and divide algebraic fractions with monomial denominators.


5. Solve word problems that lead to algebraic fractions with monomial denominators.

Key Ideas / Rules

Simplification (Cancelling Common Factors)

If the numerator and denominator share common factors (numbers or variables), divide both by the same factor.

Example: ^6x⁄₃ = (6 ÷ 3)x = 2x.

Example: ^8xy⁄_4x = (8 ÷ 4)(xy ÷ x) = 2y.

You can only cancel if the factor is common to all parts of numerator and denominator.

Addition and Subtraction

You can only add or subtract algebraic fractions directly if they have the same denominator (monomial).

Example: ^3x⁄₅ + ^2x⁄₅ = ^{3x + 2x}⁄₅ = ^5x⁄₅ = x.

If denominators differ, make them the same before adding/subtracting.

Example: ^x⁄₃ + ^2x⁄₆
LCM of 3 and 6 = 6
^x⁄₃ = ^2x⁄₆
Add: ^{2x + 2x}⁄₆ = ^4x⁄₆ = ^2x⁄₃.

Multiplication of Algebraic Fractions

Multiply numerator by numerator, denominator by denominator.

Example: ^2x⁄₃ × ⁴⁄_5x = ^8x⁄_15x. Cancel x → ⁸⁄₁₅.

Division of Algebraic Fractions

To divide by a fraction, multiply by its reciprocal.

Example: ^5x⁄₆ ÷ ^2x⁄₃ = ^5x⁄₆ × ³⁄_2x = ^15x⁄_12x → ⁵⁄₄.

Checking Domain / Restrictions

The denominator must not be zero. Any value of the variable that makes the denominator zero is not allowed.

Example: ^3x⁄_x is undefined when x = 0.

Worked Examples (Fully Step by Step)

Example 1

Simplify ^6x⁄₃

1. 6 and 3 share factor 3.


2. Divide numerator and denominator by 3 → ^2x⁄₁.


3. Simplify: 2x.

Answer: 2x

Example 2

Simplify ^8xy⁄_4x

1. Split into ⁸⁄₄ × ^xy⁄_x.


2. Simplify: ⁸⁄₄ = 2; ^xy⁄_x = y.


3. Multiply: 2y.

Answer: 2y

Example 3

Add ^3x⁄₅ + ^2x⁄₅

1. Same denominator (5).


2. Add numerators: 3x + 2x = 5x.


3. Result: ^5x⁄₅ = x.

Answer: x

Example 4

Multiply ^2x⁄₃ × ⁴⁄_5x

1. Multiply numerators: 2x × 4 = 8x.


2. Multiply denominators: 3 × 5x = 15x.


3. Cancel x → ⁸⁄₁₅.

Answer: ⁸⁄₁₅

Example 5

Divide ^5x⁄₆ ÷ ^2x⁄₃

1. Change division to multiplication by reciprocal: ^5x⁄₆ × ³⁄_2x.


2. Multiply numerators: 5x × 3 = 15x; denominators: 6 × 2x = 12x.


3. Cancel x: ¹⁵⁄₁₂ = ⁵⁄₄.

Answer: ⁵⁄₄

Example 6

Add ^x⁄₃ + ^2x⁄₆

1. LCM of 3 and 6 = 6.


2. Convert first fraction: ^x⁄₃ = ^2x⁄₆.


3. Add: ^{2x + 2x}⁄₆ = ^4x⁄₆.


4. Simplify: ^2x⁄₃.

Answer: ^2x⁄₃

Common Mistakes / Misconceptions Students Make

1. Canceling wrongly when only part of a term can be cancelled.


2. Ignoring denominator restriction (when denominator = 0).


3. Adding fractions with different denominators directly.


4. Forgetting reciprocal rule during division.


5. Not simplifying the final result.

Teaching Tips (for Teachers)

1. Start with simple denominators like 3 or 5x before harder examples.


2. Demonstrate cancellation step by step using factors.


3. Always write the restriction (denominator ≠ 0).


4. Use many worked examples and ask students to explain steps aloud.


5. Include real-life and word problems that use algebraic fractions.


6. Use diagrams or fraction bars to explain cancellation.

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Practice Exercises (with Full Solutions)

1. Example 1
   

   Simplify: 9x ÷ 3

   Step 1: Both 9 and 3 have a common factor of 3.
   Step 2: Divide both numerator and denominator by 3.
   9x ÷ 3 / 3 ÷ 3 = 3x / 1

   Step 3: Simplify.
   Answer: 3x



2. Example 2
   

   Simplify: 12ab ÷ 4a

   Step 1: Separate into numbers and letters: 12 ÷ 4 × ab ÷ a.
   Step 2: Simplify each part: 12 ÷ 4 = 3, ab ÷ a = b.
   Step 3: Multiply: 3 × b = 3b.
   Answer: 3b



3. Example 3
   

   Simplify: 15x² ÷ 5x

   Step 1: Split into 15 ÷ 5 × x² ÷ x.
   Step 2: Simplify numbers: 15 ÷ 5 = 3.
   Step 3: Simplify powers: x² ÷ x = x¹ = x.
   Step 4: Multiply: 3x.
   Answer: 3x



4. Example 4
   

   Add: (2x ÷ 7) + (3x ÷ 7)

   Step 1: Denominators are the same (7).
   Step 2: Add numerators: 2x + 3x = 5x.
   Step 3: Keep denominator: 5x ÷ 7.
   Answer: 5x/7



5. Example 5
   

   Add: (a ÷ 4) + (3a ÷ 8)

   Step 1: Denominators are 4 and 8.
   Step 2: LCM of 4 and 8 = 8.
   Step 3: Change first fraction to have denominator 8: a ÷ 4 = 2a ÷ 8.
   Step 4: Add numerators: 2a + 3a = 5a.
   Step 5: Result: 5a ÷ 8.
   Answer: 5a/8



6. Example 6
   

   Subtract: (7x ÷ 6) − (2x ÷ 6)

   Step 1: Denominators are the same (6).
   Step 2: Subtract numerators: 7x − 2x = 5x.
   Step 3: Result: 5x ÷ 6.
   Answer: 5x/6



7. Example 7
   

   Multiply: (4x ÷ 5) × (10 ÷ 2x)

   Step 1: Multiply numerators and denominators: (4x × 10) ÷ (5 × 2x) = 40x ÷ 10x.
   Step 2: Cancel common x: 40 ÷ 10 = 4.
   Answer: 4



8. Example 8
   

   Divide: (9y ÷ 12) ÷ (3 ÷ 4)

   Step 1: Change division to multiplication by reciprocal: (9y ÷ 12) × (4 ÷ 3).
   Step 2: Multiply: (9y × 4) ÷ (12 × 3) = 36y ÷ 36.
   Step 3: Simplify: y.
   Answer: y



9. Example 9
   

   Simplify: (8xy² ÷ 4y)

   Step 1: Split into parts: 8 ÷ 4 × xy² ÷ y.
   Step 2: Simplify numbers: 8 ÷ 4 = 2.
   Step 3: Simplify variables: xy² ÷ y = xy¹ = xy.
   Step 4: Multiply: 2xy.
   Answer: 2xy



10. Example 10
    

    Simplify: (18a²b ÷ 9ab)

    Step 1: Split into 18 ÷ 9 × a²b ÷ ab.
    Step 2: Simplify numbers: 18 ÷ 9 = 2.
    Step 3: Simplify letters: a²b ÷ ab = a¹ = a.
    Step 4: Multiply: 2a.
    Answer: 2a