Grade 9 Algebraic Fractions: Short Notes

9 ශ්‍රේණිය · 26 පරිච්ඡේදය · වීජීය භාග

30%

of Grade 9 term tests include it (12 of 40)

marks in past papers (15 questions)

9 min

full lesson with quick checks

🎯 What you'll learn

Recognise an algebraic fraction
Add and subtract algebraic fractions with the same number as denominator
Add and subtract algebraic fractions with different number denominators using the LCM
Add and subtract algebraic fractions with the same algebraic denominator
Use brackets correctly when subtracting

The class trip fund

Your class collects Rs. x for a trip to Sigiriya. A quarter goes on the bus, x/4, and a sixth on lunch, x/6. How much is that altogether, and how much is left for the entrance tickets? You don't know x yet, but you can still answer with one neat fraction.

What is an algebraic fraction?

A fraction whose numerator or denominator contains an algebraic term:x/5, (a + 2)/3, 4/x, 7/(y − 1)They follow exactly the same rules as number fractions. If you can do 2/7 + 3/7 and 1/3 + 1/4, you can do these too.

Same denominator

When the denominators are equal, add or subtract the numerators and keep the denominator.x/7 + 3x/7 = (x + 3x)/7 = 4x/7Put each numerator in brackets first, then collect like terms. Finally, simplify if the answer can be cancelled.

✏️ Same number denominator

Simplify (i) (2x + 1)/5 + (x + 3)/5 (ii) (5a + 2)/6 − (a − 4)/6

1(i) (2x + 1 + x + 3)/5 = (3x + 4)/5
2(ii) [(5a + 2) − (a − 4)]/6: the minus changes both signs in the second bracket.
3= (5a + 2 − a + 4)/6 = (4a + 6)/6
4Take out the common factor 2: 2(2a + 3)/6 = (2a + 3)/3

✅ (i) (3x + 4)/5 (ii) (2a + 3)/3

⚠️ The minus sign and the second bracket

The most common error in term tests:(5a + 2)/6 − (a − 4)/6 = (5a + 2 − a − 4)/6 ✗The minus applies to the whole second numerator, so −(a − 4) = −a + 4.Always write the brackets before you open them.

Different number denominators

1. Find the LCM of the denominators.2. Rewrite each fraction with the LCM as its denominator. Multiply the whole numerator by the same number as the denominator.3. Now the denominators are equal: combine the numerators.Example: x/3 + x/4 = 4x/12 + 3x/12 = 7x/12

✏️ Back to the trip fund

Rs. x/4 goes on the bus and Rs. x/6 on lunch. Find (i) the total spent (ii) the amount left, each as a single fraction.

1LCM of 4 and 6 is 12.
2(i) x/4 + x/6 = 3x/12 + 2x/12 = 5x/12
3(ii) Left = x − 5x/12 = 12x/12 − 5x/12 = 7x/12

✅ (i) Rs. 5x/12 (ii) Rs. 7x/12

⚠️ Two shortcuts that don't work

1. Adding the denominators: x/3 + x/4 = 2x/7 ✗. Denominators are never added; use the LCM: 7x/12.2. Multiplying only part of the numerator: writing 2(x + 2) as 2x + 2 ✗. Every term in the bracket is multiplied: 2x + 4.

Algebraic denominators

The rule is the same when the denominator is an algebraic expression. If the denominators are equal, just combine the numerators:3/x + 5/x = 8/x7/(y + 2) − 3/(y + 2) = 4/(y + 2)Keep the denominator exactly as it is; don't split (y + 2).

📌 Remember

• Algebraic fractions follow the same rules as number fractions.• Same denominator: combine the numerators, keep the denominator.• Different number denominators: change to the LCM first.• Put numerators in brackets; a minus sign changes every sign in the bracket after it.• Never add denominators, and simplify the final answer.

🧾 Quick revision sheet

What is an algebraic fraction?	A fraction with an algebraic term in its numerator or denominator, e.g. x/5 or 3/(y + 1)
a/k + b/k = ?	(a + b)/k
First step when the denominators are different numbers	Find the LCM of the denominators
−(a − 4) = ?	−a + 4
x/3 + x/4 = ?	7x/12 (not 2x/7)

Now test yourself: 4 quick checks and 2 more worked examples

The full lesson checks your answers as you go, and Gedara Sir tracks which parts you've got.

📖 Start the 9-minute lesson free ▶ Practise 5 past-paper questions