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Grade 10 Algebraic Fractions: Short Notes

10 ශ්‍රේණිය · 2 වාරය · 12 පරිච්ඡේදය · වීජීය භාග

81%

of Grade 10 term tests include it (44 of 54)

282

marks in past papers (76 questions)

10 min

full lesson with quick checks

🎯 What you'll learn

  • Simplify an algebraic fraction by cancelling common factors
  • Find the LCM of algebraic denominators
  • Add and subtract algebraic fractions with unlike denominators
  • Handle denominators like (4 − x) and (x² − 9)
  • Multiply and divide algebraic fractions

There and back

A three-wheeler driver in Galle goes 12 km to the bus stand at x km/h and comes back faster, at (x + 10) km/h. Total time = 12/x + 12/(x + 10) hours. To turn that into one neat fraction, you need algebraic fractions.

Same rules as number fractions

An algebraic fraction has letters in the numerator or denominator, like 3/(2x) or (x + 1)/4.Everything you know about 2/3 and 5/6 still works.To simplify, cancel common factors only:6x/(9x²) = (3x × 2)/(3x × 3x) = 2/(3x).

LCM of algebraic terms

To add or subtract, first make the denominators equal using the LCM (least common multiple).Do the numbers and the letters separately:• 4x and 6x² → numbers: LCM of 4 and 6 is 12; letters: highest power x² → LCM = 12x²• 2a and 3b → 6abUse the LCM, not just the product: 4x × 6x² = 24x³ makes the work harder.

✏️ Two letter denominators

Simplify 3/(2x) + 5/(3x).
  1. 1LCM of 2x and 3x is 6x.
  2. 23/(2x) = 9/(6x) (multiply top and bottom by 3).
  3. 35/(3x) = 10/(6x) (multiply top and bottom by 2).
  4. 4Add the numerators: (9 + 10)/(6x) = 19/(6x).

✅ 19/(6x)

✏️ Letters on top

Simplify (x + 1)/3 − (x − 2)/4.
  1. 1LCM of 3 and 4 is 12.
  2. 2Write as [4(x + 1) − 3(x − 2)]/12. Keep the brackets!
  3. 3Expand: 4x + 4 − 3x + 6. Note −3 × −2 = +6.
  4. 4Collect like terms: (x + 10)/12.

✅ (x + 10)/12

⚠️ The minus sign hits the whole bracket

In the last example many students write 4x + 4 − 3x − 6, getting (x − 2)/12. Wrong! The minus in front multiplies every term of the second numerator: −3(x − 2) = −3x + 6.Tip: always put the second numerator in a bracket before you expand.

Bracket denominators

For 2/(x + 3) − 1/(x − 2), the denominators share no factor, so the LCM is their product: (x + 3)(x − 2).Then each numerator is multiplied by the other denominator:2/(x + 3) = 2(x − 2)/[(x + 3)(x − 2)]1/(x − 2) = (x + 3)/[(x + 3)(x − 2)]Leave the denominator in factor form. Don't expand it.

⚠️ Never add the denominators

Examiners keep seeing 3/x + 2/y = 5/(x + y) and 1/(2x) − 1/(3x) = 0. Check with numbers: 1/2 + 1/2 is 1, not 2/4!You may add or subtract numerators only, and only after the denominators are made equal.

🧮 Multiplying and dividing

No common denominator needed! Multiply tops and bottoms. To divide, flip the second fraction (use its reciprocal) and multiply. Factorise and cancel before multiplying to keep numbers small.

a/b × c/d = ac/bd a/b ÷ c/d = a/b × d/c

📌 Remember

• Cancel common factors only, never single terms.• Add/subtract: find the LCM of the denominators, rewrite each fraction, then combine the numerators.• Bracket each numerator: a minus in front changes every sign.• Factorise denominators first (x² − 9, 4 − x = −(x − 4)).• Multiply: tops × tops, bottoms × bottoms. Divide: flip the second and multiply.

🧾 Quick revision sheet

LCM of 4x and 6x²12x²
What can you cancel in a fraction?Only common factors of the whole numerator and whole denominator.
4 − x written using (x − 4)4 − x = −(x − 4)
Factorise x² − 9(x + 3)(x − 3)
How do you divide by a fraction?Multiply by its reciprocal (flip it).
Is 3/x + 2/y = 5/(x + y)?No. It is (3y + 2x)/(xy).
−3(x − 2) expands to…−3x + 6

Now test yourself: 5 quick checks and 4 more worked examples

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

📖 Start the 10-minute lesson free▶ Practise 5 past-paper questions

Past paper questions on Algebraic Fractions

More Grade 10 notes

Least Common Multiple of Algebraic ExpressionsPercentagesData RepresentationEquationsInverse ProportionParallelograms I