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Secondary 4 G3 Mathematics Tutor Bukit Timah | Algebraic Fractions and Formula Rearrangement

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by

eduKate Asia
Three students work together at a classroom table, reviewing open textbooks and notes.
Three secondary students discussing Mathematics solutions and worksheets around a classroom table.

Did you know? A fraction can look beautifully simplified and still carry an important warning: some values of its variable remain forbidden. When Secondary 4 G3 Mathematics learners handle algebraic fractions, the most useful first question is often, ‘Could the original denominator equal zero?’

Quick answer: factorise before cancelling, cancel common factors rather than pieces of a sum, use a common denominator for addition and subtraction, and keep the original denominator restrictions. When changing the subject of a formula, perform equivalent operations on both sides and make the new subject stand alone.

These are named skills in the 2027 Singapore–Cambridge SEC G3 Mathematics K310 syllabus. Below, each worked example shows the calculation, one predictable mistake, and the short check that exposes it.

The three-minute answer

  • Factorise before cancelling, and keep the original denominator restrictions.
  • Find a common denominator before adding unlike algebraic fractions.
  • Rearrange each formula by doing equivalent operations on both sides.

Jump directly to a worked example

Before simplifying any algebraic fraction, find the forbidden values

An algebraic fraction is an expression in which letters can appear in the numerator or denominator, such as 1/(x − 3). That expression is defined for every real x except x = 3, because division by zero is not permitted.

When several denominators appear, collect all the restrictions before manipulating the expression. For instance, 1/(x − 2) + 2/(x − 3) requires x ≠ 2 and x ≠ 3.

Did you know? A cancelled factor can disappear from the final line without removing the original restriction. A complete answer respects the expression that was given, not merely the one you ended up writing.

Worked example 1: cancel factors, never unrelated terms

Simplify (6x² − 24)/(3x + 6). First factorise the numerator and denominator: 6(x − 2)(x + 2) / [3(x + 2)].

Provided x is not −2, the matching factor (x + 2) cancels and 6/3 = 2. So the simplified expression is 2(x − 2), with x ≠ −2.

Check by choosing x = 0: the original is (−24)/6 = −4, and the simplified expression is 2(−2) = −4. But when x = −2 the original is 0/0, which is undefined. You cannot restore x = −2 simply because a factor disappeared.

Why cancelling the x in x + 2 is a common trap

Look at (x + 2)/(x + 3). You cannot cancel the x symbols to obtain 2/3; the numerator and denominator contain sums, not a shared multiplicative factor.

Try x = 1 to expose the incorrect cancellation: the actual value is 3/4, not 2/3. To cancel correctly, an entire common factor must multiply the rest of the numerator and denominator.

A good tutor asks the learner to draw brackets around each factor before striking anything out. If there is no identical factor, there is nothing to cancel.

Worked example 2: multiplying algebraic fractions

Multiply (3a/4b²) × (8b/9a), with a and b nonzero. Numerators multiply to 24ab; denominators multiply to 36ab².

Cancel the common factors to obtain 2/(3b). You can reach the same result by cancelling a across the two fractions and one b from b/b² before multiplying the remaining numbers.

The original fractions require a ≠ 0 and b ≠ 0. Although a no longer appears in the simplified expression, the restriction a ≠ 0 still belongs to the original expression.

Worked example 3: dividing fractions means multiplying by the reciprocal

Evaluate (2x/3) ÷ (4x²/9), assuming x ≠ 0. Flip the fraction being divided by: (2x/3) × (9/4x²).

Multiply, then simplify: 18x/(12x²) = 3/(2x). Check at x = 2: the original is (4/3) ÷ (16/9) = 3/4, and the result 3/(2 × 2) is also 3/4.

The numerator of the divisor cannot be zero because division by a zero-valued fraction is not defined. That is why x = 0 remains excluded.

Worked example 4: adding fractions with different linear denominators

Add 1/(x − 2) + 2/(x − 3). A suitable common denominator is (x − 2)(x − 3), provided x ≠ 2 and x ≠ 3.

Rewrite the first numerator as (x − 3) and the second as 2(x − 2). Combining gives [(x − 3) + 2(x − 2)] / [(x − 2)(x − 3)].

Expand the numerator carefully: x − 3 + 2x − 4 = 3x − 7. The result is (3x − 7)/[(x − 2)(x − 3)].

Do not add the denominators to write 3/(2x − 5). Fractions are added by expressing the parts in the same-sized units, which is what a common denominator accomplishes.

Worked example 5: a quadratic denominator needs factorisation first

Simplify 1/(x² − 9) + 2/(x − 3). The difference of squares factorises as x² − 9 = (x − 3)(x + 3). The restrictions are x ≠ 3 and x ≠ −3.

Using (x − 3)(x + 3) as common denominator, the numerator becomes 1 + 2(x + 3) = 2x + 7. The combined fraction is (2x + 7)/[(x − 3)(x + 3)].

This is where factorising and adding fractions work together. It is not a new rule; it is two familiar rules applied in the correct sequence.

Worked example 6: solve a simple fractional equation and check the denominator

Suppose (x + 1)/(x − 2) = 3. We must first exclude x = 2. Multiplying both sides by x − 2 gives x + 1 = 3(x − 2).

Expand and rearrange: x + 1 = 3x − 6, so 7 = 2x and x = 7/2 = 3.5. Check in the original fraction: 4.5/1.5 = 3.

Not every solution produced after clearing denominators is automatically valid. A value that makes any original denominator zero must be rejected, even if it happens to satisfy an equation formed later.

Changing the subject of a formula in two clear steps

The subject is the variable standing alone on one side. In A = bh/2, A is the subject. To make h the subject, multiply both sides by 2 to get 2A = bh, then divide by nonzero b.

The resulting formula is h = 2A/b. Check with A = 24 and b = 6: h = 48/6 = 8, and 6 × 8/2 = 24 as required.

Changing the subject is not moving symbols with invisible sign changes. Write the mathematical operation explicitly so the equality stays true at every stage.

Worked example 7: rearrange the area of a circle formula

For a circle, A = πr². Divide by π to get r² = A/π. Since radius is non-negative, take the non-negative square root: r = √(A/π).

If A = 49π cm², then r = √49 = 7 cm. Writing ±7 cm would not make sense as a physical radius; context can rule out the negative algebraic branch.

A formula rearrangement question can therefore test both symbolic work and interpretation. Ask what the variable represents before deciding which possible sign is appropriate.

Worked example 8: rearrange a volume formula

For a cylinder V = πr²h, to make h the subject, divide both sides by πr²: h = V/(πr²), assuming the radius r is nonzero.

If V = 200π cm³ and r = 5 cm, h = 200π/(25π) = 8 cm. Substitute back: π(5²)(8) = 200π cm³.

This is a useful connection between algebra and mensuration. Clear symbolic rearrangement reduces the chances of using a remembered but incorrectly inverted formula.

A one-week algebraic fractions repair plan

Day 1: list invalid denominator values for ten short expressions. Day 2: factorise and cancel only common factors. Day 3: multiply and divide two algebraic fractions.

Day 4: find common denominators for sums. Day 5: combine a linear and a factorised quadratic denominator. Day 6: rearrange three geometry or speed formulas. Day 7: solve new mixed questions without notes and explain every excluded value.

Keep an error ledger with categories factorisation, cancellation, least common denominator, sign, restriction, equivalence and context. Ask whether the same incorrect first line appears in the next unseen problem.

Common parent questions about algebraic fractions

Why did a cancelled factor still matter? The initial expression was undefined at a value that made the cancelled factor zero.

Can I add the numerators directly? Yes when denominators are the same; otherwise first express the fractions over an appropriate common denominator.

Must the simplest form have no brackets? Not necessarily. A factored form can be clearer and more helpful for checking restrictions.

When is a formula rearrangement complete? When the requested subject is alone and the operations have been valid for the stated conditions.

Why check a solved fraction equation? Clearing denominators can hide forbidden values; substitution into the original exposes them.

Is this part of G3 Mathematics rather than Additional Mathematics? Yes, these algebraic expression and formula skills are explicitly described in the 2027 K310 G3 Mathematics syllabus.

What this means for a Bukit Timah Mathematics tutorial

At eduKateSG, the unchanged Mathematics small-group programme describes three-pupil sessions lasting 1.5 hours weekly, deliberate practice and close review of each learner’s reasoning. The teaching venue identified in the original reference is 8 Fourth Avenue, Singapore 268674, near Sixth Avenue MRT. This page does not claim a separate branch or guarantee a particular available place.

A tutor can begin with the learner’s unprompted attempt, identify the first incorrect decision, rebuild the concept and try an unfamiliar question without hints. A student who makes a notation mistake needs a different correction from one who has chosen the wrong mathematical model. The point of a small group is to keep these individual differences visible.

Ask eduKate Singapore about a Mathematics consultation and bring one recent independent school script. Class suitability and availability can be discussed directly.

Check the actual examination year and subject

The SEAB 2027 school-candidate list identifies Mathematics as SEC G3 K310 and its earlier O-Level reference as 4052. The 2027 K310 syllabus includes the techniques discussed above. For a student sitting the 2026 O-Level, use the applicable 2026 Mathematics 4052 examination documents, not an assumption that every detail of the 2027 SEC paper applies.

Official syllabus and the original teaching reference

Read the other guides in this learning series

Reader note: Worked numbers here are original illustrative practice examples. They are not leaked questions, specimen papers or a promise of results.