Algebraic fractions, and proving an identity

Simplifying to a single fraction, and showing two expressions are equal — both graded on showing every line of method.

The method for combining fractions

Find a common denominator by factorising each denominator FIRST — a shared factor is easy to miss if the denominators are left unfactorised.
Rewrite each fraction over the common denominator, expand only the numerators, and combine into ONE fraction.
Factorise the resulting numerator if possible, and cancel any factor shared with the denominator — this is usually the final simplification step expected.
For "write in the form A + B/x": divide the numerator by the denominator (or split it algebraically) so ONE term has no fraction and the remainder is a single fraction.

Worked example Simplifying 1/t + (t − 5)/t² − 1/(3t) as a single fraction Combine into one fraction. Common denominator is 3t² (the lowest common multiple of t, t², and 3t). Rewrite each term: 1/t = 3t/3t², (t−5)/t² = 3(t−5)/3t², 1/(3t) = t/3t². Combine: [3t + 3(t − 5) − t] / 3t² = [3t + 3t − 15 − t] / 3t² = (5t − 15) / 3t². Factorise the numerator: 5(t − 3) / 3t² — no common factor with the denominator, so this is fully simplified. ✓ 5(t − 3) / 3t². Factorising the denominators before combining (here, recognising t² needs 3t² not 3t·t²) avoids an unnecessarily large common denominator. ✕

Proving an identity by working both sides towards each other at once

When asked to "show that" one expression equals another, work with ONLY the more complicated side, simplifying it step by step until it matches the simpler side exactly. Manipulating both sides at once (or worse, starting from the answer) does not constitute a proof and loses the method marks even if the final line is correct.

Task wording	What to do
"Simplify fully"	Combine to one fraction, then cancel any shared factor with the denominator
"Write in the form A + B/x"	Split the numerator so one term divides exactly, leaving a smaller remainder fraction
"Show that … equals …"	Simplify ONLY the more complicated side until it matches the other exactly

Practice for this standard covers the whole topic — flashcards and quiz.