Recurring decimal → fraction needs algebra
Let equal the decimal, multiply by a power of so the recurring tails line up, subtract to remove the recurring part, then solve for and simplify.
Move between fractions, decimals and %
Fraction → decimal: divide. Decimal → percentage: . Percentage → fraction: put over and simplify.
Drawn from real examiner reports.
Not using algebra for a recurring decimal
When told to "use algebra to show" a recurring decimal equals a fraction, a non-algebraic answer scores at most 1 mark.
A commonly reported examiner note records non-algebraic attempts were capped at 1 mark.
Subtracting the wrong multiples of x
Choosing and (or other mismatched powers) so the recurring tails do NOT cancel — you must pick powers whose decimal parts are identical before subtracting.
Students who scored zero on a recent higher-tier paper used incorrect combinations to subtract.
Jumping to the calculator with no working
On the non-calculator paper, leaping to the final fraction (e.g. straight to ) with no improper-fraction or cancelling step shown loses method marks — and can hide errors.
A commonly reported examiner note records: answers appeared with no intermediate $\tfrac{420}{112}$ or cancelling.
Dividing the wrong way for a decimal
For divide top by bottom: . Doing inverts the fraction; a proper fraction giving a result above is a clear sign you divided the wrong way.
Leaving a decimal in the numerator
Converting a percentage such as and stopping at leaves a decimal on top. Scale top and bottom by : , then simplify fully.
Always give the fraction in its simplest form
After the algebra, cancel down (e.g. ) — an un-simplified fraction can cost the final mark.
Convert to one form before comparing
To order or compare a mix of fractions, decimals and percentages, first turn every value into the same form (usually decimals), compare, then give the answer back in the original forms.
Keep exact fractions for later working
A recurring decimal carried as its exact fraction (e.g. ) cancels cleanly later; a rounded decimal only gives an approximate answer and loses "exact value" marks.
Convert a decimal to a percentage — how?
Write as a fraction in its simplest form.
