Stage 9 - Q1 - 2 marks
Will this fraction terminate as a decimal? Justify.
Hint
The test applies after all common factors have been cancelled.
Stage 9 - Q2 - 2 marks
Will this fraction terminate as a decimal? Justify.
Hint
The test applies after all common factors have been cancelled.
Stage 9 - Q3 - 2 marks
Will this fraction terminate as a decimal? Justify.
Hint
The test applies after all common factors have been cancelled.
Stage 9 - Q4 - 2 marks
Will this fraction terminate as a decimal? Justify.
Hint
The test applies after all common factors have been cancelled.
Stage 9 - Q5 - 4 marks
Find all positive integer numerators smaller than 30 for which \(\frac n{30}\) terminates.
Hint
Determine which denominator factor must disappear on reduction.
Stage 9 - Q6 - 3 marks
A pupil says \(\frac{12}{48}\) recurs because its denominator has a factor 3. Explain the mistake.
Hint
Look for the greatest common factor first.
Stage 9 - Q7 - 3 marks
Is every fraction with denominator 40 terminating? Give a proof or a counterexample.
Hint
Test the prime factors and use numerator 1 if you need a counterexample.
Stage 9 - Q8 - 4 marks
Give two proper fractions with denominator 48: one terminating and one recurring. Explain.
Hint
Choose a numerator that removes the unwanted factor, then one that does not.
Stage 9 - Q9 - 4 marks
A fraction with denominator 18 terminates. Must its numerator be even? Justify with a counterexample or proof.
Hint
Factors of 2 are allowed in a terminating denominator.
Stage 9 - Q10 - 5 marks
Challenge: The denominator is \(840=2^{3}\times5\times3\times7\). Find the least positive integer numerator giving a terminating decimal, then count all such proper fractions before reducing them.
Hint
Cancel exactly the prime factors that a power of ten cannot contain.