Number 7

4 marks · Hard

Express the recurring decimal as a fraction in its simplest form. Show all your working.

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Also accept: Any correct algebraic justification yielding

Common mistakes: Using 100 instead of 1000 as multiplier by miscounting non-repeating digits; Forgetting to simplify the fraction

Show mark scheme and worked solution

Marking points

M1 · 1 mark · Let x = 0.0838383... and multiply by 1000 and 10 as appropriate.

M2 · 1 mark · Correct setup of subtraction (e.g., 1000x - 10x = ...)

A1 · 1 mark · Obtains equation: 990x = 83 (follow-through)

A2 · 1 mark · Final simplified fraction: (follow-through)

Worked solution

Let

Since two digits repeat after one non-repeating digit:

Multiply both sides by 1000 to shift the decimal point past one full cycle:

Multiply by 10 (since there is 1 non-recurring digit) to align recurring parts:

Subtract: