Number 7

4 marks · Hard

Explain how to write and as fractions in their simplest form. Describe the differences in your working and the forms of the answers.

Work this out on paper, then compare with the answer and worked solution below. With a free account you can submit written answers for marking.

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Must show: For : correct fraction and process ( or correct working), For : clear recurring decimal to fraction method, ( in simplest form with working), Comparison of methods and results

Also accept: Any equivalent clear and accurate explanations showing the correct conversions and differences

Common mistakes: Attempting to use the recurring decimal method for a terminating decimal; Not expressing answers in simplest form; Incorrectly setting up algebraic equation for recurring decimal

Show mark scheme and worked solution

Marking points

M1 · 1 mark · Correctly writes 0.875 as and simplifies to .

A1 · 1 mark · Gives the final answer for 0.875: . (follow-through)

M2 · 1 mark · Correctly sets up and solves the algebraic equation for to find .

A2 · 1 mark · Describes/compares difference between methods and between types of decimals (terminating vs. recurring).

Worked solution

For 0.875:

This is a terminating decimal.

Divide numerator and denominator by 125:

For 0.8\overline{7}:

Let (where 7 repeats)

Multiply both sides by 10:

Subtract the original x:

But 0.8\overline{7} = 0.87777... Multiply x by 10 and then by 10 again (so, by 100):

Let , then 10x = 8.77777..., 100x = 87.77777...

Difference: 0.875 is terminating and is written over a power of 10. 0.8\overline{7} is recurring: we use an algebraic method involving multiplying to align the recurring digits and subtracting before simplifying.