Number 7
3 marks · EasyThe decimal expansion of 0.\overline{123} is 0.123123123\ldots
Write down the first six digits after the decimal point for 0.\overline{123}.
Describe the pattern you see in the decimal expansion.
Write 0.\overline{123} as a fraction in its simplest form.
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(a) 123123
(b) The digits 1, 2, 3 repeat in the same order.
(c) or in simplest form
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Marking points
B1 (a) · 1 mark · Writes the first six decimal digits correctly as 123123.
B1 (b) · 1 mark · States that the digits 1,2,3 repeat in order or describes the recurring pattern.
M1 (c) · 1 mark · Correctly converts 0.\overline{123} to a fraction ( or ).
Worked solution
(a) 0.123 123 123... The first six digits after the decimal point are: 1, 2, 3, 1, 2, 3, so 123123.
(b) The sequence '1, 2, 3' repeats over and over.
(c) Let x = 0.123123123..., then 1000x = 123.123123..., so 1000x - x = 123, giving 999x = 123, so x = 123/999 = 41/333.