Value of 0.4 in base 7

Express the base‑7 numeral $0. 4_{7}$ as a fraction in ordinary (base‑10) notation.

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H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. VII. Scales of Notation, Examples VII.b, No. 19 (part 1)

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Checked against the answer in H. S. Hall & S. R. Knight, Higher Algebra (1891).