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Question
Express the following recurring decimal as a rational number:
`51.0bar(2)`
Solution
`51.0bar(2)` = 51 + 0.02 + 0.002 + 0.0002 + ... ...(1)
These terms after the first term form a G.P. whose first term is a = 0.02 and common ratio = r = 0.1
Since |r| = |0.1| = 0.1 < 1, the sum to infinity of this G.P. exists and
S = `"a"/(1 - "r")`
= `0.02/(1 - 0.1)`
= `0.02/0.9`
= `2/90`
= `1/45`
∴ from (1), `51.0bar(2) = 51 + 1/45`
= `(2295 + 1)/45`
= `2296/45`
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