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Question
Express the following recurring decimal as a rational number:
`3.dot5`
Solution
`3.dot5`
= 3.555…
= 3 + 0.5 + 0.05 + 0.005 + …
Here, 0.5, 0.05, 0.005, … are in G.P. with a = 0.5 and r = 0.1.
Since |r| = |0.1| < 1
∴ Sum to infinity exists.
∴ Sum to infinity = `"a"/(1 - "r")`
= `0.5/(1 - (0.1))`
= `0.5/0.9`
= `5/9`
∴ `3.dot5 = 3 + 5/9 = 32/9`
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