Question

Express the following recurring decimals as a rational number:

`0.3bar45`

Answer 1

`0.3bar45` = 0.3454545…

= 0.3 + 0.045 + 0.00045 + 0.0000045 + …

Here, 0.045, 0.00045, 0.0000045, … are in
G.P. with a = 0.045, r = 0.01

Since, | r | = |0.01| < 1

∴ Sum to infinity exists.

∴ Sum to infinity = `"a"/(1 - "r") = 0.045/(1 - 0.01)`

= `0.045/0.99 = 45/990`

∴ `0.3bar45 = 0.3 + 45/990`

= `3/10 + 1/22`

= `(33 + 5)/110`

= `38/110`

= `19/55`

Alternate Method:

`0.3bar45 = (3.bar45)/10`

= `(3 + 0.45 + 0.0045 + 0.000045 + ...)/10`

Here, 0.45, 0.0045, 0.000045… are in G.P.
with a = 0.45 and r = 0.01
Since, | r | = |0.01| < 1
∴ Sum to infinity exists.

∴ Sum to infinity  = `"a"/(1 - "r") = 0.45/(1 - 0.01)`

= `0.45/0.99 = 45/99 = 5/11`

∴ `0.3bar45 = (3 + 5/11)/10 = (38/11)/10 = 19/55`.