Question

Express the following recurring decimals as a rational number:

`3.4bar56`

Answer 1

`3.4bar56` = 3.4565656 …

= 3.4 + 0.056 + 0.00056 + 0.0000056 + ….

Here, 0.056, 0.00056, 0.0000056, … are in

G.P. with a = 0.056 and r = 0.01

Since, | r| = |0.01| < 1

∴ Sum to infinity exists.

∴ Sum to infinity = `"a"/(1 - "r")`

= `0.056/(1 - 0.01)`

= `0.056/0.99`

= `56/990`

∴ `3.4bar56 = 3.4 + 56/990`

= `34/10 + 56/990`

= `(3366 + 56)/990`

= `3422/990`

= `1711/495`.