Here, we try to find a fraction equivalent to the given recurring decimal.
Divide the decimal into 2 parts; a finite decimal (0.39), and recurring decimal (0.007777...),so
Sol.1: 0.397777...=0.39+0.007777... While,
1/9=0.1111..., 1/90=0.01111..., 1/900=0.001111... Thus,
0.007777...= 7x0.001111...=7x(1/900)=7/900 And,
0.39=39/100=(39x9)/900=351/900 Therefore,
0.397777...= 0.39+0.007777...=351/900 + 7/900=358/900=(2x179)/(2x450)=179/450
The answer is: 0.397777...=179/450
Sol.2: 0.397777...=0.39+0.007777... Let 0.39=A and 0.007777...=B, so 0.397777...=A+B
10B=0.07777.7.., so 10B-B=0.077777... - 0.007777...=0.07
That is: 9B=0.07 We have: B=0.07/9=7/900 While,
A=0.39=39/100=351/900 Therefore,
0.397777...=A+B=351/900 + 7/900=358/900=179/450
The answer is: 0.397777...=179/450