Let x = 0.6
Multiply by 10 on L.H.S and R.H.S,
10x = 6
x = 6/10
x = 3/5
So, the p/q form of 0.6 = 3/5
Let y = 0.77777…
Multiply by 10 on L.H.S and R.H.S,
10y = 7.7777…
10y – y = 7.7777777……. – 0.7777777…………..
9y = 7
y = 7/9
So the p/q form of 0.7777… = 7/9
Let z = 0.47777…
Multiply by 10 on L.H.S and R.H.S,
10z = 4.7777…
10z – z = 4.7777777… – 0.47777777…
9z = 4.2999
z ≈ 4.3/9
z = 43/90
So the p/q form of 0.4777… = 43/90
Therefore, p/q form of \(0.6 + 0.\overline7 + 0.4\overline7\) is,
x+y+z = 3/5 + 7/9 + 43/90
= (54 + 70 + 43)/90
= 167/90