Express \(0.6 + 0.\overline7 + 0.4\overline7\) in the form p/q, where

1.	Express \(0.6 + 0.\overline7 + 0.4\overline7\) in the form p/q, where p and q are integers and q ≠ 0.
Answer» 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

Express \(0.6 + 0.\overline7 + 0.4\overline7\) in the form p/q, where p and q are integers and q ≠ 0.

Answer»

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