Question

Express `0.6 + 0.bar7 + 0.4bar7` in the form `p/q`, where p and q are integers and q ≠ 0.

Answer 1

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.bar7 + 0.4bar7` is,

x + y + z = `3/5 + 7/9 + 43/90`

= `(54 + 70 + 43)/90`

= `167/90`