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`