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