Question

What fraction is equal to .534 repeating?

Answer 1

See a solution process below:

Note: Assuming the entire decimal `.534` is repeating

Explanation:

First, we can write:

`x = 0.bar534`

Next, we can multiply each side by `1000` giving:

`1000x = 534.bar534`

Then we can subtract each side of the first equation from each side of the second equation giving:

`1000x - x = 534.bar534 - 0.bar534`

We can now solve for `x` as follows:

`1000x - 1x = (534 + 0.bar534) - 0.bar534`

`(1000 - 1)x = 534 + 0.bar534 - 0.bar534`

`999x = 534 + (0.bar534 - 0.bar534)`

`999x = 534 + 0`

`999x = 534`

`(999x)/color(red)(999) = 534/color(red)(999)`

`(color(red)(cancel(color(black)(999)))x)/cancel(color(red)(999)) = (3 xx 178)/color(red)(3 xx 333)`

`x = (color(red)(cancel(color(black)(3))) xx 178)/color(red)(color(black)(cancel(color(red)(3))) xx 333)`

`x = 178/333`

Answer 2

Assuming that all the numbers are repeating
`x=0.bar(534)`......(1)
`1000x=534.bar(534)`.......(2)
Subtract equation 1 from 2
`1000x-x=534.534534534-0.534534534`
`999x=534`
`x=534/999`
`x=178/333`

Answer 3

Assuming that only `4` is repeating
`x=0.53bar4`
`100x=53.bar4`.......(1)
`1000x=534.bar4`....(2)
Subtract equation 1 from 2
`1000x-100x=534.444-53.444`
`900x=481`
`x=481/900`