Question

What is `0.46` repeating as a fraction ?

Answer 1

`46/99`

Explanation:

`"create 2 equations with 0.46 repeating in both"`

`"let "x=0.bar(46)to(1)`

`"multiply both sides by "100`

`100x=46.bar(46)to(2)`

`"subtracting "(1)" from "(2)`

`100x-x=46.bar(46)-0.bar(46)`

`rArr99x=46`

`rArrx=46/99`

`rArr0.bar(46)=46/99`

Answer 2

`0.bar(46) = 46/99`

`0.4bar(6) = 7/15`

Explanation:

Case `bb(0.bar(46))`

If you intend `0.464646.... = 0.bar(46)` then:

`0.bar(46) = 46/99 * 0.bar(9) = 46/99 * 1 = 46/99`

Note that `46` and `99` have no common factors larger than `1`, so this is in simplest terms.

Case `bb(0.4bar(6))`

If you intend `0.4666... = 0.4bar(6)` then:

`0.4bar(6) = 0.bar(6) - 0.2`

`color(white)(0.4bar(6)) = 6/9*0.bar(9) - 0.2`

`color(white)(0.4bar(6)) = 6/9-1/5`

`color(white)(0.4bar(6)) = 2/3-1/5`

`color(white)(0.4bar(6)) = 10/15-3/15`

`color(white)(0.4bar(6)) = 7/15`

Note that `7` and `15` have no common factors larger than `1`, so this is in simplest terms.