Question

Question #42257

Answer 1

`11/30`

Explanation:

Let's take `r=0.3dot6`.

To cancel out the recurring part of the decimal we must multiply by 10, so `10r=3.dot6`.

`9r=10r-r=3.dot6-0.3dot6=3.3`

To make it a whole number we multiply by 10 again, `90r=33`.

Now just divide both sides by 90 to get r, `r=33/90=11/30`

Answer 2

See a solution process below:

Explanation:

First, we can let:

`a = 0.3bar6`

We can then multiply each side of the equation by `10` to give:

`10 * a = 10 * 0.3bar6`

`10a = 3.bar6`

We can next subtract the left sides of the equation and right sides of the equation to give:

`10a - a = 3.bar6 - 0.3bar6`

`10a - 1a = 3.bar6 - 0.3bar6`

`(10 - 1)a = 3.bar6 - 0.3bar6`

`9a = 3.bar6 - 0.3bar6`

We can then rewrite the right side of the equation as:

`9a = (3.6 + 0.0bar6) - (0.3 + 0.0bar6)`

`9a = 3.6 + 0.0bar6 - 0.3 - 0.0bar6`

`9a = 3.6 - 0.3 + 0.0bar6 - 0.0bar6`

`9a = (3.6 - 0.3) + (0.0bar6 - 0.0bar6)`

`9a = 3.3 + 0`

`9a = 3.3`

Next, divide each side of the equation by `color(red)(9)` to solve for `a` while keeping the equation balanced:

`(9a)/color(red)(9) = 3.3/color(red)(9)`

`(color(red)(cancel(color(black)(9)))a)/cancel(color(red)(9)) = 3.3/9`

`a = 3.3/9`

We can now convert the fraction to the simplest form by multiplying the fraction by the appropriate form of `1` and then cancelling common terms in the numerator and denominator:

`a = (10/10 xx 3.3/9)`

`a = (10 xx 3.3)/(10 xx 9)`

`a = 33/90`

`a = (11 xx 3)/(30 xx 3)`

`a = (11 xx color(red)(cancel(color(black)(3))))/(30 xx color(red)(cancel(color(black)(3))))`

`a = 11/30`

Therefore:

`0.3bar6 = 11/30`