Question

Question #c4442

Answer 1

Let x equal the number.
Use multiples of powers of 10 to write 2 equations that allow you to subtract away the repeating part.
Solve the remaining equation.

Explanation:

Please notice that the bar over the 2 indicates an infinite repetition.

let `x =0.0bar2`

Multiply by both 10, because that leave only the repetitive part to the right of the decimal and mark as equation [1]:

`10x = 0.bar2" [1]"`

Multiply equation [1] by 10, because that will move 1 repetition to the left of the decimal and mark as equation [2]:

`100x = 2.bar2" [2]"`

Subtract equation [1] from equation [2]:

`100x - 10x = 2.bar2-0.bar2`

Please observe that the infinite repeating 2s to the right of the decimal sum to 0:

`90x = 2`

Solve for x by dividing both sides by 90:

`x = 2/90`

Reduce the fraction:

`x = 1/45`

Check with a calculator:

`x = 0.0bar2`

This checks.

Answer 2

`1/45`

Explanation:

`"we have "0.0bar(2)`

`"where "0.0bar(2)-=0.02222....`

`"we require to create equations with only the repeated"`
`"digit appearing after the decimal point"`

`"let "x=0.0bar(2)`

`"then "10x=0.bar(2)to(1)`

`"and "100x=2.bar(2)to(2)`

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

`rArr100x-10x=2.bar(2)-0.bar(2)`

`rArr90x=2`

`rArrx=2/90=1/45larrcolor(blue)"in simplest form"`

`rArr0.0bar(2)-=1/45`