Question #c4442

2 Answers
Nov 24, 2017

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.

Nov 24, 2017

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