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#