Question #48afa

1 Answer
Dec 2, 2017

The fraction of the repeating #2.bar(31)# decimal is #229/99#.

Explanation:

Step 1 :

Let #x# be equal to the repeating decimal you are trying to convert to a fraction. Here, #x# is #2.31# or

#x = 2.31313131...#

as an equation.

Step 2:

Examine the repeating decimal to find the repeating digit(s). The repeating digits here are #31#.

Step 3:

Place the repeating digit(s) to the left of the decimal point. Move the repeating digits #[31]# to the left of the decimal point by multiplying #100# (to move both digits to the left) to both sides of the equation in step 1. Thus,

#100x=231.313131...#

Step 4:

Place the repeating digit(s) to the right of the decimal point. Look at the equation in step 1 again. In this example, the repeating digit is already to the right, so there is nothing else to do.

#x = 2.31313131#

Step 5:

Your two equations are:

#100x=231.313131#

#x = 2.31313131#

Subtract the left sides of the two equations.

#100x - x= 99x#

Then, subtract the right sides of the two equations

#231.313131-2.31313131=228.99999969#

As you subtract, just make sure that the difference is positive for both sides

# => 99x=228.99999969#

Then round both sides up to whole numbers. In this case, #99# is already a whole number so only #228# needs to be rounded to #229# because the digit after it is #9# (Because #9# is greater or equal to #5#, the unit digit #8# of #228# needs to plus itself by #1#).

#=> 99x=229#

Step 6:

Now, you can find the fraction #x# of the repeating decimal #2.31# by dividing #99# to both sides of the equations

#99x =229#

#=> x=229/99#