How to convert non-terminating and repeating decimals to fractions

Noteworthy Notes:
1) All non-terminating and non-repeating decimals are Irrational numbers.
2) All non-terminating and repeating decimals are Rational numbers.
Non-terminating and repeating decimals are also called recurring decimals. Sometimes these are expressed by a line over the repeating digits.
For examples :
2.7777…… = 2.7
1.05693693693693….. = 1.05693
Students sometimes find very tricky to work with them. As they are rational numbers , then by definition they can be expressed in the form of p/q.
We will find out how to do that in this post.
Remember , that our target is to eliminate the digits after the decimal. Just notice how…
Suppose we are given to convert the following two decimals to their fractional equivalent:
ex-1) 0.6666…..  or 0.6         ex-2) 1.05693693693693….. or  1.05693
Just follow the steps below with the given two examples:

Step-1:  Write an equation, x = the given decimal.

Ex-1:
x
= 0.6666…..
Ex-2:
x
= 1.05693693693693…..

Step-2: Count the number of digits after decimal, the repeated pattern starts. Suppose it is equal to y. Then, multiply x with 10y .

Ex-1 :
x
= 0.6666…..
Here the repeated pattern is “6”, which gets repeated just after the decimal. Therefore here, y = 0  and 10o = 1
Hence, 1 × x = 1 × 0.66666….. = 0.66666….
Ex-2 :
x
= 1.05693693693693…..
Here the repeated pattern is “693”, which gets repeated  after two digits (0 and 5) of the decimal. Therefore here, y = 2 and 102 = 100
Hence, 100 × x  =
100 × 1.05693693693693….. = 105. 693693693…..

Step-3: Count the number of digits in the repeated pattern, let this be, z . Now multiply the given decimal, x by 10y+z .

Ex-1 :
x
= 0.6666…..
1 × x  = 0.6666…
Here the number of digits in the repeated pattern is “1”. Therefore here, z = 1==> y+z= 0+1 = 1
so , 1o1 = 10
Hence, 10 × x = 10× 0.66666….. = 6.666666…
Ex-2 :
x
= 1.05693693693693…..
100x = 105.693693693693…..
Here the number of digits in the repeated pattern is “3”. Therefore here, z = 3==> y+z= 2+3 = 5
so , 1o5 = 100000
Hence, 100000 × x =
100000× 1.05693693693693…..
= 105693.693693693…..

Step-4: Subtract the equation of step-2 from the equation of step-3 and solve for x , to get the value in the form of p/q .

Ex-1 :
x
= 0.6666…..
equation of step-2 : x = 0.6666…
equation of step-3: 10x = 6.666666…
Calculate differences of the LHS and RHS of the two equations and solve for x :
10x – x = 6.6666… – 0.6666….
9x = 6
x = 6/9
or reduce further to get
x = 2/3
Ex-2 :
x
= 1.05693693693693…..
equation of step-2 : 100x = 105.693693693693…..
equation of step-3:  100000x = 105693.693693693…..
Calculate differences of the LHS and RHS of the two equations and solve for x :
100000x -100 x =105693.693693693….. – 105.693693693693…..
99900x = 105588
x = 105588/99900
or reduce further to get
x =52794/49950 =26397 /24975

Now , take out your calculator to reverse-calculate and check if you answer is correct :

Ex-1 :
x
=6/9 =  2/3 = 0.666…. = 0.6
Ex-2 :
x
=105588/99900 = 52794/49950 =26397 /24975 = 1.05693693693…. = 1.05693

Initially , the steps may seem to be daunting but if you follow and practice converting some repeating decimal to their fractional equivalents , you will understand that it’s all about eliminating the digits after the decimal and finding p/q.
Good Luck !!

The Reader’s challenge :
Convert the following decimals to their fractional equivalent and comment with your answers:
1) 10.0474747….. or 10.047
2) 0.212121…. or 0.21

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