## 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:= 0.6666…..x | Ex-2: = 1.05693693693693…..x |

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

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

Ex-1 : = 0.6666…..x 1 × x = 0.6666…Here the number of digits in the repeated pattern is “1”. Therefore here, z = 1==> y+z= 0+1 = 1so , 1o ^{1} = 10Hence, 10 × x = 10× 0.66666….. = 6.666666… | Ex-2 : = 1.05693693693693…..x 100x = 105.693693693693….. Here the number of digits in the repeated pattern is “3”. Therefore here, z = 3==> y+z= 2+3 = 5so , 1o ^{5} = 100000Hence, 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 : = 0.6666…..x equation of step-2 : x = 0.6666…equation of step-3: 10 x = 6.666666…Calculate differences of the LHS and RHS of the two equations and solve for x :10x – x = 6.6666… – 0.6666…. 9 x = 6x = 6/9or reduce further to get x = 2/3 | Ex-2 : = 1.05693693693693…..x equation of step-2 : 100 x = 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….. 99900 x = 105588x = 105588/99900or 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 :=6/9 = 2/3 = 0.666…. = 0.6x | 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 10.0472) 0.212121…. or 0.21 **