Question

Convert 0.19 (19 repeating) to a fraction.

Answer 1

Hint:The number to be converted can be assumed as an unknown variable for easy calculations. 0.19 is a repeating decimal number with 19 repeating. The formula to convert any repeating decimal number to a fraction is\[\dfrac{{\left( {DN \times F} \right) - NRP}}{D}\]. Another way can be assuming the number as a variable and subtracting the former from latter.

Complete step by step solution:
We know that,
\[\dfrac{{\left( {DN \times F} \right) - NRP}}{D}\]
Where,
DN= Decimal Number
F= 10 if one repeating number, 100 if two repeating numbers, 1000 if three repeating numbers, etc.
NRP= Non-repeating part of decimal number.
D= 9 if one repeating number, 99 if two repeating numbers, 999 if three repeating numbers, etc.
Substituting the values we have,
\[
\dfrac{{\left( {0.19 \times 100} \right) - 0}}{{99}} \\
\Rightarrow \dfrac{{19}}{{99}} \\
\]
Hence, the answer in the simplest possible form is\[\dfrac{{19}}{{99}}\].
Another method for solving this is as follows,
Let 0.19 (19 repeating) to a fraction be assumed as a variable x.
Then we have,
\[x = 0.1919191919.........\]
Multiplying 100 both sides we have,
\[ \Rightarrow 100x = 19.191919........\]
Subtracting former from latter we get
\[ \Rightarrow 99x = 19\]
\[ \Rightarrow x = \dfrac{{19}}{{99}}\]
Hence \[0.1919191919...... = \dfrac{{19}}{{99}}\]

Note: Such repeating decimals are called recurring decimals. Here the figures are periodic and the infinitely repeated portion is not zero. It can be shown that the number is rational if and only if its decimal representation is repeating or terminating that is all except finitely many digits are zero.
Whenever there is a conversion of recurring decimal or fraction then that can be solved easily by assuming it as a variable, multiplying both sides with 100 for easy evaluation and subtracting the former from the latter.