Question

What is \[1.9791666...\] as a fraction?

Answer 1

Hint: To convert a repeating decimal to a fraction we will write it as an equation in terms of \[x\] . After that we will multiply the equation on both sides by \[{10^n}\] until we get the repeating digit before decimal. Then we will subtract the last two equations to remove the repeating decimal and solve the equation to get the decimal as a fraction.

Complete step by step answer:
We have given the decimal number as \[1.9791666...\]. Now let us write this decimal number as a fraction.For this, let’s assume the number to be \[x\].
i.e., \[x = 1.9791666...{\text{ }} - - - \left( 1 \right)\]
Now we will multiply the equation on both sides by \[{10^n}\] until we get the repeating digit before decimal. So, firstly on multiplying by \[10\] we get
\[10x = 19.791666...{\text{ }} - - - \left( 2 \right)\]
Again, multiplying by \[10\] we get
\[100x = 197.91666...{\text{ }} - - - \left( 3 \right)\]
Again, multiplying by \[10\] we get
\[1000x = 1979.1666...{\text{ }} - - - \left( 4 \right)\]

Again, multiplying by \[10\] we get
\[10000x = 19791.666...{\text{ }} - - - \left( 5 \right)\]
Again, multiplying by \[10\] we get
\[100000x = 197916.666...{\text{ }} - - - \left( 6 \right)\]
Now as we get the repeating digit before decimal, so we will now subtract the equation \[\left( 5 \right)\] and \[\left( 6 \right)\] we get
\[100000x - 10000x = 197916.666.... - 19791.666… \\ \]
\[ \Rightarrow 90000x = 178125 \\ \]
\[\therefore x = \dfrac{{178125}}{{90000}}\]
On converting it into simple fraction, we get
\[\therefore x = \dfrac{{178125}}{{90000}} = \dfrac{{95}}{{48}}\]

Therefore, the fractional form of the decimal number \[1.9791666...\] is equal to \[\dfrac{{95}}{{48}}\].

Note: In this question, we were given a non-terminating repeating decimal number. And whenever we have these types of decimal numbers, we write it using a bar over the repeating digits to indicate that they are repeating. So, the given number can also be written as \[1.9791\overline 6 \] . Also note that always remember to check the position of the recurring digits while solving this type of question and then multiply by \[{10^n}\] accordingly.