Question

Write the recurring decimal \[0.3\dot 2\] as a fraction. [\[0.3\dot 2\] means \[0.3222...\]]

Answer 1

Hint:
We shall denote the recurring decimal by a variable. Then we will multiply by 10 on both sides to get only the recurring digits after the decimal point. We will multiply this by 10 again to get a similar expression. Finally, we will subtract the resulting equations to get the required fraction.

Complete step by step solution:
Let us denote the given recurring decimal by a variable \[x\]i.e.,
\[x = 0.3\dot 2 = 0.3222...\] ……….\[(1)\]
In the decimal part of \[0.32222...\], there is only one digit that is recurring i.e., 2. In this case, we will multiply both sides of equation (1) by 10 to get a new recurring decimal. Thus, we get,
\[10x = 3.22222...\] ............ \[(2)\]
We need another equation similar to equation \[(2)\], i.e., with the decimal part containing only 2. We do this by multiplying both sides of equation \[(2)\] by 10. We get,
\[100x = 32.2222...\] ………..\[(3)\]
Now, we will subtract equation \[(2)\] from equation \[(3)\]. In doing so, we will subtract the LHS of both equations on the LHS, and the RHS of both equations on the RHS. Hence,
\[100x - 10x = 32.222... - 3.222...\]
On the LHS, we get \[90x\] and on the RHS, we get \[29\]. So,
\[90x = 29\]
Taking 90 on the LHS to the denominator in RHS, we get,
\[x = \dfrac{{29}}{{90}}\] ……….\[(4)\]
We see from equation \[(1)\] and equation \[(4)\] that the LHS is the same. So,
\[x = 0.32222... = \dfrac{{29}}{{90}}\]

Therefore, the recurring decimal \[0.32222...\] is represented as \[\dfrac{{29}}{{90}}\] in the fractional form.

Note:
While solving this problem, we should be careful to place the decimal point right before the repeating digits. If the number of recurring digits is \[n\], then the recurring decimal should be multiplied by \[{10^n}\]. A recurring decimal number is a number in which the decimal part keeps repeating the same digit. It is also known as a non-terminating number.