Question

The value of $n=0.5757...$ is equivalent to;
A.$\dfrac{9}{33}$ \[\]
B.$\dfrac{57}{99}$\[\]
C.$\dfrac{19}{33}$\[\]
D. None of these \[\]

Answer 1

Hint: We recall the recurring decimal where digits are repeated and how to convert a recurring decimal into fraction. We find the repeating term in the given decimal number and make it the numerator of the fraction and make $9999...\left( k\text{ times} \right)$where $k$the number of digits is in the repeating term. We find the equivalent fractions of the obtained fraction to choose the correct options. \[\]

Complete step-by-step answer:
We call a decimal number terminating decimal if it has a terminating digit in the fractional part, for example in 10.12 the digit 2 is the terminating digit and hence 10.12 is a terminating decimal. If the decimal number does not have a terminating digit; for example in 10.122222... it is called non-terminating decimal because 2 repeats itself infinite times . \[\]
We can convert the non-terminating decimal in the form of $\dfrac{a}{b}$ where $a$ is the repeating term and $b=9999...\left( k\text{ times} \right)$ where $k$ is the number of digits in repeating terms. \[\]
We have been given the non-terminating decimal $n=0.5757...$. We see that the repeating term here is $0.57$ and the number of digits in the repeating term $0.57$ is 2. So $k=2$. So in the fraction form $n=0.5757...$ will have in the numerator 0.57 and in the denominator 9 repeated twice which means
\[n=0.5757...=\dfrac{57}{99}\]
Now we need to find its equivalent fractions. So we divide the numerator and denominator of the obtained fraction $\dfrac{57}{99}$ by common divisors of the numerator 57 and denominator 99. We divide 57 and 99 by 3 to get
\[\dfrac{57}{99}=\dfrac{\dfrac{57}{3}}{\dfrac{99}{3}}=\dfrac{19}{33}\]
So we have;
\[n=0.5757...=\dfrac{57}{99}=\dfrac{19}{33}\]

So, the correct answer is “Option B and C”.

Note: We note that we can represent a recurring or non-terminating decimal with a bar symbol on the repeating terms$0.5757...=0.\overline{57}$. We can find an equivalent fraction by dividing by a common divisor using divisibility rules or multiply by any integer except 0. If the number is in form $0.ab\overline{cd}$ where $cd$ is the repeating part the numerator will be $abcd-ab$ and the denominator will be 9900.