Question

How do you convert \[0.07\] (\[7\] being repeated) to a fraction?

Answer 1

Hint: To solve this question, we will write the given number in the form of an equation by equation the given number to an unknown variable. Then we need to multiply the equation repeatedly by \[10\] until we get a non-zero digit before the decimal. On subtracting the original equation from the multiplied equation, the number will become simplified and the required fractional form will be obtained.

Complete step-by-step answer:
According to the question, the number given in the question is \[0.07\] and also it is stated that \[7\] is being repeated. So this means that the given number is actually \[0.0\overline 7 \]. Let us write this number the equation as
\[n = 0.0\overline 7 \]…………………….\[\left( 1 \right)\]
For converting the given number to a fraction, we need to multiply the number by \[10\] until we obtain a non zero digit before the decimal.
Therefore, we multiply the above equation by \[10\] to get
\[\begin{array}{l} \Rightarrow 10n = 0.0\overline 7 \times 10\\ \Rightarrow 10n = 0.\overline 7 \end{array}\]
We note that now also, we haven’t got a non zero digit before the decimal. So we again multiply the above equation by \[10\] to get
\[ \Rightarrow 100n = 0.\overline 7 \times 10\]
\[ \Rightarrow 100n = 7.\overline 7 \]…………………………..\[\left( 2 \right)\]
Subtracting equation \[\left( 1 \right)\] from \[\left( 2 \right)\], we get
\[\begin{array}{l} \Rightarrow 100n - n = 7.\overline 7 - 0.0\overline 7 \\ \Rightarrow 99n = 7.7\overline 7 - 0.0\overline 7 \end{array}\]
On simplifying the RHS, we get
\[ \Rightarrow 99n = 7.7\]
Dividing both sides by \[99\] we get
\[\begin{array}{l} \Rightarrow n = \dfrac{{7.7}}{{99}}\\ \Rightarrow n = \dfrac{7}{{90}}\end{array}\]
Hence, the given number is converted to the fraction of \[\dfrac{7}{{90}}\].

Note: The bar sign over a digit indicates that there are an infinite number of digits. So while subtracting the number \[0.0\overline 7 \] from the number \[7.\overline 7 \], we could write \[7.\overline 7 \] as \[7.7\overline 7 \]. There are three main types of fractions i.e. proper fractions, improper fractions, and mixed fractions. A proper fraction is a fraction having the numerator less or lowers in degree, than the denominator. The value of proper fraction after simplification is always less than 1. An improper fraction is a fraction where the numerator is greater than or equals to the denominator, then it is known as an improper fraction. A mixed Fraction is the combination of a natural number and fraction.