Chapter 9.4, Problem 32E

Solution

To calculate: A fraction of the rational number $5.93939393\cdots$ .

The resultant fraction is $\frac{196}{33}$ .

Given information:

The given rational number is $5.93939393\cdots$ .

Formula used: The geometric series ${\displaystyle {\sum }_{k=1}^{\infty }a\cdot r^{k-1}}$ , converges if and only if $\left|r\right|<1$ .If it does converge, the sum: $S=\frac{a_1}{1-r}$ .

Calculation:

Consider the given rational number $5.93939393\cdots$ ,

Rewrite $5.93939393\cdots$ as,

  $\begin{array}{c}5.93939393\cdots =5+\left(0.93939393\cdots \right)\\ =5+\left(0.93+0.0093+0.000093+\cdots \right)\end{array}$

Now $\left(0.93+0.0093+0.000093+\cdots \right)$ has become a geometric series. The terms are:

  $a_1=0.93,a_2=0.0093,a_3=0.000093$

The common ratio is:

  $r=\frac{0.0093}{0.93},r=0.01$

Use the formula for sum $\frac{a_1}{1-r}$ , and substitute $0.93$ for $a_1$ and 0.01 for r ,

  $\begin{array}{c}\frac{a_1}{1-r}=\frac{0.93}{1-0.01}\\ =\frac{0.93}{0.99}\\ =\frac{93}{99}\\ =\frac{31}{33}\end{array}$

Now find the value of $5+\left(0.93+0.0093+0.000093+\cdots \right)$ ,

  $\begin{array}{c}5+\left(0.93+0.0093+0.000093+\cdots \right)=5+\frac{31}{33}\\ =\frac{5\cdot 33+31}{33}\\ =\frac{165+31}{33}\\ =\frac{196}{33}\end{array}$

Thus, the repeating decimals $5.93939393\cdots$ can be written in fraction as $\frac{196}{33}$ .