Question

Write the rational number $\mathrm{5.31313131....}$ as the ratio of two integers.

• A. $\frac{526}{99}$
• B. $\frac{527}{99}$
• C. $\frac{555}{99}$
• D. None of the above

Answer 1

Answer: (A)

$\frac{526}{99}$

We first write the given rational number as an infinite sum as follows:

$\mathrm{5.313131...}=5+0.31+0.0031+0.000031+....$

The terms making $0.31+0.0031+0.000031...$ are those of a geometric sequence with $a_1=0.31$ and $r=0.01$.

Hence, the use of the formula for an infinite sum of a geometric sequence.

$S=\frac{a_1}{(1-r)}=\frac{0.31}{(1-0.01)}=\frac{0.31}{0.99}=\frac{31}{99}$.

We now write $\mathrm{5.313131...}$ as follows:

$\mathrm{5.313131...}=5+\frac{31}{99}=\frac{526}{99}$ .