Express 0.23̄ (only the 3 repeats: 0.2333…) as a fraction in simplest form.

Class 9Number SystemsNumber Systems

Step-by-Step Solution

Let

x = 0.2333

…. Then

10 x = 2.333

… and

100 x = 23.333

…. Subtracting,

90 x = 21

, so

x = \frac{21}{90} = \frac{7}{30}

. Using 99 as the denominator would be right only if the whole block after the point repeated.

Answer: A)

\frac{7}{30}

**Rational ↔ terminating OR non-terminating recurring decimal.**
**Irrational ↔ non-terminating non-recurring decimal.**

Example:

\frac{1}{4} = 0.25

(terminating → rational).
Example:

\frac{1}{3} = 0.333

… (recurring → rational).
Example:

2 = 1.41421356

… (non-terminating non-recurring → irrational).

**When does

\frac{p}{q}

terminate?** Exactly when the denominator (in simplest form) has only 2's and 5's as prime factors. So

\frac{1}{8} (= 1/ 2^{3})

terminates: 0.125. But

\frac{1}{6} (= 1/ (2 \times 3))

doesn't: 0.1666…

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