Which of these fractions does NOT have a terminating decimal expansion?

Class 9Number SystemsNumber Systems

Step-by-Step Solution

Reduce first!

\frac{9}{15} = \frac{3}{5} = 0.6

terminates even though 15 contains a 3.

\frac{11}{40} = 0.275

and

\frac{13}{50} = 0.26

terminate. But

12 = 2^{2} \times 3

keeps its factor 3, so

\frac{7}{12} = 0.58333

… is non-terminating recurring.

Answer: C)

\frac{7}{12}

**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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