Terminating and Non-terminating Decimal Expansions
Video Explanation
Question
Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion:
(i) \( \frac{23}{8} \)
(ii) \( \frac{125}{441} \)
(iii) \( \frac{35}{50} \)
(iv) \( \frac{77}{210} \)
(v) \( \frac{129}{22 \times 5^7 \times 7^{17}} \)
(vi) \( \frac{987}{10500} \)
Important Result
A rational number \( \frac{p}{q} \), when written in its simplest form, has:
- a terminating decimal expansion if the prime factorisation of \(q\) has only the factors \(2\) and/or \(5\).
- a non-terminating repeating decimal expansion if the prime factorisation of \(q\) has any prime factor other than \(2\) or \(5\).
Solution
(i) \( \frac{23}{8} \)
\(8 = 2^3\)
Only the prime factor \(2\) is present.
⇒ Terminating decimal expansion
(ii) \( \frac{125}{441} \)
\(441 = 3^2 \times 7^2\)
Prime factors other than \(2\) and \(5\) are present.
⇒ Non-terminating repeating decimal expansion
(iii) \( \frac{35}{50} \)
\[ \frac{35}{50} = \frac{7}{10} \]
\(10 = 2 \times 5\)
Only prime factors \(2\) and \(5\) are present.
⇒ Terminating decimal expansion
(iv) \( \frac{77}{210} \)
\[ \frac{77}{210} = \frac{11}{30} \]
\(30 = 2 \times 3 \times 5\)
Prime factor \(3\) is present.
⇒ Non-terminating repeating decimal expansion
(v) \( \frac{129}{22 \times 5^7 \times 7^{17}} \)
Denominator contains the prime factor \(7\).
⇒ Non-terminating repeating decimal expansion
(vi) \( \frac{987}{10500} \)
\[ 10500 = 2^2 \times 3 \times 5^3 \times 7 \]
Prime factors other than \(2\) and \(5\) are present.
⇒ Non-terminating repeating decimal expansion
Conclusion
| Rational Number | Nature of Decimal Expansion |
|---|---|
| \(\frac{23}{8}\) | Terminating |
| \(\frac{125}{441}\) | Non-terminating repeating |
| \(\frac{35}{50}\) | Terminating |
| \(\frac{77}{210}\) | Non-terminating repeating |
| \(\frac{129}{22 \times 5^7 \times 7^{17}}\) | Non-terminating repeating |
| \(\frac{987}{10500}\) | Non-terminating repeating |
\[ \therefore \quad \text{The required classifications are obtained without long division.} \]