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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) ** ** (ii) ** ** (iii) ** ** (iv) ** ** (v) **

**(vi) ** ** ****(vii) **** (viii) ** ** (ix) **** (x)**** **

Answer:

**Solutions: ****We know that ** x = **be a rational number, such that the prime factorization of q is of the form ** **, where n, m are non-negative integers. Then, x** ** has a decimal expansion which terminates.**

**And ** x= ** ** **be a rational number, such that the prime factorization of q is not of the form ** ** , where n, m are non-negative integers. Then, x** ** has a decimal expansion which non-terminating repeating.**

**(i) **** **

**If we factorize the denominator, we get**

**3125 = 5 × 5 × 5 × 5 × 5 = **

**So, the denominator is in the form of ** **so, โ ** **is terminating ****decimal expansion****.**

**(ii) ** ** **

**If we factorize the denominator, we get**

**8 = 2 × 2 × 2 = **

**So, the denominator is in the form of ** ** so, ****โ** ** ** **is terminating decimal expansion.**

**(iii) **** **

**If we factorize the denominator, we get**

**455 ****= ****5 ****× ****7 ****× ****13**

**So, the denominator is not in the form of ** ** so,****โ** ** **

** is non-terminating repeating decimal expansion.**

**(iv) **** **

**If we factorize the denominator, we get**

**1600 ****= ****2 ****× ****2 ****× ****2 ****× ****2 ****× ****2 ****× ****2 ****× ****5 ****× ****5 ****= **

**So, the denominator is in the form of ** **. So, ****is terminating decimal expansion.**

**(v) **

**If we factorize the denominator, we get**

**343 ****= ****7 ****× ****7 ****× ****7 ****=**

**So, the denominator is not in the form of ** . **So, ** **is non-terminating repeating decimal expansion.**

**(vi) **** **

**Here, the denominator is in the form of ** ** ** . **So, ** **is terminating decimal expansion.**

**(vii) **

**Here, the denominator is not in the form of ** ** only. So, ** **is non-terminating repeating decimal expansion.**

**(viii) **** **

**If we divide nominator and denominator both by 3 we get **

**So, the denominator is in the form of .** ** So, ** **is terminating decimal expansion. **

**(ix) **** **

**If we divide nominator and denominator both by 5, we get **

**If** **we factorize the denominator, we get**

**10 ****= ****2 ****× ****5**

**So, the denominator is in the form of ** **. So, ** ** **** is terminating decimal expansion.**

** (x)** ** **

**If we divide nominator and denominator both by 7, we get **

**If** **we factorize the denominator, we get**

**30 ****= ****2 ****× ****3 ****× ****5**

**So, the denominator is not in the form of ** **. So, ** ** is non-terminating repeating decimal expansion****.**

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