NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers Ex 1.4 are part of NCERT Solutions for Class 10 Maths. Here are we have given Chapter 1 **Real Numbers Class 10 NCERT Solutions Ex 1.4. **

- Real Numbers Class 10 Ex 1.1
- Real Numbers Class 10 Ex 1.2
- Real Numbers Class 10 Ex 1.3
- Real Numbers Class 10 Ex 1.4

Board |
CBSE |

Textbook |
NCERT |

Class |
Class 10 |

Subject |
Maths |

Chapter |
Chapter 1 |

Chapter Name |
Real Numbers |

Exercise |
Ex 1.4 |

Number of Questions Solved |
3 |

Category |
NCERT Solutions |

## NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers Ex 1.4

Page No: 17

**Question 1**

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) 13/3125

(ii) 17/8

(iii) 64/455

(iv) 15/1600

(v) 29/343

(vi) 23/2^{3 }× 5^{2
}(vii) 129/2^{2 }× 5^{7 }× 7^{5
}(viii) 6/15

(ix) 35/50

(x) 77/210

**Solution:**

3125 = 5^{5}

The denominator is of the form 5^{m}.

8 = 2^{3}

The denominator is of the form 2^{m}.

455 = 5 x 7 x 13

Since the denominator is not in the form 2^{m} x 5^{n}, and it also contains 7 and 13 as its factors, its decimal expansion will be non-terminating repeating.

1600 = 2^{6} × 5^{2}

The denominator is of the form 2^{m} x 5^{n}.

343 = 7^{3}

Since the denominator is not in the form 2^{m} x 5^{n}, and it has 7 as its factor,

The denominator is of the form 2^{m} x 5^{n}. Hence, the decimal expansion of is terminating.

(vii)

Since the denominator is not of the form 2^{m} 5^{n}, and it also has 7 as its

The denominator is of the form 5^{n}.

10 = 2 x 5

The denominator is of the form 2^{m} x 5^{n}.

30 = 2 x 3 x 5

Since the denominator is not of the form 2^{m} × 5^{n}, and it also has 3 as its factors,

Concept Insight: The concept used in this problem is that The decimal expansion of rational number p/q where p and q are coprime numbers, terminates if and only if the prime factorization of q is of the form 2^{n}5^{m}, where n and m are non negative integers. Do not forget that 0 is also a non negative integer so n or m can take value 0.

Generally, mistake is committed in identifying terminating decimals when either of the two prime numbers 2 or 5 is appearing in the prime factorization.

Page No: 18

**Question 2**

Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions.

**Solution:**

**Concept Insight:
**This is based on performing the long division and expressing the rational number in the decimal form learned in lower classes.

**Question 3**

The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form p, *q* you say about the prime factors of *q*?

(i) 43.123456789

(ii) 0.120120012000120000…

(iii) 43.123456789

**Solution:
**(i) 43.123456789

**Since this number has a terminating decimal expansion, it is a rational number of the form p\q and q is of the form 2**

^{m}x 5

^{n},

i.e., the prime factors of q will be either 2 or 5 or both.

(ii) 0.120120012000120000…

The decimal expansion is neither terminating nor recurring. Therefore, the given number is an irrational number.

(iii) 43.123456789

Since the decimal expansion is non-terminating recurring, the given number is a rational number of the form p/q and q is not of the form 2^{m} x 5^{n} i.e., the prime factors of q will also have a factor other than 2 or 5.

Concept Insight: The concept used in this problem is that, If the decimal expansion of rational number p\q, [where p and q are coprime numbers] terminates, then prime factorization of q is of the form 2^{n}5^{m}, where n and m are non negative integers.

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