﻿ Class 10 NCERT Math Solution
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TOPICS
Exercise - 1.4

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) 23/2³5², (vii) 129/2²5775,  (viii) 6/15, (ix) 35/50, (x) 77/210

Solution :-
```(i) 13/3125
3125 = 55
The denominator is of the form 5m.
Hence, the decimal expansion of 13/3125 is terminating.
```
```(ii) 17/8
17/8 = 2³
The denominator is of the form 2m.
Hence, the decimal expansion of 17/8 is terminating.
```
```(iii) 64/455
455 = 5 x 7 x 13
Since the denominator is not in the form 2m × 5n , and it also contains 7 and 13 as its factors,
its decimal expansion will be non-terminating repeating.
```
```(iv) 15/1600
1600 = 26 × 5²
The denominator is of the form 2m × 5n.
Hence, the decimal expansion of 15/1600 is terminating.
```
```(v)  29/343
343 = 7³
Since the denominator is not in the form 2m × 5n , and it also contains 7 as its factors,
its decimal expansion will be non-terminating repeating.
```
```(vi) 23/2³5²
The denominator is of the form 2m × 5n.
Hence, the decimal expansion of 23/2³5² is terminating.
```
```(vii) 129/2²5775
Since the denominator is not in the form 2m × 5n , and it also contains 7 as its factors,
its decimal expansion will be non-terminating repeating.
```
```(viii) 6/15
6/15 = (2 x 3)/(3 x 5) = 2/5
The denominator is of the form 5m.
Hence, the decimal expansion of 6/15 is terminating.
```
```(ix) 35/50
35/50 = (5 x 7)/(2 x 5 x 5) = 7/(2 x 5)
The denominator is of the form 2m × 5n.
Hence, the decimal expansion of 35/50 is terminating.
```
```(x)  77/210
77/210 = (11 x 7)/(30 x 7) = 11/30 = 11/(2 x 3 x 5)
Since the denominator is not in the form 2m × 5n , and it also contains 3 as its factors,
its decimal expansion will be non-terminating repeating.
```

Question-2 :-  Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions.
(i) 13/3125,  (ii) 17/8, (iv) 15/1600,  (vi) 23/2³5², (viii) 6/15, (ix) 35/50

Solution :-
``` ```

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 , what can you say about the prime factor 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 2m × 5n
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 2m × 5n
i.e., the prime factors of q will also have a factor other than 2 or 5.
```
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