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:
Note: - If x be a rational number, then x can be expressed in the form p/q where p and q are Co- primes. Then, if the prime factorization of denominator (q) is in the form of 2m × 5n where n, m are non- negative integers, then x has a decimal expansion which terminates. If the prime factorization is not in this form of 2m × 5n then x has a decimal expansion which is non terminating.
(i) 13 / 3125
= 5 × 5 × 5 × 5 × 5 = 5⁵ (on factorization of q)
Since, its factorization contains only power of 5.
Therefore, it has a terminating decimal expansion.
(ii) 17 / 5
= 2 × 2 × 2 = 2³ (on factorization of q)
Since, its factorization contains only power of 2.
Therefore, it has a terminating decimal expansion.
(iii) 64 / 455
= 5 × 7 × 13 = 5¹ × 7 × 13 (on factorization of q)
Since the factorization of not in the 2m ×5n
Therefore, it has a non -terminating decimal expansion.
(iv) 15 / 1600
= 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5 = 2⁶ × 5² (on factorization of q)
Since the factorization of q is in the form 2m ×5n
Therefore, it has a terminating decimal expansion
(V) 29 / 343
= 7 × 7 × 7 = 7³ (on factorization of q)
Since the factorization of denominator is not in the form 2m × 5n
Therefore it has a non -terminating repeating decimal
(vi) 23 / 23 52
Since factorization of q already given and it is in the form 2m × 5n
Therefore it has a terminating decimal expansion.
(vii) 129 / 22 57 75
Since factorization of q already given but it is not in the form 2m × 5n
Therefore it has a terminating decimal expansion.
(viii) 6 / 15 = 2/5
5 = 5 × 1 (on factorization of q)
Since 5 is the only factor in denominator.
Therefore it has a non -terminating repeating decimal expansion.
(ix). 35 / 50 = 7 / 10
10 = 2 × 5 = 2¹ × 5 (on factorization of q)
Since the factorization of denominator is in the form 2m × 5n
Therefore it has a terminating decimal expansion.
(x) 77 / 210 = 11/30
30 = 2 × 3 × 5 × 7 = 2¹ × 3 × 5 × 7. (on factorization of q)
Since the factorization of denominator is not in the form 2m × 5n
Therefore it has a non -terminating repeating decimal expansion.
Popular Questions of Class 10 Mathematics
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A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in rupees) was 3 more than twice the number of articles produced on that day. If the total cost of production on that day was Rs 90, find the number of articles produced and the cost of each article.
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Refer to Example 13. (i) Complete the following table:
(ii) A student argues that ‘there are 11 possible outcomes 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12. Therefore, each of them has a probability
Do you agree with this argument? Justify your answer.
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Is the following situation possible? If so, determine their present ages.
The sum of the ages of two friends is 20 years. Four years ago, the product of their ages in years was 48. - Q:-
Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial:
(i) t2 – 3, 2t4 + 3t3 – 2t2 – 9t – 12(ii) x2 + 3x + 1, 3x4 + 5x3 – 7x2 + 2x + 2
(iii) x3 – 3x + 1, x5 – 4x3 + x2 + 3x + 1
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