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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) 23/8
(ii) 125/441
(iii) 35/50
(iv) 77/210
(i) The given number is 23/8
Here, 8 = 23 and 2 is not a factor of 23.
So, the given number is in its simplest form.
Now, 8 = 23 is of the form 2m × 5n, where m = 3 and n = 0.
So, the given number has a terminating decimal expansion.
(ii) The given number is 125/441
Here, 441 = 32 × 72 and none of 3 and 7 is a factor of 125.
Now, 441 = 32 × 72 is not of the form 2m × 5n
So, the given number has a non-terminating repeating decimal expansion.
(iii) The given number is 35/50 and HCF (35, 50) = 5.
Here, 7/10 is in its simplest form.
Now, 10 = 2 × 5 is of the form 2m × 5n, where in = 1 and n = 1.
(iv) The given number is 77/210 and HCF (77, 210) = 7.
Here, 11/30 is in its simplest form. 30
Now, 30 = 2 × 3 × 5 is not of the form 2m × 5n.
(v) The given number isClearly, none of 2, 5 and 7 is a factor of 129.
Write down the decimal expansions of the following rational numbers by writing their denominators in the form of 2m × 5n, where m, and n, are the non- negative integers.
(i) 3/8
(ii) 13/125
(iii) 7/80
(iv) 14588/625
(i) The given number is 3/8
Clearly, 8 = 23 is of the form 2m × 5n, where m = 3 and n = 0.
So, the given number has terminating decimal expansion.
(ii) The given number is 13/125
Clearly, 125 = 53 is of the form 2m × 5″, where m = 0 and n = 3.
(iii) The given number is 7/80.
Clearly, 80 = 24 × 5 is of the form 2m × 5n, where m = 4 and n = 1.
(iv) The given number is 14588/625
Clearly, 625 = 54 is of the form 2m × 5n, where m = 0 and n = 4.
So, the given number has terminating decimal expansion?
(v) The given number is Clearly, 22 × 57 is of the form 2m × 5n, where in = 2 and n = 7.
What can you say about the prime factorization of the denominators of the following rational?
(i) 43.123456789
(iv) 0.120120012000120000
(i) Since 43.123456789 has terminating decimal expansion. So, its denominator is of the form 2m × 5n, where m, n are non-negative integers.
(ii) Since has non-terminating decimal expansion. So, its denominator has factors other than 2 or 5.
(iii) Since has non-terminating decimal expansion. So, its denominator has factors other than 2 or 5.
(iv) Since 0.120120012000120000 … has non-terminating decimal expansion. So, its denominator has factors other than 2 or 5.
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