The Correct Answer and Explanation is:
Of course, here are the decimal representations for the given rational numbers, along with a detailed explanation.
Correct Answer:
(i) -12/13 = -0.923076
(ii) -1525/50 = -30.5
(iii) -127/7 = -18.142857
(iv) -539/80 = -6.7375
Explanation:
To find the decimal representation of a rational number, which is a number expressed as a fraction, we perform the division of the numerator by the denominator. The resulting decimal will either be terminating (it ends) or non-terminating repeating (it has a sequence of digits that repeats forever). A fraction results in a terminating decimal if the prime factors of its denominator, once the fraction is in its simplest form, are only 2s and 5s.
(i) For the fraction -12/13, the denominator is 13. Since 13 is a prime number other than 2 or 5, the decimal will be repeating. Performing the long division of 12 by 13, we find a repeating pattern. The division yields 0.923076923076…, where the block of digits “923076” repeats indefinitely. We use a bar over the repeating block to represent this. Therefore, the decimal representation is -0.923076.
(ii) For the fraction -1525/50, we first examine the denominator, 50. Its prime factorization is 2 × 5², which contains only the primes 2 and 5. This indicates the decimal will terminate. We can simplify the fraction by dividing both the numerator and denominator by their greatest common divisor, 25, which gives -61/2. Dividing -61 by 2 is straightforward and results in -30.5. The division ends, confirming it is a terminating decimal.
(iii) The fraction -127/7 has a denominator of 7, which is a prime number that is not 2 or 5. Consequently, we expect a non-terminating, repeating decimal. By dividing 127 by 7, we get 18 with a remainder. Continuing the division on the remainder gives a repeating sequence of digits. The result is 18.142857142857…, with the block “142857” repeating. So, the decimal representation is -18.142857.
(iv) Lastly, for the fraction -539/80, the denominator is 80. The prime factorization of 80 is 2⁴ × 5. Since the only prime factors are 2 and 5, the decimal representation will be terminating. Dividing -539 by 80 using long division gives the exact value of -6.7375. The division process concludes with a remainder of zero, producing a finite number of decimal digits.
