Question

What is the decimal expansion of the rational number $\dfrac{{14587}}{{1250}}$?

Answer 1

Hint: For a decimal, the denominator will be in terms of powers of 10. So convert the denominator of the given fraction into powers of 10. Prime factorize the denominator and the exponents of primes 2 and 5 should be equal to be in terms of powers of 10. So if the exponents are not equal, then multiply the required respective 2’s and 5’s in the numerator and denominator of the rational number.

Complete step-by-step answer:
We are given a rational number $\dfrac{{14587}}{{1250}}$ and we have to write it in the decimal expansion.
Take the denominator 1250 and prime factorize it.
1250 can be written as two times of 625.
$1250 = 2 \times 625$
625 can be written as five times of 125.
$1250 = 2 \times 5 \times 125$
125 can be written as five times of 25.
$1250 = 2 \times 5 \times 5 \times 25$
25 can be written as five times of 5.
$
  1250 = 2 \times 5 \times 5 \times 5 \times 5 \\
  1250 = 2 \times {5^4} \\
 $
The exponent of 5 is 4; and 2 is 1. The exponents of 2 and 5 must be equal to be in terms of powers of 10.
So multiple ${2^3}$ in the numerator and denominator of $\dfrac{{14587}}{{1250}}$
$
  \dfrac{{14587}}{{1250}} \\
   = \dfrac{{14587 \times {2^3}}}{{2 \times {5^4} \times {2^3}}} \\
   = \dfrac{{14587 \times 8}}{{{2^4} \times {5^4}}} \\
   = \dfrac{{116696}}{{{{10}^4}}} \\
   = \dfrac{{116696}}{{10000}} \\
   = 11.6696 \\
 $
Therefore, the decimal expansion of $\dfrac{{14587}}{{1250}}$ is 11.6696.

Note: The decimal expansion of a number is a representation in base 10. The decimal expansion has a decimal point and each place after the decimal point consists of a digit from 0-9 and the powers of the digits after the decimal point are negative and decrease gradually. There can be an actual decimal point in a number or we can imply a decimal point if not.