Question

The decimal expansion of $\dfrac{{51}}{{{2^3} \times {5^2}}}$ will terminate after how many decimal places?

Answer 1

Hint: In the given integer, if the factors of denominator of the given rational number (p/q) is in the form of ${2^m}{5^n}$, where m and n are non-negative integer, then the decimal expression of the rational number is terminating otherwise they will not terminating and they repeat continuously.

Complete step by step solution:
Euclid Division Lemma: It states that if we have two positive integers a and b, then there exist unique integers q and r which satisfies the condition $a = bq + r$, where $\text{0 < r < b}$.
Given
$ = \dfrac{{51}}{{{2^3} \times {5^2}}}$
Or we can write as
$ = \dfrac{{51}}{{8 \times 25}}$
$ = \dfrac{{51}}{{200}}$
Further solving it. we get,
$ = 0.255$
Hence, $\dfrac{{51}}{{{2^3} \times {5^2}}}$ will terminate after three decimals.

Additional information:
Decimal system: The decimal system has been extended to infinite decimals for representing any real number, by using an infinite sequence of digits after the decimal separator. In this context, the decimal numerals with a finite number of non-zero digits after the decimal separator are sometimes called terminating decimals. A repeating decimal is an infinite decimal that, after some place, repeats indefinitely the same sequence of digits.

Note:
The rational number which terminates after a finite number of steps is known as terminating decimals. Which are not terminated after the finite number is known as non-terminating decimals. A repeating decimal which repeats itself after the finite period. Non-terminating, non-repeating decimals is a decimal that continues endlessly.