Question

Write decimal expansion to \[\dfrac{{17}}{8}\].

Answer 1

Hint:By observing the given question, we can say that \[\dfrac{{17}}{8}\]is in the fraction form and it is asked to convert into decimal expansion. By dividing the given numerator with the given denominator, we can change the fraction form into a decimal form. By following the below step by step process, you can get a clear solution.

Complete step-by-step answer:
“The way of writing the numbers where each of the given digits represents the different power of 10 is called decimal.”
Generally, the decimal form has two parts.
1.Fractional part
2.Integral part
These two parts are separated by a dot which is called a decimal point.
We have to divide the numerator with the given denominator, first we have to calculate how many multiples are in $17$. There are two multiples of $8$ and $16$ in $17$. So subtract the multiple $16$ from numerator $17$ the remainder is $1$. Write the quotient value above the division root and write the remainder below the line.Put point after the value of quotient and write zero next to remainder and Continue the above process again till getting remainder value $0$.

\[\begin{array}{l}8\mathop{\left){\vphantom{1{17}}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{{17}}}}
\limits^{\displaystyle\,\,\, {2.125}}\\\begin{array}{*{20}{c}}{ - 16}\\{\_\_\_\_\_\_\_\_\_}\\{10}\\{ - 8}\\{\_\_\_\_\_\_\_\_}\\{20}\\{ - 16}\\{\_\_\_\_\_\_\_\_}\\{40}\\{ - 40}\\{\_\_\_\_\_\_\_\_}\\0\end{array}\end{array}\]

Therefore, \[\dfrac{{17}}{8}\] can be written as \[2.125\] which is the decimal form.

Note:A rational number can have two types of decimal expansions. One is terminating decimal expansion and the other is non - terminating but repeating decimal expansion.
Terminating decimal expansion: the terminating decimal expansion can be defined as the decimal expansion or representation terminates after a certain number of digits.
For example:
\[\dfrac{1}{{16}}{\text{ }} = {\text{ 0}}{\text{.0625}}\]
Here the decimal expansion terminates after \[4\] digits.
In the terminating decimal expansion, the prime factorization of the denominator has only \[2\] and \[5\] as factors.
Non-terminating but repeating decimal expansion: The non – terminating but repeating decimal expansion has a repetitive pattern that has an infinite number of series.
For example:
\[\dfrac{1}{7}{\text{ = 0}}{\text{.142857}}\]
In non-terminating but repeating decimal expansion, the prime factorization of the denominator can have other factors than \[2\] and \[5\].
These are the two types of decimal expansion.