Question

What is the fraction \[\dfrac{17}{7}\] as a repeating decimal?

Answer 1

Hint: In order to find out the repeating decimal of \[\dfrac{17}{7}\], firstly we have to check out if it is a repeating or non-repeating decimal. After finding out the category it belongs to, then we can find out the repeating decimal by simplifying it and then finding out the repeating decimal.

Complete step-by-step solution:
Now let us find whether \[\dfrac{17}{7}\] is a repeating decimal or not, in order to find out-
Firstly, find the denominator into its lowest term.
The GCF of \[17\] and \[7\] is one. Now convert \[\dfrac{17}{7}\] into simplest form by dividing it with one.
We get, \[\dfrac{17\div 1}{7\div 1}=\dfrac{17}{7}\]
The denominator in its lowest form is \[7\].
Now, find the prime factors of the lowest term i.e. \[7\]
Since\[7\]itself is a prime number, the prime factor is \[7\] itself.
Now, we will be determining if \[\dfrac{17}{7}\] is a repeating or non-repeating decimal.
A fraction is a repeating decimal if the prime factors of the denominator of the fraction in its lowest form and do not only contain \[2s\] and/or \[5s\] or do not have any prime factors at all.
Hence our fraction \[\dfrac{17}{7}\] is repeating.
Now let us find the repeating decimal.
On simplifying the fraction, we get
\[\dfrac{17}{7}=2.4285714385714285714285.....\]
This can be truncated for pre-fixing non-repeat strings to get the form \[\dfrac{17}{7}\].
Truncating helps in approximating the numbers. This is easier than rounding but does not give the best approximation always to the original number. This can be obtained by the method of approximating.
\[\dfrac{17}{7}=2.4285+10-4\left( .714285X(1+10-6+10-12+10-18+... \right)\]
\[\therefore \] The repeating decimal is \[\dfrac{17}{7}=2.4285714385714285714285.....\]

Note: A rational number can only be shown as decimal only if it is repeating or non-repeating. The repeating part of the decimal can be represented by placing dots or by placing the line over the pattern. The repeating part is called a period.