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Classify the decimal form of the given rational number into terminating non-terminating recurring type.
A. \[\dfrac{13}{5}\]
B. \[\dfrac{2}{11}\]
C. \[\dfrac{29}{16}\]
D. \[\dfrac{17}{125}\]
E. \[\dfrac{11}{6}\]

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Last updated date: 28th Feb 2024
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IVSAT 2024
Answer
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Hint:To find whether the decimal is terminating or non-terminating type, we first need to check if the decimal is finite or infinite type as terminating decimal means that the decimal value will come to stop after certain numbers of number and non-terminating means that the number after decimal is infinite and never stops.

Complete step by step solution:
Let us check for the first option, to check whether the number is terminating or non-terminating type we first solve the fraction in terms of prime number 2 and 5 in denominator as the first option is solved below:
\[\Rightarrow \dfrac{13}{5}=\dfrac{13}{{{5}^{1}}\times {{2}^{0}}}\]
To check if the number is terminating or non-terminating we check if the number follows the pattern of \[{{5}^{y}}\times {{2}^{x}}\] i.e. number is in terms of power of 2 and 5.
By seeing the denominator as \[{{5}^{1}}\times {{2}^{0}}\] which is equal to \[5\] and is in power of \[5\]
thereby making the fraction terminating type.
Now for the second option, to check whether the number is terminating or non-terminating type we first solve the fraction in terms of prime number 2 and 5, in denominator as the second option is solved below:
\[\Rightarrow \dfrac{2}{11}=\dfrac{2}{11\times {{5}^{0}}\times {{2}^{0}}}\]
To check if the number is terminating or non-terminating we check if the number follows the pattern of \[{{5}^{y}}\times {{2}^{x}}\] i.e. number is in terms of power of 2 and 5. By seeing the denominator as \[11\], we can say that the number is not in form of \[{{5}^{y}}\times {{2}^{x}}\]
Hence, a non- terminating type decimal.
For the third option, to check whether the number is terminating or non-terminating type we first solve the fraction in terms of prime number 2 and 5, as solved below:
\[\Rightarrow \dfrac{29}{16}=\dfrac{29}{16\times {{5}^{0}}\times {{2}^{0}}}\]
To check if the number is terminating or non-terminating we check if the number follows the pattern of \[{{5}^{y}}\times {{2}^{x}}\] i.e. number is in terms of power of 2 and 5.
By seeing the denominator as \16\, we can say that the number does follow the pattern
\[{{5}^{y}}\times {{2}^{x}}\] as \[{{5}^{0}}\times {{2}^{4}}\].
Therefore, the number is the terminating type.
For the fourth option, to check whether the number is terminating or non-terminating type we first solve the fraction in terms of prime number 2 and 5, as solved below:
\[\Rightarrow \dfrac{17}{125}=\dfrac{17}{{{5}^{3}}\times {{2}^{0}}}\]
To check if the number is terminating or non-terminating we check if the number follows the pattern of \[{{5}^{y}}\times {{2}^{x}}\] i.e. number is in terms of power of 2 and 5. With the denominator as \[125\], we can safely say that the number follows pattern \[{{5}^{y}}\times {{2}^{x}}\]as \[{{5}^{3}}\times {{2}^{0}}\]
Hence, the fraction is the terminating type.
For the fifth option, to check whether the number is terminating or non-terminating type we first solve the fraction in terms of prime number 2 and 5 as solved below:
\[\Rightarrow \dfrac{11}{6}=\dfrac{11}{6\times {{5}^{0}}\times {{2}^{0}}}\]
To check if the number is terminating or non-terminating we check if the number follows the pattern of \[{{5}^{y}}\times {{2}^{x}}\] i.e. number is in terms of power of 2 and 5.
By seeing the denominator as 6, we can say that the number is not in form of \[{{5}^{y}}\times {{2}^{x}}\]
Hence, a non-terminating type decimal.

Note: The term rational number means numbers which have finite numbers after decimal for example \[0.125,\text{ }0.25,\text{ }0.1\] etc. like these and when the number after decimal if infinite and in a pattern like \[0.625625625625625\ldots .\]or the value of pie these numbers are called irrational similarly terminating numbers are \[0.125,\text{ }0.25,\text{ }0.1\] and non-terminating are given as \[0.625625625625625\ldots .\]. Another method to check if the fraction is terminating or non-terminating is by dividing the numerator by denominator and checking if the remainder comes zero after a certain digit in the decimal if it becomes zero then terminating otherwise non-terminating.