Question

Write down the decimal expansions of the rational numbers which have terminating decimal expansions
 i. \[\dfrac{{13}}{{3125}}\]
ii. \[\dfrac{{17}}{8}\]
iii.\[\dfrac{{15}}{{1600}}\]
iv.\[\dfrac{{23}}{{{2^3}{5^2}}}\]
v. \[\dfrac{6}{{15}}\]
vi.\[\dfrac{{35}}{{50}}\]

Answer 1

Hint: Terminating decimal numbers are the numbers that contain a finite number of digits after the decimal point. A number 2.56 can be a terminating decimal if it is represented as 2.5600000000….. where 0 is terminating.
In this question, expand the rational numbers in terminating form, then check for terminating numbers form \[{2^m} \times {5^n}\], and if the condition satisfies, then find the terminating number.


Complete step by step solution:
Factorize the denominator of the given rational numbers in the form \[{2^m} \times {5^n}\]to check the terminating numbers,
(i) \[\dfrac{{13}}{{3125}}\]
By factorising the denominator, we can write \[3125 = 5 \times 5 \times 5 \times 5 \times 5 = {5^5}\]
Multiply the factor by \[{2^0}\]which is equal to\[{2^0} = 1\]bring the factors in terminating form\[{2^0} \times {5^5}\]; hence we can say the rational number is in terminating form.
Now multiply the numerator and the denominator with\[{2^5}\],
\[
  \dfrac{{13}}{{3125}} = \dfrac{{13 \times {2^5}}}{{{2^0} \times {5^5} \times {2^5}}} \\
   = \dfrac{{13 \times 32}}{{1 \times {{\left( {2 \times 5} \right)}^5}}} \\
   = \dfrac{{416}}{{{{10}^5}}} \\
   = \dfrac{{416}}{{100000}} \\
   = 0.00416 \\
 \]
Hence the terminating decimal number is \[ = 0.00416\]
(ii)\[\dfrac{{17}}{8}\]
By factorizing the denominator, we can write \[8 = 2 \times 2 \times 2 = {2^3}\]
Multiply the factor by \[{5^0}\]which is equal to\[{5^0} = 1\]bring the factors in terminating form\[{2^m} \times {5^n}\] ; hence we can say the rational number is in terminating form.
\[\dfrac{{17}}{8} = \dfrac{{17}}{{{2^3} \times {5^0}}}\]
Now multiply the numerator and the denominator with\[{5^3}\],
\[
  \dfrac{{17}}{8} = \dfrac{{17 \times {5^3}}}{{{2^3} \times {5^0} \times {5^3}}} \\
   = \dfrac{{17 \times 125}}{{1 \times {{\left( {2 \times 5} \right)}^3}}} \\
   = \dfrac{{2125}}{{{{10}^3}}} \\
   = \dfrac{{2125}}{{1000}} \\
   = 2.125 \\
 \]
Hence the terminating decimal number is \[ = 2.215\]
(iii) \[\dfrac{{15}}{{1600}}\]
We can reduce the rational number as
\[\dfrac{{15}}{{1600}} = \dfrac{3}{{320}}\]
Now by factorizing the denominator, we can write \[320 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 5 = {2^6} \times {5^1}\]
Hence we can say the rational number is in terminating as the factor of the denominator is in\[{2^m} \times {5^n}\]form.
Now multiply the numerator and the denominator with\[{5^5}\],
\[
  \dfrac{3}{{320}} = \dfrac{{3 \times {5^5}}}{{{2^6} \times {5^1} \times {5^5}}} \\
   = \dfrac{{3 \times 3125}}{{{2^6} \times {{\left( 5 \right)}^{1 + 5}}}} \\
   = \dfrac{{9375}}{{{2^6} \times {5^6}}} \\
   = \dfrac{{9375}}{{{{\left( {2 \times 6} \right)}^6}}} \\
   = \dfrac{{9375}}{{1000000}} \\
   = 0.009375 \\
 \]
Hence the terminating decimal number is \[ = 0.009375\]
(iv)\[\dfrac{{23}}{{{2^3}{5^2}}}\]
The rational number is in terminating as the factor of the denominator is in\[{2^m} \times {5^n}\]the form
Now multiply the numerator and the denominator with\[{5^1}\],
\[
  \dfrac{{23}}{{{2^3}{5^2}}} = \dfrac{{23}}{{{2^3} \times {5^2} \times {5^1}}} \\
   = \dfrac{{23}}{{{2^3} \times {5^{2 + 1}}}} \\
   = \dfrac{{23}}{{{{\left( {2 \times 5} \right)}^3}}} \\
   = \dfrac{{23}}{{{{10}^3}}} \\
   = \dfrac{{23}}{{1000}} \\
   = 0.023 \\
 \]
Hence the terminating decimal number is \[ = 0.023\]
(v) \[\dfrac{6}{{15}}\]
We can reduce the rational number as
\[\dfrac{6}{{15}} = \dfrac{2}{5}\]
This is in the form
\[\dfrac{2}{5} = \dfrac{2}{{{2^0} \times {5^1}}}\]
The rational number is in terminating as the factor of the denominator is in\[{2^m} \times {5^n}\]the form
Now multiply the numerator and the denominator with\[{2^1}\]
\[
  \dfrac{2}{5} = \dfrac{{2 \times {2^1}}}{{{2^0} \times {2^1} \times {5^1}}} \\
   = \dfrac{4}{{\left( {2 \times 5} \right)}} \\
   = \dfrac{4}{{10}} \\
   = 0.4 \\
 \]
Hence the terminating decimal number is \[ = 0.4\]
(vi)\[\dfrac{{35}}{{50}}\]
We can reduce the rational number as
\[\dfrac{{35}}{{50}} = \dfrac{7}{{10}} = 0.7\]
Hence the terminating decimal number is \[ = 0.7\]


Note: Any rational number can be written as either a terminating or repeating decimal by dividing the numerator by denominator, and if the remainder is 0, then the number is a terminating decimal. Every fraction number can be either terminating or non-termination (or, repeating).