Question

After how many decimal places, will the decimal expansion of the number
\[\dfrac{{53}}{{{2^2}{5^3}}}\] terminate?

Answer 1

Hint: We use the concept of converting a fraction into decimal form by converting the denominator in terms of\[{10^n}\], which helps us to determine the number of places where the decimal will be placed.
Multiply the required number in numerator and denominator so as to convert the denominator into a form similar to \[{10^n}\].
* Any number that has a denominator term like \[{10^n}\]can be converted into decimal form by placing the decimal at the nth place starting from the right side. So, the term in the numerator terminates after ‘n’ places of decimals.
* Laws of exponents give us \[{x^m} \times {x^n} = {x^{m + n}}\]and \[{x^m} \times {y^m} = {(x \times
y)^m}\]

Complete step-by-step answer:
We are given the fraction\[\dfrac{{53}}{{{2^2}{5^3}}}\].
Since the denominator has the term \[{2^2}{5^3}\]
We look at the value of the denominator when multiplied
\[ \Rightarrow {2^2}{5^3} = 2 \times 2 \times 5 \times 5 \times 5\]
\[ \Rightarrow {2^2}{5^3} = 500\]
So, to convert the denominator in \[{10^n}\]form we multiply the denominator by 2
But we know the fraction remains unchanged if we multiply both numerator and denominator by the same number. So, we will multiply both numerator and denominator by 2.
\[ \Rightarrow \dfrac{{53}}{{{2^2}{5^3}}} = \dfrac{{53}}{{{2^2}{5^3}}} \times \dfrac{2}{2}\]
Multiply the term in the numerator
\[ \Rightarrow \dfrac{{53}}{{{2^2}{5^3}}} = \dfrac{{106}}{{{2^2}{5^3} \times 2}}\]
Since we know the law of exponents states that when the bases are the same, the powers can be added.
\[ \Rightarrow \dfrac{{53}}{{{2^2}{5^3}}} = \dfrac{{106}}{{{2^{2 + 1}}{5^3}}}\]
\[ \Rightarrow \dfrac{{53}}{{{2^2}{5^3}}} = \dfrac{{106}}{{{2^3}{5^3}}}\]
Now from another law of exponents we know when powers are same, base can be multiplied
\[ \Rightarrow \dfrac{{53}}{{{2^2}{5^3}}} = \dfrac{{106}}{{{{(2 \times 5)}^3}}}\]
\[ \Rightarrow \dfrac{{53}}{{{2^2}{5^3}}} = \dfrac{{106}}{{{{10}^3}}}\]
Now we have the denominator as \[{10^3}\].
On comparing the denominator with the term \[{10^n}\] we get \[n = 3\].

So, we can say the fraction will terminate after three places of decimal.

Note: To check the answer we write\[\dfrac{{53}}{{{2^2}{5^3}}} = \dfrac{{106}}{{1000}}\]
Convert the fraction in RHS to decimal form
\[ \Rightarrow \dfrac{{53}}{{{2^2}{5^3}}} = 0.106\]
Since there are three digits after the decimal, the answer is 3 decimal places.
Students might try to divide the numerator by the term in the denominator by usual long
division process which will be very lengthy and will include many complex calculations as it is difficult to remember the table of the term in denominator.