Answer
424.5k+ views
Hint: We will use the fact that every rational number with denominator of the form ${2^m} \times {5^n}$ will always be terminating. Then, we will try to make powers of 10 in the denominator by multiplying and dividing with some numbers, so that we can get a decimal easily and know its place.
Complete step-by-step answer:
Let us first understand the meaning of terminating rational numbers:
A terminating decimal is a decimal with a certain number of digits to the right of the decimal point. Examples include 3.2, 4.075, and -300.12002. All of these are rational. In simpler words, it means that if a number ends at some point after decimal, it is terminating.
Now, coming back to the question:
We have: $\dfrac{{43}}{{{2^4} \times {5^3}}}$. It has ${2^4} \times {5^3}$ in its denominator.
We know that every rational number with denominator of the form ${2^m} \times {5^n}$ will always be terminating.
Hence, $\dfrac{{43}}{{{2^4} \times {5^3}}}$ is terminating as well.
Now, consider the denominator for once.
Denominator = \[{2^4} \times {5^3} = ({2^3} \times {5^3}) \times 2 = {10^3} \times 2\].
Hence, the number given to us becomes $\dfrac{{43}}{{{2^4} \times {5^3}}} = \dfrac{{43}}{{2 \times {{10}^3}}}$.
We can multiply and divide the number by 5, it would not make any difference because we are eventually multiplying the number by 1 because of multiplying and dividing by the same number.
So, $\dfrac{{43}}{{{2^4} \times {5^3}}} = \dfrac{{43}}{{2 \times {{10}^3}}} \times \dfrac{5}{5}$.
Simplifying it, we will have with us:-
$\dfrac{{43}}{{{2^4} \times {5^3}}} = \dfrac{{43 \times 5}}{{{{10}^3} \times 2 \times 5}} = \dfrac{{215}}{{{{10}^3} \times 10}} = \dfrac{{215}}{{{{10}^4}}}$
Hence, $\dfrac{{43}}{{{2^4} \times {5^3}}} = \dfrac{{215}}{{10000}} = 0.0115$.
Hence, it terminates after 4 digits.
Hence, the answer is 4 digits.
Note: The students might make the mistake of just dividing the number as per their requirements but forgetting to multiply it as well to nullify any changes in the number.
The students also might forget to check in the start whether the number given to them is actually terminating or it is just a trick question. So, remember to check the conditions for that as well.
Always remember that all rational numbers may not be terminating. For example: $\dfrac{1}{3}$ and our most known number $\pi $ that is equal to $\dfrac{{22}}{7}$.
Complete step-by-step answer:
Let us first understand the meaning of terminating rational numbers:
A terminating decimal is a decimal with a certain number of digits to the right of the decimal point. Examples include 3.2, 4.075, and -300.12002. All of these are rational. In simpler words, it means that if a number ends at some point after decimal, it is terminating.
Now, coming back to the question:
We have: $\dfrac{{43}}{{{2^4} \times {5^3}}}$. It has ${2^4} \times {5^3}$ in its denominator.
We know that every rational number with denominator of the form ${2^m} \times {5^n}$ will always be terminating.
Hence, $\dfrac{{43}}{{{2^4} \times {5^3}}}$ is terminating as well.
Now, consider the denominator for once.
Denominator = \[{2^4} \times {5^3} = ({2^3} \times {5^3}) \times 2 = {10^3} \times 2\].
Hence, the number given to us becomes $\dfrac{{43}}{{{2^4} \times {5^3}}} = \dfrac{{43}}{{2 \times {{10}^3}}}$.
We can multiply and divide the number by 5, it would not make any difference because we are eventually multiplying the number by 1 because of multiplying and dividing by the same number.
So, $\dfrac{{43}}{{{2^4} \times {5^3}}} = \dfrac{{43}}{{2 \times {{10}^3}}} \times \dfrac{5}{5}$.
Simplifying it, we will have with us:-
$\dfrac{{43}}{{{2^4} \times {5^3}}} = \dfrac{{43 \times 5}}{{{{10}^3} \times 2 \times 5}} = \dfrac{{215}}{{{{10}^3} \times 10}} = \dfrac{{215}}{{{{10}^4}}}$
Hence, $\dfrac{{43}}{{{2^4} \times {5^3}}} = \dfrac{{215}}{{10000}} = 0.0115$.
Hence, it terminates after 4 digits.
Hence, the answer is 4 digits.
Note: The students might make the mistake of just dividing the number as per their requirements but forgetting to multiply it as well to nullify any changes in the number.
The students also might forget to check in the start whether the number given to them is actually terminating or it is just a trick question. So, remember to check the conditions for that as well.
Always remember that all rational numbers may not be terminating. For example: $\dfrac{1}{3}$ and our most known number $\pi $ that is equal to $\dfrac{{22}}{7}$.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)