Question

If \[\sqrt{3}=1.732\] then the approximate value of \[\dfrac{1}{\sqrt{3}}\] is
(a) 0.617
(b) 0.313
(c) 0.577
(d) 0.173

Answer 1

Hint: We solve this problem by using the simple division method of dividing the two numbers.
We divide the number 1 with 1.732 to get the value of \[\dfrac{1}{\sqrt{3}}\]
Here, the divisor is greater than the dividend. We use the condition that we use decimal in quotient and we increase the dividend by 10 times to use the proper division.

Complete step by step answer:
We are given that \[\sqrt{3}=1.732\]
We are asked to find the value of \[\dfrac{1}{\sqrt{3}}\]
Let us assume that the required value as
\[\begin{align}
  & \Rightarrow x=\dfrac{1}{\sqrt{3}} \\
 & \Rightarrow x=\dfrac{1}{1.732} \\
\end{align}\]
We know that the condition that if a number has \[n\] digits after the decimal point then we can remove the decimal by dividing the number with \[{{10}^{n}}\]

By using the this condition in the required value then we get
\[\begin{align}
  & \Rightarrow x=\dfrac{1}{\left( \dfrac{1732}{1000} \right)} \\
 & \Rightarrow x=\dfrac{1000}{1732} \\
\end{align}\]
Now, let us use the normal division method for the above result as follows
\[1732\overline{\left){1000}\right.}\]
Here, we can see that the divisor is greater than the dividend
We know that we can add decimal point in the quotient as that we can raise the dividend by 10 times.
By using the above condition to the required division then we get
\[1732\overset{0.}{\overline{\left){10000}\right.}}\]
Now, let us multiply the divisor with \[p\] such that \[1732\times p\le 10000\] and subtract the result from 10000 to get the remainder then we get
\[1732\overset{0.5}{\overline{\left){\begin{align}
  & 10000 \\
 & -\left( 8660 \right) \\
 & =1340 \\
\end{align}}\right.}}\]
Here, we can see that the remainder is again smaller than divisor.
So, we can raise it by 10 times after using the decimal in the quotient.
Here, we can see that we already used the decimal in quotient so that we can raise the remainder by 10 times then we get
\[1732\overset{0.5}{\overline{\left){13400}\right.}}\]
Now, let us multiply the divisor with \[p\] such that \[1732\times p\le 13400\] and subtract the result from 13400 to get the remainder then we get
\[1732\overset{0.57}{\overline{\left){\begin{align}
  & 13400 \\
 & -\left( 12124 \right) \\
 & =1276 \\
\end{align}}\right.}}\]
Now, again using the same division process for the remainder 1276 then we get
\[1732\overset{0.577}{\overline{\left){\begin{align}
  & 12760 \\
 & -\left( 12124 \right) \\
 & =636 \\
\end{align}}\right.}}\]
Here, we can see that this division has no end so that we can stop the division here as we asked to find the approximate value.
Therefore, we can conclude that the approximate value of required expression as
\[\Rightarrow x=\dfrac{1}{\sqrt{3}}=0.577\]
So, option (c) is correct answer.



Note:
Students may do mistakes in raising the dividend by 10 times in the second division.
We know that we can use the decimal point in the quotient so that we can raise the dividend by 10 times.
For the second time, there is no need to use the other decimal point that is we can raise the dividend by 10 times directly that is
\[1732\overset{0.5}{\overline{\left){13400}\right.}}\]
But students may use the second decimal point as
\[1732\overset{0.5.}{\overline{\left){13400}\right.}}\]
This is a very wrong representation because a number can never have two decimal points.