Question

If given that: \[\dfrac{{13w}}{{1 - w}} = 13\]. Then, what will be the value of \[{\left( {2w} \right)^2}\]?
A. \[\dfrac{1}{4}\]
B. \[1\]
C. Does not exist
D. None of these

Answer 1

Hint: Here, we will have to first solve the given equation ‘\[\dfrac{{13w}}{{1 - w}} = 13\]’ algebraically, so as to get the value of the unknown ‘\[w\]’. As a result, substituting the obtained value of ‘\[w\]’ in the required expression that is \[{\left( {2w} \right)^2}\]’, the significant value is earned.

Complete step-by-step answer:
Since, we have given that
\[\dfrac{{13w}}{{1 - w}} = 13\]
As we have to find value of the given expression ‘\[{\left( {2w} \right)^2}\] ‘,
First of all solving the given equation so as to get the desire value of ‘\[w\]’,
Hence, considering the given equation that is ‘\[\dfrac{{13w}}{{1 - w}} = 13\]’
Therefore, solving the respective equation, we get
That is multiplying the whole equation by ‘\[1 - w\]’, we get
 \[ \Rightarrow \left( {1 - w} \right) \times \dfrac{{13w}}{{1 - w}} = 13 \times \left( {1 - w} \right)\]
As a result, cancelling (or dividing) the certain terms in left hand side, we get
\[ \Rightarrow 13w = 13 \times \left( {1 - w} \right)\]
Now, since multiplying the bracket by ‘\[13\]’ that is present in right hand side of the respective above equation that is also known as ‘distributive law or the property which seems to be \[a \times \left( {b + c} \right) = ab + ac\]’, we get
\[ \Rightarrow 13w = 13 - 13w\]
Now,
Simplifying the equation (i.e. by adding or subtracting), we get
\[ \Rightarrow 13w + 13w = 13\]
\[ \Rightarrow 26w = 13\]
Dividing the equation by ‘\[13\]’ that is taking ‘\[13\]’ to left hand side, we get
\[ \Rightarrow w = \dfrac{{13}}{{26}}\]
Again, simplifying the equation in to the simplest form that is the common divisible at its maximum, we get
\[ \Rightarrow w = \dfrac{{13 \times 1}}{{13 \times 2}}\]
\[ \Rightarrow w = \dfrac{1}{2}\]
Now,
Substituting ‘\[w = \dfrac{1}{2}\]’ in the required expression that is ‘\[{\left( {2w} \right)^2}\]’, we get
\[ \Rightarrow {\left( {2w} \right)^2} = {\left( {2 \times \dfrac{1}{2}} \right)^2}\]
Simplifying the equation as done above, we get
\[ \Rightarrow {\left( {2w} \right)^2} = {\left( 1 \right)^2}\]
\[ \Rightarrow {\left( {2w} \right)^2} = 1\]
So, the correct answer is “Option B”.

Note: Whenever we come up with this type of problem, always remember that when a fraction is obtained, then there is a need to reduce the fraction in its simplest form. As a result, it cannot be reduced further, if it is reduced further then it needs to be checked again. Also, while doing such calculations just try to mug up the algebraic property such as associative law, commutative law, distributive law, etc. Like in this question, we have used the distributive law to multiply ‘\[13\]’ to the entire bracket i.e. ‘\[\left( {1 - w} \right)\]’ respectively.