Question

\[10\] men can make a wall in \[8\] days. How many men are required to complete the same work in half a day?
A.80
B.100
C.120
D.160

Answer 1

Hint : In order to find the number of men required to complete the wall in half a day, we must use the proportionality theorem as men and the number of days are to be considered in direct proportion. We will be assigning variables to the unknown value and then apply the proportion method and upon solving that, we obtain our required answer.


Complete step-by-step solution:
Now let us briefly discuss proportions. . Proportion is nothing but saying that two ratios are equal. Two ratios can be written in proportion in the following ways- \[\dfrac{a}{b}=\dfrac{c}{d}\] or \[a:b=c:d\]. From the second way of notation, the values on the extreme end are called as extremes and the inner ones as means. Proportions are of two types: direct proportions and indirect or inverse proportions. In the direct proportion, there would be a direct relation between the quantities. In the case of indirect proportion, there exists an indirect relation between the quantities.
Now let us find the number of men required to build the wall in half a day.
We are given that \[10\] man can make a wall in \[8\] days.
So let us say that \[{{M}_{1}}=10\] and \[{{D}_{1}}=8\]
Also we are given that \[{{D}_{2}}=\dfrac{1}{2}\]
Now, we are supposed to calculate \[{{M}_{2}}\], we get it as
We know that \[{{M}_{1}}{{D}_{1}}={{M}_{2}}{{D}_{2}}\]
Upon substituting the values we get,
\[\begin{align}
  & \Rightarrow 10\times 8={{M}_{2}}\times \dfrac{1}{2} \\
 & \Rightarrow {{M}_{2}}=10\times 8\times 2=160 \\
\end{align}\]
\[\therefore \] The number of men required to finish the wall in half a day is \[160\].
Hence, option C is the correct option.

Note: Before performing the operations, we must always check if the given quantities or comparisons are in direct or indirect proportion. The common mistake would be not expressing the ratios in simplest forms.
 We can also solve our above proportion in a horizontal method as shown below.
\[\begin{align}
  & \Rightarrow 10:8::x:\dfrac{1}{2} \\
 & \Rightarrow \dfrac{10}{8}=\dfrac{x}{\dfrac{1}{2}} \\
\end{align}\]
As we are solving it using a horizontal method, consider both the numerators and denominators and check for the multiplicand that gives us the product mentioned.
So upon considering the denominators, we can see that \[\dfrac{1}{2}\left( 16 \right)=8\]
So our required multiplicand is \[16\].
Now we will be multiplying with \[16\] to the numerator also. We get ,
\[10\times 16=160\].
\[\therefore x=160\] and the number of men required is \[160\].