Question

4 mat weavers can weave \[4\] mats in \[4\] days. At the same rate, how many mats would be woven by \[8\] mat weavers in \[8\] days?
A. \[4\]
B. \[8\]
C. \[12\]
D. \[16\]

Answer 1

Hint: This question is based on the chain rule. According to it, the number of men weaving the mats is inversely proportional to the number of days taken by them to weave the mats. The second thing is that the number of men required to weave the mats is directly proportional to the number of mats weaved by the men.
Formula Used: \[\dfrac{{{M_1} \times {D_1}}}{{{N_1}}} = \dfrac{{{M_1} \times {D_2}}}{{{N_2}}}\]=Rate (constant acc. To the question)

Complete step-by-step solution:
As we can see that this question is based on chain rule, and according to it, we will discuss:
1)Direct Proportion
Two quantities are known to be directly proportional, if on the increasing (or decreasing) of the one, the other increases (or decreases) to the similar extent.
2)Indirect Proportion
Two quantities are known to be indirectly proportional, if by increasing any one, the other also decreases to the same extent and vice-versa
Deriving the formula as such, we get:
\[\begin{align}
   &\Rightarrow M \propto \dfrac{1}{D} \\
   &\Rightarrow M \times D = const. \\
   &\Rightarrow {M_1} \times {D_1} = {M_2} \times {D_{2\,\,\,\,\,\,\,\,}}\,\,\,\, - \,equation\,1 \\
\end{align} \]
  And, we get:
\[\begin{align}
   &\Rightarrow M \propto N \\
   &\Rightarrow \dfrac{M}{N} = const \\
   &\Rightarrow \dfrac{{{M_1}}}{{{N_1}}} = \dfrac{{{M_2}}}{{{N_2}}}\,\,\,\, - \,equation\,(2) \\
\end{align} \]
Combining both equation (1) and equation (2), we get
\[\dfrac{{{M_1} \times {D_1}}}{{{N_1}}} = \dfrac{{{M_1} \times {D_2}}}{{{N_2}}}\]
Now, let the number of mats weaved by \[8\] mat weavers in \[8\] days be \[x\]. So, now applying the given values in the formula, we get:
\[ \Rightarrow \dfrac{{4 \times 4}}{4} = \dfrac{{8 \times 8}}{{{N_2}}}\]
Here, we have to find out the value of \[{N_2}\]. By further solving the question, we get:
\[ \Rightarrow {N_2} = \dfrac{{64 \times 4}}{{16}}\] (Cross-multiplication)
\[ \Rightarrow {N_2} = 16\]
So, \[16\] days will be taken by \[8\] mat weavers to weave \[8\] mats.
Therefore, option D is correct.

Note: The above method was very easy but we have to always keep in mind that while solving the problems by chain rule, we have to always compare each and every item with the term that has to be found. By doing that, it becomes easier for us to get to the final answer.