the answer given to the question that i just asked (in a camp there was provision for 12 men for 15 days . for how many men will the provision last for 30 days.)is 24 men while the answer given in my text book is 6 men.

Dear student,

Please find below the solution to the asked query:

$$\begin{gathered} \mathbf{First}\mathbf{ }\mathbf{of}\mathbf{ }\mathbf{all}\mathbf{ }\mathbf{get}\mathbf{ }\mathbf{a}\mathbf{ }\mathbf{brief}\mathbf{ }\mathbf{idea}\mathbf{ }\mathbf{about}\mathbf{ }\mathbf{Direct}\mathbf{ }\mathbf{and}\mathbf{ }\mathbf{Inverse}\mathbf{ }\mathbf{Proportions}: \\ \mathbf{ }\mathbf{Direct}\mathbf{ }\mathbf{ }\mathbf{Proportions}:\mathrm{Two} \mathrm{values} \mathrm{x} \mathrm{and} \mathrm{y} \mathrm{are} \mathrm{directly} \mathrm{proportional} \mathrm{to} \mathrm{each} \mathrm{other} \mathrm{when} \mathrm{the} \mathrm{ratio} \mathrm{x}:\mathrm{y} \mathrm{or} \frac{\mathrm{x}}{\mathrm{y}} \mathrm{is} \mathrm{a} \mathrm{constant} \\ \mathrm{that} \mathrm{is} \mathrm{it} \mathrm{always} \mathrm{remains} \mathrm{the} \mathrm{same}. \\ \mathrm{This} \mathrm{further} \mathrm{means} \mathrm{that} \mathrm{x} \mathrm{and} \mathrm{y} \mathrm{will} \mathrm{either} \mathrm{increase} \mathrm{together} \mathrm{or} \mathrm{decrease} \mathrm{together} \mathrm{by} \mathrm{an} \mathrm{amount} \\ \mathrm{that} \mathrm{would} \mathrm{not} \mathrm{change} \mathrm{the} \mathrm{ratio}. \\ \mathbf{For}\mathbf{ }\mathbf{example}: \mathrm{Earning} \mathrm{is} \mathrm{directly} \mathrm{proportional} \mathrm{to} \mathrm{working} \mathrm{hours}. \\ \mathrm{If} \mathrm{you} \mathrm{work} \mathrm{for} 2 \mathrm{hours}, \mathrm{you} \mathrm{get} \mathrm{Rs} 200 \\ \mathrm{If} \mathrm{you} \mathrm{work} \mathrm{for} 3 \mathrm{hours}, \mathrm{you} \mathrm{get} \mathrm{Rs} 300 \\ \mathbf{Indirect}\mathbf{ }\mathbf{Proportions}:\mathrm{Two} \mathrm{values} \mathrm{x} \mathrm{and} \mathrm{y} \mathrm{are} \mathrm{indirectly} \mathrm{proportional} \mathrm{to} \mathrm{each} \mathrm{other} \mathrm{when} \mathrm{the} \mathrm{product} \mathrm{xy} \mathrm{is} \mathrm{a} \mathrm{constant} \\ \mathrm{that} \mathrm{is} \mathrm{it} \mathrm{always} \mathrm{remains} \mathrm{the} \mathrm{same}. \\ \mathrm{This} \mathrm{further} \mathrm{means} \mathrm{that} \mathrm{when} \mathrm{x} \mathrm{increases}, \mathrm{then} \mathrm{y} \mathrm{decreases} \mathrm{and} \mathrm{when} \mathrm{x} \mathrm{decreases}, \mathrm{the} \mathrm{value} \\ \mathrm{of} \mathrm{y} \mathrm{increases}. \mathrm{The} \mathrm{change} \mathrm{in} \mathrm{the} \mathrm{amount} \mathrm{of} \mathrm{second} \mathrm{variable} \mathrm{is} \mathrm{such} \mathrm{that} \mathrm{the} \mathrm{product} \mathrm{xy} \mathrm{remains} \\ \mathrm{the} \mathrm{same}. \\ \mathbf{For}\mathbf{ }\mathbf{example}: \mathrm{As} \mathrm{speed} \mathrm{goes} \mathrm{up}, \mathrm{the} \mathrm{travel} \mathrm{time} \mathrm{goes} \mathrm{down} \mathrm{and} \mathrm{as} \mathrm{speed} \mathrm{goes} \mathrm{down}, \mathrm{travel} \end{gathered}$$

$$\begin{gathered} \mathrm{Here} \mathrm{with} \mathrm{the} \mathrm{increase} \mathrm{in} \mathrm{the} \mathrm{number} \mathrm{of} \mathrm{men}, \mathrm{the} \mathrm{procision} \mathrm{for} \mathrm{number} \mathrm{of} \\ \mathrm{days} \mathrm{decrease}. \\ \mathrm{This} \mathrm{means} \mathrm{that} \mathrm{the} \mathrm{given} \mathrm{case} \mathrm{is} \mathrm{indirect} \mathrm{proportion}. \\ \mathrm{Let} \mathrm{p} \mathrm{be} \mathrm{the} \mathrm{number} \mathrm{of} \mathrm{men} \mathrm{for} \mathrm{which} \mathrm{the} \mathrm{procision} \mathrm{last} \mathrm{for} 30 \mathrm{days}. \\ \mathrm{Then} \mathrm{we} \mathrm{get} \\ \mathrm{p}\times 30 =12\times 15 \\ \Rightarrow \mathrm{p}=\frac{12\times \cancel{15}}{{\cancel{30}}_2} \\ \Rightarrow \mathrm{p}=\frac{{\displaystyle \stackrel{6}{\cancel{12}}}}{\cancel{2}} \\ \Rightarrow \mathrm{p}=6 \\ \mathbf{So}\mathbf{ }\mathbf{the}\mathbf{ }\mathbf{required}\mathbf{ }\mathbf{number}\mathbf{ }\mathbf{of}\mathbf{ }\mathbf{mens}\mathbf{ }\mathbf{are}\mathbf{ }\mathbf{6} \end{gathered}$$

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