Question

\[18\] men can reap a field in \[35\] days. For reaping the same field in \[15\] days, how many men are required?

Answer 1

Hint: Multiply the number of days with the number of men needed to reap the field. Form equations for both the cases. Consider the number of men required to reap the field as x. Cross multiply the equations formed and solve for \[x\].

Complete step by step solution: Let the number of men required to reap the field be \[x\] men.
Let us form a table and enter the data that has been given to us;

Number of men	\[18\]	\[x\]
Number of days	\[35\]	\[15\]

Since the field needs to be reaped in a lesser number of time therefore we are going to need more men. Therefore here the number of days for reaping the field and the number of men needed are in inverse variation therefore the equation can be formed is as follows;
\[x = \frac{{18 \times 35}}{{15}}\]
(we have multiplied the number of men \[18\] and \[x\] with the number of days \[35\] and \[15\]; after which we cross multiplied the equations to solve for \[x\])
 \[x = \frac{{18 \times 35}}{{15}}\]
\[18\]is written as \[\left( {6 \times 3} \right)\] and \[35\] is written as \[\left( {7 \times 5} \right)\]as well as \[15\]is written as \[\left( {3 \times 5} \right)\]
\[x = \frac{{6 \times 3 \times 7 \times 5}}{{3 \times 5}}\]
\[3\]and \[5\] is cancel out from numerator and denominator we get \[x = 6 \times 7\]
The value of \[x\] is \[42\];

Therefore the number of men needed to reap the field will be \[42\] men.

Note: Inverse variation means; while one increases the other decreases and vise versa. Here the number of days are reducing to reap the field and so the number of men needed are in greater numbers. In case of inverse variation; multiplication takes place between different variables.