# If the work done by $x - 1$ men in $x + 1$ days is to the work done by $x + 2$ men in $x - 1$ days in the ratio of 9:10. Find $x$.

Hint: Convert the given ratio to a linear equation and then solve it.

Complete step-by-step answer:

We have to find the value of $x$. The ratio of the work done by different numbers of men in different numbers of days is given.

Let the work done by $x - 1$ men in $x + 1$ days be p.

$\Rightarrow p = (x - 1)(x + 1)$

Let the work done by $x + 2$ men in $x - 1$ days be q.

$\Rightarrow q = (x + 2)(x - 1)$

According to the ratio given in question,

$\Rightarrow \dfrac{p}{q} = \dfrac{9}{{10}} \to (1)$

So, now putting the expressions of p and q in equation (1), we get,

$\Rightarrow \dfrac{{\left( {x - 1} \right)\left( {x + 1} \right)}}{{\left( {x + 2} \right)\left( {x - 1} \right)}} = \dfrac{9}{{10}} \to (2) \\$

Here $ x \ne 1 $ because that would give us an indeterminate form.

In real life, that would mean either the work has to be executed by 0 men or in 0 days, both of which are impossible.

Now, equation (2) becomes,

$\Rightarrow \dfrac{{\left( {x + 1} \right)}}{{\left( {x + 2} \right)}} = \dfrac{9}{{10}}$

Cross - multiplying, we get,

$\Rightarrow 10x + 10 = 9x + 18$

Solving the above equation, we get,

$\Rightarrow x = 8$

Note: To solve such problems one must have a knowledge of the ratios and solving linear equations in one variable and also one must make sure he doesn’t end up getting an undefined or indeterminate form while solving the equations.