Question

in a company number of employees increases by k℅ and also their working hours increases by k℅ due to this reason production of company increases by 50.0625℅ find the initial number of employees in the company if total 720 new employees joined the company

Answer 1

Consider, the initial number of employees is x and their working hours is y,

So, the total production hours = xy

After increasing both number of employees and working of hours of each employee by k%,

New employees = x + k% of x

New working hours for each employee = y + k% of y

Thus, the new total production hours = (x+k% of x)(y+k% of y)

Here, the production of company is increased by 50.0625%, so, the total hours of production must be increased by 50.0625%.

That is,

(x+k% of x)(y+k% of y) = (100+50.0625)% of xy

$(x+0.01kx)(y+0.01ky)=1.500625xy$

$xy+0.01kxy+0.01kxy+0.0001k^2xy=1.500625xy$

$1+0.02k+0.0001k^2=1.500625$

$100k^2 + 20000k + 1000000 = 1500625$

$100k^2+20000k+1000000-1500625=0$

$100k^2+20000k-500625=0$

$4k^2+800k-20025=0$

$4k^2+890k-90k-20025=0$

$2k(2k+445)-45(2k+445)=0$

$(2k-45)(2k+445)=0$

$k=\frac{45}{2}\text{ or }k=-\frac{445}{2}\text{ (not possible)}$

Employees who joined the company = 45/2% of x

$=\frac{45x}{200}$

$=\frac{9x}{40}$

According to the question,

$\frac{9x}{40}=720$

$x=\frac{720\times 40}{9}=3200$

Therefore, the initial number of employees would be 3200.