Question

An employer reduces the number of employees in the ratio $8:5$ and increases their wages in the ratio $7:9$. Therefore, the overall wages bill is
A: Increased in the ratio $45:56$
B: Decreased in the ratio $56:45$
C: Increased in the ratio $13:17$
D: Decreased in the ratio $72:35$

Answer 1

Hint: The problem can be simplified by assuming the actual values (not ratios) of the quantities using variables. This will simplify the problem from mere statements into mathematical equations. Also, the total wage bill will be equal to the product of the number of employees and their wages. Using this information, the required result can be calculated easily.

Complete step-by-step answer:
It is given that the number of employees are reduced in the ratio \[8:5\]. So,
$\dfrac{{number\,of\,employees\,before}}{{number\,of\,employees\,after}}$$ = $$\dfrac{8}{5}$
It is also given that their wages(per employee) were increased in the ratio $7:9$. So,
$\dfrac{{Initial\,wages}}{{New\,wages}}$$ = $$\dfrac{7}{9}$
To approach this problem in a mathematical form, we need to assume the actual values of the number of employees and their wages using particular variables. This will simplify the question. Further, these values can be used to achieve the desired result.
Let the initial number of employees $ = $$8x$
And the final number of employees $ = $$5x$
Also, let the total initial wage $ = $$7y$
  And the new wage $ = $$9y$
Remember that here, $x$ and $y$ are constants.
The Overall Wage bill will depend on both, the number of employees and the wage of each employee. It will be equal to the product of the number of employees and the wage per person.
Therefore, overall wages bill $ = $number of employees $ \times $ wage per person
So, initial overall wages bill $ = $$8x\, \times \,7y = 56xy$
      Final overall wages bill $ = $$5x\, \times \,9y = 45xy$
Now, we can use these parameters to calculate the required ratio. Also, note that as the ratio is a fraction, the variables will cut off and leave a pure solution in the end.
$\therefore $ Ratio $ = $$\dfrac{{Initial\,overall\,wage\,bill}}{{Final\,overall\,wage\,\,bill}}$
     Ratio $ = $$\dfrac{{56xy}}{{45xy}}$$ = $$\dfrac{{56}}{{45}}$
    Ratio$ \to $$56:45$
$\therefore $The overall wages bill is decreased in the ratio $56:45$

So, the correct answer is “Option B”.

Note: The student should not confuse himself between the ratio value and original value. Therefore constants x and y are used to simplify our purpose of calculation. Also, since our final solution required is also in ratio form, it should be clear that these variables used should be eliminated using some techniques, so that we are left with the pure solution in the end. In our case, the ratio form helped to eliminate these variables since they were located on both, the numerator and denominator positions.