How to calculate how much the company saved after % increase and decrease in two departments

Question

Company X has increased the hourly rate of the accounting department by 40% from the original rate of $60 (the total hours used 4450).

AND the company decreased the hourly rate of the logistics department by 20% from the original rate of $100 (total hours used 10100). How much did the company save?

the answer was -95,200 and according to the manual, it was saving (even though the answer has a negative sign).

The way I worked it out is as follows. (1.40*$60* 4450) - (0.80 *$100*10100) = -434200. The manual used this instead 40%*60*4450 - 20%*100*10100 = -95,200 which doesn't make sense to me :/

Answer 1

There's an increase in the cost for the accounting department, and a decrease in the cost for the logistics department. If the decrease is larger in magnitude than the increase, then the company saved money.

The way you solved it includes the original amounts that the company was spending on each department, which is incorrect. You only should consider the change in the two amounts, which is what the solution gives you.

Answer 2

Without change of each rate we have that the total expense $S$ is the sum of the expense of the accounting department $S_a$ and the expense of the logistics department $S_l$ $$ S=S_a+S_l=60\times 4450+100\times 10100 $$ With change we have that $S_a$ changes of $\Delta S_a$ and $S_l$ changes of $\Delta S_l$ $$ \begin{align} S'=S+\Delta S&=(S_a+\Delta S_a)+(S_l+\Delta S_l)\\ &=(S_a+S_l)+(\Delta S_a+\Delta S_l)\\ &=S+(\Delta S_a+\Delta S_l)\\ &=60\times (1+0.4)\times 4450+100\times (1-0.2)\times 10100\\ &=(\underbrace{60\times 4450}_{S_a}+\underbrace{60\times 0.4\times 4450}_{\Delta S_a})+(\underbrace{100\times 10100}_{S_l}+\underbrace{100\times (-0.2)\times 10100}_{\Delta S_l})\\ &=\underbrace{(60\times 4450+100\times 10100)}_{S=S_a+s_l}+\underbrace{(60\times 0.4\times 4450-100\times 0.2\times 10100)}_{\Delta S=\Delta S_a+\Delta S_l} \end{align} $$ and the saving is $$ S'-S=\Delta S=\Delta S_a+\Delta S_l=60\times 0.4\times 4450-100\times 0.2\times 10100 $$ It is a saving because $\Delta S<0$ and $S' <S $ that is the final expense is less than the initial expense.