In Exercises 29–36, answer the given questions, which are related to percentages.

Percentages in Negotiations When the author was negotiating a contract for the faculty and administration at a college, a dean presented the argument that if faculty receive a 4% raise and administrators receive a 4% raise, that’s an 8% raise and it would never be approved. What’s wrong with that argument?

The raise for faculty is 4%.

The raise for administrators is 4%.

It is given that these raises imply an aggregate raise of 8%, which cannot be approved.

If two commodities have a certain level of increase from the base value, the aggregate increase is not the additive sum of the two percentages.

Let the original value for the faculty be x and for the administrators be y.

A 4% increase in x and y each implies:

\(\begin{aligned}{c}{x_{new}} = x + 4\% \;{\rm{of}}\;x\\ = x\left( {1 + \frac{4}{{100}}} \right)\\ = 1.04x\end{aligned}\)

\(\begin{aligned}{c}{y_{new}} = y + 4\% \;{\rm{of}}\;y\\ = y\left( {1 + \frac{4}{{100}}} \right)\\ = 1.04y\end{aligned}\)

The aggregate increase can be computed as:

\(\begin{aligned}{c}\frac{{{\rm{Aggregate}}\;{\rm{increase}}}}{{{\rm{Total}}\;{\rm{original}}\;{\rm{value}}}} \times 100 = \left( {\frac{{\left( {{x_{new}} - x} \right) + \left( {{y_{new}} - y} \right)}}{{x + y}}} \right) \times 100\\ = \left( {\frac{{\left( {1.04x - x} \right) + \left( {1.04y - y} \right)}}{{x + y}}} \right) \times 100\\ = \left( {\frac{{0.04x + 0.04y}}{{x + y}}} \right) \times 100\end{aligned}\)

\(\begin{aligned}{c} = \left( {0.04\frac{{\left( {x + y} \right)}}{{x + y}}} \right) \times 100\\ = 0.04 \times 100\\ = 4\% \end{aligned}\)

Thus, the given statement is wrong. The total raise is 4%, not 8%.