Question

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?

Short answer

The aggregate raise is only 4%, not 8%.

Step-by-step solution

Given information

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.

Percentage operations

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.

Computing the raised value

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

A 4% increase in x and y each implies:

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

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

Computing the percentage aggregate increase

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%.