Mathematics in Trades and Life

One of the purposes of graphing is to compare how data is changing. Here we will compare the way the raises for Vasya and Tien were calculated.

We know that Vasya’s salary rose by $5000 each year. We need to calculate how much Tien’s salary was raised each year. These calculations are in Table 3.2.2.

Notice that Tien’s raises were not the same amount each year. How did the raises change? Vasya’s raises were the same each year, so her salary changed by a constant amount. When the rate of change is constant we call the pattern linear. We will learn other patterns as well.

Note in Table 3.2.6 that the first differences are not the same. However, they increase in a suspiciously simple pattern. Checking the second differences (the differences of the 1st differences) we see a linear pattern. This turns out to be true for all quadratic data.

In Table 3.2.9 the differences are not the same nor do they show the pattern of quadratics. However, there is a pattern in the differences. Notice that the differences are exactly equal to the original data. This measn the rate is determined by the current scale. This is the pattern of data that varies exponentially.

In Table 3.2.11 note that the differences are not exactly equal to the original numbers. However, note that \(6=2 \cdot 3\) \(18=2 \cdot 9\text{,}\) and \(54=2 \cdot 27\text{.}\) The differences are double the original numbers. In general for exponential data the differences will be the original data scaled by some number.

We will first calculate the percent increase of salary each year for Tien and Vasya. Because salary numbers are exact, we will not use significant digits. Rather we will round to the nearest percent. This is in Table 3.2.14 and Table 3.2.15.

Notice that for Tien the percent increase is the same each year. It is 5%. For Vasya, the percent increase is not the same each year. How does the percent increase change for her?