The table and the given chart show the evolution of the average price of three types $A, B$, and $C$ of beans in the food exchange during the first four months of a certain year. Of these three types, those that presented, respectively, the highest and the lowest percentage increase in price over this period are:
(a) $A$ and $B$;
(b) $A$ and $C$;
(c) $B$ and $C$;
(d) $C$ and $A$;
(e) $C$ and $B$.

![](https://cdn.mathpix.com/cropped/2024_05_01_30c9a294a58a6e4b8190g-023.jpg?height=680&width=462&top_left_y=257&top_left_x=503)

|  | jan. | feb. | mar. | abr. |
| :---: | :---: | :---: | :---: | :---: |
| $A$ | 65,67 | 83,33 | 96,67 | 103,33 |
| $B$ | 73,30 | 80,50 | 99,55 | 109,50 |
| $C$ | 64,50 | 71,57 | 89,55 | 100,00 |
The correct option is (a).

If $b$ denotes the final price and $a$ the initial price of a good, then the variation is $b-a$ and the percentage increase is

$$
\frac{b-a}{a}
$$

Note that intermediate values of the good do not alter the percentage increase in variation over a certain period. Using only the data from January and April from the given table, we obtain the percentage increases of

$$
\begin{array}{ll}
\text { bean A: } & \frac{103.33-65.67}{65.67}=\frac{37.66}{65.67}=0.57=57 \% \\
\text { bean B: } & \frac{109.50-73.30}{73.30}=\frac{36.20}{73.30}=0.49=49 \% \\
\text { bean C: } & \frac{100.00-64.50}{64.50}=\frac{35.50}{64.50}=0.55=55 \%
\end{array}
$$

Therefore, the largest percentage increase in price was for bean $A$ and the smallest was for bean $B$.