Question

toward the middle of the harvesting season, peaches for canning come in three types, early, late, and extra late, depending on the expected date of ripening. during a certain week, the data to the right were recorded at a fruit delivery station. complete parts (a) through (d) below. 29 trucks went out carrying early peaches; 66 carried late peaches; 51 carried extra late peaches; 22 carried early and late; 30 carried late and extra late; 4 carried early and extra late; 2 carried all three; 8 carried only figs (no peaches at all). (a) how many trucks carried only late variety peaches? 16 trucks (type a whole number.) (b) how many carried only extra late? trucks (type a whole number.)

Explanation:

Step1: Define sets

Let $E$ be early - peaches set, $L$ be late - peaches set, and $X$ be extra - late peaches set. We use the principle of inclusion - exclusion $|A\cup B\cup C|=|A|+|B|+|C|-|A\cap B|-|A\cap C|-|B\cap C|+|A\cap B\cap C|$.

Step2: Calculate number of only extra - late peaches

We know that $|X| = 51$, $|X\cap L|=30$, $|X\cap E| = 4$, and $|X\cap E\cap L|=2$.
The number of trucks that carried only extra - late peaches is $|X|-(|X\cap L|+|X\cap E|)+|X\cap E\cap L|$.
Substitute the values: $51-(30 + 4)+2$.
First, calculate inside the parentheses: $30 + 4=34$.
Then, $51-34 + 2=19$.

Answer:

19