Question

a farmer has an orchard that covers an area of 40 acres. he grows apples on 25 acres, peaches on 7 acres, nectarines on 5 acres, and plums on 3 acres. the fruit trees are equally distributed within the orchard. a tree is chosen at random. rounded to the nearest tenth of a percent, what is the theoretical probability that the tree is not within the acres of apple trees?
25.0%
37.5%
62.5%
75.4%

Explanation:

Step1: Find non - apple acres

The total area of the orchard is 40 acres. The area with apple trees is 25 acres. So the area with non - apple trees is the sum of the areas of peaches, nectarines, and plums. That is \(7 + 5+3=15\) acres? Wait, no. Wait, the total area is 40, and apple is 25. So non - apple area is \(40 - 25=15\)? Wait, no, wait: \(7 + 5+3 = 15\), and \(25+15 = 40\), yes. But wait, the probability that a tree is not in apple trees is the area of non - apple trees divided by total area. So non - apple area is \(40 - 25=15\)? Wait, no, \(7 + 5+3=15\), correct. Then the probability is \(\frac{40 - 25}{40}=\frac{15}{40}\)? Wait, no, wait, that can't be. Wait, no, wait: the total area is 40. Apple is 25. So non - apple is \(40-25 = 15\)? Wait, but \(7 + 5+3=15\), yes. But wait, the answer choices have 62.5%, which is \(\frac{25}{40}=0.625\), wait no. Wait, no, the probability of not being in apple trees is \(1-\) probability of being in apple trees. Probability of being in apple trees is \(\frac{25}{40}=0.625\), so probability of not being in apple trees is \(1 - 0.625 = 0.375\)? Wait, that's 37.5%? Wait, no, wait, I must have messed up. Wait, no: total area is 40. Apple is 25. So non - apple is \(40 - 25=15\)? Wait, \(15\div40 = 0.375=37.5\%\)? But wait, the answer choice 62.5% is \(\frac{25}{40}\), which is the probability of being in apple trees. Wait, no, the question is "not within the acres of apple trees". So the area not apple is \(40 - 25 = 15\)? Wait, no, \(7+5 + 3=15\), yes. Then \(15\div40=0.375 = 37.5\%\). Wait, but let's recalculate: \(40-25 = 15\), \(15\div40=\frac{15}{40}=\frac{3}{8}=0.375 = 37.5\%\). So the probability is 37.5%.

Wait, but wait, maybe I made a mistake. Wait, total area is 40. Apple is 25. So non - apple is \(40 - 25=15\). Then probability is \(15/40 = 0.375=37.5\%\). So the correct answer is 37.5%.

Step1: Calculate non - apple area

The total area of the orchard is \(A = 40\) acres. The area with apple trees is \(A_{apple}=25\) acres. The area with non - apple trees is \(A_{non - apple}=40 - 25 = 15\) acres.

Step2: Calculate the probability

The probability \(P\) that a randomly chosen tree is not in the apple - tree acres is the ratio of the non - apple area to the total area. So \(P=\frac{A_{non - apple}}{A}=\frac{15}{40}=0.375\). To convert to a percentage, we multiply by 100: \(0.375\times100 = 37.5\%\).

Answer:

37.5% (which corresponds to the option "37.5%")