A science class has 3 girls and 7 boys in the seventh grade and 5 girls and 5 boys in the eighth grade. The teacher randomly selects a seventh grader and an eighth grader from the class for a competition. What is the probability that the students she selects are both boys? Write your answer as a fraction in simplest form.

7th grade ... 7/10 boys

8th grade ... 1/2 boys

P(two boys selected) = 7/10 * 1/2

To find the probability that the teacher selects two boys, we need to find the probability of selecting a boy from the seventh grade and a boy from the eighth grade and then multiply these probabilities.

Step 1: Find the probability of selecting a boy from the seventh grade.
In the seventh grade, there are a total of 3 girls and 7 boys, so the probability of selecting a boy from the seventh grade is 7 boys / (3 girls + 7 boys) = 7/10.

Step 2: Find the probability of selecting a boy from the eighth grade.
In the eighth grade, there are a total of 5 girls and 5 boys, so the probability of selecting a boy from the eighth grade is 5 boys / (5 girls + 5 boys) = 5/10 = 1/2.

Step 3: Multiply the probabilities from Step 1 and Step 2.
The probability of selecting both a boy from the seventh grade and a boy from the eighth grade is (7/10) * (1/2) = 7/20.

Therefore, the probability that the students she selects are both boys is 7/20.

To find the probability of selecting two boys, we need to determine the total number of outcomes and the number of favorable outcomes.

Total Number of Outcomes:
The teacher can select any student from the class for the first selection, and there are a total of 10 students in the seventh grade and 10 students in the eighth grade. So, the total number of outcomes for the first selection is 10 + 10 = 20.

For the second selection, since one student has already been selected, there are 9 remaining students in the seventh grade and 9 remaining students in the eighth grade. Therefore, the total number of outcomes for the second selection is 9 + 9 = 18.

As these selections are independent events, we multiply the total number of outcomes for each selection to find the total number of outcomes for both selections. Therefore, the total number of outcomes for both selections is 20 * 18 = 360.

Number of Favorable Outcomes:
To have a favorable outcome, we need to select a boy for both the seventh grade and the eighth grade. The number of boys in the seventh grade is 7, and the number of boys in the eighth grade is 5.

So, the number of favorable outcomes is 7 * 5 = 35.

Now, we can find the probability by dividing the number of favorable outcomes by the total number of outcomes:

Probability = Number of Favorable Outcomes / Total Number of Outcomes
Probability = 35 / 360

To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 5:

Probability = 7 / 72

Therefore, the probability that the teacher selects two boys is 7/72.