In a class of 43 students at least every one of them offers physics or geography. If 27 of them offers physics and 31 offers geography, how many of them offer both physics and geography

Offer? I think you mean "... 43 students ... takes physics or geography ... "

The school offers the courses. The students take the courses.

27+31-x = 43

Time to review set unions and intersections and Venn diagrams...

To find the number of students who offer both physics and geography, you can use the principle of inclusion-exclusion. This principle states that to count the total number of elements in multiple sets, you need to consider the elements that belong to each set individually, subtract the elements that belong to their intersections, and add them back to get the correct count.

In this case, we know that there are 27 students offering physics and 31 offering geography. However, we need to find the number of students who offer both physics and geography.

To do this, add the number of students offering physics and the number offering geography: 27 + 31 = 58.

Now, we need to consider the total number of students in the class, which is given as 43.

To find the number of students who offer both physics and geography, subtract the total number of students in the class from the sum of the two subjects: 58 - 43 = 15.

Therefore, 15 students offer both physics and geography.