you are about to take a test that contain questions of type A worth 4 points and of type B worth 7 points. You must answer at least 5 of type A and 3 of type B, but time restricts answering more than 10 of either type. In total, you can answer no more than 18. How many of each type of question can you answer, assuming all of your answers are correct, to maximize your score? What is the maximum score?

Answer 1

You want to answer as many of type B as possible because they're worth the most points.
10*7 = 70
That leaves eight questions of type A that you can answer.
8*4 = 32

70 + 32 = 102

Answer 2

let the number of questions answered from part A be x
let the number of questions answered from part B be y

we have 3 main conditions:
x ≤ 10
y ≤ 10 and
x+y ≤ 18

also we have x>5 and y>3

sketch this on a graph and any point with integer coordinates in the first quadrant that satisfies these conditions would be a possible "test".

Then Marks = 4x + 7y

this can be represented by a straight line with slope -4/7.
we want this line to be as far away from the origin as possible, (highest mark), but still be in contact with the region we have defined.

notice the slope of x+y=18 is -1 which is 45º to the left, while a slope of -4/7 is a line not yet 45º

so marks=4x+7y would still touch our region at the point (10,18), namely the intersection of the boundaries x+y=18 and y=10

so you should answer 8 questions form part A and 10 questions from part B
for a total score of 102