Given a math problem with context and a question and 5 answer choices, the task is to provide the correct answer choice based on the problem. You must choose one of the given answer choices by letter: a, b, c, d, or e; anything else is invalid.

[Q]: Problem: for any positive integer n, the sum of the first n positive integers equals n ( n + 1 ) / 2. what is the sum of all the integers between 100 and 200 ?
Options: a. 4545, b. 5050, c. 6525, d. 4352, e. 4652
[A]: b

[Q]: Problem: kathleen can paint a room in 4 hours, and anthony can paint an identical room in 5 hours. how many hours would it take kathleen and anthony to paint both rooms if they work together at their respective rates ?
Options: a. 8 / 15, b. 40 / 9, c. 15 / 8, d. 9 / 4, e. 15 / 4
[A]: b

[Q]: Problem: the marks obtained by vijay and amith are in the ratio 4: 5 and those obtained by amith and abhishek in the ratio of 3: 4. the marks obtained by vijay and abhishek are in the ratio of ?
Options: a. 3: 4, b. 3: 5, c. 6: 5, d. 3: 6, e. 3: 8
[A]:
b