A school has four sections A, B, C, D of Class IX students.



Which section exhibits the highest pass percentage in either of the two examinations?

First, let's understand what "pass percentage in either of the two examinations" means. A student passes in either of the examinations if they passed the half-yearly exam, or the annual exam, or both. The only students who did not pass either exam are those who "failed in both exams".
Therefore, for each section:

$$\text{Total Students}=\text{Failed in both}+\text{Failed in HY but passed in Annual}+\text{Passed in HY but failed in Annual}+\text{Passed in both}$$

$$\text{Total Passed Students (in either exam)}=\text{Failed in HY but passed in Annual}+\text{Passed in HY but failed in Annual}+\text{Passed in both}$$

$$\text{Alternatively: Total Passed Students}=\text{Total Students}-\text{Students failed in both exams}$$

Let's compute the totals and pass percentages for each section:

* Section A:

$$\text{Total Students}=28+14+6+64=112$$

$$\text{Passed Students}=14+6+64=84(\text{or }112-28=84)$$

$$\text{Pass Percentage}=\frac{84}{112}\times 100\%=75.00\%$$

* Section B:

$$\text{Total Students}=23+12+17+55=107$$

$$\text{Passed Students}=12+17+55=84(\text{or }107-23=84)$$

$$\text{Pass Percentage}=\frac{84}{107}\times 100\%\approx 78.50\%$$

* Section C:

$$\text{Total Students}=17+8+9+46=80$$

$$\text{Passed Students}=8+9+46=63(\text{or }80-17=63)$$

$$\text{Pass Percentage}=\frac{63}{80}\times 100\%=78.75\%$$

* Section D:

$$\text{Total Students}=27+13+15+76=131$$

$$\text{Passed Students}=13+15+76=104(\text{or }131-27=104)$$

$$\text{Pass Percentage}=\frac{104}{131}\times 100\%\approx 79.39\%$$

Comparing the pass percentages:

$$75.00\%<78.50\%<78.75\%<79.39\%$$

Section D has the highest pass percentage ($79.39\%$).