If the original number is x, then x + 12 is a multiple of 5. This implies that x must be one more than a multiple of 5.

The sum of the exterior angles of any polygon is 360 degrees. If each exterior angle is 45 degrees, the number of sides is 360 / 45 = 8.

If the ratio of men to women is 3:2, then for every 3 men, there are 2 women. If there are 120 men, we can set up the proportion: 3/2 = 120/x. Solving for x gives x = 80. Therefore, there are 80 women.

If the ratio of men to women is 3:4, then for every 3 men, there are 4 women. If there are 120 men, we can set up the proportion: 3/4 = 120/x. Cross-multiplying gives us 3x = 480, so x = 160. Therefore, there are 160 women.

The number of games played is given by n(n-1)/2. For 12 players, it is 12(12-1)/2 = 66 games.

The angle subtended at the circumference is half the angle subtended at the center. Therefore, the angle at the circumference is 80/2 = 40 degrees.

The angle subtended at the circumference is half of that at the center, so it is 30 degrees.

The angle subtended at the circumference is half of that at the center, so it is 30 degrees.

If the radius increases by 50%, the new radius is 1.5r. The area becomes π(1.5r)² = 2.25πr², which is an increase of 125%.

If A is next to B and C is opposite A, the only person who can be next to C is D.

Since A is next to B and C is opposite A, D must be next to C.

Since A is next to B and C is opposite A, the arrangement shows that D must be next to C.

M is between N and O, which means N is next to O. P is opposite M, so the only person next to O is N.

The arrangement is Y-X-Z. Since W is opposite Y, the only person next to W would be Z.

Since C is opposite A, and A is next to B, the only person who can be next to C is D.

The arrangement is F-G-H. Therefore, H is sitting next to F.

Since A is to the left of B, and C is opposite A, the arrangement can be visualized. If we place A at a position, B will be to the right of A, and C will be directly across from A. Therefore, D will be to the right of C.

Since A is to the left of B, B is to the right of A. C is opposite A, so the arrangement is A-B-D-C-E-F-G-H. Therefore, D is to the immediate right of B.

The arrangement is A-B-C. Therefore, E is sitting next to D.

Since L is to the right of K and M is to the left of K, N must be sitting opposite L.

The arrangement is M-N-O. If we assume a circular arrangement, the person directly opposite N would be S.

Given that X is to the left of Y and Z is to the right of Y, the only person who can be directly opposite X is A.

Since Z is opposite X, and X is next to Y, the only person who can be next to Z is W, as Y is already next to X.

To find the number of students who study only Mathematics, we use the formula: Only Mathematics = Total Mathematics - Both subjects. Thus, 18 - 10 = 8.

Let the number of boys be x and the number of girls be 30 - x. The total score of boys is 80x and that of girls is 70(30 - x). The overall average is given by (80x + 70(30 - x)) / 30 = 75. Solving this gives x = 15.

The expected number of students selected is calculated as 30 * 0.4 = 12.

The number of students who study only Mathematics is |Mathematics| - |Both| = 25 - 10 = 15.

To find the number of students who study only Hindi, we subtract those who study both from those who study Hindi: 25 - 10 = 15.

The number of students studying only English is 20 - 10 = 10, and only Mathematics is 25 - 10 = 15. Thus, total studying only one subject is 10 + 15 = 25.

The number of students who study only Science is 25 - 10 = 15.