Initial sugar = 1 part, water = 4 parts. Total = 5 parts. New sugar = 2 liters, water = 8 liters. Ratio = 2:8 = 1:4.

Initially, there is 1 part sugar and 4 parts water, totaling 5 parts. In 20 liters, there are 4 liters of sugar and 16 liters of water. After adding 2 liters of sugar, the new ratio is 6:16, which simplifies to 1:5.

In modular arithmetic, if a ≡ b (mod m), then (a - b) is divisible by m. Here, 7 - 3 = 4, which is divisible by 4.

'101' in binary is 5 in decimal. '110' in binary is 6 in decimal.

'101' in binary is 5, so '111' in binary is 7.

In base 5, '123' = 1*25 + 2*5 + 3 = 27.

'210' in base 3 is 2*3^2 + 1*3^1 + 0*3^0 = 18 + 3 + 0 = 21. Therefore, '102' in base 3 is 1*3^2 + 0*3^1 + 2*3^0 = 9 + 0 + 2 = 11.

'B' represents 11 in the same system where 'A' is 10.

'B' (18) is a multiple of 'A' (12) since 18 can be expressed as 12 multiplied by 1.5.

'A' can be 1, 2, 3, 4, 6, or 12 (factors of 12) and 'B' can be 3, 6, 9, 12, etc. The only combination that fits is A=3 and B=12, which gives us 36.

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.

'210' in base 4 is calculated as 2*4^2 + 1*4^1 + 0*4^0 = 32 + 4 + 0 = 36.

'123' in base 3 = 1*3^2 + 2*3^1 + 3*3^0 = 27. '321' in base 3 = 3*3^2 + 2*3^1 + 1*3^0 = 81.

The sum of the exterior angles of any polygon is 360 degrees. Therefore, the number of sides n = 360 / 30 = 12.

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. Solving for x gives x = 160. Therefore, there are 160 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 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.

The angle in radians is given by the formula θ = s/r, where s is the arc length. Thus, θ = 10/5 = 2 rad.

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%.

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.

To find the number of students who study only Mathematics, we subtract those who study both from those who study Mathematics: 20 - 10 = 10.

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

To find the number of students who study only History, we subtract those who study both from those who study History: 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.