💡First, I need to determine the total number of students who excelled in at least one of the tests. Since there are 50 students in total and 4 did not excel in either test, the number of students who excelled in at least one test is 50 - 4 = 46.Next, I'll use the principle of inclusion-exclusion to find out how many students excelled in both tests. The total number of students who excelled in physical ability is 40, and those who excelled in intellectual ability is 31. Adding these gives 40 + 31 = 71.However, this sum counts the students who excelled in both tests twice. To find the actual number of students who excelled in both, I'll subtract the total number of students who excelled in at least one test from this sum: 71 - 46 = 25.Therefore, 25 students excelled in both the physical and intellectual tests.

💡First, I need to determine the original area of the rectangular garden. Let the original length be ( l ) and the original breadth be ( b ). The original area is calculated by multiplying the length and breadth:[A_{text{original}} = l times b]Next, the garden is extended. The new length is twice the original length, so:[text{New length} = 2l]The new breadth is three times the original breadth, so:[text{New breadth} = 3b]Now, I calculate the new area with the extended dimensions:[A_{text{new}} = (2l) times (3b) = 6lb]To find the ratio of the original area to the new area, I divide the original area by the new area:[frac{A_{text{original}}}{A_{text{new}}} = frac{lb}{6lb} = frac{1}{6}]Therefore, the ratio of the original area to the enlarged area is (boxed{frac{1}{6}}).

💡First, I need to determine which snowballs have radii greater than 3 inches. From the given radii of 2 inches, 4 inches, 5 inches, and 6 inches, the snowballs with radii greater than 3 inches are those with radii of 4 inches, 5 inches, and 6 inches.Next, I'll calculate the volume of each of these snowballs using the formula for the volume of a sphere, which is ( V = frac{4}{3}pi r^3 ).For the snowball with a radius of 4 inches:[V_4 = frac{4}{3}pi (4)^3 = frac{4}{3}pi times 64 = frac{256}{3}pi]For the snowball with a radius of 5 inches:[V_5 = frac{4}{3}pi (5)^3 = frac{4}{3}pi times 125 = frac{500}{3}pi]For the snowball with a radius of 6 inches:[V_6 = frac{4}{3}pi (6)^3 = frac{4}{3}pi times 216 = frac{864}{3}pi]Now, I'll add the volumes of these three snowballs to find the total volume:[text{Total Volume} = V_4 + V_5 + V_6 = frac{256}{3}pi + frac{500}{3}pi + frac{864}{3}pi = frac{256 + 500 + 864}{3}pi = frac{1620}{3}pi = 540pi]

💡First, I need to simplify the expression (dfrac{9!}{108}). I know that (9! = 362880). Dividing this by 108 gives (dfrac{362880}{108} = 3360).Next, I need to find the square root of 3360. To do this, I'll factorize 3360 into its prime factors. Breaking it down, I get:[3360 = 2^5 times 3 times 5 times 7]Now, I'll take the square root of each prime factor:[sqrt{3360} = sqrt{2^5 times 3 times 5 times 7} = 2^{2} times sqrt{2 times 3 times 5 times 7} = 4 times sqrt{210}]Therefore, the square root of (dfrac{9!}{108}) is (4sqrt{210}).

💡First, I need to determine the greatest common divisor (GCD) of the three coin totals: 1050, 1260, and 210. The GCD will give me the largest number of coins that can be evenly distributed into each box.I'll start by finding the GCD of the first two numbers, 1050 and 1260. Using the Euclidean algorithm:1260 divided by 1050 equals 1 with a remainder of 210.1050 divided by 210 equals 5 with a remainder of 0.So, the GCD of 1050 and 1260 is 210.Next, I'll find the GCD of this result (210) with the third number, 210:210 divided by 210 equals 1 with a remainder of 0.Therefore, the GCD of 210 and 210 is 210.Since the GCD of all three numbers is 210, the greatest number of coins that George could put in each box is 210.

💡First, I need to identify all the even integers between 50 and 150 whose digits sum to 10. I'll start by listing all numbers in this range where the sum of the digits equals 10. These numbers are 55, 64, 73, 82, 91, 100, 109, 118, 127, 136, and 145.Next, I'll filter out the even numbers from this list. The even numbers are 64, 82, 100, 118, 136, and 145, which gives me a total of 6 clever integers.Now, I need to determine which of these clever integers are divisible by 11. Checking each number, I find that 55, 100, and 118 are divisible by 11. This means there are 3 clever integers that meet the divisibility criteria.Finally, to find the fraction of clever integers that are divisible by 11, I'll divide the number of integers divisible by 11 by the total number of clever integers. This gives me 3 out of 6, which simplifies to 1/2.