💡First, I need to identify all two-digit numbers where the product of their digits is a perfect square. A two-digit number can be represented as (10a + b), where (a) is the tens digit (ranging from 1 to 9) and (b) is the units digit (ranging from 0 to 9).Next, I'll list the perfect squares that are possible products of two digits. The maximum product of two digits is (9 times 9 = 81), so the relevant perfect squares are (1, 4, 9, 16, 25, 36, 49, 64,) and (81).For each perfect square, I'll find all pairs of digits (a) and (b) such that (a times b) equals the perfect square:- **1:** (1 times 1 = 1)- **4:** (1 times 4 = 4), (2 times 2 = 4)- **9:** (1 times 9 = 9), (3 times 3 = 9)- **16:** (2 times 8 = 16), (4 times 4 = 16)- **25:** (5 times 5 = 25)- **36:** (4 times 9 = 36), (6 times 6 = 36)- **49:** (7 times 7 = 49)- **64:** (8 times 8 = 64)- **81:** (9 times 9 = 81)Now, I'll convert these digit pairs into two-digit numbers:- **1:** 11- **4:** 14, 22, 41- **9:** 19, 33, 91- **16:** 28, 44, 82, 24, 42- **25:** 55- **36:** 34, 43, 66- **49:** 77- **64:** 48, 84, 88- **81:** 99Finally, I'll count all the unique two-digit numbers obtained from the above list. There are a total of 21 such numbers.

💡Okay, so I have this problem about three pipes filling up a tank, and I need to figure out how long it will take for the tank to overflow when all three pipes are open at the same time. Let me try to break this down step by step.First, let's understand what each pipe does. Pipe A can fill the tank in 24 minutes. That means, in one minute, Pipe A fills 1/24 of the tank. Okay, that makes sense. So, the rate of Pipe A is 1/24 per minute.Now, Pipe B is said to be 6 times faster than Pipe A. If Pipe A fills 1/24 of the tank in a minute, then Pipe B should fill 6 times that amount in a minute. So, let me calculate that: 6 * (1/24) = 6/24 = 1/4. So, Pipe B fills 1/4 of the tank in one minute. Got it.Next, Pipe C is a bit more complicated because it alternates between two different speeds every minute. It says Pipe C alternates between a speed of 3 times and half the speed of Pipe A every minute. So, in the first minute, Pipe C is 3 times faster than Pipe A, and in the next minute, it's half the speed of Pipe A, and this pattern repeats every two minutes.Let me try to figure out how much Pipe C fills in each minute. In the first minute, it's 3 times Pipe A's rate, which is 3 * (1/24) = 3/24 = 1/8. So, in the first minute, Pipe C fills 1/8 of the tank. In the next minute, it's half the speed of Pipe A, which is (1/2) * (1/24) = 1/48. So, in the second minute, Pipe C fills 1/48 of the tank.Since Pipe C alternates every minute, its rate changes every minute. This means that over two minutes, Pipe C fills 1/8 + 1/48 of the tank. Let me calculate that: 1/8 is equal to 6/48, so 6/48 + 1/48 = 7/48. Therefore, in two minutes, Pipe C fills 7/48 of the tank. To find the average rate per minute, I can divide this by 2: (7/48) / 2 = 7/96. So, Pipe C's average rate is 7/96 per minute.Now, I need to find the combined rate of all three pipes working together. Let's add up their rates:- Pipe A: 1/24 per minute- Pipe B: 1/4 per minute- Pipe C: 7/96 per minuteTo add these fractions, I need a common denominator. The denominators are 24, 4, and 96. The least common multiple of these numbers is 96. So, I'll convert each fraction to have 96 as the denominator:- Pipe A: 1/24 = 4/96- Pipe B: 1/4 = 24/96- Pipe C: 7/96 remains the sameNow, adding them up: 4/96 + 24/96 + 7/96 = (4 + 24 + 7)/96 = 35/96.So, the combined rate of all three pipes is 35/96 of the tank per minute. To find out how long it takes to fill the entire tank, I need to take the reciprocal of this rate. That is, time = 1 / (35/96) = 96/35 minutes.Calculating 96 divided by 35 gives approximately 2.74 minutes. So, it will take about 2.74 minutes for the tank to overflow when all three pipes are open simultaneously.Wait, let me double-check my calculations to make sure I didn't make any mistakes. Starting with Pipe C's rate: in the first minute, it's 3 times Pipe A, which is 3/24 = 1/8, and in the second minute, it's half of Pipe A's rate, which is 1/48. Adding these gives 1/8 + 1/48 = 6/48 + 1/48 = 7/48 over two minutes. Dividing by 2 gives 7/96 per minute. That seems correct.Adding up all the rates: 1/24 + 1/4 + 7/96. Converting to 96 denominators: 4/96 + 24/96 + 7/96 = 35/96. Reciprocal is 96/35, which is approximately 2.74 minutes. Yeah, that seems right.I think I've got it. The tank will overflow in about 2.74 minutes when all three pipes are open at the same time.

💡Okay, let me try to figure out this problem step by step. So, Bětka wrote down thirty consecutive integers and added them all together. Then she erased the second, fifth, eighth, and every following third number. After that, she added the remaining numbers and found that this new sum is 265 less than the original sum. We need to determine the first number that Bětka erased.First, I need to understand what exactly is happening here. She starts with thirty consecutive integers. Let me denote the first integer as 'a'. So, the sequence would be: a, a+1, a+2, ..., up to a+29, since there are thirty numbers in total.Next, she erases every third number starting from the second number. That means she erases the 2nd, 5th, 8th, 11th, and so on. Let me list out the positions she erases: 2, 5, 8, 11, 14, 17, 20, 23, 26, 29. Wait, that's ten numbers in total because 30 divided by 3 is 10, but since she starts at the second number, it's every third number from there.So, the numbers she erases are: a+1, a+4, a+7, a+10, a+13, a+16, a+19, a+22, a+25, a+28. That's ten numbers, as expected.Now, the sum of the original thirty numbers is the sum of an arithmetic series. The formula for the sum of an arithmetic series is S = n/2 * (first term + last term). Here, n = 30, first term = a, last term = a + 29. So, the original sum S_original is:S_original = 30/2 * (a + (a + 29)) = 15 * (2a + 29) = 30a + 435.Now, the sum of the erased numbers is the sum of the ten numbers I listed earlier: a+1, a+4, a+7, ..., a+28. This is also an arithmetic series with first term a+1, last term a+28, and common difference 3. The number of terms is 10. So, the sum of the erased numbers S_erased is:S_erased = 10/2 * ((a + 1) + (a + 28)) = 5 * (2a + 29) = 10a + 145.According to the problem, the new sum (after erasing) is 265 less than the original sum. So, the new sum S_new = S_original - S_erased = (30a + 435) - (10a + 145) = 20a + 290.But it's also given that S_new = S_original - 265. So,20a + 290 = (30a + 435) - 265.Let me compute the right side:30a + 435 - 265 = 30a + 170.So, the equation becomes:20a + 290 = 30a + 170.Subtracting 20a from both sides:290 = 10a + 170.Subtracting 170 from both sides:120 = 10a.Dividing both sides by 10:a = 12.So, the first number in the original sequence is 12. Therefore, the first number that Bětka erased is the second number in the sequence, which is a + 1 = 12 + 1 = 13.Wait, let me double-check my calculations to make sure I didn't make any mistakes.Original sum: 30a + 435.Sum of erased numbers: 10a + 145.New sum: 30a + 435 - (10a + 145) = 20a + 290.Given that new sum is 265 less than original sum:20a + 290 = 30a + 435 - 265 = 30a + 170.So, 20a + 290 = 30a + 170.Subtract 20a: 290 = 10a + 170.Subtract 170: 120 = 10a.Divide by 10: a = 12.Yes, that seems correct. So, the first number erased is 13.I think that makes sense. Let me just verify by plugging the numbers back in.If a = 12, then the original sequence is from 12 to 41 (since 12 + 29 = 41). The sum of these numbers is 30/2 * (12 + 41) = 15 * 53 = 795.The erased numbers are: 13, 16, 19, 22, 25, 28, 31, 34, 37, 40. Let's sum these:13 + 16 = 29,29 + 19 = 48,48 + 22 = 70,70 + 25 = 95,95 + 28 = 123,123 + 31 = 154,154 + 34 = 188,188 + 37 = 225,225 + 40 = 265.So, the sum of the erased numbers is 265, which matches the given information that the new sum is 265 less than the original sum.Therefore, the first number erased is indeed 13.

💡To determine the quantity of rose petals Mr. Blue expects from his flower bed, we need to follow these steps:1. **Convert the pacing measurements to feet:** - Length: (18) paces (times 1.5) feet/pace = (27) feet - Width: (24) paces (times 1.5) feet/pace = (36) feet2. **Calculate the area of the flower bed:** [ text{Area} = text{Length} times text{Width} = 27 text{ feet} times 36 text{ feet} = 972 text{ square feet} ]3. **Determine the total yield of rose petals:** [ text{Total Yield} = text{Area} times text{Yield per square foot} = 972 text{ square feet} times 0.4 text{ pounds/square foot} = 388.8 text{ pounds} ]Thus, Mr. Blue expects to gather (boxed{textbf{(D)} 388.8}) pounds of rose petals from his flower bed.

💡First, I need to determine the total number of carrot sticks James consumed. He ate 22 carrot sticks before dinner and 15 more after dinner, which adds up to 37 carrot sticks eaten in total.Next, I should account for the carrot sticks he gave away. James gave away 8 carrot sticks during dinner.Now, I'll calculate the total number of carrot sticks that are no longer with James by adding the eaten and given away carrot sticks: 37 + 8 = 45 carrot sticks.Finally, to find out how many carrot sticks James has left, I'll subtract the total number of carrot sticks no longer with him from the original number he had: 50 - 45 = 5 carrot sticks.

💡First, I need to determine the total amount of money John had. Let's denote this amount as ( X ).John spent:- ( frac{1}{4} ) of his money on fresh fruits and vegetables,- ( frac{1}{3} ) on meat products, and- ( frac{1}{6} ) on bakery products.Adding these fractions together:[frac{1}{4} + frac{1}{3} + frac{1}{6}]To add these fractions, I'll find a common denominator, which is 12:[frac{1}{4} = frac{3}{12}, quad frac{1}{3} = frac{4}{12}, quad frac{1}{6} = frac{2}{12}]Now, add them up:[frac{3}{12} + frac{4}{12} + frac{2}{12} = frac{9}{12} = frac{3}{4}]This means John spent ( frac{3}{4} ) of his money on fresh fruits and vegetables, meat products, and bakery products. Therefore, he had ( frac{1}{4} ) of his money left, which he spent on candy.Given that John spent 24 in total at the supermarket, and ( frac{3}{4} ) of his money equals 24, I can find out how much ( frac{1}{4} ) is by dividing 24 by 3:[frac{24}{3} = 8]Therefore, John spent 8 on candy.