💡Okay, so I've got this problem here: "A scale is divided into 5 equal parts. The length of each part is 25 inches. What is the total length of the scale?" Hmm, let's see. Alright, first, I need to understand what the problem is asking. It's about a scale that's been divided into five equal sections, and each of those sections is 25 inches long. I need to find the total length of the entire scale.Alright, so if the scale is divided into five equal parts, that means each part is the same size, right? And each of those parts is 25 inches long. So, if I think about it, if I have five pieces, each 25 inches, and I put them all together, the total length should be five times 25 inches. That makes sense because if each part is 25 inches, then adding them all up should give me the total length.Let me write that down to make it clearer. So, the total length would be 5 multiplied by 25 inches. Mathematically, that would be:Total length = 5 × 25 inchesNow, let's do the multiplication. 5 times 25. Well, 5 times 20 is 100, and 5 times 5 is 25, so adding those together gives me 125. So, 5 × 25 inches equals 125 inches.Wait, does that make sense? Let me double-check. If I have five equal parts, each 25 inches, then adding them up should give me the total length. So, 25 + 25 + 25 + 25 + 25. Let's add them one by one:25 + 25 = 5050 + 25 = 7575 + 25 = 100100 + 25 = 125Yep, that adds up to 125 inches. So, that seems consistent.Is there another way to think about this? Maybe using division? Hmm, if I know the total length and I divide it into five equal parts, each part would be the total length divided by 5. But in this case, I know the length of each part and need to find the total length, so multiplication makes more sense here.Another way to visualize it could be to imagine the scale as a straight line divided into five segments. Each segment is 25 inches. So, if I were to lay out five of these segments end to end, the entire length from one end to the other would be 125 inches.Let me also consider units to make sure I'm not making a mistake. The length of each part is given in inches, and I'm multiplying by the number of parts, which is unitless. So, the result should indeed be in inches, which it is—125 inches.Is there any chance I could have misread the problem? For example, maybe the scale is divided into five parts, but not necessarily equal parts? But the problem specifically says "5 equal parts," so that shouldn't be an issue.Also, could there be any trick or catch to this problem? Sometimes problems are designed to seem straightforward but have a hidden complexity. But in this case, it seems pretty straightforward. It's just a simple multiplication problem.Just to be thorough, let's think about real-world applications. If I have a ruler that's 12 inches long and I divide it into five equal parts, each part would be 12 divided by 5, which is 2.4 inches. But in this problem, it's the opposite. We know the length of each part and need to find the total length.So, in summary, the scale is divided into five equal parts, each 25 inches long. To find the total length, I multiply the number of parts by the length of each part: 5 × 25 inches = 125 inches. That seems correct.I think I've covered all the bases here. I've done the multiplication, checked my work by adding the parts individually, considered the units, and even thought about real-world applications to ensure my approach makes sense. I don't see any errors in my reasoning, so I'm confident that the total length of the scale is 125 inches.

💡First, I need to determine the total number of votes cast in the election. Let's denote the total votes as ( V ).The winning candidate received 62% of the total votes, which is ( 0.62V ). The losing candidate received the remaining 38% of the votes, which is ( 0.38V ).The difference in votes between the winning and losing candidates is 312 votes. This can be expressed as:[0.62V - 0.38V = 312]Simplifying the equation:[0.24V = 312]To find ( V ), I divide both sides by 0.24:[V = frac{312}{0.24} = 1300]Now that I have the total number of votes, I can calculate the number of votes cast for the winning candidate by finding 62% of 1300:[0.62 times 1300 = 806]

💡First, I will start with the given quadratic equation:[ x^2 - 8x - 1 = 0 ]My goal is to rewrite this equation in the form:[ (x - 4)^2 = m ]To do this, I need to complete the square. I'll begin by isolating the quadratic and linear terms on one side of the equation:[ x^2 - 8x = 1 ]Next, I'll complete the square by adding the square of half the coefficient of ( x ) to both sides. The coefficient of ( x ) is (-8), so half of that is (-4), and its square is ( 16 ):[ x^2 - 8x + 16 = 1 + 16 ]This simplifies to:[ (x - 4)^2 = 17 ]Therefore, the value of ( m ) is ( 17 ), which corresponds to option A.

💡Okay, so I have this problem: I need to find how many integer values of m between 1 and 500 inclusive make the decimal representation of m/980 terminate. Hmm, I remember something about terminating decimals from math class. Let me try to recall.First, I think a fraction will have a terminating decimal if, after simplifying the fraction, the denominator only has prime factors of 2 and 5. Is that right? So, for example, fractions like 1/2, 1/4, 1/5, 1/8, 1/10, etc., all have terminating decimals because their denominators are powers of 2 or 5. But fractions like 1/3, 1/6, 1/7, etc., don't terminate because their denominators have other prime factors.So, applying that to this problem, I need to look at the denominator 980 and see what its prime factors are. Let me factorize 980. Let's see, 980 divided by 2 is 490, divided by 2 again is 245. 245 divided by 5 is 49, and 49 is 7 squared. So, putting that all together, 980 factors into 2 squared times 5 times 7 squared. So, 980 = 2² × 5 × 7².Now, for the decimal to terminate, after simplifying m/980, the denominator should only have 2s and 5s. That means any factors of 7 in the denominator need to be canceled out by factors of 7 in the numerator, which is m. So, m must contain enough factors of 7 to cancel out the 7² in the denominator.Therefore, m must be a multiple of 7², which is 49. So, m has to be a multiple of 49. Now, I need to find how many multiples of 49 are there between 1 and 500 inclusive.To find that, I can divide 500 by 49 and take the floor of that result. Let me calculate 500 divided by 49. 49 times 10 is 490, which is less than 500. 49 times 11 is 539, which is more than 500. So, 500 divided by 49 is approximately 10.204. Taking the floor of that gives me 10. So, there are 10 multiples of 49 between 1 and 500.Wait, let me double-check that. If I list the multiples of 49: 49, 98, 147, 196, 245, 294, 343, 392, 441, 490. That's 10 numbers. The next multiple would be 539, which is beyond 500, so yes, 10 is correct.So, each of these multiples of 49 will cancel out the 7² in the denominator, leaving the simplified fraction with only 2² and 5 as prime factors, which means the decimal will terminate.Therefore, the number of integer values of m between 1 and 500 inclusive for which m/980 has a terminating decimal is 10.I think that's it. I don't see any mistakes in my reasoning. I considered the prime factors, determined what m needs to be, counted the multiples, and verified the count. Seems solid.

💡Okay, so I have this problem here: compute the square root of (34)(33)(32)(31) plus 1. Hmm, that looks a bit complicated at first glance, but maybe there's a pattern or a trick to simplify it. Let me think about how I can approach this.First, I notice that the numbers 34, 33, 32, and 31 are four consecutive integers in descending order. That might be useful. I remember sometimes when dealing with products of consecutive numbers, especially in problems involving square roots, there's a way to express them as a perfect square. Maybe I can find a way to rewrite this product so that when I add 1, it becomes a perfect square, making the square root straightforward.Let me denote one of these numbers as a variable to make it easier. Let's choose 32 as the middle number because it's in the middle of 31 and 33. So, let x = 32. Then, the numbers can be expressed in terms of x:- 34 is x + 2- 33 is x + 1- 32 is x- 31 is x - 1So, the product becomes (x + 2)(x + 1)(x)(x - 1). That looks like (x - 1)x(x + 1)(x + 2). Maybe I can pair them in a way that makes it easier to multiply. Let me try pairing the outer terms and the inner terms:First pair: (x - 1)(x + 2) = x^2 + 2x - x - 2 = x^2 + x - 2Second pair: x(x + 1) = x^2 + xSo now, the product is (x^2 + x - 2)(x^2 + x). Let me denote y = x^2 + x for simplicity. Then, the product becomes (y - 2)y = y^2 - 2y.So, the original expression is (y^2 - 2y) + 1. Let's write that out:y^2 - 2y + 1Hmm, that looks familiar. That's a quadratic in y. Let me see if it factors:y^2 - 2y + 1 = (y - 1)^2Yes, that's a perfect square! So, substituting back for y:(y - 1)^2 = (x^2 + x - 1)^2Therefore, the expression inside the square root is (x^2 + x - 1)^2. Taking the square root of that would just give us x^2 + x - 1.Now, let's compute x^2 + x - 1 where x = 32:x^2 = 32^2 = 1024x = 32So, x^2 + x - 1 = 1024 + 32 - 1 = 1055Therefore, the square root of (34)(33)(32)(31) + 1 is 1055.Wait, let me double-check my steps to make sure I didn't make a mistake. I set x = 32, expressed the product in terms of x, paired them as (x - 1)(x + 2) and x(x + 1), which gave me (x^2 + x - 2)(x^2 + x). Then I set y = x^2 + x, so the product became y(y - 2) = y^2 - 2y. Adding 1 gives y^2 - 2y + 1, which factors to (y - 1)^2. Substituting back, that's (x^2 + x - 1)^2. Plugging in x = 32, I got 1055. Seems solid.Just to be thorough, maybe I can compute the original product and add 1, then take the square root to confirm.Compute (34)(33)(32)(31):First, multiply 34 and 31: 34 * 31 = 1054Then, multiply 33 and 32: 33 * 32 = 1056Now, multiply those two results: 1054 * 1056Hmm, that's a big number. Let me compute that.I know that 1054 * 1056 can be expressed as (1055 - 1)(1055 + 1) = 1055^2 - 1So, 1054 * 1056 = 1055^2 - 1Therefore, (34)(33)(32)(31) = 1055^2 - 1Adding 1 gives us 1055^2 - 1 + 1 = 1055^2So, the square root of that is 1055. Yep, that confirms it.I think I'm confident now that the answer is 1055.**Final Answer**boxed{1055}

💡Alright, so I've got this problem here: Bryan had 8 precious stones in his collection, and he sold them to his friend from the jewelry store. Each stone was sold for 1785 dollars. The question is asking, in total, how much money did Bryan get from selling all those stones?Okay, let's break this down. First, I need to understand what's being asked. It's about calculating the total amount Bryan received after selling all his stones. Since he sold each stone for a specific amount, I think I need to multiply the number of stones by the price per stone to get the total.So, the number of stones is 8, and each stone was sold for 1785 dollars. That means for each stone, Bryan got 1785 dollars. To find out how much he got in total, I need to add up the amount he got for each stone. But instead of adding 1785 eight times, which would take a while, I can just multiply 1785 by 8. That should give me the total amount.Let me write that out: Total amount = Number of stones × Price per stone. Plugging in the numbers, that would be Total amount = 8 × 1785 dollars.Now, I need to do the multiplication. Let's see, 8 times 1785. Hmm, I'm not sure off the top of my head what that equals. Maybe I can break it down to make it easier. Let's think about 1785 as 1000 + 700 + 80 + 5. Then, I can multiply each of those parts by 8 and add them up.So, 8 × 1000 is 8000. 8 × 700 is 5600. 8 × 80 is 640. And 8 × 5 is 40. Now, if I add all those together: 8000 + 5600 is 13600. Then, 13600 + 640 is 14240. Finally, 14240 + 40 is 14280.Wait, let me double-check that to make sure I didn't make a mistake. 8 × 1785. Another way to do it is to multiply 1785 by 8 directly. Let's see:1785× 8-------?Starting from the right, 5 × 8 is 40. I write down 0 and carry over 4. Next, 8 × 8 is 64, plus the carried over 4 is 68. I write down 8 and carry over 6. Then, 7 × 8 is 56, plus the carried over 6 is 62. I write down 2 and carry over 6. Finally, 1 × 8 is 8, plus the carried over 6 is 14. So, putting it all together, I get 14280.Okay, that matches what I got earlier when I broke it down. So, it seems like the total amount Bryan got is 14280 dollars.Just to be extra sure, maybe I can use another method, like the distributive property. Let's see, 1785 × 8 can be thought of as (1700 + 85) × 8. Then, I can distribute the multiplication:1700 × 8 = 1360085 × 8 = 680Now, adding those together: 13600 + 680 = 14280.Yep, that still gives me the same result. So, I'm pretty confident that the total amount Bryan received is 14280 dollars.Another way to verify is by using estimation. If each stone is roughly 1700 dollars, then 8 stones would be about 13600 dollars. But since each stone is actually 1785, which is 85 dollars more than 1700, the total should be a bit more than 13600. Specifically, 85 × 8 = 680, so adding that to 13600 gives 14280. That makes sense.I think I've covered all the ways to approach this multiplication, and each method leads me to the same answer. So, I'm pretty sure that Bryan got a total of 14280 dollars from selling his 8 precious stones.