💡First, I need to determine the ratio of the investments made by A, B, and C. A invested Rs. 8000, B invested Rs. 10000, and C invested Rs. 12000. To simplify the ratio, I'll divide each investment by the greatest common divisor, which is 2000. This gives me the ratio:A : B : C = 4 : 5 : 6Next, I know that B's share of the profit is Rs. 1800. Since the profit is divided according to the investment ratio, I can determine the value of one part of the ratio. B's ratio is 5, so:5 parts = Rs. 18001 part = Rs. 1800 / 5 = Rs. 360Now, I can calculate the total number of parts in the ratio:Total parts = 4 (A's share) + 5 (B's share) + 6 (C's share) = 15 partsThe total profit is:Total profit = 15 parts * Rs. 360/part = Rs. 5400Now, I'll find A's and C's share of the profit:A's share = 4 parts * Rs. 360/part = Rs. 1440C's share = 6 parts * Rs. 360/part = Rs. 2160Finally, the difference between the profit shares of A and C is:Difference = C's share - A's share = Rs. 2160 - Rs. 1440 = Rs. 720

💡Okay, so I have this problem about regular polygons with 6, 7, 8, and 9 sides inscribed in the same circle. The conditions are that no two polygons share a vertex, and no three sides intersect at the same point. I need to find how many points inside the circle where two sides intersect.Hmm, let me think. First, all these polygons are inscribed in the same circle, so they all have the same circumradius. Since they are regular, their vertices are equally spaced around the circle. But they don't share any vertices, which probably means their vertices are at different points on the circle.Now, the key here is that sides from different polygons can intersect inside the circle. I remember that when two regular polygons are inscribed in the same circle, their sides can intersect at points inside the circle. The number of intersections depends on the number of sides each polygon has.Wait, how do two sides from different polygons intersect? Each side is a chord of the circle. So, two chords can intersect inside the circle if they are not overlapping and don't share an endpoint. Since the polygons don't share any vertices, their sides won't share endpoints, so any two sides from different polygons could potentially intersect.But how many times do they intersect? I think each pair of sides from two different polygons can intersect at most once inside the circle. So, if I can count how many pairs of sides from different polygons intersect, that should give me the total number of intersection points.Let me break it down. There are four polygons: 6, 7, 8, and 9 sides. So, the number of pairs of polygons is C(4,2) = 6. Each pair will contribute some number of intersections.For each pair of polygons, say one with n sides and the other with m sides, how many times do their sides intersect? I think the formula is 2 * n * m / gcd(n, m), but wait, that might be for something else. Or maybe it's just 2 * n * m? No, that seems too high.Wait, actually, when two regular polygons are inscribed in the same circle, the number of intersection points inside the circle is 2 * n * m / gcd(n, m). Is that right? Let me think.No, that formula might be for the number of times their edges cross each other, but considering that each edge is a chord. Wait, actually, each pair of edges from different polygons can intersect at most once inside the circle, right? So, for two polygons with n and m sides, the number of intersection points would be n * m, but only if none of their edges are overlapping or something.But wait, in reality, not all pairs of edges will intersect. Because depending on the number of sides, the edges might be arranged in such a way that some pairs don't cross. Hmm, this is getting complicated.Wait, maybe I should think about the general formula for the number of intersection points inside a circle when you have two regular polygons with n and m sides. I think it's 2 * n * m / gcd(n, m). Let me check that.If n and m are coprime, gcd(n, m) = 1, so the number of intersections would be 2 * n * m. But that seems too high because, for example, a hexagon and a heptagon (7 sides) inscribed in the same circle, how many intersections would they have?Wait, a hexagon has 6 sides, a heptagon has 7. If they don't share any vertices, each side of the hexagon can potentially intersect with two sides of the heptagon? Or is it once?Wait, no, each side of the hexagon can intersect with two sides of the heptagon because as you go around the circle, each side of the hexagon will cross two sides of the heptagon. So, for each side of the hexagon, there are two intersections with the heptagon's sides. Therefore, total intersections would be 6 * 2 = 12.Similarly, for the hexagon and octagon (8 sides), each side of the hexagon would intersect two sides of the octagon, so 6 * 2 = 12 intersections.Wait, but hold on, is that correct? Because if you have a hexagon and an octagon, each side of the hexagon might intersect two sides of the octagon, but actually, depending on their rotational positions, maybe it's different.Wait, no, actually, since they are regular and don't share any vertices, their sides will be evenly spaced. So, for each side of the hexagon, it will intersect two sides of the heptagon, two sides of the octagon, and two sides of the nonagon (9 sides). So, for each polygon, the number of intersections with another polygon would be 2 * number of sides of the smaller polygon.Wait, no, that might not be correct. Let me think again.When you have two regular polygons with n and m sides inscribed in the same circle, the number of intersection points is 2 * n * m / gcd(n, m). So, for example, for a hexagon (6 sides) and a heptagon (7 sides), since gcd(6,7)=1, the number of intersections would be 2 * 6 * 7 = 84? That seems too high because the answer choices are only up to 88.Wait, but that can't be right because 84 is one of the options, but let's see.Wait, no, actually, I think the formula is 2 * n * m / gcd(n, m). So, for n=6 and m=7, it's 2*6*7 /1=84. But that's the number of intersection points? That seems too high because each pair of sides can only intersect once.Wait, no, actually, each pair of sides can intersect at most once, so the total number of intersections between two polygons would be n * m, but considering that some sides might not intersect because they are adjacent or something.Wait, no, in reality, for two regular polygons with n and m sides inscribed in the same circle, the number of intersection points is 2 * n * m / gcd(n, m). So, for n=6 and m=7, it's 84, but that seems too high because each polygon only has 6 and 7 sides.Wait, maybe I'm confusing something. Let me think differently.Each side of the hexagon can intersect with multiple sides of the heptagon. Since the hexagon has 6 sides and the heptagon has 7, each side of the hexagon will intersect with two sides of the heptagon. So, total intersections would be 6 * 2 = 12.Similarly, each side of the hexagon will intersect with two sides of the octagon, so another 12 intersections, and two sides of the nonagon, another 12. So, for the hexagon, total intersections with the other three polygons would be 12 * 3 = 36.Similarly, for the heptagon, each side can intersect with two sides of the octagon and two sides of the nonagon. So, 7 * 2 = 14 intersections with the octagon, and 7 * 2 = 14 intersections with the nonagon. So, total for the heptagon would be 14 * 2 = 28.Then, for the octagon, each side can intersect with two sides of the nonagon, so 8 * 2 = 16 intersections.Adding them all up: 36 (from hexagon) + 28 (from heptagon) + 16 (from octagon) = 80.Wait, that seems to be one of the answer choices, option C) 80.But let me verify this because I might be double-counting. For example, when I count the intersections from the hexagon with the heptagon, and then later from the heptagon with the hexagon, am I counting the same intersections twice?Wait, no, because when I count the hexagon's intersections with the heptagon, I'm considering the hexagon as the smaller polygon, and each side intersects two sides of the larger polygon. Similarly, when I count the heptagon's intersections with the octagon, I'm considering the heptagon as the smaller polygon, and each side intersects two sides of the octagon. So, each intersection is only counted once because each pair of polygons is considered once in this way.Therefore, the total number of intersections is 36 + 28 + 16 = 80.So, the answer should be 80, which is option C.**Final Answer**boxed{C}

💡First, I need to determine the total number of car selections made by all clients. Since there are 15 cars and each car was selected exactly 3 times, the total number of selections is 15 cars multiplied by 3 selections per car, which equals 45 selections.Next, each client selects 3 cars. To find out how many clients there are, I divide the total number of selections by the number of cars each client selects. So, 45 selections divided by 3 selections per client equals 15 clients.Therefore, the number of clients who visited the garage is 15.

💡Okay, so I have this triangle ABC, and I know that angle A is π/6, which is 30 degrees. The sides opposite angles A, B, and C are a, b, and c respectively. There's also this equation given: (1 + √3) sin B = 2 sin C. Part (1) asks me to prove that b = √2 a. Hmm, okay. So, I remember from the Law of Sines that in any triangle, a/sin A = b/sin B = c/sin C. Maybe I can use that here. Given that A is π/6, sin A is 1/2. So, a/(1/2) = 2a. That means b = 2a sin B and c = 2a sin C. Now, the given equation is (1 + √3) sin B = 2 sin C. Maybe I can express sin C in terms of sin B. Since in a triangle, A + B + C = π, so C = π - A - B. That means C = π - π/6 - B = (5π/6 - B). So, sin C = sin(5π/6 - B). I can use the sine subtraction formula: sin(π - x) = sin x, but here it's sin(5π/6 - B). Let me recall the sine of a difference: sin(x - y) = sin x cos y - cos x sin y. So, sin(5π/6 - B) = sin(5π/6) cos B - cos(5π/6) sin B. I know that sin(5π/6) is 1/2, and cos(5π/6) is -√3/2. So, substituting these values in, we get: sin C = (1/2) cos B - (-√3/2) sin B = (1/2) cos B + (√3/2) sin B. So, sin C = (1/2) cos B + (√3/2) sin B. Now, going back to the given equation: (1 + √3) sin B = 2 sin C. Let's substitute sin C with the expression we just found: (1 + √3) sin B = 2[(1/2) cos B + (√3/2) sin B]. Let me simplify the right-hand side: 2*(1/2 cos B) is cos B, and 2*(√3/2 sin B) is √3 sin B. So, the equation becomes: (1 + √3) sin B = cos B + √3 sin B. Hmm, let's subtract √3 sin B from both sides: (1 + √3) sin B - √3 sin B = cos B. Simplify the left side: 1 sin B + √3 sin B - √3 sin B = sin B. So, sin B = cos B. That's interesting. So, sin B = cos B. When does that happen? Well, sin B = cos B implies that tan B = 1, because tan B = sin B / cos B. So, tan B = 1. So, B must be π/4, since tan(π/4) = 1. So, angle B is 45 degrees. Now, since I know angles A and B, I can find angle C. Since A + B + C = π, then C = π - π/6 - π/4. Let me calculate that: π is 12/12 π, π/6 is 2/12 π, π/4 is 3/12 π. So, 12/12 - 2/12 - 3/12 = 7/12 π. So, angle C is 7π/12. But maybe I don't need angle C for part (1). Let's see. I need to find the relationship between sides a and b. From the Law of Sines, a/sin A = b/sin B. We know sin A is 1/2, and sin B is sin(π/4) which is √2/2. So, a/(1/2) = b/(√2/2). Simplify this: a/(1/2) is 2a, and b/(√2/2) is (2b)/√2. So, 2a = (2b)/√2. Multiply both sides by √2: 2a√2 = 2b. Divide both sides by 2: a√2 = b. So, b = √2 a. Okay, that proves part (1). Now, part (2) says that the area of triangle ABC is 2 + 2√3, and we need to find b. I remember that the area of a triangle can be calculated using the formula: (1/2)ab sin C. But in this case, maybe it's better to express it in terms of sides a and b and the angle between them. Wait, actually, since we know two sides and the included angle, but in this case, we might need to express the area in terms of a and b. Let me think. We can use the formula: Area = (1/2)ab sin C. But we need to find sin C. Alternatively, since we know angles A, B, and C, maybe we can express the area in terms of a or b. Wait, from part (1), we have b = √2 a. So, if I can express the area in terms of a, then I can solve for a and then find b. Let me try that. So, Area = (1/2)ab sin C. We know that a is a, b is √2 a, and sin C we can compute. We already found angle C is 7π/12, so sin(7π/12). Let me compute sin(7π/12). 7π/12 is equal to π/3 + π/4. So, sin(π/3 + π/4). Using the sine addition formula: sin(x + y) = sin x cos y + cos x sin y. So, sin(π/3 + π/4) = sin(π/3) cos(π/4) + cos(π/3) sin(π/4). We know sin(π/3) is √3/2, cos(π/4) is √2/2, cos(π/3) is 1/2, and sin(π/4) is √2/2. So, substituting these in: √3/2 * √2/2 + 1/2 * √2/2 = (√6)/4 + (√2)/4 = (√6 + √2)/4. So, sin C = (√6 + √2)/4. Now, plug this back into the area formula: Area = (1/2) * a * (√2 a) * (√6 + √2)/4. Let me compute this step by step. First, multiply (1/2) and (√2 a): (1/2) * √2 a = (√2 / 2) a. Then, multiply that by (√6 + √2)/4: (√2 / 2) a * (√6 + √2)/4 = (√2 (√6 + √2)) / 8 * a^2. Let me compute √2 * √6: that's √12 = 2√3. And √2 * √2 = 2. So, the numerator becomes 2√3 + 2. So, the area is (2√3 + 2)/8 * a^2. Simplify this: factor out 2 in the numerator: 2(√3 + 1)/8 = (√3 + 1)/4. So, Area = (√3 + 1)/4 * a^2. But we are given that the area is 2 + 2√3. So, set up the equation: (√3 + 1)/4 * a^2 = 2 + 2√3. Multiply both sides by 4: (√3 + 1) a^2 = 8 + 8√3. Now, solve for a^2: a^2 = (8 + 8√3)/(√3 + 1). Let me rationalize the denominator. Multiply numerator and denominator by (√3 - 1): a^2 = [ (8 + 8√3)(√3 - 1) ] / [ (√3 + 1)(√3 - 1) ] Compute denominator: (√3)^2 - (1)^2 = 3 - 1 = 2. Compute numerator: 8*(√3 - 1) + 8√3*(√3 - 1) First term: 8√3 - 8 Second term: 8√3*√3 - 8√3*1 = 8*3 - 8√3 = 24 - 8√3 So, total numerator: (8√3 - 8) + (24 - 8√3) = 8√3 - 8 + 24 - 8√3 Simplify: 8√3 - 8√3 = 0, -8 + 24 = 16. So, numerator is 16. Thus, a^2 = 16 / 2 = 8. So, a^2 = 8, which means a = √8 = 2√2. But we need to find b. From part (1), b = √2 a. So, b = √2 * 2√2 = 2 * 2 = 4. So, b is 4. Wait, let me double-check my calculations to make sure I didn't make a mistake. Starting from the area formula: (√3 + 1)/4 * a^2 = 2 + 2√3. Multiply both sides by 4: (√3 + 1) a^2 = 8 + 8√3. Then, a^2 = (8 + 8√3)/(√3 + 1). Multiply numerator and denominator by (√3 - 1): Numerator: (8 + 8√3)(√3 - 1) = 8√3 - 8 + 8*3 - 8√3 = 8√3 - 8 + 24 - 8√3 = 16. Denominator: (√3 + 1)(√3 - 1) = 3 - 1 = 2. So, a^2 = 16/2 = 8, so a = 2√2. Then, b = √2 * 2√2 = 4. Yes, that seems correct. So, the value of b is 4.

💡Okay, so I have this problem here where the sequence {a_n} is defined by a_n = n² + 3n + 2. I need to find the sum of the first 10 terms of the sequence {1/a_n}. The options given are A: 1/3, B: 5/12, C: 1/2, and D: 7/12. Hmm, let me think about how to approach this.First, I remember that when dealing with sums of sequences, especially when the terms are fractions, partial fraction decomposition might be useful. Let me try that. So, I have 1/a_n, which is 1/(n² + 3n + 2). I can factor the denominator to make it easier to decompose.Factoring n² + 3n + 2, I look for two numbers that multiply to 2 and add up to 3. That would be 1 and 2. So, the denominator factors into (n + 1)(n + 2). Therefore, 1/(n² + 3n + 2) becomes 1/[(n + 1)(n + 2)]. Now, I want to express this as a difference of two fractions. Let me set it up as A/(n + 1) + B/(n + 2). To find A and B, I can write:1 = A(n + 2) + B(n + 1)Expanding the right side, I get:1 = (A + B)n + (2A + B)Since this must hold for all n, the coefficients of like terms must be equal on both sides. So, the coefficient of n on the left side is 0, and the constant term is 1. Therefore, I can set up the following system of equations:A + B = 0 (coefficient of n)2A + B = 1 (constant term)Subtracting the first equation from the second, I get:(2A + B) - (A + B) = 1 - 0A = 1Plugging A = 1 back into the first equation:1 + B = 0B = -1So, the partial fractions decomposition is:1/(n + 1) - 1/(n + 2)Therefore, 1/a_n = 1/(n + 1) - 1/(n + 2). Now, I need to find the sum of the first 10 terms of {1/a_n}, which is:Sum from n = 1 to 10 of [1/(n + 1) - 1/(n + 2)]Let me write out the terms to see if there's a pattern or cancellation:For n = 1: 1/2 - 1/3For n = 2: 1/3 - 1/4For n = 3: 1/4 - 1/5...For n = 10: 1/11 - 1/12So, when I add all these up, most of the intermediate terms should cancel out. Let me write them all out:(1/2 - 1/3) + (1/3 - 1/4) + (1/4 - 1/5) + ... + (1/11 - 1/12)Looking at this, the -1/3 cancels with the +1/3, the -1/4 cancels with the +1/4, and so on, all the way up to -1/11 canceling with +1/11. So, what's left is the first term of the first expression and the last term of the last expression.That is, 1/2 - 1/12.Now, I need to compute 1/2 - 1/12. To subtract these fractions, they need a common denominator. The least common denominator of 2 and 12 is 12.Converting 1/2 to twelfths: 1/2 = 6/12.So, 6/12 - 1/12 = 5/12.Therefore, the sum of the first 10 terms of the sequence {1/a_n} is 5/12.Looking back at the options, B is 5/12, so that should be the correct answer.I think this makes sense because telescoping series often result in simplifications where most terms cancel out, leaving only the first and last terms. In this case, the telescoping nature of the partial fractions decomposition led to the cancellation of all intermediate terms, resulting in a straightforward subtraction of the first and last terms.Just to double-check, let me compute the sum step by step without relying on the telescoping property, just to ensure I didn't make a mistake.Compute each term:n=1: 1/(1+1) - 1/(1+2) = 1/2 - 1/3 ≈ 0.5 - 0.333 ≈ 0.1667n=2: 1/3 - 1/4 ≈ 0.333 - 0.25 ≈ 0.0833n=3: 1/4 - 1/5 ≈ 0.25 - 0.2 ≈ 0.05n=4: 1/5 - 1/6 ≈ 0.2 - 0.1667 ≈ 0.0333n=5: 1/6 - 1/7 ≈ 0.1667 - 0.1429 ≈ 0.0238n=6: 1/7 - 1/8 ≈ 0.1429 - 0.125 ≈ 0.0179n=7: 1/8 - 1/9 ≈ 0.125 - 0.1111 ≈ 0.0139n=8: 1/9 - 1/10 ≈ 0.1111 - 0.1 ≈ 0.0111n=9: 1/10 - 1/11 ≈ 0.1 - 0.0909 ≈ 0.0091n=10: 1/11 - 1/12 ≈ 0.0909 - 0.0833 ≈ 0.0076Now, adding all these approximate decimal values:0.1667 + 0.0833 = 0.250.25 + 0.05 = 0.30.3 + 0.0333 ≈ 0.33330.3333 + 0.0238 ≈ 0.35710.3571 + 0.0179 ≈ 0.3750.375 + 0.0139 ≈ 0.38890.3889 + 0.0111 ≈ 0.40.4 + 0.0091 ≈ 0.40910.4091 + 0.0076 ≈ 0.4167Converting 5/12 to decimal: 5 ÷ 12 ≈ 0.4167. So, the approximate decimal sum is about 0.4167, which matches 5/12. Therefore, my earlier conclusion seems correct.I think I've considered it from both the telescoping series perspective and by directly computing the approximate decimal values. Both methods lead me to the same answer, so I feel confident that 5/12 is the correct sum of the first 10 terms of the sequence {1/a_n}.**Final Answer**The sum of the first 10 terms of the sequence is boxed{B}.

💡First, I need to determine what percentage of the total games the soccer team won. The team played a total of 158 games and won 40% of them.To find the number of games won, I will calculate 40% of 158. This can be done by multiplying 158 by 0.4.158 multiplied by 0.4 equals 63.2. Since a team cannot win a fraction of a game, I will round 63.2 to the nearest whole number, which is 63.Therefore, the soccer team won 63 games.