💡Okay, so I have this geometry problem here. Let me try to understand it step by step. The problem says that point O is the center of the circle circumscribed around triangle DEF. That means O is the circumcenter of triangle DEF, right? So, OD, OE, and OF are all radii of the circumscribed circle. They give me two angles: angle DOF is 110 degrees, and angle EOF is 130 degrees. I need to find the measure of angle EDF. Hmm, angle EDF is one of the angles of triangle DEF. Since O is the circumcenter, I remember that the angles at the center (the central angles) relate to the angles of the triangle itself. I think the key here is to use the fact that the central angle is twice the inscribed angle subtended by the same arc. So, if I can find the central angle that corresponds to angle EDF, I can then find angle EDF by taking half of that central angle. First, let me visualize the circle with center O and triangle DEF inscribed in it. Points D, E, and F are on the circumference, and O is inside the triangle. The angles given, angle DOF and angle EOF, are central angles. So, angle DOF is 110 degrees, which is the central angle between points D and F. Angle EOF is 130 degrees, which is the central angle between points E and F. I need to find angle EDF, which is the angle at point D between points E and F. Wait, angle EDF is an inscribed angle. The inscribed angle theorem says that an inscribed angle is half the measure of its central angle. So, if I can find the central angle that corresponds to angle EDF, I can just divide it by two to get angle EDF. But what is the central angle corresponding to angle EDF? Angle EDF is at point D, so it's subtended by the arc EF. Therefore, the central angle subtended by arc EF would be angle EOF, which is given as 130 degrees. Wait, hold on. If angle EDF is subtended by arc EF, then the central angle for arc EF is angle EOF, which is 130 degrees. So, angle EDF should be half of 130 degrees, which is 65 degrees. But that doesn't seem right because the answer I remember from before was 60 degrees. Maybe I'm missing something here.Let me think again. Maybe I need to consider all the central angles around point O. Since O is the center, the sum of all central angles around O should be 360 degrees. They've given me angle DOF as 110 degrees and angle EOF as 130 degrees. So, if I add those together, 110 + 130 is 240 degrees. That means the remaining central angle, which is angle EOD, should be 360 - 240, which is 120 degrees.Ah, so angle EOD is 120 degrees. Now, angle EOD is the central angle subtended by arc ED. Therefore, the inscribed angle subtended by arc ED would be angle EFD. Wait, no, angle EDF is subtended by arc EF, not ED. Hmm, maybe I need to clarify.Wait, angle EDF is at point D, so it's subtended by arc EF. The central angle for arc EF is angle EOF, which is 130 degrees. So, angle EDF should be half of 130, which is 65 degrees. But earlier, I thought the answer was 60 degrees. Maybe I made a mistake in my initial thought.Alternatively, perhaps I need to consider the triangle's angles in relation to the central angles. Let me recall that in a triangle inscribed in a circle, the measure of an angle is half the measure of the central angle that subtends the same arc. So, angle EDF subtends arc EF, which is 130 degrees. Therefore, angle EDF should be 65 degrees.Wait, but maybe I'm confusing the arcs. Let me draw a diagram mentally. Points D, E, F on the circle, O is the center. Angle DOF is 110 degrees, which is the central angle between D and F. Angle EOF is 130 degrees, which is the central angle between E and F. So, the arcs subtended by these angles are arc DF (110 degrees) and arc EF (130 degrees). Since the total circumference is 360 degrees, the remaining arc DE would be 360 - 110 - 130, which is 120 degrees. So, arc DE is 120 degrees. Now, angle EDF is an inscribed angle that subtends arc EF. Arc EF is 130 degrees, so angle EDF is half of that, which is 65 degrees. But wait, earlier I thought the answer was 60 degrees. Maybe I was confusing angle EDF with another angle. Let me double-check. If angle EOD is 120 degrees, which is the central angle subtended by arc ED, then the inscribed angle subtended by arc ED would be angle EFD, which is at point F. So, angle EFD would be half of 120 degrees, which is 60 degrees. But the question asks for angle EDF, not angle EFD. So, angle EDF is subtended by arc EF, which is 130 degrees, so it should be 65 degrees. Hmm, I'm getting conflicting thoughts here. Maybe I need to approach this differently.Let me consider the triangle DEF and its circumcircle. The central angles are given for arcs DF and EF. So, arc DF is 110 degrees, arc EF is 130 degrees, and arc DE is 120 degrees. Now, in triangle DEF, the angles at the vertices are half the measure of the arcs opposite them. So, angle at D (angle EDF) is half the measure of arc EF, which is 130 degrees, so angle EDF is 65 degrees. Similarly, angle at E would be half of arc DF, which is 110 degrees, so 55 degrees. Angle at F would be half of arc DE, which is 120 degrees, so 60 degrees. Wait, but if I add up these angles, 65 + 55 + 60, that's 180 degrees, which makes sense for a triangle. So, angle EDF is 65 degrees. But earlier, I thought the answer was 60 degrees. Maybe I was confusing angle EDF with angle EFD. Alternatively, perhaps the problem is referring to angle EDF as the angle at D, which is subtended by arc EF. So, yes, that should be 65 degrees. But I need to make sure I'm interpreting the problem correctly. Wait, let me check the problem statement again: "Determine the degree measure of angle EDF." So, angle EDF is at point D, between points E and F. So, it's subtended by arc EF, which is 130 degrees. Therefore, angle EDF is half of 130, which is 65 degrees. But I'm still confused because I thought the answer was 60 degrees. Maybe I made a mistake in my initial thought process. Let me try another approach. Since O is the circumcenter, triangles DOE, EOF, and FOD are all isosceles because OD, OE, and OF are radii. So, in triangle DOE, sides OD and OE are equal, so the base angles are equal. Similarly for the other triangles. Given angle DOF is 110 degrees, which is at point O between D and F. So, in triangle DOF, sides OD and OF are equal, so the base angles at D and F are equal. Let me denote angle ODF as x. Then, angle OFD is also x. So, the sum of angles in triangle DOF is 110 + x + x = 180. So, 2x = 70, so x = 35 degrees. Therefore, angle ODF is 35 degrees. Similarly, in triangle EOF, angle EOF is 130 degrees, so sides OE and OF are equal. Let me denote angle OEF as y. Then, angle OFE is also y. So, 130 + y + y = 180, so 2y = 50, so y = 25 degrees. Therefore, angle OEF is 25 degrees. Now, in triangle DEF, angle EDF is the angle at D. To find angle EDF, I can consider the angles at D from triangles DOF and DEF. Wait, angle EDF is part of triangle DEF, but I also have angle ODF from triangle DOF, which is 35 degrees. Similarly, at point E, angle DEF is part of triangle DEF, and angle OEF is 25 degrees from triangle EOF. Wait, maybe I can find angle EDF by considering the angles around point D. At point D, we have angle EDF, which is part of triangle DEF, and angle ODF, which is 35 degrees from triangle DOF. But I'm not sure how these relate directly. Alternatively, perhaps I can use the fact that the sum of angles in triangle DEF is 180 degrees. If I can find the other two angles, I can subtract them from 180 to find angle EDF. From triangle DEF, angle at E is angle DEF, which is subtended by arc DF, which is 110 degrees. So, angle DEF is half of 110, which is 55 degrees. Similarly, angle at F is angle DFE, which is subtended by arc DE, which is 120 degrees, so angle DFE is 60 degrees. Therefore, angle EDF is 180 - 55 - 60 = 65 degrees. So, that confirms my earlier calculation. Wait, but earlier I thought the answer was 60 degrees. Maybe I was confusing angle EDF with angle DFE. Let me make sure. Angle EDF is at D, so it's opposite arc EF, which is 130 degrees, so it's 65 degrees. Angle DFE is at F, opposite arc DE, which is 120 degrees, so it's 60 degrees. So, the measure of angle EDF is 65 degrees. But I'm still a bit confused because I thought the answer was 60 degrees. Maybe I need to double-check my calculations. Let me go through it again. The central angles are angle DOF = 110 degrees, angle EOF = 130 degrees. The remaining central angle is angle EOD = 360 - 110 - 130 = 120 degrees. Now, angle EDF is an inscribed angle subtended by arc EF, which is 130 degrees. So, angle EDF = 1/2 * 130 = 65 degrees. Alternatively, if I consider the triangle DEF, the sum of its angles should be 180 degrees. If angle EDF is 65 degrees, angle DEF is 55 degrees, and angle DFE is 60 degrees, then 65 + 55 + 60 = 180, which checks out. So, I think the correct measure of angle EDF is 65 degrees. Maybe I was mistaken earlier when I thought it was 60 degrees. Wait, but in the initial problem, the user provided an answer of 60 degrees. Let me see if that's correct. In the initial problem, the user wrote: "The point O is the center of the circle circumscribed about triangle DEF, with angle DOF = 110 degrees and angle EOF = 130 degrees. Determine the degree measure of angle EDF."Then, the user provided an answer: "Since angle DOF = 110 degrees and angle EOF = 130 degrees, the remaining central angle in the triangle can be found by: 360 - 110 - 130 = 120 degrees. Now, using the Central Angle Theorem, angle EDF = 1/2 * 120 = 60 degrees."Wait, so in this case, the user is considering angle EOD as 120 degrees, which is the central angle subtended by arc ED. Then, angle EDF is half of that, which is 60 degrees. But that would mean angle EDF is subtended by arc ED, which is not correct because angle EDF is at point D, so it's subtended by arc EF, not ED. So, I think the user made a mistake in their initial answer by considering the wrong arc. They took the central angle for arc ED (120 degrees) and halved it to get angle EDF, but angle EDF is subtended by arc EF, which is 130 degrees, so it should be 65 degrees. Therefore, the correct measure of angle EDF is 65 degrees, not 60 degrees. Wait, but let me make sure. Maybe I'm misapplying the inscribed angle theorem. Let me recall: the measure of an inscribed angle is half the measure of its intercepted arc. So, angle EDF intercepts arc EF, which is 130 degrees, so angle EDF is 65 degrees. Alternatively, if I consider the central angles, angle DOF = 110 degrees, angle EOF = 130 degrees, and angle EOD = 120 degrees. Then, in triangle DEF, the angles are half of these central angles, so angle DEF = 1/2 * 110 = 55 degrees, angle DFE = 1/2 * 120 = 60 degrees, and angle EDF = 1/2 * 130 = 65 degrees. Yes, that makes sense. So, the measure of angle EDF is 65 degrees. But wait, in the initial problem, the user provided an answer of 60 degrees, which is incorrect. So, I need to make sure that I'm not making the same mistake. Let me try to think of another way. Maybe using coordinates or trigonometry. Let me place point O at the origin. Let me assign coordinates to points D, E, F such that OD, OE, OF are radii of the circle. Let me assume the circle has radius 1 for simplicity. Let me place point D at (1, 0). Then, angle DOF is 110 degrees, so point F is at an angle of 110 degrees from D. So, the coordinates of F would be (cos(110°), sin(110°)). Similarly, angle EOF is 130 degrees, so point E is at an angle of 130 degrees from F. Wait, no, angle EOF is the central angle between E and F, so if F is at 110 degrees from D, then E is at 110 + 130 = 240 degrees from D? Wait, no, that might not be correct. Alternatively, since angle DOF is 110 degrees, and angle EOF is 130 degrees, the total angle around O is 360 degrees, so the remaining angle EOD is 120 degrees. So, point E is at 120 degrees from D. Wait, maybe I'm complicating things. Let me try to assign coordinates properly. Let me place point D at (1, 0). Then, since angle DOF is 110 degrees, point F is at an angle of 110 degrees from D, so its coordinates are (cos(110°), sin(110°)). Similarly, angle EOF is 130 degrees, so from point F, moving 130 degrees towards E. Wait, no, angle EOF is the central angle between E and F, so point E is at an angle of 110 + 130 = 240 degrees from D? Wait, no, that's not necessarily correct because the angles are at O, not along the circumference. Wait, maybe I should consider the positions of E and F relative to D. Let me think of point D at (1, 0). Then, angle DOF is 110 degrees, so point F is at an angle of 110 degrees from D, which would be at (cos(110°), sin(110°)). Similarly, angle EOF is 130 degrees, so from point F, the angle between E and F is 130 degrees. Wait, no, angle EOF is the central angle between E and F, so point E is at an angle of 110 + 130 = 240 degrees from D? Wait, no, that's not correct because the angles are measured at O, not along the circumference. Wait, perhaps I should consider the positions of E and F such that angle DOF is 110 degrees and angle EOF is 130 degrees. So, starting from D at (1, 0), moving 110 degrees to F, and then from F, moving another 130 degrees to E, but that would make the total angle from D to E as 110 + 130 = 240 degrees. But since the total around a circle is 360 degrees, the remaining angle from E back to D would be 360 - 240 = 120 degrees, which matches the earlier calculation of angle EOD being 120 degrees. So, point E is at 240 degrees from D, which is (cos(240°), sin(240°)) = (-0.5, -√3/2). Now, with points D at (1, 0), E at (-0.5, -√3/2), and F at (cos(110°), sin(110°)), I can calculate the coordinates and then find angle EDF. But this might be a bit involved, but let's try. First, let's find the coordinates of F. cos(110°) is approximately cos(110) ≈ -0.3420, and sin(110) ≈ 0.9397. So, F is approximately (-0.3420, 0.9397). Point E is at (-0.5, -√3/2) ≈ (-0.5, -0.8660). Point D is at (1, 0). Now, angle EDF is the angle at point D between points E and F. To find this angle, I can use the vectors from D to E and from D to F and find the angle between them. Vector DE is E - D = (-0.5 - 1, -0.8660 - 0) = (-1.5, -0.8660). Vector DF is F - D = (-0.3420 - 1, 0.9397 - 0) = (-1.3420, 0.9397). Now, the angle between vectors DE and DF can be found using the dot product formula: cos(theta) = (DE . DF) / (|DE| |DF|) First, calculate the dot product: DE . DF = (-1.5)(-1.3420) + (-0.8660)(0.9397) ≈ (1.5 * 1.3420) - (0.8660 * 0.9397) Calculating each term: 1.5 * 1.3420 ≈ 2.013 0.8660 * 0.9397 ≈ 0.813 So, DE . DF ≈ 2.013 - 0.813 ≈ 1.2 Now, calculate |DE| and |DF|: |DE| = sqrt((-1.5)^2 + (-0.8660)^2) = sqrt(2.25 + 0.75) = sqrt(3) ≈ 1.732 |DF| = sqrt((-1.3420)^2 + (0.9397)^2) ≈ sqrt(1.801 + 0.883) ≈ sqrt(2.684) ≈ 1.639 So, cos(theta) ≈ 1.2 / (1.732 * 1.639) ≈ 1.2 / 2.846 ≈ 0.421 Therefore, theta ≈ arccos(0.421) ≈ 65 degrees. So, this confirms that angle EDF is approximately 65 degrees. Wait, but earlier I thought the answer was 60 degrees, but now both methods give me 65 degrees. So, I think the initial answer provided by the user was incorrect. Therefore, the correct measure of angle EDF is 65 degrees. But wait, let me check one more time. Maybe I made a mistake in the coordinate calculation. Let me recalculate the dot product more accurately. DE = (-1.5, -0.8660) DF = (-1.3420, 0.9397) Dot product = (-1.5)(-1.3420) + (-0.8660)(0.9397) = 1.5 * 1.3420 + (-0.8660 * 0.9397) 1.5 * 1.3420 = 2.013 0.8660 * 0.9397 ≈ 0.813 So, dot product ≈ 2.013 - 0.813 = 1.2 |DE| = sqrt( (-1.5)^2 + (-0.8660)^2 ) = sqrt(2.25 + 0.75) = sqrt(3) ≈ 1.732 |DF| = sqrt( (-1.3420)^2 + (0.9397)^2 ) ≈ sqrt(1.801 + 0.883) ≈ sqrt(2.684) ≈ 1.639 So, cos(theta) = 1.2 / (1.732 * 1.639) ≈ 1.2 / 2.846 ≈ 0.421 arccos(0.421) ≈ 65 degrees. Yes, that seems correct. Alternatively, using the law of cosines in triangle DEF. Wait, but I don't have the lengths of the sides, but since all points are on a unit circle, I can calculate the distances. Wait, if the circle has radius 1, then the distance between D and E is 2 * sin(theta/2), where theta is the central angle. So, the distance DE is 2 * sin(120°/2) = 2 * sin(60°) = 2 * (√3/2) = √3 ≈ 1.732 Similarly, distance DF is 2 * sin(110°/2) = 2 * sin(55°) ≈ 2 * 0.8192 ≈ 1.638 Distance EF is 2 * sin(130°/2) = 2 * sin(65°) ≈ 2 * 0.9063 ≈ 1.812 Now, in triangle DEF, sides are DE ≈ 1.732, DF ≈ 1.638, EF ≈ 1.812 Using the law of cosines to find angle EDF: cos(angle EDF) = (DE^2 + DF^2 - EF^2) / (2 * DE * DF) Plugging in the values: DE^2 ≈ (1.732)^2 ≈ 3 DF^2 ≈ (1.638)^2 ≈ 2.684 EF^2 ≈ (1.812)^2 ≈ 3.283 So, cos(angle EDF) ≈ (3 + 2.684 - 3.283) / (2 * 1.732 * 1.638) Calculating numerator: 3 + 2.684 = 5.684; 5.684 - 3.283 = 2.401 Denominator: 2 * 1.732 * 1.638 ≈ 2 * 2.846 ≈ 5.692 So, cos(angle EDF) ≈ 2.401 / 5.692 ≈ 0.421 Therefore, angle EDF ≈ arccos(0.421) ≈ 65 degrees. This confirms again that angle EDF is approximately 65 degrees. So, I think the initial answer provided by the user was incorrect because they considered the wrong arc. They took the central angle for arc ED (120 degrees) and halved it to get angle EDF, but angle EDF is subtended by arc EF, which is 130 degrees, so it should be 65 degrees. Therefore, the correct measure of angle EDF is 65 degrees.

💡First, I need to determine June's biking rate based on the information provided. She can ride 2 miles in 8 minutes. To find her rate in minutes per mile, I'll divide the total time by the distance.Next, I'll use this rate to calculate the time it would take June to ride 6 miles to Bernard's house. By multiplying the distance by her biking rate, I can find the total time required for the trip.

💡Alright, so I've got this geometry problem here about planes and lines in space. Let me try to wrap my head around it step by step. The problem says:"In space, α, β represent planes, and m represents a line. Given α ∩ β = l, which of the following statements is correct?A: If m is parallel to l, then m is parallel to both α and β.B: If m is parallel to both α and β, then m is parallel to l.C: If m is on a different plane from l, then m intersects both α and β.D: If m intersects both α and β, then m is on a different plane from l."Okay, so first off, α and β are two planes that intersect along the line l. So, l is the line of intersection of these two planes. Now, we have a line m, and we need to evaluate these four statements about its relationship to α, β, and l.Let me start by recalling some basic concepts in three-dimensional geometry. When two planes intersect, their intersection is a line. So, l is that line. Now, a line can be parallel to a plane, lie on the plane, or intersect the plane at a point.Let's tackle each statement one by one.**Statement A: If m is parallel to l, then m is parallel to both α and β.**Hmm, okay. So, if line m is parallel to line l, which is the intersection of α and β, does that mean m is parallel to both planes α and β? Well, let's think about it. If m is parallel to l, then m doesn't intersect l. But since l is in both α and β, does that mean m doesn't intersect α or β? Not necessarily. Because m could be parallel to l but still lie in one of the planes, say α, but then it wouldn't be parallel to α. Wait, that doesn't make sense. If m is in α, then it's not parallel to α. So, if m is parallel to l, it can't lie in either α or β because l is in both. So, m must be parallel to both planes. Hmm, maybe Statement A is correct.But wait, hold on. If m is parallel to l, it could also be that m is parallel to one plane but not the other. Or maybe it's parallel to both. I'm not entirely sure. Let me visualize this. Imagine two planes intersecting along line l. If I have another line m that's parallel to l, it could be in a different plane altogether, not intersecting either α or β. Or it could be in one of the planes, but then it wouldn't be parallel to that plane. So, if m is parallel to l and not in either plane, then it's parallel to both α and β. But if m is in one of the planes, it's not parallel to that plane. So, the statement says "if m is parallel to l, then m is parallel to both α and β." But m could be in one of the planes, so it's not necessarily parallel to both. Therefore, Statement A is not necessarily true.**Statement B: If m is parallel to both α and β, then m is parallel to l.**Alright, so if m is parallel to both planes α and β, does that mean it's parallel to their line of intersection l? Let's think. If m is parallel to α, it doesn't intersect α, and similarly, it doesn't intersect β. Since l is the intersection of α and β, m doesn't intersect l either because it doesn't intersect either plane. So, m must be parallel to l. That makes sense. So, Statement B seems correct.**Statement C: If m is on a different plane from l, then m intersects both α and β.**Okay, so if m is on a different plane from l, does that mean it intersects both α and β? Hmm. Let's see. If m is on a different plane, say γ, which is not α or β, then m could intersect α and β at some points. But it's not necessarily the case. For example, m could be parallel to both α and β, in which case it wouldn't intersect either. Or m could intersect one plane and be parallel to the other. So, just because m is on a different plane from l doesn't guarantee that it intersects both α and β. Therefore, Statement C is not necessarily true.**Statement D: If m intersects both α and β, then m is on a different plane from l.**Alright, so if m intersects both α and β, does that mean it's on a different plane from l? Well, l is the intersection of α and β, so any line that intersects both α and β could either lie on l or be on a different plane. If m intersects both α and β but is not l itself, then it must lie on a different plane. But wait, could m intersect both α and β and still lie on the same plane as l? If m lies on the same plane as l, which is either α or β, then m would intersect l, right? But l is the intersection of α and β, so if m is on α and intersects β, it must intersect l. Similarly, if m is on β and intersects α, it must intersect l. So, if m intersects both α and β, it could either intersect l or not. If it doesn't intersect l, then it must be on a different plane. But if it does intersect l, then it's on the same plane as l. So, the statement says "if m intersects both α and β, then m is on a different plane from l." But m could intersect both α and β and still be on the same plane as l, by intersecting l. Therefore, Statement D is not necessarily true.So, after going through each statement, it seems like only Statement B is necessarily correct.

💡Okay, so I have this problem about a parallelogram. The sides are 3s and s units long, and they form a 30-degree angle. The area is given as 9 times the square root of 3. I need to find the value of s. Hmm, let's see. First, I remember that the area of a parallelogram is calculated by the formula: base times height. But wait, since it's a parallelogram, the height isn't just the other side; it's the side multiplied by the sine of the angle between them. So, maybe the formula is base times side times sine of the angle. Let me write that down.So, area = base × side × sin(angle). In this case, the base is 3s, the side is s, and the angle is 30 degrees. Plugging those in, the area should be 3s × s × sin(30°). I know that sin(30°) is 0.5, so that simplifies things a bit. Let me calculate that. 3s times s is 3s², and then multiplied by 0.5 is 1.5s². So, the area is 1.5s². But the area is given as 9√3. So, setting up the equation: 1.5s² = 9√3.Hmm, to solve for s², I can divide both sides by 1.5. Let's see, 9 divided by 1.5 is 6, so s² = 6√3. Then, to find s, I take the square root of both sides. So, s = √(6√3). Wait, that seems a bit complicated. Maybe I can simplify √(6√3) further. Let me think. I know that √(a√b) can be written as (a√b)^(1/2), which is a^(1/2) × b^(1/4). So, applying that here, √(6√3) is equal to 6^(1/2) × 3^(1/4). But 6 is 2 times 3, so 6^(1/2) is √6, and 3^(1/4) is the fourth root of 3. So, putting it all together, s = √6 × ³√3^(1/4). Wait, no, that's not quite right. Let me correct that. It's √6 times the fourth root of 3, which can be written as 6^(1/2) × 3^(1/4). Alternatively, I can express this as 3^(1/4) × √6. That seems like a more standard way to write it. So, s = 3^(1/4) × √6. Let me double-check my steps to make sure I didn't make a mistake. Starting from the area formula: base × side × sin(angle). Plugging in the values: 3s × s × sin(30°). Sin(30°) is 0.5, so that gives 1.5s². Setting that equal to 9√3, solving for s² gives 6√3, and then s is the square root of that. Yes, that seems right. So, s = √(6√3), which simplifies to 3^(1/4) × √6. I think that's the simplest radical form. Wait, another thought: maybe I can rationalize or express it differently? Let's see. If I write √(6√3) as (6√3)^(1/2), that's the same as 6^(1/2) × (√3)^(1/2). Which is √6 × (3)^(1/4). Yeah, that's consistent with what I had before. So, I think that's as simplified as it gets. Therefore, the value of s is 3^(1/4) times the square root of 6.

💡First, I need to determine how many months after Praveen started the business Hari joined as a partner. I know that Praveen started with Rs. 3500 and Hari joined later with Rs. 9000. The profit after one year is divided in the ratio 2:3, which means the ratio of their investments should also be 2:3.Praveen's investment is for the entire year, so his investment is 3500 multiplied by 12 months. Hari's investment is for (12 - x) months, where x is the number of months after Praveen started that Hari joined.Setting up the ratio of their investments:(3500 * 12) / (9000 * (12 - x)) = 2 / 3Simplifying this equation:(3500 * 12) / (9000 * (12 - x)) = 2 / 3Cross-multiplying to solve for x:(3500 * 12) * 3 = (9000 * (12 - x)) * 2Calculating the values:3500 * 36 = 18000 * (12 - x)Simplifying further:126000 = 18000 * (12 - x)Dividing both sides by 18000:7 = 12 - xSolving for x:x = 12 - 7x = 5Therefore, Hari joined the business 5 months after Praveen started.

💡Okay, so I need to figure out what binom{n}{2} is for any positive integer n. I remember that binomial coefficients are used in combinations, right? So, binom{n}{k} represents the number of ways to choose k elements from a set of n elements without considering the order. In this case, k is 2, so I'm looking at how many ways I can choose 2 items from n items.First, let me recall the formula for combinations. The general formula for binom{n}{k} is:binom{n}{k} = frac{n!}{k!(n - k)!}So, substituting k = 2 into this formula, I get:binom{n}{2} = frac{n!}{2!(n - 2)!}Now, I need to simplify this expression. I know that n! means the factorial of n, which is the product of all positive integers up to n. Similarly, 2! is just 2 × 1, which equals 2. The (n - 2)! term is the factorial of (n - 2).Let me write out the factorials to see if I can cancel out some terms. So, n! is n × (n - 1) × (n - 2) × ... × 1, and (n - 2)! is (n - 2) × (n - 3) × ... × 1. When I divide n! by (n - 2)!, most of the terms cancel out:frac{n!}{(n - 2)!} = n × (n - 1)So, substituting this back into the combination formula, I have:binom{n}{2} = frac{n × (n - 1)}{2!}Since 2! = 2, this simplifies to:binom{n}{2} = frac{n × (n - 1)}{2}Let me check if this makes sense. If I choose 2 items from n, the number of ways should be the same as the number of unique pairs I can form. For example, if n = 3, then binom{3}{2} = 3, which matches the formula because frac{3 × 2}{2} = 3. Similarly, if n = 4, then binom{4}{2} = 6, and frac{4 × 3}{2} = 6. This seems to hold true.I also remember that binom{n}{2} is the formula for the number of edges in a complete graph with n vertices, which makes sense because each edge connects two vertices, and there are frac{n(n - 1)}{2} such connections.Another way to think about it is that for each of the n items, there are (n - 1) other items it can pair with. However, since each pair is counted twice (once for each item in the pair), we divide by 2 to avoid double-counting. This reasoning also leads to the formula frac{n(n - 1)}{2}.Let me test this with a small value of n to ensure it's correct. Suppose n = 5. Then, binom{5}{2} = frac{5 × 4}{2} = 10. Listing out all possible pairs: (1,2), (1,3), (1,4), (1,5), (2,3), (2,4), (2,5), (3,4), (3,5), (4,5). That's 10 pairs, which matches the formula.I think this makes sense now. The formula frac{n(n - 1)}{2} correctly calculates the number of ways to choose 2 items from n without regard to order, which is exactly what binom{n}{2} represents.