💡First, let's denote the radius of the larger circle as ( R ) and the radius of the smaller circle as ( r ). According to the problem, ( r = frac{R}{2} ).The longest chord of a circle is the diameter. Therefore, the diameter of the larger circle is 24, which means its radius ( R ) is half of that:[R = frac{24}{2} = 12]Since the smaller circle is tangent to the midpoint of the chord of the larger circle, it will fit exactly into the larger circle, touching the larger circle at one point (since they are externally tangent) and the chord at its midpoint.Thus, the radius of the larger circle is ( 12 ) units.

💡First, I need to establish the relationships between the variables x, y, and z based on the given information.I know that y is 40% less than z. This means that y is 60% of z. Mathematically, this can be expressed as:[ y = 0.60z ]Additionally, it's given that x is 78% of z:[ x = 0.78z ]Now, I want to find the percentage by which x is more than y. To do this, I'll set up the equation:[ x = y + left(frac{P}{100}right)y ]Substituting the expressions for x and y in terms of z into this equation:[ 0.78z = 0.60z + left(frac{P}{100}right)0.60z ]Next, I'll simplify the equation by dividing both sides by 0.60z:[ frac{0.78z}{0.60z} = 1 + frac{P}{100} ][ 1.30 = 1 + frac{P}{100} ]Subtracting 1 from both sides to isolate the term with P:[ 0.30 = frac{P}{100} ]Finally, I'll solve for P by multiplying both sides by 100:[ P = 30 ]This means that x is 30% more than y.

💡Alright, so I've got this geometry problem here, and I need to prove that the ratio |CD| : |DE| doesn't depend on the choice of the secant through point A. Let me try to unpack this step by step.First, let me visualize the problem. There are three circles, k₁, k₂, and k₃, all passing through points A and B. So, A and B are common points to all three circles. Now, there's a secant line passing through A, and this secant intersects each of the circles again at points C, D, and E respectively. So, starting from A, the secant goes out and hits k₁ at C, then k₂ at D, and finally k₃ at E. The goal is to show that the ratio of the lengths CD to DE remains constant, no matter how we choose the secant line through A. That is, even if we rotate the secant around A, changing where it intersects the circles, the ratio |CD| : |DE| stays the same. Interesting.I remember that when dealing with circles and secants, power of a point is a useful concept. The power of a point with respect to a circle is defined as the product of the lengths from that point to the points where a line through it intersects the circle. So, for a point P outside a circle, the power is PA * PB, where PA and PB are the lengths from P to the points of intersection A and B on the circle.In this case, since all three circles pass through A and B, maybe I can use the power of point A with respect to each circle. But wait, A is on all three circles, so the power of A with respect to each circle would be zero, right? Because if a point is on the circle, its power is zero. Hmm, that might not be directly helpful.Alternatively, maybe I should consider another point. Let's think about point D. Point D is on both k₂ and k₁? Wait, no, D is on k₂, and C is on k₁, E is on k₃. So, D is on k₂, but not necessarily on k₁ or k₃. Hmm.Wait, maybe I should think about the power of point D with respect to k₁ and k₃. Since D is on k₂, which passes through A and B, perhaps there's a relationship there.Let me recall that if two circles are intersecting, the line through their intersection points is called the radical axis. The radical axis is the set of points that have equal power with respect to both circles. So, for circles k₁ and k₂, their radical axis is line AB, since both circles pass through A and B. Similarly, for k₂ and k₃, the radical axis is also AB, and for k₁ and k₃, it's AB as well. So, all three circles share the same radical axis, AB.This means that AB is the radical axis for all pairs of these circles. Therefore, any point on AB has equal power with respect to all three circles. But how does that help me with points C, D, and E?Wait, points C, D, and E are all on the same secant line through A. So, they lie on a line that passes through A, which is one of the intersection points of all three circles. Maybe I can use the power of point A with respect to each circle, but since A is on each circle, its power is zero. That doesn't seem immediately helpful.Perhaps I should consider the power of another point. Let me think about point D. Since D is on k₂, the power of D with respect to k₂ is zero. But what about its power with respect to k₁ and k₃?The power of D with respect to k₁ would be DA * DC, because D lies on the secant that intersects k₁ at A and C. Similarly, the power of D with respect to k₃ would be DA * DE, since the same secant intersects k₃ at A and E.So, power of D with respect to k₁ is DA * DC, and power with respect to k₃ is DA * DE. But since D is on k₂, which also passes through A and B, the power of D with respect to k₂ is zero. Wait, but all three circles share the radical axis AB, so the power of D with respect to k₁ and k₂ should be equal? Or is it the other way around? Let me recall: the radical axis is the set of points with equal power with respect to both circles. So, for any point on AB, the power with respect to k₁ and k₂ is equal. But D is not necessarily on AB, unless the secant is AB itself, which is a special case.Hmm, maybe I need a different approach. Let me think about inversion. Inversion is a powerful tool in circle geometry, but I'm not sure if it's necessary here. Maybe I can use similar triangles or some properties of intersecting chords.Wait, let's consider the power of point D with respect to k₁ and k₃. As I mentioned earlier, the power with respect to k₁ is DA * DC, and with respect to k₃ is DA * DE. Since D is on k₂, which also passes through A and B, perhaps there's a relationship between these powers.But how? Maybe if I can relate the power of D with respect to k₁ and k₃ through some constant factor.Alternatively, maybe I can use cross ratios or something related to projective geometry. Since all the points lie on a secant line through A, which is a common point, perhaps the cross ratio is preserved.Wait, cross ratio is invariant under projection, so maybe that's the key here. If I consider the cross ratio of points A, C, D, E on the secant line, it might be related to some fixed points on the circles.But I'm not sure. Let me try to write down the power expressions more formally.Let me denote:- For circle k₁: Power of D with respect to k₁ is DA * DC.- For circle k₂: Power of D with respect to k₂ is zero, since D is on k₂.- For circle k₃: Power of D with respect to k₃ is DA * DE.But since all three circles share the radical axis AB, the power of D with respect to k₁ and k₂ must satisfy some condition. Wait, actually, the radical axis AB is the locus of points with equal power with respect to k₁ and k₂. So, for any point on AB, power with respect to k₁ equals power with respect to k₂. But D is not on AB, unless the secant is AB itself.Hmm, maybe I'm overcomplicating this. Let me think about the ratio CD : DE.I need to find |CD| / |DE| and show that it's constant, regardless of the secant.Let me express CD and DE in terms of DA, DC, and DE.Wait, CD is the distance from C to D, which is |DC|, and DE is the distance from D to E, which is |DE|. So, the ratio is |DC| / |DE|.But from the power of point D with respect to k₁ and k₃, we have:Power(D, k₁) = DA * DCPower(D, k₃) = DA * DESo, the ratio of the powers is (DA * DC) / (DA * DE) = DC / DE.Therefore, DC / DE = Power(D, k₁) / Power(D, k₃)So, if I can show that Power(D, k₁) / Power(D, k₃) is constant, regardless of the position of D, then DC / DE is constant.But how can I show that Power(D, k₁) / Power(D, k₃) is constant?Since D lies on k₂, which passes through A and B, maybe there's a relationship between the powers of D with respect to k₁, k₂, and k₃.Wait, since D is on k₂, Power(D, k₂) = 0.But Power(D, k₁) and Power(D, k₃) can be related through the radical axis.Wait, the radical axis of k₁ and k₂ is AB, so for any point on AB, Power(D, k₁) = Power(D, k₂). But D is not on AB, unless the secant is AB.Hmm, maybe I need to use the fact that all three circles are coaxial. Coaxial circles share the same radical axis. So, since all three circles pass through A and B, they are coaxial with radical axis AB.In that case, for any point D not on AB, the power with respect to each circle can be expressed as:Power(D, k₁) = PA * PB, where PA and PB are the lengths from D to A and B.Wait, no, that's not quite right. The power of D with respect to a circle is equal to the square of the tangent from D to the circle. But since all circles pass through A and B, the power of D with respect to each circle is equal to DA * DB, because DA and DB are the lengths from D to the points where the secant intersects the circle.Wait, actually, for any circle passing through A and B, the power of D with respect to that circle is DA * DB, because DA and DB are the lengths from D to the intersection points A and B.But hold on, if that's the case, then for all three circles k₁, k₂, k₃, the power of D is DA * DB. But that can't be, because D is on k₂, so Power(D, k₂) = 0. But DA * DB is not zero unless D is A or B.Wait, I think I'm confusing something here. Let me clarify.The power of a point D with respect to a circle is defined as the square of the length of the tangent from D to the circle. If D lies on the circle, then the power is zero. If D is outside, it's positive, and if D is inside, it's negative.In this case, since all three circles pass through A and B, the power of D with respect to each circle can be expressed as DA * DB, but only if the line AB is the radical axis. Wait, no, that's not quite accurate.Actually, for any circle passing through A and B, the power of a point D with respect to that circle is equal to DA * DB if and only if D lies on the radical axis of the circle and some other circle. Wait, this is getting confusing.Let me try a different approach. Let me consider the power of point D with respect to k₁ and k₃.Power(D, k₁) = DA * DCPower(D, k₃) = DA * DESo, the ratio Power(D, k₁) / Power(D, k₃) = (DA * DC) / (DA * DE) = DC / DESo, DC / DE = Power(D, k₁) / Power(D, k₃)Now, since all three circles are coaxial with radical axis AB, the power of D with respect to each circle can be expressed in terms of DA and DB.Wait, for any circle in the coaxial system, the power of D is given by DA * DB + c, where c is a constant depending on the circle. But since all three circles pass through A and B, the power of D with respect to each circle is DA * DB.Wait, no, that can't be, because if D is on k₂, then Power(D, k₂) = 0, but DA * DB is not necessarily zero.Wait, perhaps I need to recall that for coaxial circles, the power of a point with respect to each circle differs by a constant. So, if I have two circles in a coaxial system, the power of a point D with respect to the first circle is equal to the power with respect to the second circle plus some constant.But in this case, all three circles pass through A and B, so they are coaxial with radical axis AB. Therefore, for any point D not on AB, the power with respect to each circle can be expressed as:Power(D, k₁) = DA * DB + c₁Power(D, k₂) = DA * DB + c₂Power(D, k₃) = DA * DB + c₃But since D is on k₂, Power(D, k₂) = 0. Therefore,0 = DA * DB + c₂ => c₂ = -DA * DBSimilarly, for k₁ and k₃,Power(D, k₁) = DA * DB + c₁Power(D, k₃) = DA * DB + c₃But c₁ and c₃ are constants for each circle, independent of D.Wait, but D is a variable point depending on the secant. So, c₁ and c₃ must be constants that don't depend on D.Therefore, the ratio Power(D, k₁) / Power(D, k₃) = (DA * DB + c₁) / (DA * DB + c₃)But DA * DB is the same for all three circles, right? Because all three circles pass through A and B, so for any point D, DA * DB is fixed?Wait, no, DA and DB depend on the position of D on the secant. So, DA * DB varies as D moves along the secant.But wait, if I consider the ratio Power(D, k₁) / Power(D, k₃) = (DA * DB + c₁) / (DA * DB + c₃)If I can show that this ratio is constant, independent of DA * DB, then DC / DE is constant.But how?Wait, let me think about the constants c₁ and c₃. Since k₁, k₂, and k₃ are coaxial, the constants c₁, c₂, c₃ are related to the distances from the centers of the circles to the radical axis AB.But I'm not sure if that helps directly.Alternatively, maybe I can use the fact that the ratio of the powers is equal to the ratio of the lengths DC / DE, which we need to show is constant.Wait, let me consider two different secants through A, say secant 1 and secant 2, intersecting the circles at C₁, D₁, E₁ and C₂, D₂, E₂ respectively.For secant 1:Power(D₁, k₁) = DA₁ * DC₁Power(D₁, k₃) = DA₁ * DE₁So, DC₁ / DE₁ = Power(D₁, k₁) / Power(D₁, k₃)Similarly, for secant 2:Power(D₂, k₁) = DA₂ * DC₂Power(D₂, k₃) = DA₂ * DE₂So, DC₂ / DE₂ = Power(D₂, k₁) / Power(D₂, k₃)If I can show that Power(D₁, k₁) / Power(D₁, k₃) = Power(D₂, k₁) / Power(D₂, k₃), then DC₁ / DE₁ = DC₂ / DE₂, meaning the ratio is constant.So, how can I show that Power(D, k₁) / Power(D, k₃) is the same for any D on any secant through A?Well, since all three circles are coaxial, the power of any point D with respect to each circle can be expressed as:Power(D, k₁) = DA * DB + c₁Power(D, k₃) = DA * DB + c₃But since D is on the secant through A, DA is the distance from D to A along the secant, and DB is the distance from D to B, which is fixed if B is fixed.Wait, no, DB is not fixed because D is moving along the secant, so DA and DB both vary as D moves.But wait, if I consider the ratio Power(D, k₁) / Power(D, k₃) = (DA * DB + c₁) / (DA * DB + c₃)If this ratio is constant for all D on the secant, then DC / DE is constant.But how can (DA * DB + c₁) / (DA * DB + c₃) be constant?Let me denote x = DA * DB. Then the ratio becomes (x + c₁) / (x + c₃). For this to be constant for all x, the function (x + c₁)/(x + c₃) must be constant, which is only possible if c₁ = c₃, which would make the ratio 1, or if the function is a constant function, which it isn't unless c₁ = c₃.But that can't be, because k₁ and k₃ are different circles, so their constants c₁ and c₃ are different.Wait, maybe I'm missing something here. Let me think differently.Since all three circles are coaxial, the centers lie on a line perpendicular to AB. Let me denote this line as the line of centers. The radical axis is AB, so the line of centers is perpendicular to AB.Now, for any point D on the secant through A, the power with respect to each circle is given by:Power(D, k₁) = DA * DCPower(D, k₂) = 0 (since D is on k₂)Power(D, k₃) = DA * DEBut since k₁, k₂, k₃ are coaxial, the power of D with respect to k₁ and k₃ can be related through the line of centers.Wait, maybe I can use the fact that the power difference is constant.Power(D, k₁) - Power(D, k₂) = constantBut Power(D, k₂) = 0, so Power(D, k₁) = constant.Similarly, Power(D, k₃) = constant.But that can't be, because Power(D, k₁) = DA * DC, which varies as D moves.Wait, perhaps the difference in powers is constant.Power(D, k₁) - Power(D, k₃) = constantBut Power(D, k₁) - Power(D, k₃) = DA * DC - DA * DE = DA (DC - DE)But DC - DE is equal to CD - DE = -ED - DE = -2DE if CD = DE, but that's not necessarily the case.Hmm, this seems messy.Wait, maybe I should consider the cross ratio. Since all points lie on a line through A, and we're dealing with ratios of segments, cross ratio might be preserved.The cross ratio of four points A, C, D, E on a line is given by (AC/AD) / (AE/AD) = (AC/AE) * (AD/AD) = AC/AE.Wait, no, cross ratio is (AC/AD) / (AE/AD) = AC/AE.But cross ratio is preserved under projection, but I'm not sure how that helps here.Alternatively, maybe I can use Menelaus' theorem or something similar.Wait, Menelaus' theorem applies to transversals cutting through triangles, but I'm not sure if that's directly applicable here.Let me think about similar triangles. If I can find two similar triangles that relate CD and DE, then their ratio would be preserved.Wait, perhaps triangles formed by the centers of the circles and points A, B, C, D, E.But I don't know the positions of the centers, so that might not be straightforward.Wait, another idea: since all three circles pass through A and B, the angles subtended by AB at points C, D, E are related.Specifically, the angle ACB is equal to the angle ADB, which is equal to the angle AEB, because they all subtend the same chord AB.Wait, no, that's not necessarily true unless the points are on the same circle. But C is on k₁, D on k₂, E on k₃, which are different circles.Hmm, maybe not.Wait, but all three circles are coaxial, so they share the same radical axis AB. Therefore, the angles subtended by AB at any point on the radical axis are equal. But D is not on AB, so that might not help.Wait, perhaps I can use the property that for any point D, the angles ∠ADB is equal for all circles, but I'm not sure.Alternatively, maybe I can use the fact that the cross ratio (A, B; C, E) is constant, but I'm not sure.Wait, let me try to think about inversion. If I invert the figure with respect to a circle centered at A, then the circles k₁, k₂, k₃ passing through A will invert to lines, since inversion maps circles through the center to lines.Let me try that. Let's invert the figure with respect to a circle centered at A with some radius r. After inversion, circles k₁, k₂, k₃ passing through A will become straight lines, since inversion maps circles through the center to lines.Let me denote the images after inversion as follows:- Circle k₁ inverts to line k₁'- Circle k₂ inverts to line k₂'- Circle k₃ inverts to line k₃'The point B, which is on all three circles, will invert to some point B' on all three lines k₁', k₂', k₃'.The secant through A, which intersects k₁ at C, k₂ at D, and k₃ at E, will invert to a circle passing through A (which inverts to infinity, since we're inverting with respect to A) and the other points C', D', E'.Wait, actually, inversion maps lines not passing through A to circles passing through A, and lines passing through A to lines.But in this case, the secant passes through A, so after inversion, it will still be a line.Wait, no, inversion maps lines not passing through A to circles passing through A, and lines passing through A to lines.So, the secant passes through A, so after inversion, it remains a line.Therefore, the images of points C, D, E on the secant will lie on a straight line after inversion.Moreover, since k₁, k₂, k₃ invert to lines k₁', k₂', k₃', and the secant inverts to a line, the intersections C', D', E' will lie on both the inverted secant line and the inverted circles (which are lines).Therefore, C' is the intersection of the inverted secant with k₁', D' with k₂', and E' with k₃'.Now, since k₁', k₂', k₃' are lines, and they all pass through B', the inverted point of B.So, we have three lines k₁', k₂', k₃' passing through B', and a secant line passing through A' (which is at infinity) intersecting these lines at C', D', E'.Wait, but A inverts to infinity, so the inverted secant is a line passing through infinity, which is just a straight line.So, in the inverted figure, we have three lines k₁', k₂', k₃' passing through B', and a line (the inverted secant) intersecting them at C', D', E'.Now, the ratio |CD| : |DE| in the original figure corresponds to some ratio in the inverted figure.But since inversion preserves cross ratios, the cross ratio of four points on a line is preserved.Wait, but in this case, we have three points C', D', E' on a line, and we're interested in the ratio |C'D'| : |D'E'|.But cross ratio involves four points, so maybe I need to include another point.Alternatively, since inversion preserves ratios of lengths along a line when the center of inversion is not on the line. Wait, in this case, the center of inversion is A, which inverts to infinity, so the line of the secant passes through A, which is the center.Wait, inversion from a center on the line will not preserve ratios, because inversion from a point on a line maps the line to itself, but scales the distances.Wait, actually, inversion from a point on a line maps the line to itself, but inverts the orientation and scales the distances.So, if I invert with respect to A, which is on the secant line, then the secant line maps to itself, but points on the secant are inverted with respect to A.Therefore, the distances from A are inverted, meaning that if a point P is at distance x from A, its image P' is at distance r² / x from A, where r is the radius of inversion.Therefore, the ratio of distances along the secant is inverted, but the cross ratio might still be preserved.Wait, cross ratio is preserved under inversion, so the cross ratio of four points on a line is the same before and after inversion.But in our case, we have three points C, D, E on the secant, and A is the center of inversion. So, the cross ratio (A, C; D, E) is preserved.But cross ratio is defined as (AC/AD) / (AE/AD) = (AC/AE).Wait, no, cross ratio is (AC/AD) / (AE/AD) = (AC/AE).But in our case, since A is the center of inversion, the cross ratio might not be directly applicable.Wait, maybe I'm overcomplicating this again.Let me think about the inverted figure. After inversion, the three circles become lines k₁', k₂', k₃' passing through B', and the secant becomes a line passing through infinity (i.e., a straight line) intersecting these lines at C', D', E'.Now, in the inverted figure, the ratio |C'D'| : |D'E'| corresponds to the ratio |CD| : |DE| in the original figure, scaled by some factor due to inversion.But since inversion preserves the cross ratio, the ratio |C'D'| : |D'E'| is equal to |CD| : |DE| multiplied by some constant factor depending on the inversion.But since we're only interested in the ratio, not the absolute lengths, the scaling factor cancels out, so the ratio |C'D'| : |D'E'| is equal to |CD| : |DE|.Therefore, if I can show that in the inverted figure, the ratio |C'D'| : |D'E'| is constant, then the original ratio is also constant.Now, in the inverted figure, we have three lines k₁', k₂', k₃' passing through B', and a secant line intersecting them at C', D', E'. The ratio |C'D'| : |D'E'| is the ratio of segments on the secant line between the intersections with k₁', k₂', k₃'.But since k₁', k₂', k₃' are fixed lines passing through B', and the secant is any line through infinity (i.e., any direction), the ratio |C'D'| : |D'E'| should be constant, because the positions of C', D', E' are determined by the fixed lines k₁', k₂', k₃'.Wait, but how?Let me think about the pencil of lines from B'. The three lines k₁', k₂', k₃' form a pencil from B'. The secant line intersects this pencil at C', D', E'. The ratio |C'D'| : |D'E'| is determined by the angles between the lines in the pencil.But since the pencil is fixed, the ratio should be constant, regardless of the direction of the secant.Wait, that makes sense. Because the pencil of lines from B' is fixed, the cross ratio of the intersections with any secant line is constant.Therefore, the ratio |C'D'| : |D'E'| is constant, which implies that |CD| : |DE| is constant in the original figure.Therefore, the ratio |CD| : |DE| does not depend on the choice of the secant through A.I think that makes sense. By inverting the figure, we transformed the problem into one involving lines and their intersections, where the ratio of segments is preserved as a cross ratio, which is invariant under projection. Therefore, the ratio remains constant regardless of the secant chosen.So, putting it all together, the key steps are:1. Recognize that the three circles are coaxial with radical axis AB.2. Use inversion to transform the circles into lines, simplifying the problem.3. Observe that the ratio of segments on the inverted secant corresponds to the original ratio.4. Use the invariance of cross ratio under projection to conclude that the ratio is constant.Therefore, the ratio |CD| : |DE| is indeed constant, independent of the secant chosen through A.

💡Okay, so I need to round the number 24.6375 to the nearest tenth. Hmm, let me think about how to do this step by step. First, I remember that rounding to the nearest tenth means I need to look at the number in the tenths place and decide whether to keep it the same or round it up based on the digit that follows.Alright, let's break down the number 24.6375. The number is 24.6375, so the digits after the decimal point are 6, 3, 7, and 5. The tenths place is the first digit after the decimal, which is 6. The next digit is the hundredths place, which is 3. Now, I recall that when rounding to the nearest tenth, I should look at the hundredths digit to decide whether to round up or keep the tenths digit the same.So, the rule is: if the hundredths digit is 5 or greater, I round the tenths digit up by one. If it's less than 5, I leave the tenths digit as it is. In this case, the hundredths digit is 3, which is less than 5. That means I don't need to round up the tenths digit. Therefore, the tenths digit remains 6.Wait, but there are more digits after the hundredths place—7 and 5. Do I need to consider those? I think once I've looked at the hundredths digit to decide whether to round up the tenths place, the digits beyond that don't affect the rounding anymore. So, even though there's a 7 and a 5 after the 3, they don't change the fact that the hundredths digit is 3, which is less than 5.So, putting it all together, since the hundredths digit is 3, which is less than 5, I keep the tenths digit as 6 and drop all the digits that follow. Therefore, 24.6375 rounded to the nearest tenth is 24.6.Let me double-check to make sure I didn't make any mistakes. The number is 24.6375. Tenths place is 6, hundredths is 3. Since 3 is less than 5, I don't round up. So, yes, it should be 24.6. I think that's correct.

💡Okay, so I have this probability problem here. It says that events A and B are independent, P(A) is 0.6, and P(AB) is 0.42. I need to find P(A+B). Hmm, let me think about how to approach this.First, I remember that when two events are independent, the probability of both happening is the product of their individual probabilities. So, P(AB) should be equal to P(A) times P(B). They've given me P(AB) as 0.42 and P(A) as 0.6. Maybe I can use this to find P(B).Let me write that down: P(AB) = P(A) * P(B). Plugging in the numbers, 0.42 = 0.6 * P(B). To find P(B), I can divide both sides by 0.6. So, P(B) = 0.42 / 0.6. Let me calculate that. 0.42 divided by 0.6 is... 0.7. Okay, so P(B) is 0.7.Now, I need to find P(A+B). I think P(A+B) refers to the probability of either A or B happening, or both. I remember there's a formula for that. It's P(A) + P(B) - P(AB). This makes sense because if I just add P(A) and P(B), I'm double-counting the overlap where both A and B happen, so I need to subtract that once.So, plugging in the numbers: P(A+B) = P(A) + P(B) - P(AB). That would be 0.6 + 0.7 - 0.42. Let me do the math. 0.6 plus 0.7 is 1.3, and then subtracting 0.42 gives me 0.88. So, P(A+B) should be 0.88.Wait, let me double-check to make sure I didn't make any mistakes. I found P(B) by dividing 0.42 by 0.6, which is 0.7. Then, using the formula for the union of two events, I added 0.6 and 0.7 to get 1.3, and subtracted the overlap of 0.42 to get 0.88. That seems right.I also remember that for independent events, knowing one doesn't affect the probability of the other. So, since they're independent, the calculations should hold. I don't think I missed anything here. Yeah, I think 0.88 is the correct answer.

💡First, I need to determine the maximum marks based on the information provided. The student needs to obtain 33% of the total marks to pass. He scored 125 marks and fell short by 73 marks.I'll start by defining the maximum marks as ( M ).The passing marks are 33% of ( M ), which can be expressed as:[0.33M]The student's score plus the marks he fell short by equals the passing marks:[125 + 73 = 0.33M]Simplifying the left side:[198 = 0.33M]To find ( M ), I'll divide both sides by 0.33:[M = frac{198}{0.33} = 600]Therefore, the maximum marks are 600.