💡Okay, so I have this problem about clock hands and interesting moments. Let me try to understand what it's asking. An interesting moment X is when there exists another moment Y such that the hour hand at Y is where the minute hand is at X, and the minute hand at Y is where the hour hand is at X. Hmm, that sounds a bit abstract. Maybe I can visualize it.First, I know that the hour hand moves slower than the minute hand. The hour hand completes a full circle (360 degrees) in 12 hours, while the minute hand does it in 1 hour. So, their speeds are different. The hour hand moves at 30 degrees per hour, and the minute hand moves at 360 degrees per hour.Let me denote the time X as t hours past 12:00. Similarly, Y would be some other time, say s hours past 12:00. At time X, the hour hand is at 30t degrees, and the minute hand is at 360(t - floor(t)) degrees, where floor(t) is the integer part of t, giving the number of complete hours. Similarly, at time Y, the hour hand is at 30s degrees, and the minute hand is at 360(s - floor(s)) degrees.According to the problem, at time Y, the hour hand should be where the minute hand was at X, and the minute hand should be where the hour hand was at X. So, we have two equations:1. 30s = 360(t - floor(t))2. 360(s - floor(s)) = 30tLet me simplify these equations. From the first equation:30s = 360(t - floor(t))Divide both sides by 30:s = 12(t - floor(t))From the second equation:360(s - floor(s)) = 30tDivide both sides by 30:12(s - floor(s)) = tSo now we have:s = 12(t - floor(t)) ...(1)t = 12(s - floor(s)) ...(2)Let me substitute equation (1) into equation (2). t = 12(s - floor(s)) But s = 12(t - floor(t)), so substitute that in:t = 12(12(t - floor(t)) - floor(12(t - floor(t))))Simplify:t = 12[12(t - floor(t)) - floor(12(t - floor(t)))]Let me denote u = t - floor(t). Since t is in hours, u is the fractional part of t, representing the minutes. So u is between 0 and 1.Then, s = 12uAnd the equation becomes:t = 12[12u - floor(12u)]But t = floor(t) + u, so:floor(t) + u = 12[12u - floor(12u)]Let me denote v = 12u. Since u is between 0 and 1, v is between 0 and 12.So, floor(t) + u = 12[v - floor(v)]But v = 12u, so u = v/12.Therefore:floor(t) + v/12 = 12[v - floor(v)]Let me denote w = floor(v). Since v is between 0 and 12, w can be 0, 1, 2, ..., 11.So, v = w + f, where f is the fractional part of v, between 0 and 1.Therefore:floor(t) + (w + f)/12 = 12[(w + f) - w] = 12fSo:floor(t) + (w + f)/12 = 12fMultiply both sides by 12 to eliminate denominators:12floor(t) + w + f = 144fRearrange:12floor(t) + w = 144f - f12floor(t) + w = 143fBut f is the fractional part of v, which is v - w. Since v = 12u, and u = t - floor(t), we have f = 12u - w.But u = t - floor(t), so f = 12(t - floor(t)) - w.But from earlier, s = 12u = 12(t - floor(t)) = w + f.Wait, this is getting a bit convoluted. Maybe I can approach it differently.From the two equations:s = 12(t - floor(t))t = 12(s - floor(s))Let me denote t = a + x, where a is the integer part (hours) and x is the fractional part (minutes). Similarly, s = b + y, where b is the integer part and y is the fractional part.Then, from the first equation:s = 12xSo, b + y = 12xFrom the second equation:t = 12ySo, a + x = 12yNow, since s = b + y = 12x, and t = a + x = 12y, we can substitute.From t = a + x = 12y, so y = (a + x)/12But s = 12x = b + y = b + (a + x)/12Multiply both sides by 12:144x = 12b + a + xRearrange:144x - x = 12b + a143x = 12b + aBut x is the fractional part of t, so 0 ≤ x < 1. Therefore, 0 ≤ 143x < 143.Similarly, a is the integer part of t, so a is from 0 to 11 (since we're considering 12 hours). Similarly, b is the integer part of s, which is 12x, so since x < 1, 12x < 12, so b can be from 0 to 11.So, 143x = 12b + aBut x = (12b + a)/143Since x must be less than 1:(12b + a)/143 < 112b + a < 143But a is from 0 to 11, and b is from 0 to 11.So, 12b + a can range from 0 to 12*11 + 11 = 143.But since 12b + a < 143, the maximum value is 142.So, for each a from 0 to 11, and b from 0 to 11, such that 12b + a < 143, we can have a solution.But wait, 12b + a can be up to 12*11 + 11 = 143, but since x < 1, 12b + a < 143, so 12b + a ≤ 142.Therefore, for each a and b, where a is 0 to 11, b is 0 to 11, and 12b + a ≤ 142, we have a valid x.But we need to count the number of valid (a, b) pairs.Let's see:For each a from 0 to 11, b can range from 0 to floor((142 - a)/12).Let me compute this:For a = 0:b ≤ (142 - 0)/12 ≈ 11.833, so b = 0 to 11 (12 values)For a = 1:b ≤ (142 - 1)/12 ≈ 11.75, so b = 0 to 11 (12 values)Similarly, for a = 2 to a = 11:For a = 11:b ≤ (142 - 11)/12 ≈ 10.916, so b = 0 to 10 (11 values)Wait, let's check:For a = 0:12b + 0 ≤ 142 → b ≤ 142/12 ≈ 11.833 → b = 0 to 11 (12 values)For a = 1:12b + 1 ≤ 142 → 12b ≤ 141 → b ≤ 141/12 ≈ 11.75 → b = 0 to 11 (12 values)Similarly, for a = 2:12b + 2 ≤ 142 → 12b ≤ 140 → b ≤ 140/12 ≈ 11.666 → b = 0 to 11 (12 values)...For a = 10:12b + 10 ≤ 142 → 12b ≤ 132 → b ≤ 11 → b = 0 to 11 (12 values)For a = 11:12b + 11 ≤ 142 → 12b ≤ 131 → b ≤ 131/12 ≈ 10.916 → b = 0 to 10 (11 values)So, for a = 0 to 10, b ranges from 0 to 11 (12 values each), and for a = 11, b ranges from 0 to 10 (11 values).Therefore, total number of solutions is:For a = 0 to 10: 11 a's * 12 b's = 132For a = 11: 11 b'sTotal: 132 + 11 = 143But wait, each solution corresponds to a unique t, right? Because t = a + x, where x = (12b + a)/143.So, each (a, b) pair gives a unique t in [a, a+1).But we need to make sure that t is within 00:01 to 12:01, which is essentially 0 < t < 12.But since a ranges from 0 to 11, and x is positive, t ranges from just above 0 to just below 12.But the problem says from 00:01 to 12:01, which is t from 1/60 to 12 + 1/60. But since we're considering t up to 12, and the next moment after 12:00 is 12:01, which is t = 12 + 1/60, but our solutions are up to t approaching 12 from below.So, the count remains 143.But wait, let me think again. Each (a, b) pair gives a unique t, but do all these t's fall within 00:01 to 12:01?Well, t starts from just above 0 (which is 00:00:00, but we start from 00:01, which is t = 1/60 ≈ 0.0167 hours). So, the first solution would be t = (12*0 + 0)/143 ≈ 0, which is before 00:01, so we need to exclude that.Similarly, the last solution would be t approaching 12 from below, which is just before 12:00, but we need to include up to 12:01, which is t = 12 + 1/60. But our solutions don't reach 12:01, so we need to check if the last solution is before 12:01.Wait, actually, t is less than 12, so all solutions are within 0 ≤ t < 12. But the problem is from 00:01 to 12:01, which is t from 1/60 to 12 + 1/60. So, we need to check if any of the solutions fall outside this range.The first solution is t = 0, which is before 00:01, so we need to exclude it. Similarly, the last solution is t approaching 12, which is before 12:01, so it's included.But wait, in our count, for a = 0, b = 0, we have t = (12*0 + 0)/143 = 0, which is 00:00, so we need to exclude this one.Similarly, for a = 11, b = 10, t = (12*10 + 11)/143 = (120 + 11)/143 = 131/143 ≈ 0.916 hours, which is about 54.97 minutes, so t = 11 + 0.916 ≈ 11.916 hours, which is 11:55:00 approximately, which is within 00:01 to 12:01.Wait, no, t = a + x, where a is from 0 to 11, and x is (12b + a)/143.So, for a = 0, b = 0: t = 0 + 0 = 0 (excluded)For a = 0, b = 1: t = 0 + (12*1 + 0)/143 ≈ 0.0838 hours ≈ 5 minutes, which is within 00:01 to 12:01.Similarly, for a = 11, b = 10: t = 11 + (12*10 + 11)/143 ≈ 11 + 131/143 ≈ 11.916 hours, which is within 00:01 to 12:01.So, the only solution we need to exclude is t = 0, which is 00:00. So, out of 143 solutions, one is before 00:01, so we have 142 interesting moments.But wait, let me double-check. The problem says from 00:01 to 12:01, which is 12 hours and 1 minute. So, t ranges from 1/60 to 12 + 1/60.But our solutions are t from 0 to 12, with t = 0 being excluded. So, the number of solutions in [0,12) is 143, but t = 0 is excluded, so 142.But wait, actually, t = 0 is the only solution outside the range [1/60, 12 + 1/60). All other solutions are within [0,12), and since 12 is included in the problem's upper limit (12:01), but our solutions don't reach 12:01, they only approach it.Wait, no, t = 12 is not included in our solutions because s = 12x, and x < 1, so s < 12. So, t = 12 would require s = 12x = 12, which would mean x = 1, but x < 1, so t cannot be 12.Therefore, all 143 solutions are within [0,12), but t = 0 is before 00:01, so we exclude it, leaving 142.But wait, let me think again. The problem says "from 00:01 to 12:01", which is 12 hours and 1 minute. So, technically, t can be up to 12 + 1/60. But our solutions only go up to t approaching 12, not including 12:01.But since t = 12 is not a solution, and the next moment after 12:00 is 12:01, which is t = 12 + 1/60, but our solutions don't reach that.Therefore, the number of interesting moments is 143, but excluding t = 0, which is before 00:01, so 142.Wait, but in our earlier count, we had 143 solutions, including t = 0. So, if we exclude t = 0, we have 142.But let me check if t = 0 is the only solution outside the range. For a = 0, b = 0: t = 0.For a = 0, b = 1: t ≈ 0.0838 hours ≈ 5 minutes, which is within 00:01.Similarly, for a = 11, b = 10: t ≈ 11.916 hours, which is within 00:01 to 12:01.So, only t = 0 is outside the range, so 143 - 1 = 142.But wait, let me think again. The problem says "from 00:01 to 12:01", which includes 12:01. So, t = 12:01 is t = 12 + 1/60 ≈ 12.0167 hours.But our solutions are t < 12, so none of them reach 12:01. Therefore, all 143 solutions are within [0,12), but t = 0 is before 00:01, so we exclude it, leaving 142.But wait, another thought: when a = 0, b = 0, t = 0, which is 00:00, which is before 00:01, so we exclude it.But for a = 0, b = 1: t ≈ 0.0838 hours ≈ 5 minutes, which is 00:05, which is after 00:01, so it's included.Similarly, for a = 11, b = 10: t ≈ 11.916 hours ≈ 11:55, which is before 12:01, so it's included.Therefore, total interesting moments are 143 - 1 = 142.But wait, let me check the initial equations again. When a = 0, b = 0, t = 0, which is 00:00, excluded. So, 143 total solutions, minus 1, gives 142.But I'm a bit confused because the problem says "from 00:01 to 12:01", which is 12 hours and 1 minute. So, t ranges from 1/60 to 12 + 1/60.But our solutions are t in [0,12). So, t = 0 is excluded, but t = 12 is not a solution, so all other solutions are within [0,12), but t = 0 is the only one outside the desired range.Therefore, the number of interesting moments is 143 - 1 = 142.But wait, I'm not sure if t = 12 is considered. Since 12:01 is included, but t = 12 is not a solution, so it's fine.Alternatively, maybe t = 12 is considered as t = 0 in a 12-hour clock, but since we're considering 00:01 to 12:01, which is 12 hours and 1 minute, t = 12 is equivalent to t = 0, but we already excluded t = 0.Wait, no, t = 12 is 12:00, which is the same as 00:00, but in our case, we're considering up to 12:01, which is t = 12 + 1/60.But our solutions don't reach t = 12, so t = 12 is not a solution.Therefore, the total number of interesting moments is 143, but excluding t = 0, which is before 00:01, so 142.But I'm still a bit unsure. Maybe I should think about it differently.Each solution corresponds to a unique t in [0,12), and we have 143 solutions. Since the problem is from 00:01 to 12:01, which is t from 1/60 to 12 + 1/60, but our solutions are t in [0,12). So, t = 0 is excluded, and t = 12 is not a solution, so we have 143 - 1 = 142.Alternatively, maybe t = 12 is considered as t = 0, but since we're excluding t = 0, we don't need to worry about t = 12.Wait, but t = 12 is 12:00, which is the same as t = 0 in a 12-hour clock, but in our case, we're considering t up to 12:01, which is t = 12 + 1/60. But our solutions don't reach t = 12, so t = 12 is not a solution.Therefore, the number of interesting moments is 143, but excluding t = 0, which is before 00:01, so 142.But wait, another thought: when a = 11, b = 11, t = (12*11 + 11)/143 = (132 + 11)/143 = 143/143 = 1, which is t = 11 + 1 = 12, but t = 12 is 12:00, which is the same as t = 0, but we have t = 12 as a solution? Wait, no, because when a = 11, b = 11, t = (12*11 + 11)/143 = 143/143 = 1, but t = a + x = 11 + 1 = 12, which is 12:00, which is equivalent to t = 0, but in our case, t = 12 is not a solution because s = 12x = 12*1 = 12, which would mean b = 12, but b is only up to 11. So, actually, when a = 11, b = 11, t = 12, but b cannot be 12, so this solution is invalid.Wait, no, when a = 11, b = 11, t = (12*11 + 11)/143 = 143/143 = 1, so t = 11 + 1 = 12, but s = 12x = 12*1 = 12, which would mean b = 12, but b is only up to 11, so this solution is invalid. Therefore, when a = 11, b = 11, it's invalid because b cannot be 12. So, we need to exclude this case as well.Therefore, for a = 11, b can only go up to 10, as we initially thought, giving 11 solutions for a = 11.So, total solutions are 11*12 + 11 = 143, but excluding t = 0 and t = 12, which are both invalid, so 143 - 2 = 141.Wait, but t = 12 is not a solution because b cannot be 12, so we only need to exclude t = 0, giving 142.I'm getting confused here. Let me try to summarize:- Total solutions: 143 (including t = 0 and t = 12)- t = 0 is before 00:01, so exclude it.- t = 12 is not a solution because b cannot be 12, so it's already excluded.Therefore, total interesting moments are 143 - 1 = 142.But wait, when a = 11, b = 11, t = 12, which is invalid because b cannot be 12, so we have to exclude this case as well. So, for a = 11, b can only go up to 10, giving 11 solutions, not 12. So, total solutions are 11*12 + 11 = 143, but when a = 11, b = 11 is invalid, so total solutions are 143 - 1 = 142, and then excluding t = 0, which is another solution, so 142 - 1 = 141.Wait, no, because when a = 11, b = 11 is invalid, so we already have 142 solutions (143 - 1). Then, t = 0 is another solution to exclude, so 142 - 1 = 141.But I'm not sure if t = 12 is considered as a solution or not. Since t = 12 is equivalent to t = 0, which we've already excluded, so maybe we don't need to exclude it again.Alternatively, maybe t = 12 is not a solution because b cannot be 12, so it's already excluded in the count.I think the correct approach is:- Total solutions: 143 (including t = 0 and t = 12)- Exclude t = 0 because it's before 00:01- Exclude t = 12 because it's invalid (b cannot be 12)- So, total interesting moments: 143 - 2 = 141But I'm not entirely sure. Maybe I should think about it differently.Each solution corresponds to a unique t in [0,12), and we have 143 solutions. Since the problem is from 00:01 to 12:01, which is t from 1/60 to 12 + 1/60, but our solutions are t in [0,12). So, t = 0 is excluded, and t = 12 is not a solution, so we have 143 - 1 = 142.Alternatively, since t = 12 is not a solution, we don't need to exclude it, so 143 - 1 = 142.I think the correct answer is 143, but excluding t = 0, so 142.But wait, another way to think about it: the problem is asking for interesting moments from 00:01 to 12:01, which is 12 hours and 1 minute. So, t ranges from 1/60 to 12 + 1/60.Our solutions are t in [0,12), so t = 0 is excluded, but t = 12 is not a solution, so we have 143 solutions in [0,12), but t = 0 is excluded, so 142.Therefore, the number of interesting moments is 142.But wait, I'm still not sure. Maybe I should look for another approach.Alternatively, think about the problem as solving for t where the hands swap positions. This is similar to the problem where the hands overlap or are symmetric.In general, the hands swap positions 11 times every 12 hours, but that's when they overlap. But in this case, it's a different condition.Wait, no, swapping positions is different. The hands swap positions 11 times in 12 hours, but I'm not sure.Wait, actually, the number of times the hands are in positions that can swap is 11 times in 12 hours, but I'm not sure.Wait, no, the number of solutions to the equation where the hands swap positions is 11 times in 12 hours, but I'm not sure.Wait, let me think about it. The hands swap positions when the hour hand is where the minute hand was, and vice versa. This happens 11 times in 12 hours because the minute hand laps the hour hand 11 times.But in our case, we're looking for moments where such a swap is possible, which might be more frequent.Wait, actually, the number of solutions is 143, but considering the range from 00:01 to 12:01, which is 12 hours and 1 minute, so 143 solutions in 12 hours, but we need to adjust for the extra minute.But I think the initial count of 143 is correct, but we need to exclude t = 0, so 142.But I'm still not entirely confident. Maybe I should look for a formula or a known result.Wait, I recall that the number of times the hands are in positions that can swap is 143 in 12 hours. So, the answer is 143.But considering the problem is from 00:01 to 12:01, which is 12 hours and 1 minute, so 143 solutions, but t = 0 is excluded, so 142.But I'm not sure. Maybe the answer is 143.Alternatively, maybe the answer is 143 because the problem includes 12:01, which is t = 12 + 1/60, but our solutions are up to t approaching 12, so 143 solutions.I think the correct answer is 143.Wait, but in our earlier count, we had 143 solutions, including t = 0, which is before 00:01, so we need to exclude it, giving 142.But I'm still unsure. Maybe I should think about it as follows:Each hour, the hands swap positions once, except for the 11th hour, where it happens twice. So, total 13 times in 12 hours.But that's not matching with 143.Wait, no, that's for overlaps. Swapping positions is different.Wait, actually, the number of times the hands are in positions that can swap is 143 in 12 hours, as per the equations we solved earlier.Therefore, the answer is 143.But considering the problem is from 00:01 to 12:01, which is 12 hours and 1 minute, so 143 solutions, but t = 0 is excluded, so 142.But I'm not sure. Maybe the answer is 143.I think I need to go with 143, as that's the number of solutions in 12 hours, and the problem includes up to 12:01, which is just after 12:00, so t = 12 is not a solution, but t approaching 12 is included.Therefore, the number of interesting moments is 143.

💡Okay, so I have this math problem here that defines a special operation called "#". It says that for any two numbers a and b, the operation a # b is equal to a minus a divided by b. So, in symbols, that's a # b = a - (a/b). The question is asking me to find the value of 10 # 5 using this operation.Alright, let's break this down step by step. First, I need to understand what this operation does. It takes two numbers, a and b, and then subtracts the result of dividing a by b from a itself. So, it's like taking the first number and then subtracting a portion of it based on the second number.Let me write this out to make it clearer. If a is 10 and b is 5, then according to the definition:10 # 5 = 10 - (10 / 5)Now, I need to compute what 10 divided by 5 is. I know that 10 divided by 5 equals 2 because 5 times 2 is 10. So, substituting that back into the equation:10 # 5 = 10 - 2Now, subtracting 2 from 10 gives me 8. So, the result of 10 # 5 is 8.Wait, let me double-check to make sure I didn't make any mistakes. Sometimes when I'm doing math quickly, I might skip a step or miscompute something. So, starting over:Given a # b = a - (a/b), and a = 10, b = 5.First, compute a/b: 10 / 5 = 2.Then subtract that result from a: 10 - 2 = 8.Yes, that seems correct. I don't think I made any errors there.Just to be thorough, let me think about what this operation represents. It's taking a number and subtracting a fraction of itself based on the second number. So, in this case, since b is 5, we're subtracting 1/5 of a from a itself. Since a is 10, 1/5 of 10 is 2, and subtracting that from 10 gives 8.Another way to look at it is to factor out a from the expression. Let's see:a # b = a - (a/b) = a(1 - 1/b)So, for a = 10 and b = 5:10 # 5 = 10(1 - 1/5) = 10(4/5) = 8That's another way to compute it, and it still gives me 8. So, that confirms my earlier result.I think I've got it right. The operation is straightforward once you break it down into its components. It's just a matter of carefully substituting the values and performing the arithmetic correctly.Just to make sure, let me try another example with different numbers to see if the operation works as expected. Let's say a = 6 and b = 3. According to the operation:6 # 3 = 6 - (6/3) = 6 - 2 = 4Alternatively, using the factored form:6 # 3 = 6(1 - 1/3) = 6(2/3) = 4Yep, that works too. So, the operation seems consistent.Going back to the original problem, I feel confident that 10 # 5 is indeed 8. I don't see any mistakes in my calculations, and the alternative method using factoring also leads to the same answer. So, I think I'm all set.

💡To solve the problem, we start with the given equation involving the complex number ( z ):[ z(1 + i) = 4 ]Our goal is to find the imaginary part of ( z ). To do this, we need to express ( z ) in the standard form of a complex number, which is ( a + bi ), where ( a ) is the real part and ( b ) is the imaginary part.First, we isolate ( z ) by dividing both sides of the equation by ( 1 + i ):[ z = frac{4}{1 + i} ]To simplify this expression, we multiply the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of ( 1 + i ) is ( 1 - i ). This step helps to eliminate the imaginary unit ( i ) from the denominator:[ z = frac{4}{1 + i} times frac{1 - i}{1 - i} ]Multiplying the numerators and the denominators:[ z = frac{4(1 - i)}{(1 + i)(1 - i)} ]Next, we simplify the denominator using the difference of squares formula:[ (1 + i)(1 - i) = 1^2 - i^2 = 1 - (-1) = 2 ]So, the expression for ( z ) becomes:[ z = frac{4(1 - i)}{2} ]Dividing both terms in the numerator by 2:[ z = 2(1 - i) = 2 - 2i ]Now, ( z ) is expressed in the standard form ( a + bi ), where ( a = 2 ) and ( b = -2 ). Therefore, the imaginary part of ( z ) is ( -2 ).**Final Answer**The imaginary part of the complex number ( z ) is boxed{D}.

💡Okay, so I need to figure out how many degrees are in each interior angle of a regular decagon. A decagon has 10 sides, right? I remember that for polygons, there's a formula to find the sum of all interior angles. Let me try to recall that formula. I think it's something like 180 times the number of sides minus something. Oh yeah, I think it's 180 times (n minus 2), where n is the number of sides. So, for a decagon, n is 10.Let me write that down: Sum of interior angles = 180 × (10 - 2). That simplifies to 180 × 8. Hmm, 180 times 8 is 1440 degrees. Okay, so all the interior angles together add up to 1440 degrees.But the question is asking for each interior angle, not the total. Since it's a regular decagon, all the angles are equal. So, I need to divide the total sum by the number of angles, which is the same as the number of sides, so 10.Let me calculate that: 1440 divided by 10. That's straightforward; 1440 ÷ 10 = 144 degrees. So, each interior angle in a regular decagon is 144 degrees.Wait, let me double-check my steps to make sure I didn't make a mistake. First, the formula for the sum of interior angles is correct: 180(n - 2). Plugging in 10 for n gives 180 × 8, which is indeed 1440. Then, dividing by 10 gives 144. Yeah, that seems right.I think I've got it. Each interior angle in a regular decagon is 144 degrees.

💡First, I need to simplify the complex fraction (frac{4+3i}{2-i}). To do this, I'll multiply both the numerator and the denominator by the conjugate of the denominator, which is (2+i).Multiplying the numerator:[(4+3i)(2+i) = 4 times 2 + 4 times i + 3i times 2 + 3i times i = 8 + 4i + 6i + 3i^2]Since (i^2 = -1), this simplifies to:[8 + 10i + 3(-1) = 8 + 10i - 3 = 5 + 10i]Multiplying the denominator:[(2-i)(2+i) = 2 times 2 + 2 times i - i times 2 - i times i = 4 + 2i - 2i - i^2]Again, since (i^2 = -1), this simplifies to:[4 - (-1) = 4 + 1 = 5]Now, the fraction becomes:[frac{5 + 10i}{5} = 1 + 2i]Therefore, the correct answer is (1 + 2i), which corresponds to option B.

💡First, I need to determine the amount of the raise John received. The percentage increase is given as 10.76923076923077%.To find the raise amount, I'll convert the percentage to a decimal by dividing it by 100:10.76923076923077% ÷ 100 = 0.1076923076923077Next, I'll multiply John's original weekly salary by this decimal to calculate the raise:65 × 0.1076923076923077 ≈ 7.00Finally, I'll add the raise to the original salary to find the new weekly salary:65 + 7.00 = 72.00Therefore, John now makes 72.00 per week after the raise.