💡First, I'll denote the present ages of Shekhar, Shobha, and Kapil as ( S ), ( Sh ), and ( K ) respectively.According to the problem, the ratio of their ages is ( 4:3:2 ). This means:[frac{S}{Sh} = frac{4}{3} quad text{(1)}][frac{S}{K} = frac{4}{2} quad text{(2)}]From equation (2), we can say that ( S = 2K ).We are also given that in 10 years, Kapil's age will be equal to Shekhar's present age. So, we can write:[K + 10 = S quad text{(3)}]Since we know ( S = 2K ) from equation (2), we can substitute ( S ) in equation (3):[K + 10 = 2K][10 = 2K - K][10 = K]So, Kapil's present age is 10 years.Now, we can find Shekhar's present age using the relationship ( S = 2K ):[S = 2 times 10 = 20]Shekhar's present age is 20 years.We are also given that Shekhar's age will be 30 years in 8 years. This confirms our calculation because:[20 + 8 = 28]However, there seems to be a slight discrepancy in the information provided. The problem states that Shekhar will be 30 years old in 8 years, which would make him 22 years old now, not 20. Let's correct this:If Shekhar will be 30 in 8 years, then his current age is:[30 - 8 = 22]Now, let's find Shobha's present age using the ratio of Shekhar's age to Shobha's age (( 4:3 )). Since Shekhar is now 22 years old, we can write:[frac{22}{Sh} = frac{4}{3}]Cross-multiplying gives us:[3 times 22 = 4 times Sh][66 = 4Sh][Sh = frac{66}{4} = 16.5]So, Shobha's present age is 16.5 years.In summary, the present ages are:- Shekhar: 22 years- Shobha: 16.5 years- Kapil: ( boxed{10} ) years

💡Okay, so I need to figure out Tom's average speed for his entire trip. He traveled 100 miles in total. The first half of the trip, which is 50 miles, he went at a constant speed of 20 miles per hour. Then, for the remaining 50 miles, he increased his speed to 50 miles per hour. Hmm, average speed can be a bit tricky because it's not just the average of the two speeds; it's the total distance divided by the total time taken.Alright, let me start by recalling the formula for average speed. I think it's:[text{Average Speed} = frac{text{Total Distance}}{text{Total Time}}]So, I need to find the total distance and the total time Tom took for his trip.The total distance is straightforward—it's 100 miles. Now, I need to calculate the total time he took. Since he traveled two different segments at two different speeds, I'll calculate the time taken for each segment separately and then add them together.For the first segment, he traveled 50 miles at 20 miles per hour. To find the time taken for this part, I can use the formula:[text{Time} = frac{text{Distance}}{text{Speed}}]Plugging in the numbers:[text{Time}_1 = frac{50 text{ miles}}{20 text{ mph}} = 2.5 text{ hours}]Okay, so the first part took him 2.5 hours.Now, for the second segment, he also traveled 50 miles, but at a faster speed of 50 miles per hour. Using the same formula:[text{Time}_2 = frac{50 text{ miles}}{50 text{ mph}} = 1 text{ hour}]So, the second part took him 1 hour.To find the total time for the entire trip, I'll add the two times together:[text{Total Time} = text{Time}_1 + text{Time}_2 = 2.5 text{ hours} + 1 text{ hour} = 3.5 text{ hours}]Now that I have the total distance and the total time, I can plug these into the average speed formula:[text{Average Speed} = frac{100 text{ miles}}{3.5 text{ hours}} approx 28.57 text{ mph}]Wait, let me double-check my calculations to make sure I didn't make any mistakes. For the first segment, 50 divided by 20 is indeed 2.5 hours. For the second segment, 50 divided by 50 is 1 hour. Adding them gives 3.5 hours total. Dividing 100 by 3.5 gives approximately 28.57 mph. That seems right.I think another way to look at it is by using the harmonic mean since we're dealing with two different speeds over the same distance. The harmonic mean formula for two speeds is:[text{Average Speed} = frac{2 times v_1 times v_2}{v_1 + v_2}]Plugging in the values:[text{Average Speed} = frac{2 times 20 times 50}{20 + 50} = frac{2000}{70} approx 28.57 text{ mph}]Okay, that confirms my earlier calculation. So, Tom's average speed for the entire trip is approximately 28.57 miles per hour.

💡First, calculate the total area of the living room:[text{Total Area} = 18 text{ feet} times 24 text{ feet} = 432 text{ square feet}]Next, determine the area that will be covered by the two-foot by two-foot border tiles. The border is two feet wide along all edges, so the inner area to be covered by one-foot by one-foot tiles will have reduced dimensions:[text{Inner Length} = 24 text{ feet} - 2 times 2 text{ feet} = 20 text{ feet}][text{Inner Width} = 18 text{ feet} - 2 times 2 text{ feet} = 14 text{ feet}][text{Inner Area} = 20 text{ feet} times 14 text{ feet} = 280 text{ square feet}]The area covered by the two-foot by two-foot border tiles is the total area minus the inner area:[text{Border Area} = 432 text{ square feet} - 280 text{ square feet} = 152 text{ square feet}]Each two-foot by two-foot tile covers:[text{Area per Border Tile} = 2 text{ feet} times 2 text{ feet} = 4 text{ square feet}][text{Number of Border Tiles} = frac{152 text{ square feet}}{4 text{ square feet per tile}} = 38 text{ tiles}]The number of one-foot by one-foot tiles needed for the inner area is equal to the inner area:[text{Number of Inner Tiles} = 280 text{ tiles}]Finally, add the number of border tiles and inner tiles to find the total number of tiles used:[text{Total Tiles} = 38 text{ tiles} + 280 text{ tiles} = 318 text{ tiles}]

💡First, I need to evaluate the expression ( 8 div 2 - 3 - 10 + 3 cdot 9 ).According to the order of operations, I should perform multiplication and division before addition and subtraction. Calculating the division:[8 div 2 = 4]Calculating the multiplication:[3 cdot 9 = 27]Now, substitute these values back into the expression:[4 - 3 - 10 + 27]Next, I'll perform the subtraction and addition from left to right:[4 - 3 = 1][1 - 10 = -9][-9 + 27 = 18]Therefore, the final answer is 18.

💡Okay, so I have this equation: ( z^8 = -frac{1}{sqrt{2}} + frac{i}{sqrt{2}} ). I need to find the sum of the angles ( theta_1 + theta_2 + dots + theta_8 ) where each root is given by ( text{cis } theta_k ).First, I remember that complex numbers can be expressed in polar form as ( text{cis } theta ), which stands for ( cos theta + i sin theta ). So, if ( z = text{cis } theta ), then ( z^8 = text{cis } 8theta ).Now, the right side of the equation is ( -frac{1}{sqrt{2}} + frac{i}{sqrt{2}} ). I need to express this in polar form as well. To do that, I should find the magnitude and the angle.The magnitude ( r ) is calculated as:[r = sqrt{left(-frac{1}{sqrt{2}}right)^2 + left(frac{1}{sqrt{2}}right)^2} = sqrt{frac{1}{2} + frac{1}{2}} = sqrt{1} = 1]So, the magnitude is 1, which means it's on the unit circle.Next, the angle ( theta ). Since the real part is negative and the imaginary part is positive, the complex number is in the second quadrant. The reference angle can be found using:[tan alpha = left|frac{text{Imaginary part}}{text{Real part}}right| = left|frac{frac{1}{sqrt{2}}}{-frac{1}{sqrt{2}}}right| = 1]So, ( alpha = 45^circ ). Therefore, the angle ( theta ) is ( 180^circ - 45^circ = 135^circ ).Thus, ( -frac{1}{sqrt{2}} + frac{i}{sqrt{2}} = text{cis } 135^circ ).So, the equation becomes:[z^8 = text{cis } 135^circ]Which means:[text{cis } 8theta = text{cis } 135^circ]For two complex numbers in polar form to be equal, their magnitudes must be equal, and their angles must differ by an integer multiple of ( 360^circ ). Since both sides have magnitude 1, we can equate the angles:[8theta = 135^circ + 360^circ k quad text{for some integer } k]Solving for ( theta ):[theta = frac{135^circ + 360^circ k}{8}]Now, since ( theta ) must be between ( 0^circ ) and ( 360^circ ), I need to find all distinct values of ( theta ) by choosing appropriate integers ( k ).Let me compute the values for ( k = 0 ) to ( k = 7 ):- For ( k = 0 ): ( theta = frac{135}{8} = 16.875^circ )- For ( k = 1 ): ( theta = frac{135 + 360}{8} = frac{495}{8} = 61.875^circ )- For ( k = 2 ): ( theta = frac{135 + 720}{8} = frac{855}{8} = 106.875^circ )- For ( k = 3 ): ( theta = frac{135 + 1080}{8} = frac{1215}{8} = 151.875^circ )- For ( k = 4 ): ( theta = frac{135 + 1440}{8} = frac{1575}{8} = 196.875^circ )- For ( k = 5 ): ( theta = frac{135 + 1800}{8} = frac{1935}{8} = 241.875^circ )- For ( k = 6 ): ( theta = frac{135 + 2160}{8} = frac{2295}{8} = 286.875^circ )- For ( k = 7 ): ( theta = frac{135 + 2520}{8} = frac{2655}{8} = 331.875^circ )Wait, let me check ( k = 7 ):( 135 + 7*360 = 135 + 2520 = 2655 ). Divided by 8: 2655 / 8 = 331.875. Yes, that's correct.So, all eight roots are at angles ( 16.875^circ, 61.875^circ, 106.875^circ, 151.875^circ, 196.875^circ, 241.875^circ, 286.875^circ, 331.875^circ ).Now, I need to find the sum of these angles. Let me write them down:1. 16.875°2. 61.875°3. 106.875°4. 151.875°5. 196.875°6. 241.875°7. 286.875°8. 331.875°I can notice that these angles are equally spaced around the unit circle. Since it's an 8th root, the spacing between each angle should be ( frac{360°}{8} = 45° ). Let me check:61.875 - 16.875 = 45°106.875 - 61.875 = 45°151.875 - 106.875 = 45°196.875 - 151.875 = 45°241.875 - 196.875 = 45°286.875 - 241.875 = 45°331.875 - 286.875 = 45°And the last one wraps around: 360° - 331.875° + 16.875° = 45°, so yes, equally spaced by 45°.Since they are equally spaced, the sum can be found using the formula for the sum of an arithmetic series. The formula is:[S = frac{n}{2} times (a_1 + a_n)]Where ( n ) is the number of terms, ( a_1 ) is the first term, and ( a_n ) is the last term.Here, ( n = 8 ), ( a_1 = 16.875° ), ( a_n = 331.875° ).So, plugging in:[S = frac{8}{2} times (16.875 + 331.875) = 4 times 348.75 = 1395°]Wait, that's different from the initial thought. Hmm, maybe I made a mistake.Wait, no, actually, let's think about it differently. Since all the roots are symmetrically placed around the circle, their angles should average out to the angle of the original complex number divided by the root. But wait, the original angle is 135°, and we're taking 8th roots, so each root's angle is 135°/8 plus multiples of 360°/8.But when we sum all the angles, maybe there's a smarter way than adding them up individually.Alternatively, I remember that for the roots of a complex number, the sum of their angles can be related to the angle of the original number. But I'm not sure about that.Wait, another approach: Each root is ( text{cis } theta_k ), and when we take the product of all roots, it should be equal to the original complex number raised to the power of 1/8, but considering all roots, the product is related to the constant term of the polynomial.Wait, maybe that's complicating things. Let me go back.I have 8 angles, each 45° apart, starting from 16.875°. So, the angles are 16.875°, 61.875°, 106.875°, 151.875°, 196.875°, 241.875°, 286.875°, 331.875°.If I add them up:16.875 + 61.875 = 78.7578.75 + 106.875 = 185.625185.625 + 151.875 = 337.5337.5 + 196.875 = 534.375534.375 + 241.875 = 776.25776.25 + 286.875 = 1063.1251063.125 + 331.875 = 1395°So, the total sum is 1395°. But wait, earlier I thought the answer was 1575°, but that was incorrect.Wait, let me check the initial approach again. The user's initial solution said the sum is 1575°, but when I calculated, I got 1395°. Hmm, perhaps I made a mistake in the initial approach.Wait, in the initial solution, they used the formula for the sum of an arithmetic series:Sum = (number of terms)/2 * (first term + last term)Which is correct. So, let's apply that.First term: 135/8 = 16.875°Last term: (135 + 7*360)/8 = (135 + 2520)/8 = 2655/8 = 331.875°Number of terms: 8So, sum = 8/2 * (16.875 + 331.875) = 4 * 348.75 = 1395°So, why did the initial solution say 1575°? Maybe there was a miscalculation.Wait, let me check the initial solution again.In the initial solution, they wrote:Sum = (1/2) * (135/8 + (135 + 7*360)/8) * 8Which is (1/2)*( (135 + 2655)/8 ) *8 = (1/2)*(2790/8)*8 = (1/2)*2790 = 1395°Wait, but in the initial solution, they wrote 1575°, which is incorrect. So, the correct sum should be 1395°, not 1575°.Wait, but let me think again. Maybe I'm missing something.Alternatively, perhaps the sum of the angles of the roots of unity multiplied by some factor.Wait, another approach: The sum of the angles of the nth roots of a complex number is n times the angle of the original complex number divided by n, but that doesn't make sense.Wait, actually, for the roots of unity, the sum of the angles is 0 because they are symmetrically distributed. But in this case, the roots are not roots of unity, but roots of a specific complex number.Wait, but in general, for the equation ( z^n = w ), the sum of the arguments of the roots is equal to the argument of w plus k*360°, but I'm not sure.Wait, let me think about the product of the roots. The product of the roots of ( z^8 = w ) is ( w ) times (-1)^8, which is ( w ). But the product of the roots in polar form is the product of their magnitudes times ( text{cis } ) the sum of their angles.Since all roots have magnitude 1 (because w has magnitude 1), the product of the magnitudes is 1. Therefore, the product is ( text{cis } (theta_1 + theta_2 + dots + theta_8) ).But the product is also equal to ( w ), which is ( text{cis } 135° ).Therefore:[text{cis } (theta_1 + theta_2 + dots + theta_8) = text{cis } 135°]Which implies that:[theta_1 + theta_2 + dots + theta_8 = 135° + 360° k quad text{for some integer } k]But since each ( theta_k ) is between 0° and 360°, the sum is between 0° and 8*360° = 2880°. So, 135° + 360° k must be in that range.But wait, the sum of the angles is 1395°, which is 135° + 3*360° = 135° + 1080° = 1215°, which is less than 1395°. Wait, no, 135° + 3*360° = 1215°, 135° + 4*360°= 1575°, which is more than 1395°. So, 1395° is not equal to 135° + 360° k.Hmm, this seems contradictory. So, perhaps my initial approach is wrong.Wait, maybe I made a mistake in assuming the product of the roots is ( text{cis } ) the sum of the angles. Actually, the product of the roots in polar form is ( prod_{k=1}^8 text{cis } theta_k = text{cis } (theta_1 + theta_2 + dots + theta_8) ). But the product of the roots of the equation ( z^8 = w ) is ( w ) times (-1)^8, which is ( w ). So, ( text{cis } (theta_1 + theta_2 + dots + theta_8) = text{cis } 135° ). Therefore, the sum of the angles is congruent to 135° modulo 360°. But the actual sum is 1395°, which is 135° + 3*360° = 135° + 1080° = 1215°, but 1395° is 1215° + 180°, which is not a multiple of 360°. So, this seems conflicting.Wait, maybe I'm misunderstanding something. Let me think again.The product of the roots of the equation ( z^8 = w ) is ( w ) if the leading coefficient is 1. So, yes, the product is ( w ). Therefore, the product of the roots is ( text{cis } 135° ). On the other hand, the product of the roots in polar form is ( prod_{k=1}^8 text{cis } theta_k = text{cis } (theta_1 + theta_2 + dots + theta_8) ). Therefore, ( text{cis } (theta_1 + theta_2 + dots + theta_8) = text{cis } 135° ). So, the sum of the angles must be congruent to 135° modulo 360°, but the actual sum can be 135° + 360°*k.But when I calculated the sum as 1395°, which is 135° + 3*360° + 180°, which is not a multiple of 360°. Wait, 1395° divided by 360° is 3.875, which is 3 + 15/16. So, it's not an integer multiple.Wait, perhaps I made a mistake in calculating the sum. Let me recalculate the sum:16.875 + 61.875 = 78.7578.75 + 106.875 = 185.625185.625 + 151.875 = 337.5337.5 + 196.875 = 534.375534.375 + 241.875 = 776.25776.25 + 286.875 = 1063.1251063.125 + 331.875 = 1395°Yes, that's correct. So, the sum is 1395°, which is 3*360° + 315°, which is 135° + 3*360° + 180°, which is not matching the product.Wait, maybe the product is not just ( text{cis } ) the sum of the angles, but considering that each root is ( text{cis } theta_k ), the product is ( prod text{cis } theta_k = text{cis } sum theta_k ). So, that should be equal to ( text{cis } 135° ). Therefore, ( sum theta_k = 135° + 360° k ).But 1395° is 135° + 3*360° + 180°, which is 135° + 360°*3 + 180°, which is 135° + 1080° + 180° = 1395°, which is not 135° + 360°*k. So, this is conflicting.Wait, perhaps I made a mistake in the product. Let me think again.The equation is ( z^8 = w ), where ( w = text{cis } 135° ). The polynomial is ( z^8 - w = 0 ). The product of the roots is ( (-1)^8 * (-w) = -w ). Wait, no, the product of the roots of ( z^n + a = 0 ) is ( (-1)^n a ). So, in this case, the polynomial is ( z^8 - w = 0 ), so the product of the roots is ( (-1)^8 * (-w) = -w ). Therefore, the product is ( -w = -text{cis } 135° = text{cis } (135° + 180°) = text{cis } 315° ).Wait, so the product of the roots is ( text{cis } 315° ), not ( text{cis } 135° ). Therefore, ( text{cis } (theta_1 + theta_2 + dots + theta_8) = text{cis } 315° ). Therefore, the sum of the angles is 315° + 360° k.But when I calculated the sum, I got 1395°, which is 315° + 3*360° = 315° + 1080° = 1395°. So, that matches.Therefore, the sum of the angles is 1395°, which is 315° + 3*360°, so it's congruent to 315° modulo 360°, but the actual sum is 1395°.Wait, but earlier I thought the product was ( w ), but it's actually ( -w ). So, that's where the confusion was.Therefore, the correct sum is 1395°, not 1575°. So, the initial solution was incorrect.But wait, let me double-check. The polynomial is ( z^8 - w = 0 ). The product of the roots is ( (-1)^8 * (-w) = -w ). So, yes, the product is ( -w ), which is ( text{cis } 315° ). Therefore, the sum of the angles is 315° + 360° k. Since the sum is 1395°, which is 315° + 3*360°, that's correct.Therefore, the correct answer is 1395°, not 1575°.Wait, but in the initial solution, they said 1575°, which is 135° + 4*360° = 135° + 1440° = 1575°, which is different from 1395°. So, perhaps the initial solution was wrong.Alternatively, maybe I made a mistake in calculating the sum. Let me check again.The angles are:1. 16.875°2. 61.875°3. 106.875°4. 151.875°5. 196.875°6. 241.875°7. 286.875°8. 331.875°Adding them up:16.875 + 61.875 = 78.7578.75 + 106.875 = 185.625185.625 + 151.875 = 337.5337.5 + 196.875 = 534.375534.375 + 241.875 = 776.25776.25 + 286.875 = 1063.1251063.125 + 331.875 = 1395°Yes, that's correct. So, the sum is indeed 1395°, which is 315° + 3*360°, so it's congruent to 315° modulo 360°, but the actual sum is 1395°.Therefore, the correct answer should be 1395°, not 1575°.Wait, but the initial solution said 1575°, which is 135° + 4*360°, which is 135° + 1440° = 1575°. So, why is that?Wait, perhaps the initial solution considered the sum as 8 times the average angle, which is 8*(135° + 360°*k)/8 = 135° + 360°*k, which is 135°, 495°, 855°, etc. But that's not correct because the sum is not just 8 times the average, but the average is (first + last)/2, and the sum is 8*(first + last)/2 = 4*(first + last).Wait, in the initial solution, they wrote:Sum = (1/2)*(135/8 + (135 + 7*360)/8)*8 = (1/2)*(2790/8)*8 = (1/2)*2790 = 1395°But in the initial solution, they wrote 1575°, which is incorrect. So, the correct sum is 1395°.Therefore, the answer should be 1395°, not 1575°.Wait, but let me think again. Maybe the initial solution was considering the sum of the angles in a different way.Alternatively, perhaps the initial solution was wrong because they added 135° eight times, which is 1080°, and then added 360°*7/2, which is 1260°, but that's not correct.Wait, no, the initial solution used the formula for the sum of an arithmetic series correctly, but then they wrote the answer as 1575°, which is incorrect. So, the correct answer is 1395°.Therefore, the sum of the angles is 1395°.

💡First, I need to find the original number of students in the class. Let's denote this number as ( x ).The average age of the original class is 40 years, so the total age of all the students in the original class is ( 40x ).When 18 new students join the class, the total number of students becomes ( x + 18 ).The average age of the new students is 32 years, so the total age of the new students is ( 18 times 32 = 576 ) years.After the new students join, the average age of the entire class decreases by 4 years, making the new average age 36 years. Therefore, the total age of the class after the new students join is ( (x + 18) times 36 ).I can set up the equation:[40x + 576 = 36(x + 18)]Expanding the right side:[40x + 576 = 36x + 648]Subtracting ( 36x ) from both sides:[4x + 576 = 648]Subtracting 576 from both sides:[4x = 72]Dividing both sides by 4:[x = 18]So, the original strength of the class was 18 students.