answer:Part (a) To prove that the sum of the lengths of any two medians of a triangle is not greater than 3/4 of the perimeter of the triangle, frac{3}{4}P, where P is the perimeter of the triangle. 1. **Define the Triangle and Medians**: Let a, b, and c be the lengths of the sides of a triangle. Let m_a, m_b, and m_c be the medians corresponding to sides a, b, and c, respectively. 2. **Express Perimeter**: The perimeter P of the triangle is given by: [ P = a + b + c ] 3. **Median Properties**: Recall that the medians of a triangle intersect at the centroid, which divides each median into a ratio of 2:1. For simplicity, we consider the medians intersect at a single point dividing the triangle into smaller segments. 4. **Apply the Triangle Inequality**: Consider the triangle inequalities for the segments: [ 3m_a < 2a + 2c ] [ 3m_b < 2b + 2c ] [ 3m_c < a + b ] 5. **Sum of Two Medians**: We add the above inequalities: [ 3m_a + 3m_b < 2a + 2c + 2b + 2c ] Simplifying, [ 3(m_a + m_b) < 2(a + b + c) ] 6. **Final Multiplication to Conclude**: Divide both sides by 2: [ m_a + m_b < frac{2}{3}(a + b + c) ] Since frac{3}{4}P = frac{3}{4}(a + b + c), it follows that, [ m_a + m_b le frac{3}{4}P ] Therefore, the sum of any two medians is not greater than 3/4 of the triangle's perimeter. Conclusion: [ boxed{text{The sum of any two medians is not greater than } frac{3}{4}P.} ] Part (b) To prove that the sum of the lengths of any two medians of a triangle is not less than 3/4 of the semi-perimeter of the triangle, frac{3}{4}p, where p is the semi-perimeter of the triangle. 1. **Express Semi-Perimeter**: The semi-perimeter p is given by: [ p = frac{a + b + c}{2} ] 2. **Apply the Triangle Inequality**: Consider another set of triangle inequalities: [ a < 2m_a + m_b ] [ b < m_a + 2m_b ] [ c < m_a + m_b ] 3. **Sum of One Median and Two Medians**: We add these inequalities to get: [ a + b + c < 2m_a + m_b + m_a + 2m_b + m_a + m_b ] Simplifying, [ a + b + c < 4(m_a + m_b) ] 4. **Final Multiplication to Conclude**: Divide both sides by 4: [ frac{a + b + c}{4} < m_a + m_b ] Which implies, [ frac{a + b + c}{2} times frac{3}{4} < m_a + m_b ] Therefore, [ frac{3}{4}p < m_a + m_b ] Since frac{3}{4}p = frac{3}{8}(a + b + c) is the semi-perimeter. Conclusion: [ boxed{text{The sum of any two medians is not less than } frac{3}{4}p.} ]

answer:First, let's calculate the total cost of the blankets with known prices: Cost of 3 blankets at Rs. 100 each = 3 * 100 = Rs. 300 Cost of 2 blankets at Rs. 150 each = 2 * 150 = Rs. 300 Now, we know the total cost of the two blankets with the unknown rate is Rs. 900. So, the total cost of all the blankets is: Total cost = Cost of 3 blankets at Rs. 100 each + Cost of 2 blankets at Rs. 150 each + Cost of 2 blankets at unknown rate Total cost = Rs. 300 + Rs. 300 + Rs. 900 Total cost = Rs. 1500 The total number of blankets purchased is: Total blankets = 3 blankets at Rs. 100 each + 2 blankets at Rs. 150 each + 2 blankets at unknown rate Total blankets = 3 + 2 + 2 Total blankets = 7 Now, to find the average price of all the blankets, we divide the total cost by the total number of blankets: Average price = Total cost / Total blankets Average price = Rs. 1500 / 7 Average price = Rs. 214.29 (rounded to two decimal places) Therefore, the average price of all the blankets is Rs. boxed{214.29} .

answer:Let's start by calculating the total number of white t-shirts Darren bought. Since each pack of white t-shirts contains 6 t-shirts and he bought 5 packs, we can multiply 6 by 5 to get the total number of white t-shirts: 6 t-shirts/pack * 5 packs = 30 white t-shirts Now we know that Darren bought a total of 57 t-shirts, and we've just calculated that 30 of those are white. To find out how many blue t-shirts he bought, we subtract the number of white t-shirts from the total number of t-shirts: 57 t-shirts in total - 30 white t-shirts = 27 blue t-shirts Now we need to find out how many blue t-shirts are in each pack. We know that Darren bought 3 packs of blue t-shirts, so we divide the total number of blue t-shirts by the number of packs to find out how many are in each pack: 27 blue t-shirts / 3 packs = 9 blue t-shirts/pack Therefore, there are boxed{9} blue t-shirts in each pack.

answer:1. **Assumptions and Setup**: Assume there are 100 students at Central College for simplicity. Since 50% like dancing, 50 students like it and 50 students do not. 2. **Distribution of Preferences**: - Of the 50 students who like dancing, 70% say they like it, giving (0.7 times 50 = 35) students. - The remaining 30% who like dancing say they dislike it, which amounts to (0.3 times 50 = 15) students. - Among the 50 students who dislike dancing, 80% rightly say they dislike it, which is (0.8 times 50 = 40) students. - The remaining 20% wrongly say they like it; hence (0.2 times 50 = 10) students. 3. **Tabulation**: [ begin{array}{|c|c|c|c|} hline & text{Likes dancing} & text{Doesn't like dancing} & text{Total} hline text{Says they like dancing} & 35 & 10 & 45 hline text{Says they don't like dancing} & 15 & 40 & 55 hline text{Total} & 50 & 50 & 100 hline end{array} ] 4. **Finding the specific fraction**: - The number of students who say they dislike dancing but actually like it: 15. - Total students saying they dislike dancing: 55. - Calculate the fraction: (frac{15}{55} = frac{3}{11}). 5. **Conclusion**: - The fraction of students who say they dislike dancing but actually like it is (frac{3}{11}), which cannot be accurately expressed as a percentage. The correct answer (final content) is therefore (frac{3{11}}). The final answer is The correct answer, given the choices, is boxed{textbf{(C)} frac{3}{11}}.