Mathematics

The average age of all the boys and girls combined is calculated by taking the total age of all boys (m × b) plus the total age of all girls (n × c) and dividing by the total number of children (m + n). Hence, the formula is (mb + nc) / (m + n).

The combined average is calculated by taking the total marks of both classes divided by the total number of students: ((30×40) + (50×60)) / (30+50) = (1200 + 3000) / 80 = 4200 / 80 = 52.5.

The first 10 even numbers are 2, 4, 6, 8, 10, 12, 14, 16, 18, and 20. Their sum is 110, and the average is 110 divided by 10, which equals 11. However, option D is 11, so the correct answer is D, not C. Upon rechecking, option D is 11, which is the correct average. Therefore, the correct answer is D.

The average of the first 10 natural numbers (1 to 10) is calculated as the sum of these numbers divided by 10. The sum is (10 × 11) / 2 = 55, so the average is 55 ÷ 10 = 5.5.

The total of five results is 5 × 46 = 230. The total of the first four results is 4 × 45 = 180. Therefore, the fifth result is 230 - 180 = 50.

The total average age of the three boys is 15 years, so their total combined age is 15 × 3 = 45 years. The ages are in the ratio 3:5:7, which sums to 15 parts. Each part equals 45 ÷ 15 = 3 years. Therefore, the youngest boy's age is 3 × 3 = 9 years.

The average charge per person is calculated by taking the weighted average of charges based on attendance ratios: (15×2 + 7.5×5 + 2.5×13) / (2 + 5 + 13) = (30 + 37.5 + 32.5) / 20 = 100 / 20 = Rs.5. However, the calculation shows Rs.5, which corresponds to option C, but the original question's options and answer need rechecking. Recalculating carefully: total charge = (15×2) + (7.5×5) + (2.5×13) = 30 + 37.5 + 32.5 = 100; total attendance = 2 + 5 + 13 = 20; average = 100/20 = 5. So the correct answer is Rs.5, option C. The original correct_answer is correct, but the explanation was missing and difficulty was not set. Difficulty is medium because it involves weighted averages and ratios.

Prime numbers are natural numbers greater than 1 that have no divisors other than 1 and themselves. This means they can only be divided evenly by 1 and the number itself, which matches option C.

Let the average weight of the first 25 persons be x kg. After adding the new person weighing 46 kg, the average decreases by 5 kg, so the new average is x - 5. The total weight of the first 25 persons is 25x, and the total weight after adding the new person is 25x + 46. The new average is (25x + 46) / 26 = x - 5. Solving this equation gives x = 62 kg.

First, find the cost price (CP) using the selling price (SP) of Rs. 50,000 and profit of 17%. CP = SP / (1 + profit%) = 50000 / 1.17 ≈ Rs. 42735. Then, calculate profit when SP = Rs. 47000: Profit = 47000 - 42735 = Rs. 4265. Profit percentage = (4265 / 42735) × 100 ≈ 9.98%. However, this matches option B, so let's re-check calculations: Actually, the profit percentage when sold at Rs. 47000 is less than 17%, so option B is correct. The initial explanation was incorrect; the correct answer is B.

Since Rs. 3300 is 75% of the man's monthly income, the full monthly income can be calculated by dividing 3300 by 0.75. This gives 3300 ÷ 0.75 = Rs. 4400. Therefore, the correct monthly income is Rs. 4400, which corresponds to option B.

The total marks for each batch are calculated by multiplying the number of students by their average marks: (55×50)=2750, (60×55)=3300, and (45×60)=2700. The total number of students is 55+60+45=160. The overall average marks for all students combined is the total marks divided by the total number of students: (2750+3300+2700)/160 = 8750/160 = 54.6875, which rounds to 54.68. Among the given options, 54.68 (option B) is the correct answer.

The total sum of 11 results is 11 × 50 = 550. The sum of the first six results is 6 × 49 = 294, and the sum of the last six results is 6 × 52 = 312. Since the sixth result is included in both the first six and last six results, adding these sums gives 294 + 312 = 606, which counts the sixth result twice. Therefore, the sixth result = 606 - 550 = 56.

The total sum of 9 observations is 9 × 9 = 81. The sum of the first 5 observations is 5 × 10 = 50, and the sum of the last 5 observations is 5 × 8 = 40. The 5th observation is included in both the first and last 5 observations, so adding these sums gives 50 + 40 = 90, which counts the 5th observation twice. Therefore, the 5th observation = 90 - 81 = 9. However, since the sum of all observations is 81, the 5th observation must be 9 to satisfy both averages. But this contradicts the calculation, so rechecking: sum(first 5) + sum(last 5) = sum(all) + 5th observation, so 50 + 40 = 81 + x, thus x = 9. Therefore, the 5th observation is 9. The correct answer is A.

The total sum of 11 numbers is 11 × 10.9 = 119.9. The sum of the first six numbers is 6 × 10.5 = 63, and the sum of the last six numbers is 6 × 11.4 = 68.4. Since the sixth number is included in both the first six and last six, adding these sums counts the sixth number twice. Therefore, sixth number = (63 + 68.4) - 119.9 = 11.5. However, this calculation yields 11.5, which matches option D. Rechecking the arithmetic: 63 + 68.4 = 131.4; 131.4 - 119.9 = 11.5. So the sixth number is 11.5, which corresponds to option D. The original correct_answer was D, which is correct. The explanation was missing and is now provided. Difficulty is medium as it requires understanding of averages and overlapping sums.

The total cost of known blankets is (3×100) + (5×150) = 300 + 750 = Rs.1050. Let the unknown rate be x. The average price is given by total cost divided by total blankets: (1050 + 2x) / 10 = 150. Solving gives 1050 + 2x = 1500, so 2x = 450, and x = 225. However, option B is Rs.225, so correct answer is B, not C. Rechecking options: A=150, B=225, C=250, D=None. So correct answer is B.

The first ten odd prime numbers are 3, 5, 7, 11, 13, 17, 19, 23, 29, and 31. Their sum is 158. To find the average, divide the sum by 10, which gives 158 ÷ 10 = 15.8. Therefore, the correct answer is 15.8, corresponding to option C.

Let the average expenditure of all nine men be x. Then the ninth man spends x + 2. The other eight men spend 3 each, so total expenditure is 8*3 + (x + 2) = 24 + x + 2 = x + 26. Since average is total/9, x = (x + 26)/9. Solving gives x = 3.25. Total expenditure = 9 * 3.25 = Rs.29.25. However, this contradicts the initial calculation, so rechecking: 8 men spend 3 each = 24, ninth spends x + 2, average x = (24 + x + 2)/9 = (26 + x)/9, so x = (26 + x)/9 => 9x = 26 + x => 8x = 26 => x = 3.25. Ninth man spends 3.25 + 2 = 5.25. Total = 24 + 5.25 = 29.25. So total money spent is Rs.29.25, which matches option A. Therefore, correct answer is A.

Let the average before the 17th innings be x. Total runs before 17th innings = 16x. After scoring 85 in the 17th innings, the new average is x + 3. So, (16x + 85)/17 = x + 3. Solving gives x = 33, so the new average is 36.

Let the original average expenditure per student be x. Initially, total expenditure = 100x. After accommodating 20 more students, average decreases by Rs.5, so new average = x - 5. New total expenditure = 120(x - 5) = 120x - 600. Given total expenditure increased by Rs.400, so 120x - 600 = 100x + 400. Solving gives x = Rs.50. Therefore, new total expenditure = 120 * (50 - 5) = 120 * 45 = Rs.5400.

Isaac Newton, along with Gottfried Wilhelm Leibniz, is credited as a founder of calculus. Newton developed the fundamental principles of calculus independently, earning him the title 'Father of Calculus.'

1 inch is exactly equal to 2.54 centimeters, which is the standard conversion factor used internationally.

Doubling each student's marks doubles the total sum of marks, so the average also doubles. Since the original average is 36, the new average is 36 × 2 = 72.

The original average is 23. Increasing each of the 10 numbers by 4 increases the total sum by 10 × 4 = 40. The new average is (original total + 40) ÷ 10 = (10 × 23 + 40) ÷ 10 = (230 + 40) ÷ 10 = 270 ÷ 10 = 27.

Three years ago, the total age of the six members was 6 × 19 = 114 years. Today, each member is 3 years older, so total age of original six members is 114 + 6 × 3 = 132 years. Let the boy's age be x. The average age today is (132 + x) / 7. Since the average remains 19, (132 + x) / 7 = 19, so 132 + x = 133, giving x = 1 year. However, this contradicts the calculation. Rechecking: 3 years ago average = 19, so total age = 114. Today, original members aged 3 years each, total 114 + 18 = 132. Average remains 19 with 7 members: total age = 7 × 19 = 133. So, boy's age = 133 - 132 = 1 year. Therefore, the correct answer is A (1 year). The initial answer was A, which is correct. The explanation clarifies the calculation.