Average and Calculations Quiz

Average

• The average of a set of numbers is the sum of the numbers divided by the number of numbers
• Finding the seventh number: Given the average of the first 7 numbers is 5 and the average of the first 6 numbers is 4, we can calculate the seventh number
• The sum of the first 7 numbers is 7 * 5 = 35
• The sum of the first 6 numbers is 6 * 4 = 24
• Therefore, the seventh number is 35 - 24 = 11
• Adding a number to the average: The average of 11 numbers is 24. If we add 108, the total sum becomes 11 * 24 + 108 = 330
• The new average is 330/12 = 27.5
• Maximum average weight: The average weight of 40 students is 40 kgs.
• The total weight of the 40 students is 40 * 40 = 1600 kgs
• If m new students join with average weight n kgs, the total weight increases by m * n kgs
• Since m + n = 50, the maximum value of m * n occurs when m and n are as close as possible
• Therefore, the maximum value of m * n is 25 * 25 = 625
• The total weight of the class after the new students join is 1600 + 625 = 2225 kgs
• The maximum possible average weight is 2225 / (40 + 25) = 40.49 kgs
• Average of averages: The average of the averages of all possible groups of 4 numbers will not be equal to the average of the 7 numbers
• Example: Imagine the first 4 numbers are all very large, and the last 3 numbers are all very small.
• The average of the first 4 numbers would be very high
• The average of the last 3 numbers would be very low
• The average of all the averages of groups of 4 would be closer to the high values, and not equal to the overall average of the 7 numbers
• Maximum possible score: The average score of the remaining 5 students is 60 - 5 = 55
• The total score of the remaining 5 students is 5 * 55 = 275
• The total score of all 10 students is 10 * 60 = 600
• The total score of the top 5 students is 600 - 275 = 325
• Since each of the top 5 scorers had distinct integral scores, the maximum possible score of the topper is 100 + 99 + 98 + 97 + 96 = 490
• The total score of the top 5 students cannot exceed 490, so their maximum possible total score is 490
• The maximum possible score of the topper is 490 - (99 + 98 + 97 + 96) = 98

Ratio

• Ratio of wheat: To achieve a mixture worth 3 Rs/kg from wheat at 3.20 Rs/kg and 2.90 Rs/kg, the ration must be calculated
• Assume the fraction of wheat at 3.20 Rs/kg is x and the fraction of wheat at 2.90 Rs/kg is 1-x
• Considering price per kg:
• 3 = 3.20 * x + 2.90 *(1-x)
• Solving for x:
• 3 = 3.20 * x + 2.90 - 2.90 * x
• 0.30 = 0.30 * x
• x = 1
• Therefore, to mix the wheat at 3.20 Rs/kg with wheat at 2.90 Rs/kg to get a mixture worth 3 Rs/kg, the ratio is 1:0

Profit

• Ratio of tea: To achieve a profit of 20% when selling the mixed tea at 194.40 Rs/kg, the ratio of the two types of tea must be calculated
• Given one type of tea costs 192 Rs/kg and the other costs 150 Rs/kg, let x be the fraction of tea costing 192 Rs/kg and 1-x be the fraction of tea costing 150 Rs/kg
• Considering profit: After adding the profit of 20% , the cost price of the mixture becomes 194.40 * (100/120) = 162
• Considering price per kg:
• 162 = 192 * x + 150 * (1-x)
• Solving for x:
• 162 = 192 * x + 150 - 150 * x
• 12 = 42 * x
• x = 2/7
• Therefore, the ratio of the two types of tea should be 2:5, meaning the mixture should contain 2 parts of the tea costing 192 Rs/kg and 5 parts of the tea costing 150 Rs/kg

This quiz explores concepts related to averages, including calculating the average of groups of numbers and the impact of adding new values. It covers finding specific numbers based on average conditions and determining new averages after changes in data sets. Test your understanding of these averaging principles!