Average - Cricket Based Problems on Average (Part 2)
Duration: 10 min
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AI Summary
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This video is a lecture on solving average-related problems, presented by an instructor in a virtual classroom setting. The lecture begins with a problem about Yuvraj Singh's cricket innings, where the average of 40 innings is 50, and the difference between the highest and lowest runs is 172. The average of the remaining 38 innings is 48. The instructor uses the formula for average (Sum = Average × Number) to find the total sum of runs for 40 innings (2000) and for 38 innings (1824). By subtracting these, the sum of the highest and lowest runs is found to be 176. This is combined with the given difference of 172 to form a system of equations: x + y = 176 and x - y = 172, where x is the highest run and y is the lowest. Solving this system yields x = 174 and y = 2, making the highest run 174. The video then transitions to a second problem about the average age of an Indian cricket team. It states that Hardik Pandya is 25, KL Rahul is 3 years older (28), and the average age of the remaining 9 players is x. The total sum of ages is 11x + 25 + 28 = 11x + 53. When Hardik and Rahul are removed, the average of the remaining 9 players is x-1, so their total sum is 9(x-1) = 9x - 9. The equation 11x + 53 = 9x - 9 is solved to find x = -31, which is an impossible age, indicating a data inconsistency. The final problem involves 8 observations with an average of 20, so the total sum is 160. The average of the first two is 15.5, so their sum is 31. The average of the next three is 21.3, so their sum is 64. The sixth observation is 4 less than the seventh, and the seventh is 7 less than the eighth. The instructor sets up the equation 31 + 64 + (x-4) + x + (x+7) = 160, which simplifies to 3x + 98 = 160, leading to 3x = 62 and x = 20.67. The last observation is x+7 = 27.67. The video concludes with a 'Thanks for Watching' screen.
Chapters
0:00 – 2:00 00:00-02:00
The video opens with a title card displaying the word 'AVERAGE' in a futuristic, glowing green font against a dark, digital background. This transitions to a lecture screen where the first problem is presented. The problem states: 'The average runs of 40 innings of Yuvraj Singh is 50. The difference between the highest and lowest runs scored by him in any innings is 172. If the average run of the remaining 38 innings is 48. What is the highest run scored by Yuvraj in any inning?'. The options are (a) 172, (b) 174, (c) 176, (d) Data Insufficient. The instructor begins to solve the problem by writing the formula 'Avg = Sum / n' on the blackboard. He calculates the total sum of runs for 40 innings as 40 * 50 = 2000.
2:00 – 5:00 02:00-05:00
The instructor continues solving the first problem. He calculates the total sum of runs for the 38 remaining innings as 38 * 48 = 1824. He then subtracts this from the total sum of 40 innings to find the sum of the highest and lowest runs: 2000 - 1824 = 176. He writes the equation 'x + y = 176' on the board, where x is the highest run and y is the lowest. He then uses the given difference, 'x - y = 172'. He adds the two equations: (x + y) + (x - y) = 176 + 172, which simplifies to 2x = 348. He solves for x, getting x = 174. He concludes that the highest run scored is 174, which corresponds to option (b). The instructor then moves to the next problem.
5:00 – 10:00 05:00-10:00
The second problem is presented: 'In Indian Cricket team, the age of Hardik Pandya is 25 years. Wicket Keeper KL Rahul is 3 years older than him. If we leave these two from the team, then the average age of the remaining team is decreased by 1 year from the average age of the whole Indian team. What is the average age of the whole team?'. The options are (a) 19, (b) 20, (c) 21, (d) 22. The instructor sets up the problem by letting the average age of the remaining 9 players be 'x'. The total sum of ages for the 11 players is 11x + 25 + 28 = 11x + 53. The sum of the ages of the remaining 9 players is 9(x-1) = 9x - 9. He sets up the equation 11x + 53 = 9x - 9. He simplifies this to 2x = -62, which gives x = -31. He notes that this is an impossible age, indicating a data inconsistency. He then moves to the third problem.
10:00 – 10:24 10:00-10:24
The third problem is presented: 'The average of 8 observations is 20. The average of first two and next three observations is 15 1/2 and 21 1/3. If the 6th observation is 4 less than the 7th and 7th is 7 less than 8th observation, then find the last observation?'. The options are (a) 25, (b) 30, (c) 27, (d) Data Insufficient. The instructor calculates the total sum of 8 observations as 8 * 20 = 160. The sum of the first two observations is 2 * 15.5 = 31. The sum of the next three is 3 * 21.333... = 64. He lets the 7th observation be 'x'. Then the 6th is 'x-4' and the 8th is 'x+7'. He sets up the equation: 31 + 64 + (x-4) + x + (x+7) = 160. He simplifies this to 3x + 98 = 160, which gives 3x = 62 and x = 20.67. The last observation is x+7 = 27.67. The video ends with a 'Thanks for Watching' screen.
The video presents a structured lesson on solving average problems, progressing from a straightforward calculation to more complex scenarios involving systems of equations. The core concept is the fundamental formula for average, Sum = Average × Number, which is applied to find total sums and then used to set up equations. The first problem demonstrates how to find the sum of two unknown values (highest and lowest runs) by subtracting the sum of the remaining values from the total sum. This is then combined with a difference to form a system of two linear equations, which is solved to find the individual values. The second problem introduces a logical inconsistency, where the data leads to an impossible negative age, highlighting the importance of checking the validity of solutions. The third problem uses a similar approach, setting up a system of equations based on relationships between variables (observations) to find a specific value. The overall teaching method is to break down each problem into clear, logical steps, using the blackboard to show the derivation process.