Question 1

The average weight of 8 men is increased by 1.5 kg when one of the men, who weighs 65 kg is replaced by a new man. The weight of the new man is

Accepted Answer

Concept & Strategy: When a member of a group is replaced and the overall average increases, the new member must weigh more than the departing member. Specifically, the new member must weigh enough to match the departed person's weight, *plus* the extra weight required to increase the average of the entire group. New\ Value = Replaced\ Value + (Number\ of\ Entities × Increase\ in\ Average) Step-by-Step Solution: • Determine the total weight increase: The average weight of all 8 men increases by 1.5 kg. Total\ Increase = 8 × 1.5 kg = 12 kg • Calculate the new man's weight: The new man must cover the 65 kg of the man he replaced, plus the additional 12 kg that raised the group's average. New\ Man = 65 kg + 12 kg New\ Man = 77 kg Exam Strategy & Shortcut: Problems involving a 1:1 replacement don't require you to know the original average. Use the direct deviation formula instantly: New Weight = Old Weight + (Total Members × Average Change). 65 + (8 × 1.5) = 77. You can calculate this mentally in seconds. Common Pitfall: Adding the average increase (1.5 kg) directly to the replaced man's weight (65 + 1.5 = 66.5 kg). Remember that the 1.5 kg increase applies to *every single one* of the 8 men, so you must multiply it by the total group size first. Final Answer: Therefore, the correct answer is 77 kg.

Question 2

The average height of 25 boys is 1.4 m. When 5 boys leave the group, then the average height increases by 0.15 m. What is the average height of the 5 boys who leave ?

Accepted Answer

Concept & Strategy: When a subgroup leaves and the overall average *increases*, it means the departing subgroup had a lower average than the main group. They were pulling the average down. The total amount the average increased across the remaining members is equal to the "weight" the departing members left behind. Total\ Sum = Average × Number\ of\ Entities Step-by-Step Solution: • Calculate the initial total height: For the 25 boys, the average is 1.4 m. Sum_{initial} = 25 × 1.4 = 35 m • Calculate the new total height: 5 boys leave, so 20 boys remain. The new average increases by 0.15 m, making it 1.4 + 0.15 = 1.55 m. Sum_{new} = 20 × 1.55 = 31 m • Find the total height of the 5 boys who left: Subtract the new total from the initial total. Sum_{left} = 35 - 31 = 4 m • Calculate their average height: Divide their total height by 5. Average_{left} = (4/5) = 0.8 m Exam Strategy & Shortcut: Use the Deviation Method to avoid multiplying large decimals. The 5 boys who left gave up 0.15 m to each of the remaining 20 boys. Total height they "gave away" to the group = 20 × 0.15 = 3 m. This means the 5 boys were 3 m *short* of maintaining the old average for their subgroup. If they had the old average, their sum would be 5 × 1.4 = 7 m. Since they were 3 m short, their actual sum is 7 - 3 = 4 m. Average = (4/5) = 0.8 m. One-line formula: Avg of Left = Old Avg - \frac{Remaining Members × Increase in Avg}{Leaving Members} Avg = 1.4 - (20 × 0.15/5) = 1.4 - 0.6 = 0.8 m Common Pitfall: Calculating the total increase (20 × 0.15 = 3) and mistakenly subtracting it directly from the original average 1.4 m without distributing it across the 5 departing boys. The 3 m deficit belongs to all 5 boys collectively. Final Answer: Therefore, the correct answer is 0.8 m.

Question 3

The average of 11 players of a cricket team is decreased by 2 months when two of them aged 17 years and 20 years are replaced by two new players. The average age of the new players is

Accepted Answer

Concept & Strategy: When members are replaced and the average decreases, the incoming members must weigh less (or be younger) than the outgoing members. The total difference in age between the old and new players equals the total decrease in the group's average. Step-by-Step Solution: • Calculate the total age decrease: The average of all 11 players decreases by 2 months. Total\ Decrease = 11 × 2 months = 22 months • Calculate the total age of the departing players: Player 1 is 17 years old, Player 2 is 20 years old. Sum_{old} = 17 + 20 = 37 years • Calculate the total age of the new players: The new players must be 22 months younger collectively than the players they replaced. Sum_{new} = 37 years - 22 months To subtract properly, convert 1 year from the 37 years into months: 37 years = 35 years + 24 months Sum_{new} = (35 years + 24 months) - 22 months Sum_{new} = 35 years and 2 months • Calculate the average age of the two new players: Divide their total age by 2. Average_{new} = \frac{35 years 2 months}{2} Convert the odd year to months to make division by 2 perfectly clean: 35 years 2 months = 34 years + 14 months Average_{new} = 17 years and 7 months Exam Strategy & Shortcut: Instead of converting to large totals of months, handle the years and the deviation separately. The average of the departing players is (37/2) = 18.5 years, which is 18 years 6 months. The total deficit they bring is 22 months, meaning *each* new player is individually responsible for an 11-month deficit from that outgoing average. 18 years 6 months - 11 months = 17 years 7 months. Common Pitfall: Struggling with the base-12 time conversion. Students often treat months as base-10 decimals, incorrectly assuming 35 years 2 months divided by 2 is 17.5 years 1 month. Always borrow 12 months when carrying over years. Final Answer: Therefore, the correct answer is 17 years 7 months.

Question 4

When 15 is included in a list of natural numbers, their mean is increased by 2. When 1 is included in this new list, the mean of the numbers in the new list is decreased by 1. How many numbers were there in the original list?

Accepted Answer

Concept & Strategy: When a new number alters the mean, the difference between that incoming number and the resulting new mean equals the total change distributed across all previous items. We can build two linked equations from the two successive additions. Step-by-Step Solution: Let the original number of items be N and their original mean be M. • Analyze the First Addition (15 is added): The new list has N+1 items. The mean becomes M+2. The sum equation is: N \cdot M + 15 = (N + 1)(M + 2) N \cdot M + 15 = N \cdot M + 2N + M + 2 15 = 2N + M + 2 2N + M = 13\ \ --- (Equation 1) • Analyze the Second Addition (1 is added): The current list has N+1 items and a mean of M+2. We add a 1, so the new list has N+2 items. The mean decreases by 1, so it becomes (M+2) - 1 = M+1. The sum equation is: (N+1)(M+2) + 1 = (N+2)(M+1) (N \cdot M + 2N + M + 2) + 1 = N \cdot M + N + 2M + 2 2N + M + 3 = N + 2M + 2 N + 1 = M\ \ --- (Equation 2) • Solve the System: Substitute M = N + 1 into Equation 1: 2N + (N + 1) = 13 3N + 1 = 13 3N = 12 N = 4 Exam Strategy & Shortcut: Use the Deviation Spread mental framework to build the equations without expanding algebraic binomials. First addition: 15 comes in, making the new mean M+2. The excess value 15 brings is 15 - (M+2). This excess is what gave +2 to the original N numbers. N × 2 = 15 - (M+2) \implies 2N + M = 13. Second addition: 1 comes into a list that currently has N+1 members and a mean of M+2. The new mean drops by 1, becoming M+1. The deficit that 1 brings is 1 - (M+1) = -M. This deficit is what caused the -1 drop for the current N+1 members. (N+1) × (-1) = -M \implies -N - 1 = -M \implies M = N+1. Substitute M = N+1 directly into 2N + M = 13 \implies 3N + 1 = 13 \implies N = 4. Common Pitfall: Misreading "decreased by 1" in the second condition. Students often subtract 1 from the *original* mean M (writing M-1) instead of subtracting 1 from the *current* mean of the new list (which is M+2, making it M+1). Pay close attention to relative baselines in successive problems. Final Answer: Therefore, the correct answer is 4.

Question 5

A batsman makes a score of $84$ runs in the 21st inning and thus increases his average by $2$ runs. His average after 21st inning is

Accepted Answer

Concept & Logic: The batting average is mathematically defined as the total runs scored divided by the number of innings played. Batting Average = \frac{Total Runs}{Total Innings} Step-by-Step Solution: • Given: • Score in the 21st inning = 84 • Increase in overall average = 2 • Calculation: Let the average after 20 innings be x. Total runs after 20 innings = 20x Total runs after 21 innings = 20x + 84 New average after 21 innings = x + 2 Since the total runs after 21 innings can also be expressed as 21(x + 2), we set up the equation: 20x + 84 = 21(x + 2) 20x + 84 = 21x + 42 x = 42 The question specifically asks for the average *after* the 21st inning, which is x + 2. 42 + 2 = 44 Exam Strategy & Shortcut: You can solve this mentally in under 10 seconds. The batsman increased his average by 2 runs over 21 innings. This means he needed 21 × 2 = 42 extra runs above his old average to pull the overall average up by 2. Old Average = Runs scored - Extra runs = 84 - 42 = 42. New Average = Old Average + 2 = 42 + 2 = 44. Common Pitfall: The most frequent mistake is stopping at solving for x = 42 and selecting that as the answer. Always re-read the final sentence of the question to ensure you are finding the "new" average, not the previous one. Final Answer: Therefore, the correct answer is 44.

Question 6

A team of $8$ persons joins in a shooting competition. The best marksman scored $85$ points. If he had scored $92$ points, the average score for the team would have been $84$. The number of points, the team scored was

Accepted Answer

Concept & Formula: The total sum of observations is related to the average by the following baseline formula: Total Score = Average Score × Number of Persons Step-by-Step Solution: • Given: • Number of persons in the team = 8 • Actual score of the best marksman = 85 • Hypothetical score of the best marksman = 92 • Hypothetical average score of the team = 84 • Calculation: Let the actual total score of the team be T. If the marksman scores 92 instead of 85, the net increase in the team's total score would be: 92 - 85 = 7 points Therefore, the hypothetical total score becomes (T + 7). We are given that this hypothetical total score results in an average of 84 for the 8 members: T + 7 = 84 × 8 T + 7 = 672 T = 672 - 7 = 665 Exam Strategy & Shortcut: Calculate the hypothetical total directly: 84 × 8 = 672. Since this total is based on an assumed score that is 7 points higher (92 - 85 = 7) than what was actually achieved, simply subtract the difference from the hypothetical total to find the actual score: 672 - 7 = 665. Common Pitfall: A common error is to accidentally add the 7-point difference to 672 instead of subtracting it, leading to the incorrect option of 672 or a completely wrong value. Always remember that the hypothetical condition yields a *higher* total than reality. Final Answer: Therefore, the correct answer is 665.

Question 7

The average age of husband, wife and their child $3$ years ago was $27$ years and that of wife and the child $5$ years ago was $20$ years. The present age of the husband is

Accepted Answer

Concept & Strategy: Break the problem down by bringing different historical data points to the present timeline separately. Once all sums are in the present tense, finding the missing individual is simple subtraction. Step-by-Step Solution: • Given: • Average age of Husband (H), Wife (W), and Child (C) 3 years ago = 27 years • Average age of Wife (W) and Child (C) 5 years ago = 20 years • Calculation: Step 1: Process the 3-member group Sum of ages of H, W, and C (3 members) 3 years ago: 27 × 3 = 81 years Present sum of H, W, C (add 3 years for each of the 3 members): Present Sum (H+W+C) = 81 + (3 × 3) = 81 + 9 = 90 years Step 2: Process the 2-member group Sum of ages of W and C (2 members) 5 years ago: 20 × 2 = 40 years Present sum of W and C (add 5 years for each of the 2 members): Present Sum (W+C) = 40 + (2 × 5) = 40 + 10 = 50 years Step 3: Isolate the Husband's age H = (H + W + C) - (W + C) H = 90 - 50 = 40 years Exam Strategy & Shortcut: Operate entirely on the present averages. Present average of (H+W+C) = 27 + 3 = 30. Total = 30 × 3 = 90. Present average of (W+C) = 20 + 5 = 25. Total = 25 × 2 = 50. Husband's age = 90 - 50 = 40. This mental math method takes less than 15 seconds. Common Pitfall: Mixing up the timeframes when adding the elapsed years. Students sometimes add 3 × 2 = 6 years to the second group instead of 5 × 2 = 10, misreading the conditions. Always pair the correct historical timeframe with the correct group. Final Answer: Therefore, the correct answer is 40 years.

Question 8

Four years ago, the average age of a family of four persons was 18 years. During this period, a baby was born. Today if the average age of the family is still 18 years, the age of the baby is

Accepted Answer

Concept & Formula: When time passes, every living member's age increases by the same number of years. Total Age = Average Age × Number of Members --- Step-by-Step Solution: • Given: • 4 years ago, 4 members had an average age of 18 years. • Total age 4 years ago = 4 × 18 = 72 years. • Calculation for Today: • In 4 years, each of the 4 original members aged by 4 years. • Present total age of these 4 members = 72 + (4 × 4) = 88 years. • Today, there are 5 members (including the baby), and the average age is still 18 years. • Present total age of the 5-member family = 5 × 18 = 90 years. • Finding the Baby's Age: • Age of the baby = (Total age of 5 members today) - (Total age of 4 original members today) • Age of the baby = 90 - 88 = 2 years. --- Exam Strategy & Shortcut: • If no baby was born, the average age of the 4 members today would increase by 4 years, becoming 18 + 4 = 22 years. • The deficit in the total sum caused by the baby maintaining the average at 18 is: 4 × (22 - 18) = 16 years • The baby brings down the average, meaning the baby's age is: 18 - 16 = 2 years --- Common Pitfall: A frequent error is forgetting to add 4 years to each of the 4 original family members, leading to an incorrect total present age calculation. --- Final Answer: Therefore, the correct answer is 2 hours.

Question 9

After replacing an old member by a new member, it was found that the average age of five members of a club is the same as it was 3 years ago. What is the difference between the ages of the replaced and the new member?

Accepted Answer

Concept & Logic: If no member was replaced, the average age of the 5 members today would have increased by 3 years compared to 3 years ago. Since the replacement keeps the average exactly the same as 3 years ago, the new member must be younger than the old member by the total value that would have accumulated over those 3 years. --- Step-by-Step Solution: • Given: * Number of members = 5 • Time duration = 3 years • Deduction: • If the same 5 members continued, the total age of the group would increase by: 5 × 3 = 15 years • Because the old member was replaced by a new member, the total age of the group dropped by exactly this cumulative increase of 15 years to bring the average back to its previous state. • Therefore, the difference between the age of the replaced member and the new member is exactly 15 years. --- Exam Strategy & Shortcut: Direct Formula for Replacement: Difference in age = Number of items × Change in average for the group Difference = 5 × 3 = 15 years This problem can be solved mentally in under 5 seconds using this direct relation. --- Common Pitfall: Do not confuse the change in average over time with a regular replacement where the average changes instantly. Here, the phrase "same as it was 3 years ago" means the total age dropped by 5 × 3 relative to what it should be today. --- Final Answer: Therefore, the correct answer is 15 years.

Question 10

The average age of 3 children in a family is 20% of the average age of the father and the eldest child. The total age of the mother and the youngest child is 39 years. If the father’s age is 26 years, what is the age of the second child?

Accepted Answer

Concept & Strategy: To find a specific unknown variable in a system of linear equations, the number of independent equations must be at least equal to the number of unknown variables. --- Step-by-Step Solution: • Given Variables: Let the three children be C_1 (eldest), C_2 (second), and C_3 (youngest). Let the Father be F and Mother be M. • Given Constraints: 1. Average of 3 children = 20% of average of father and eldest child: (C_1 + C_2 + C_3/3) = 0.20 × ((F + C_1/2)) 2. Total age of mother and youngest child: M + C_3 = 39 3. Father's age: F = 26 • Deduction: • We have 5 variables: C_1, C_2, C_3, F, M. • We are provided with only 3 independent pieces of information/equations. • Substituting F = 26 into the first equation yields: (C_1 + C_2 + C_3/3) = 0.10 × (26 + C_1) C_1 + C_2 + C_3 = 0.3(26 + C_1) 0.7C_1 + C_2 + C_3 = 7.8 • We still have one equation containing M and C_3, and another containing C_1, C_2, C_3. There is insufficient data to isolate C_2. --- Exam Strategy & Shortcut: Scan the variables versus the equations provided. You have 5 unknown individuals (F, M, C_1, C_2, C_3) and only 3 mathematical conditions. Since no other relationships link the middle child uniquely to the other known values, the system is underdetermined. Mark Cannot be determined immediately without doing algebraic expansions. --- Common Pitfall: Do not waste time trying to solve the algebraic equations by guessing values. If the number of variables significantly exceeds the number of independent equations, look out for the "Cannot be determined" option. --- Final Answer: Therefore, the correct answer is Cannot be determined.

Question 11

Mr. Joe's family consists of six people—himself, his wife and their four children. It is known that the average age of the family immediately after the birth of the first, second, third and fourth child was 16, 15, 16 and 15 years respectively. Find…

Accepted Answer

Concept & Logic: To solve this problem, we use the principle of Total Age = Average Age × Number of Members. We must track the total age of the family at each given milestone (birth of each child). Note that the age of a newborn is exactly 0 years. Step-by-step Solution: • At 1st child's birth: Family size = 3. Average age = 16. Total age = 3 × 16 = 48 years. (This is simply the sum of the parents' ages, as the newborn is 0). • At 2nd child's birth: Let the gap between the 1st and 2nd child be x years. During this time, the parents and the first child all age by x years. Total age = 48 + 3x. Given average = 15, Family size = 4. 48 + 3x = 4 × 15 = 60 \Rightarrow 3x = 12 \Rightarrow x = 4 years. (The first child is now 4 years old). • At 3rd child's birth: Let the gap between the 2nd and 3rd child be y years. Total age = 60 + 4y. Given average = 16, Family size = 5. 60 + 4y = 5 × 16 = 80 \Rightarrow 4y = 20 \Rightarrow y = 5 years. • At 4th child's birth: Let the gap between the 3rd and 4th child be z years. Total age = 80 + 5z. Given average = 15, Family size = 6. 80 + 5z = 6 × 15 = 90 \Rightarrow 5z = 10 \Rightarrow z = 2 years. At this point, the eldest son's age is the sum of the gaps: x + y + z = 4 + 5 + 2 = 11 years. • Present Day: Let t years have passed since the 4th child's birth. Total present age = 90 + 6t. Given present average = 16, Family size = 6. 90 + 6t = 6 × 16 = 96 \Rightarrow 6t = 6 \Rightarrow t = 1 year. The present age of the eldest son is his age at the 4th birth plus 1 year: 11 + 1 = 12 years. Exam Strategy & Shortcut: Instead of complex equations, just track the total age increment mentally. Start: Total age 48 (3 people). To reach average 15 for 4 people, total must be 60. Difference is 12, shared by 3 existing people \Rightarrow 4 years gap. To reach average 16 for 5 people, total must be 80. Difference is 20, shared by 4 existing people \Rightarrow 5 years gap. To reach average 15 for 6 people, total must be 90. Difference is 10, shared by 5 existing people \Rightarrow 2 years gap. Total gap = 4+5+2 = 11 years. Present total is 96, so gap is 96-90 = 6, shared by 6 people \Rightarrow 1 year gap. Final Age = 11+1 = 12. This logic is lightning-fast on paper. Common Pitfall: Students often forget that the parents *and all existing children* age during the gap years. They might divide the age difference incorrectly. Always divide the total age increase by the number of people who were alive *during* that time gap. Final Answer: Therefore, the correct answer is 12 years.

Question 12

A certain factory employed 600 men and 400 women and the average wage was ₹ 25.50 per day. If a woman got ₹ 5 less than a man, then what are their daily wages?

Accepted Answer

Concept & Logic: This relies on the principle of the weighted average. The total wages paid to the entire workforce is simply the sum of wages paid to all men plus the sum of wages paid to all women. Step-by-Step Solution: • Given Information: Total men = 600 Total women = 400 Total workforce = 1000 Overall average wage = ₹ 25.50 • Let the daily wage of a man be M. Let the daily wage of a woman be W. We are given that W = M - 5. • Calculation: Total daily wage of the factory = 1000 × 25.50 = 25500. Set up the equation for total wages: 600M + 400W = 25500 Substitute W with (M - 5): 600M + 400(M - 5) = 25500 600M + 400M - 2000 = 25500 1000M = 27500 M = 27.50 Now, find the woman's wage: W = 27.50 - 5 = 22.50 Exam Strategy & Shortcut: Use the method of deviations or "assume a base". Assume all 1000 workers were paid the woman's lower wage, W. The 600 men get an "extra" ₹ 5 each, creating a total surplus of 600 × 5 = 3000. If we distribute this surplus evenly over all 1000 workers, it increases the overall average by 3000 / 1000 = 3. So, Average Wage = W + 3. We know Average Wage = 25.50. W + 3 = 25.50 \Rightarrow W = 22.50. Consequently, M = 22.50 + 5 = 27.50. This method solves the problem mentally in under 15 seconds. Common Pitfall: Students often try to average the difference directly, thinking the average of M and M-5 is the overall average. You must account for the weights (600 vs 400); an unweighted average of the two individual wages will lead to an incorrect answer. Final Answer: Therefore, the correct answer is Man : ₹ 27.50, Woman : ₹ 22.50.

Question 13

The arithmetic mean of the scores of a group of students in a test was 52. The brightest 20% of them secured a mean score of 80 and the dullest 25% a mean score of 31. The mean score of remaining 55% is

Accepted Answer

Concept & Strategy: This is a classic weighted average problem. To find the mean of the remaining group, subtract the total values of the known subgroups (the top and bottom percentiles) from the overall total value, then divide by the remaining percentage of students. Step-by-Step Solution: • Assume a base: Let the total number of students be 100. Overall mean score = 52. Total score of all students = 100 × 52 = 5200. • Subgroup 1 (Brightest 20%): Number of students = 20. Mean score = 80. Total score of this group = 20 × 80 = 1600. • Subgroup 2 (Dullest 25%): Number of students = 25. Mean score = 31. Total score of this group = 25 × 31 = 775. • Subgroup 3 (Remaining 55%): Remaining students = 100 - (20 + 25) = 55. Remaining total score = 5200 - (1600 + 775) = 5200 - 2375 = 2825. • Calculation: Mean of the remaining students = 2825 / 55 \approx 51.36. This is approximately 51.4. Exam Strategy & Shortcut: Use the concept of Net Deviation from the average. Overall average is 52. Top 20% deviation = +28 per student. Total deviation = 20 × 28 = +560. Bottom 25% deviation = -21 per student. Total deviation = 25 × (-21) = -525. Net deviation of these two groups combined = +560 - 525 = +35. For the overall average to remain 52, the remaining 55 students must balance this +35 deviation with an exact -35 deviation. Deviation per student for the remaining 55 = -35 / 55 = -0.636. Therefore, their average is 52 - 0.636 = 51.364, which safely rounds to 51.4. Common Pitfall: A common mistake is simply averaging the averages (i.e., taking the straight mean of 80 and 31) without accounting for the respective weights (percentages) of the groups. Always use the weighted sum to preserve accuracy. Final Answer: Therefore, the correct answer is 51.4 approx.

Question 14

A coaching institute has students in 3 batches - X, Y and Z. In a certain examination, the average marks obtained by these batches are 72, 60 and 50 respectively. The average marks of batches X and Y taken together is 69. If the ratio of the number …

Accepted Answer

Concept & Logic: This problem requires the application of Alligation or weighted averages in two stages to find the true ratio of students in all three batches. Once the combined ratio n_x : n_y : n_z is unified, we apply the standard weighted average formula to find the final aggregate average. Step-by-Step Solution: • Step 1: Find the ratio of students in Batches X and Y. Average of X = 72 Average of Y = 60 Combined Average of X and Y = 69 Using Alligation (Cross difference method): Ratio of X : Y = (69 - 60) : (72 - 69) = 9 : 3 = 3 : 1. • Step 2: Combine the ratios. We found X : Y = 3 : 1 We are given Y : Z = 6 : 7 To merge these, make the value of Y consistent in both ratios. Multiply the first ratio by 6: X : Y = 18 : 6 Therefore, the cleanly combined ratio of students is X : Y : Z = 18 : 6 : 7. • Step 3: Calculate the overall average. Let the number of students be 18k, 6k, and 7k. (Total parts = 31k). Total marks = (18 × 72) + (6 × 60) + (7 × 50) Total marks = 1296 + 360 + 350 = 2006. Overall Average = 2006 / 31 \approx 64.709. Exam Strategy & Shortcut: Using Alligation to find the X : Y ratio visually is the fastest start. Instead of calculating massive numbers like 18 × 72, you can shift the base to 60 (the middle value). Deviations from 60: X has a deviation of +12 for 18 parts = +216. Y has a deviation of 0 for 6 parts = 0. Z has a deviation of -10 for 7 parts = -70. Net deviation = 216 - 70 = 146. Distribute this net deviation evenly over all 31 parts (18+6+7). 146 / 31 \approx 4.7. Add this to the assumed base of 60: 60 + 4.7 = 64.7. This requires much smaller calculations and slashes the chance of multiplication errors. Common Pitfall: Failing to align the ratios correctly before combining them. If you blindly add the parts or use unaligned variables (e.g., mixing up the Y value of 1 and 6), the final weighted average will be entirely warped and lead to a trap answer. Final Answer: Therefore, the correct answer is 64.7.

Question 15

Ten years ago, the ages of the members of a joint family of eight people added up to 231 years.  Three years later, one member died at the age of 60 years and a child was born during the same year.  After another three years, one more member died, a…

Accepted Answer

Concept & Logic: To find the current average age, we must track the total age of the family over time. The key principle is that for every passing year, the total age of the family increases by the number of living members. Step-by-Step Solution: • 10 Years Ago (T = -10): Family size = 8. Total age = 231. • 7 Years Ago (T = -7): (3 years later) Total age increases by 8 × 3 = 24 years. Total age before changes = 231 + 24 = 255. One member dies (-60) and a newborn is added (+0). New total age = 255 - 60 + 0 = 195. (Family size remains 8). • 4 Years Ago (T = -4): (Another 3 years later) Total age increases by 8 × 3 = 24 years. Total age before changes = 195 + 24 = 219. Another member dies (-60) and a newborn is added (+0). New total age = 219 - 60 + 0 = 159. (Family size remains 8). • Present Day (T = 0): (4 years later) Total age increases by 8 × 4 = 32 years. Current total age = 159 + 32 = 191. Current average = 191 / 8 = 23.875 years. Exam Strategy & Shortcut: Instead of calculating the total age at every step, track the net change over the entire 10-year period. Because a birth instantly replaces every death, the family size is a constant 8 people. Think of the family as 8 "slots" that age continuously over the 10 years. Total natural aging for these 8 slots = 8 × 10 = 80 years. During this time, two people left the family, taking their accumulated ages with them at the exact time of their departure. Net Total Age = Initial Total + Total Natural Aging - Ages at Death Net Total = 231 + 80 - 60 - 60 = 191. Current Average = 191 / 8 = 23.875. This is nearest to 24. This powerful logic bypasses tracking the individual newborns entirely and solves the problem in seconds! Common Pitfall: Students often get tangled up trying to calculate the current ages of the newborns (who are 7 and 4 years old today). While calculating their specific individual ages works if you meticulously track every living member, it frequently leads to arithmetic errors and wastes valuable time. Leveraging the constant family size of 8 to find the net total is much faster and safer. Final Answer: Therefore, the correct answer is 24 years.

Question 16

The average salary of all the workers in a workshop is ₹ 8000. The average salary of 7 technicians is ₹ 12000 and the average salary of the rest is ₹ 6000. The total number of workers in the workshop is

Accepted Answer

Concept & Strategy: This problem is a classic application of the Weighted Average or Rule of Alligation. It involves mixing two distinct groups (technicians and the rest) to form a combined whole (all workers). Step-by-Step Solution: • Given Data: Overall average salary = ₹ 8000 Average salary of technicians (Group 1) = ₹ 12000 Number of technicians = 7 Average salary of the rest (Group 2) = ₹ 6000 • Algebraic Approach: Let the total number of workers be x. The total salary of all workers = 8000x. The total salary of the 7 technicians = 7 × 12000 = 84000. The number of remaining workers = x - 7. The total salary of the remaining workers = 6000 × (x - 7). Equating the total salaries: 8000x = 84000 + 6000(x - 7) 8000x = 84000 + 6000x - 42000 8000x - 6000x = 42000 2000x = 42000 x = (42000/2000) = 21 Exam Strategy & Shortcut: The fastest way to solve this is using the Rule of Alligation. Set up the averages of the two groups and the mean in the middle: Technicians (₹ 12000) ----------- Rest (₹ 6000) \ / Mean (₹ 8000) / \ (8000 - 6000) ----------- (12000 - 8000) Ratio \rightarrow 2000 : 4000 = 1 : 2 This means the ratio of Technicians to Rest is 1 : 2. If 1 part represents 7 technicians, then the total parts (1 + 2 = 3 parts) represent the total workers. Total workers = 3 × 7 = 21. Common Pitfall: A common mistake is finding the number of "rest" workers (14) and forgetting to add back the 7 technicians to find the *total* number of workers as the question asks. Always double-check what the final variable represents! Final Answer: Therefore, the correct answer is 21.

Question 17

In an engineering college the average salary of all engineering graduates from Mechanical trade is ₹ 2.45 lacs per annum and that of the engineering graduates from Electronics trade is ₹ 3.56 lacs per annum. The average salary of all Mechanical and …

Accepted Answer

Concept & Logic: This problem uses the Rule of Alligation to find the ratio of graduates in the two trades. The crucial secondary concept is understanding that the number of graduates must be a whole integer. Finding the "least number" requires simplifying the resulting ratio to its lowest coprime terms. Step-by-Step Solution: • Given Data: Mechanical Average = 2.45 lacs Electronics Average = 3.56 lacs Combined Average = 3.12 lacs • Applying Alligation: Mech (2.45) ------------- Elec (3.56) \ / Mean (3.12) / \ (3.56 - 3.12) ----------- (3.12 - 2.45) Ratio of Mech : Elec = 0.44 : 0.67 Multiply both sides by 100 to remove decimals: Ratio of Mech : Elec = 44 : 67 • Deducing the Least Number: The ratio of Mechanical to Electronics graduates is 44 : 67. This means for every 44 Mechanical graduates, there are 67 Electronics graduates. To find the *least* possible actual number of graduates, we check if the ratio 44/67 can be simplified further. The number 67 is a prime number, meaning 44 and 67 are coprime (they share no common factors other than 1). Therefore, the ratio cannot be reduced to smaller whole numbers. The smallest integer values that satisfy this ratio are 44 Mechanical and 67 Electronics graduates. Exam Strategy & Shortcut: Perform the cross-subtraction directly mentally: 356 - 312 = 44 (Mech parts) 312 - 245 = 67 (Elec parts) Ratio is 44 : 67. Recognizing immediately that 67 is prime confirms that the ratio is fully simplified, locking in 67 as the absolute minimum number of Electronics students. Common Pitfall: The primary trap here is option (d) "Cannot be determined". Because the exact total number of students isn't provided, students quickly assume there is missing information. The key lies in the word "least". While the *actual* number could be 134 (67 × 2) or 201 (67 × 3), the *least possible* number is the prime ratio base itself. Final Answer: Therefore, the correct answer is 67.

Question 18

My Scooty gives an average of 40 kmpl of petrol. But after recent filling at the new petrol pump, its average dropped to 38 kmpl.   I investigated and found out that it was due to adulterated petrol. Petrol pumps add kerosene, which is $\frac{2}{3}$…

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Concept & Logic: This problem requires a two-step approach: first using the Rule of Alligation to find the volumetric ratio of petrol to kerosene in the mixture, and then calculating the cost price of this mixture to determine the profit made by substituting cheaper fuel. Step-by-Step Solution: • Given: • Original pure petrol average = 40 km/L • Adulterated mixture average = 38 km/L • Kerosene average = 18 km per 900 ml • Pure petrol cost = ₹ 30 per Litre • Kerosene is (2/3) cheaper than petrol. • Calculation / Deduction: 1. Find the per-litre average of kerosene: If it gives 18 km for 900 ml (0.9 L), then for 1000 ml (1 L), it gives: (18/0.9) = 20 km/L. 2. Find the mixture ratio using alligation: Petrol average = 40 Kerosene average = 20 Mixture average = 38 Ratio of Petrol to Kerosene = (38 - 20) : (40 - 38) = 18 : 2 = 9 : 1. This means in 1 Litre of mixture, petrol is (9/10) L and kerosene is (1/10) L. 3. Calculate the cost price for the pump owner: Cost of pure petrol = ₹ 30/L. Kerosene is (2/3) cheaper, meaning its cost is reduced by (2/3). Cost of kerosene = 30 - ((2/3) × 30) = 30 - 20 = ₹ 10/L. Cost of 1 Litre of the adulterated mixture = (Cost of (9/10) L petrol) + (Cost of (1/10) L kerosene) Mixture Cost = ((9/10) × 30) + ((1/10) × 10) = 27 + 1 = ₹ 28. 4. Calculate the additional amount made: He sells it at the pure petrol price (₹ 30) but his actual cost is ₹ 28. Additional amount = 30 - 28 = ₹ 2. Exam Strategy & Shortcut: Standardize all your units first (convert 900 ml to 1 Litre immediately). Once you use alligation to find the 9:1 ratio, simply calculate the cost difference per portion. The owner replaces (1/10) of ₹ 30 petrol with ₹ 10 kerosene. The savings (extra profit) is exactly (1/10) of the price difference between the two fuels: (1/10) × (30 - 10) = (20/10) = ₹ 2. Common Pitfall: Students often misinterpret "(2/3) cheaper" as meaning the cost is (2/3) of the original. "Cheaper by" implies subtraction, so the actual cost is 1 - (2/3) = (1/3) of the petrol price. Missing this nuance will lead to the wrong mixture cost. Final Answer: Therefore, the correct answer is ₹ 2.

Question 19

Average score of a class of 60 students, in an exam, was 43.   Average score of the students who had passed is 52 and the average score of students who had failed is 16.   How many failed the exam?

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Concept & Strategy: This problem is best solved using the Rule of Alligation. Alligation helps us quickly find the ratio in which two distinct groups (passed and failed students) are mixed to form a combined group (the whole class) with a known overall average. \frac{Quantity of Group 1}{Quantity of Group 2} = \frac{Average of Group 2 - Overall Average}{Overall Average - Average of Group 1} Step-by-Step Solution: • Given: • Total students = 60 • Overall average score = 43 • Average score of passed students = 52 • Average score of failed students = 16 • Calculation / Deduction: 1. Set up the alligation cross: Passed Average (52) -------------- Failed Average (16) \ / \ / Overall (43) / \ / \ (43 - 16) ------------------------ (52 - 43) = 27 = 9 2. Find the ratio of Passed to Failed students: Ratio = 27 : 9 = 3 : 1 3. Calculate the number of failed students: The total number of parts is 3 + 1 = 4 parts. 4 parts correspond to the total class size of 60 students. 1 part = (60/4) = 15 students. Since the ratio of failed students is 1 part, the number of failed students is 15. Exam Strategy & Shortcut: Alligation is the ultimate shortcut for "mixture" average problems. If you used the standard algebraic equation 52x + 16(60-x) = 43 × 60, you would waste precious seconds multiplying and shifting large numbers. The alligation cross gives you the 3:1 ratio almost instantly, turning a complex multiplication problem into a simple division problem (60 \div 4). Common Pitfall: A frequent mistake is matching the final ratio to the wrong group. In the alligation cross, the difference calculated on the left side corresponds to the quantity of the group on the right side. Always remember to cross the values properly so you don't accidentally conclude that 45 students failed. Final Answer: Therefore, the correct answer is 15.

Question 20

The average of 11 results is 60.   If the average of first six results is 58 and that of last six is 63, find the $6^{	ext{th}}$ result.

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Concept & Formula: This is a classic Overlapping Averages problem. When a central item is counted twice (once in the first half and once in the second half), you can find its exact value by subtracting the total sum of all distinct items from the combined sum of the overlapping subsets. Value of Overlapping Item = (Sum of Part 1 + Sum of Part 2) - Total Sum Step-by-Step Solution: • Given: • Average of all 11 results = 60 • Average of the first 6 results = 58 • Average of the last 6 results = 63 • Calculation / Deduction: 1. Calculate the sum of all 11 results: Total Sum = 11 × 60 = 660 2. Calculate the sum of the first 6 results: Sum of First 6 = 6 × 58 = 348 3. Calculate the sum of the last 6 results: Sum of Last 6 = 6 × 63 = 378 4. Determine the 6^{th} result: Notice that 6 results + 6 results = 12 results. The 6^{th} result is counted in both groups. 6^{th} Result = (Sum of First 6 + Sum of Last 6) - Total Sum 6^{th} Result = (348 + 378) - 660 6^{th} Result = 726 - 660 = 66 Exam Strategy & Shortcut: Instead of calculating massive sums, use the Net Deviation Method. The overall average is 60. The first 6 results deviate by -2 each (58 - 60). Total deviation = 6 × (-2) = -12. The last 6 results deviate by +3 each (63 - 60). Total deviation = 6 × (+3) = +18. Net deviation = -12 + 18 = +6. The 6^{th} result must be the overall average plus this net deviation: 60 + 6 = 66. This mental math completely avoids large multiplications like 6 × 63. Common Pitfall: Students often get confused by the overlap and subtract the overlapping subsets from each other rather than summing them and subtracting the master total. Always visualize the set: \{1..6\} and \{6..11\}. Adding them gives \{1..11\} + \{6\}. Subtracting the master sum \{1..11\} leaves just \{6\}. Final Answer: Therefore, the correct answer is 66.

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