Question 1

The average of two numbers is 8 and the average of another three numbers is 3. What is the average of all five numbers together?

Accepted Answer

Introduction / Context: This question examines how to combine averages from two different groups of numbers to find the overall average of the combined group. It is a common type of problem in aptitude tests involving weighted averages. Given Data / Assumptions: - There are two groups of numbers. - First group: 2 numbers with average 8. - Second group: 3 numbers with average 3. - We need the average of all 5 numbers together. Concept / Approach: The overall average of combined groups is not simply the average of the two averages. Instead, it is computed using total sums. For each group, total sum = average * number of terms. We compute the total of both groups, add them, and then divide by the total number of terms, which is 5. This is a straightforward application of the definition of average. Step-by-Step Solution: Step 1: Let the two numbers in the first group be a and b. Step 2: Average of a and b is 8, so (a + b) / 2 = 8. Step 3: Therefore, a + b = 16. Step 4: Let the three numbers in the second group be c, d and e. Step 5: Average of c, d and e is 3, so (c + d + e) / 3 = 3. Step 6: Therefore, c + d + e = 9. Step 7: Total sum of all five numbers = (a + b) + (c + d + e) = 16 + 9 = 25. Step 8: Total number of numbers = 2 + 3 = 5. Step 9: Average of all five numbers = total sum / total count = 25 / 5. Step 10: Compute the result: 25 / 5 = 5. Step 11: So, the overall average is 5. Verification / Alternative check: As a quick check, note that the two averages 8 and 3 are being combined with weights 2 and 3 respectively. The overall average is (2 * 8 + 3 * 3) / (2 + 3) = (16 + 9) / 5 = 25 / 5 = 5. This is a direct application of the weighted average formula and confirms the earlier result. Why Other Options Are Wrong: Option A (2) and B (3): Too low and ignore the influence of the larger numbers in the first group. Option D (6): Might come from averaging 8 and 4 incorrectly; it does not match the combined total. Option E (4): Could come from averaging 8 and 3 directly, but that is incorrect because the groups have different sizes. Common Pitfalls: A common error is to average the two given averages themselves, such as (8 + 3) / 2 = 5.5, without considering the number of values in each group. While sometimes this may accidentally give a close value, it is not a reliable method. Always convert averages to totals, combine the totals, and then divide by the total number of values. Final Answer: The average of all five numbers together is 5.

Question 2

If a + 1/a = 2, then what is the value of a^5 + 1/a^5?

Accepted Answer

Introduction / Context: This question tests algebraic manipulation and the use of identities involving a variable and its reciprocal. When a + 1/a has a simple value, it often allows us to deduce powers like a^2, a^3 and higher without solving complicated equations repeatedly. Given Data / Assumptions: - We are given that a + 1/a = 2. - We assume a is non zero so that 1/a is defined. - We are asked to find a^5 + 1/a^5. Concept / Approach: When a + 1/a has a simple value, a common trick is to square or manipulate this relation to find a^2 + 1/a^2, a^3 + 1/a^3 and so on. However, in this particular case, the equation a + 1/a = 2 is especially simple and can be interpreted as a quadratic equation. Multiplying both sides by a gives a^2 + 1 = 2a, which can be rearranged into a standard quadratic form. Solving this will show that a is actually equal to 1, which makes higher powers easy to compute. Step-by-Step Solution: Step 1: Start from the given equation: a + 1/a = 2. Step 2: Multiply both sides by a (a is non zero): a^2 + 1 = 2a. Step 3: Rearrange terms to form a quadratic: a^2 - 2a + 1 = 0. Step 4: Recognize the left side as a perfect square: (a - 1)^2 = 0. Step 5: Solve for a: a - 1 = 0, so a = 1. Step 6: Now compute a^5 + 1/a^5 using a = 1. Step 7: a^5 = 1^5 = 1. Step 8: 1/a^5 = 1 / 1^5 = 1. Step 9: Therefore, a^5 + 1/a^5 = 1 + 1 = 2. Verification / Alternative check: We can also note that when a = 1, the original equation a + 1/a = 1 + 1 = 2 is satisfied, confirming that a = 1 is indeed a valid solution. Since the quadratic (a - 1)^2 = 0 has only one root, there is no other possible value of a. Therefore, any expression in powers of a and 1/a will be determined uniquely by a = 1. Substituting into a^5 + 1/a^5 yields 2, which is consistent. Why Other Options Are Wrong: Option A (0): Would require a^5 and 1/a^5 to cancel each other, which does not happen when a = 1. Option B (1): Would only occur if one of a^5 or 1/a^5 were zero or both were fractions summing to 1, which is not the case. Option C (3): Does not correspond to any logical manipulation of the given relation. Option E (4): Would require each term to be 2, which does not follow from a = 1. Common Pitfalls: Some students attempt to compute powers step by step using identities such as (a + 1/a)^2 or (a + 1/a)^3 without noticing that the equation reduces to a very simple quadratic. Others may forget to enforce that a is non zero. The quickest and safest way here is to solve for a directly from the quadratic form, then substitute in the required expression. Final Answer: The value of a^5 + 1/a^5 is 2.

Question 3

In the first 39 overs of a cricket match, the run rate was 4.6 runs per over. What run rate is required in the remaining 11 overs so that the team reaches a target of 252 runs?

Accepted Answer

Introduction / Context: This is a typical cricket run rate problem. It tests your ability to convert a given run rate into total runs scored so far and then determine the required run rate in the remaining overs to reach a specified target. Such problems are common in quantitative aptitude exams and sports based reasoning questions. Given Data / Assumptions: - Total overs in the innings = 50 overs (39 completed and 11 remaining). - Run rate for the first 39 overs = 4.6 runs per over. - Target runs to be scored = 252 runs. - We must find the required run rate in the last 11 overs. Concept / Approach: Run rate is defined as total runs scored divided by overs faced. First we compute total runs scored in the first 39 overs using the given run rate. Then we subtract this from the target to find how many runs remain to be scored. Finally, we divide the remaining runs by the remaining overs (11) to get the required run rate for those overs. This is a direct and standard application of the run rate formula. Step-by-Step Solution: Step 1: Compute runs scored in the first 39 overs. Step 2: Runs in first 39 overs = run rate * overs = 4.6 * 39. Step 3: Calculate 4.6 * 39. Note that 4.6 * 40 = 184, so 4.6 * 39 = 184 - 4.6 = 179.4 runs. Step 4: Total target runs = 252. Step 5: Remaining runs to be scored = target - runs already scored = 252 - 179.4. Step 6: Compute 252 - 179.4 = 72.6 runs. Step 7: Remaining overs = 50 - 39 = 11 overs. Step 8: Required run rate in the last 11 overs = remaining runs / remaining overs. Step 9: Required run rate = 72.6 / 11. Step 10: Compute 72.6 / 11 = 6.6 runs per over. Step 11: Therefore, the team must score at a run rate of 6.6 runs per over in the last 11 overs. Verification / Alternative check: Check the total runs if the team follows the required rate. Runs in the last 11 overs at 6.6 runs per over = 6.6 * 11 = 72.6 runs. Adding to the 179.4 runs already scored gives 179.4 + 72.6 = 252 runs, matching the target exactly. This confirms that a required run rate of 6.6 runs per over is correct. Why Other Options Are Wrong: Option A (7.2): Would yield more than the target when multiplied by 11 overs. Option B (5.7): Too low; it would leave the team short of 252 runs. Option C (6.3): Slightly low and would not reach the full target. Option E (7.0): Higher than required and not consistent with the exact arithmetic. Common Pitfalls: Mistakes often happen in the multiplication 4.6 * 39 or the division 72.6 / 11. Some may also confuse the target runs with runs still required, or use 39 instead of 11 as the denominator when computing the required run rate. Carefully keep track of what each run rate refers to and always separate runs already scored from runs yet to be scored. Final Answer: The required run rate in the last 11 overs is 6.6 runs per over.

Question 4

In a particular week, the average number of people who visited a park per day is 40. If we exclude the holidays in that week, the average number of visitors per day increases by 16. Further, if we also exclude the day on which the maximum number of …

Accepted Answer

Introduction / Context: This question combines averages with exclusion of certain days, such as holidays and a peak day with the maximum number of visitors. It tests multi step reasoning about totals and averages when different subsets of the data are considered, making it a good higher level average problem. Given Data / Assumptions: - Consider a week with 7 days. - Average visitors per day for the entire week = 40. - When holidays are excluded, the average for the remaining non holiday days increases by 16, so it becomes 56. - When we further exclude the day with the maximum 112 visitors, the average for the remaining days becomes 42. - We assume one independent maximum day with 112 visitors. - We need to find the number of holidays in that week. Concept / Approach: The key idea is to translate each average into a total. First, we find the total visitors in the week from the average 40. Then we let h be the number of holidays. The number of non holiday days is (7 - h), and we know the average for those days is 56, so we can find their total. The total visitors for holidays is the difference between the weekly total and the non holiday total. After that, we exclude both the holidays and the single peak day with 112 visitors and use the new average 42 to create another equation. Solving these equations gives us h, the number of holidays. Step-by-Step Solution: Step 1: Total visitors in the full week = average * number of days = 40 * 7 = 280 visitors. Step 2: Let the number of holidays in the week be h. Step 3: Then, number of non holiday days = 7 - h. Step 4: When holidays are excluded, the average visitors per non holiday day is 56. Step 5: So total visitors on non holiday days = 56 * (7 - h). Call this Tn. Step 6: Total visitors on holidays = total weekly visitors - non holiday visitors = 280 - Tn. Step 7: Now we also exclude the peak day with 112 visitors. This peak day is one of the non holiday days. Step 8: After excluding holidays and this peak day, the remaining number of days = (7 - h) - 1 = 6 - h. Step 9: The total visitors on the remaining days = Tn - 112. Step 10: We are told that the average for these remaining days is 42, so: (Tn - 112) / (6 - h) = 42. Step 11: Substitute Tn = 56 * (7 - h) into the equation: (56 * (7 - h) - 112) / (6 - h) = 42. Step 12: Expand the numerator: 56 * (7 - h) = 392 - 56h. Step 13: So the numerator becomes 392 - 56h - 112 = 280 - 56h. Step 14: The equation is now (280 - 56h) / (6 - h) = 42. Step 15: Multiply both sides by (6 - h): 280 - 56h = 42 * (6 - h). Step 16: Expand the right side: 42 * 6 - 42 * h = 252 - 42h. Step 17: So 280 - 56h = 252 - 42h. Step 18: Rearrange: 280 - 252 = 56h - 42h. Step 19: 28 = 14h, so h = 28 / 14 = 2. Step 20: Therefore, there are 2 holidays in the week. Verification / Alternative check: Let us verify by constructing a possible scenario. With h = 2 holidays, non holiday days = 5. Total weekly visitors = 280. Non holiday total = 56 * 5 = 280. This means holiday visitors total 0, so holidays are days with no visitors, which is acceptable. Now non holiday days total 280 visitors across 5 days, average 56. The peak day has 112 visitors. Removing this day leaves 4 days with total visitors 280 - 112 = 168. Average for those 4 days = 168 / 4 = 42, which matches the given second adjusted average. This confirms that h = 2 is consistent. Why Other Options Are Wrong: Option A (1 holiday): Would lead to different totals, and the second average of 42 could not be satisfied simultaneously. Option C (3 holidays) and D (4 holidays): Changing h alters the denominator and the total visitors in a way that breaks the equation (280 - 56h) / (6 - h) = 42. Option E (5 holidays): Leaves too few non holiday days to maintain both average conditions. Common Pitfalls: Typical mistakes include assuming that holidays must have zero visitors without checking consistency, or mixing up the number of days remaining after removing both holidays and the peak day. Another common error is to try to solve intuitively without writing equations, which often leads to confusion. Writing clear expressions for totals and averages at each stage is the safest method. Final Answer: The number of holidays in the week is 2.

Question 5

In a One Day cricket match of 50 overs, Team A scores at an average rate of 4.5 runs per over for the first 30 overs and 5.5 runs per over for the remaining 20 overs. Team B bats the full 50 overs while chasing this target but finally loses the matc…

Accepted Answer

Introduction / Context: This problem is based on the concept of average run rate in a limited overs cricket match. We are given the over wise scoring pattern of Team A and the margin of defeat for Team B. Using this information we have to determine the average runs per over scored by Team B over the full 50 overs of its innings. Given Data / Assumptions: Total overs in the match for each team = 50 overs. For Team A, average in the first 30 overs = 4.5 runs per over. For Team A, average in the last 20 overs = 5.5 runs per over. Team B bats all 50 overs and loses by 10 runs. We assume standard one day rules where every over has 6 legal balls and totals are simple sums of runs. Concept / Approach: The key ideas are: Runs scored = average run rate * number of overs. Team A total gives the target for Team B. If Team B loses by 10 runs, its total is 10 runs less than Team A. Average run rate of Team B = (total runs of Team B) / 50. We will first compute Team A total, then Team B total, and finally Team B average run rate. Step-by-Step Solution: Step 1: Compute runs scored by Team A in the first 30 overs. Runs in first 30 overs = 30 * 4.5 = 135 runs. Step 2: Compute runs scored in the last 20 overs. Runs in last 20 overs = 20 * 5.5 = 110 runs. Step 3: Compute Team A total score. Team A total = 135 + 110 = 245 runs. Step 4: Compute Team B total, given that it loses by 10 runs. Team B total = 245 - 10 = 235 runs. Step 5: Compute Team B average run rate. Average of Team B = 235 / 50 = 4.7 runs per over. Verification / Alternative check: We can check reasonableness. Team A scores 245 in 50 overs, which is an overall run rate of 245 / 50 = 4.9 runs per over. Team B scores slightly lower, 235 runs, so its average must be slightly less than 4.9 and more than 4.5. The value 4.7 fits well between these, confirming our calculation is consistent. Why Other Options Are Wrong: The option 4.4 is too low and would give a total of only 4.4 * 50 = 220 runs, which would mean a defeat margin of 25 runs, not 10. The option 5.2 is higher than Team A run rate and would give 260 runs, implying Team B wins, which contradicts the question. The option 4.6 gives 230 runs, corresponding to a 15 run defeat, again not matching the given 10 run margin. Only 4.7 produces exactly 235 runs and a 10 run defeat. Common Pitfalls: Students often make the mistake of taking the simple average of run rates, that is (4.5 + 5.5) / 2 = 5, which is wrong because the number of overs in each segment is different. Another common error is to forget that Team B total is 10 runs less than Team A, or to divide the total runs by 20 or 30 instead of the full 50 overs. Always relate runs, overs and run rate using total runs = overs * run rate. Final Answer: The average number of runs per over scored by Team B is 4.7 runs per over.

Question 6

In a class of 60 students there are 30 girls. The average weight of the girls is 58 kg and the average weight of the entire class is 63 kg. What is the average weight (in kilograms) of the boys in the class?

Accepted Answer

Introduction / Context: This question uses the concept of weighted averages to relate the average weight of a subgroup (girls), the overall class average, and the unknown average weight of the complementary subgroup (boys). Such questions are very common in aptitude tests and help to strengthen understanding of averages in mixed groups. Given Data / Assumptions: Total number of students in the class = 60. Number of girls = 30. Number of boys = 60 - 30 = 30. Average weight of girls = 58 kg. Average weight of the entire class = 63 kg. We assume all weights are measured accurately in kilograms. Concept / Approach: The central idea is that: Total weight of all students = average weight of class * number of students. Similarly, the total weight of girls is average of girls times their count, and the total weight of boys is average of boys times their count. The total class weight is also equal to the sum of the weights of girls and boys. Using these relations we can solve for the unknown average weight of the boys. Step-by-Step Solution: Step 1: Compute total weight of all 60 students. Total class weight = 63 * 60 = 3780 kg. Step 2: Compute total weight of the 30 girls. Total weight of girls = 58 * 30 = 1740 kg. Step 3: Compute total weight of the boys. Total weight of boys = total class weight - total girls weight. Total weight of boys = 3780 - 1740 = 2040 kg. Step 4: Compute average weight of the boys. Average weight of boys = 2040 / 30 = 68 kg. Verification / Alternative check: We can verify by recomputing the combined average using boys average 68 and girls average 58. Combined average = (30 * 58 + 30 * 68) / (30 + 30) = (1740 + 2040) / 60 = 3780 / 60 = 63 kg, which matches the given overall average, confirming the result is consistent. Why Other Options Are Wrong: If the boys average were 67, the total boys weight would be 30 * 67 = 2010 kg, giving class total 2010 + 1740 = 3750 kg and class average 3750 / 60 = 62.5 kg, not 63 kg. For 66 kg, the class average becomes even lower. For 65 kg, it reduces further. Only an average of 68 kg for the boys restores the overall average to 63 kg as given in the question. Common Pitfalls: A common mistake is to take the simple average of 58 and 63, which ignores the fact that both groups have equal strength here but may not in other questions. Another pitfall is confusing total and average, or attempting to directly manipulate averages without first converting them into totals. Always convert averages to totals using total = average * number of items, then combine and reconvert back into averages. Final Answer: The average weight of the boys in the class is 68 kilograms.

Question 7

The mean (arithmetic average) of 21 different observations is 40. If each observation that is greater than the median of the data is increased by 21, what will be the new mean of all 21 observations?

Accepted Answer

Introduction / Context: This question checks understanding of how the arithmetic mean changes when a fixed amount is added to some of the observations. The twist is that the modification applies only to those observations that are greater than the median, which requires careful counting of how many such observations there are in a set of 21 values. Given Data / Assumptions: Number of observations = 21. All observations are distinct. Original mean of all 21 observations = 40. Each observation greater than the median is increased by 21. Median of 21 observations is the 11th value when arranged in order. Concept / Approach: The key facts are: The sum of all observations is mean * number of observations. For 21 ordered observations, 10 values lie above the median, and 10 lie below it. Adding a constant k to n observations increases the total sum by n * k. New mean = new total sum / number of observations. We simply compute the original total, adjust it by the net increase, and divide by 21 to get the new mean. Step-by-Step Solution: Step 1: Compute original total of the 21 observations. Original total = 40 * 21 = 840. Step 2: Determine how many observations are greater than the median. With 21 ordered observations, the median is the 11th value. There are 10 observations greater than this median. Step 3: Compute the increase in total when each of these 10 values is increased by 21. Increase in total = 10 * 21 = 210. Step 4: Compute the new total. New total = 840 + 210 = 1050. Step 5: Compute the new mean. New mean = 1050 / 21 = 50. Verification / Alternative check: A simple check is to note that only 10 out of 21 observations shifted upward by the same amount. If every observation had increased by 21, the mean would also increase by 21. Here the fraction of observations increased is 10 / 21, so the mean increases by 21 * (10 / 21) = 10. Starting from original mean 40, the new mean is 40 + 10 = 50, which matches our computed answer. Why Other Options Are Wrong: An answer of 30 suggests a decrease in mean, which is impossible when certain observations are increased. Values like 50.5 or 45 correspond to incorrect counts of how many observations were changed or incorrect arithmetic. Only 50 is consistent with both methods of reasoning and the given data. Common Pitfalls: Students sometimes mistakenly think that only one value (the median) is affected, or they miscount and take 11 observations instead of 10 above the median. Another frequent error is to try to change each observation individually rather than using total sums, which leads to messy calculations. The safe strategy is always to convert means into totals, apply the changes, and then convert back to the mean. Final Answer: The new mean of all 21 observations is 50.

Question 8

The average number of fruits offered in a temple over a full week of seven days is 75. The average for six days excluding Tuesday is 72. How many fruits were offered on Tuesday?

Accepted Answer

Introduction / Context: This is a classic average problem where we know the average number of items over a full period and also the average over a subset of that period. Using these two averages, we can find the value for the missing day, which in this case is Tuesday. Such questions train you to manipulate totals and averages quickly. Given Data / Assumptions: Number of days in the week considered = 7. Average fruits offered over all 7 days = 75. Average fruits offered over 6 days excluding Tuesday = 72. We need to find the number of fruits offered on Tuesday. Concept / Approach: The basic relation is: Total fruits over a period = average fruits * number of days. Using this, we will compute: Total fruits over 7 days. Total fruits over the 6 days excluding Tuesday. Then Tuesday fruits = total for 7 days minus total for the other 6 days. Step-by-Step Solution: Step 1: Compute the total number of fruits offered over 7 days. Total over 7 days = 75 * 7 = 525 fruits. Step 2: Compute the total number of fruits offered on the 6 days excluding Tuesday. Total over 6 days = 72 * 6 = 432 fruits. Step 3: Compute fruits offered on Tuesday. Tuesday fruits = total 7 day fruits - total 6 day fruits. Tuesday fruits = 525 - 432 = 93 fruits. Verification / Alternative check: We can verify by recomputing the overall average including Tuesday. If the six days have a total of 432 fruits and Tuesday has 93 fruits, then total for 7 days is 432 + 93 = 525. The average is 525 / 7 = 75 fruits per day, which matches the given weekly average, so the value is consistent. Why Other Options Are Wrong: If Tuesday had 90 fruits, the total for the week would be 432 + 90 = 522 and the average 522 / 7 = 74.57, not 75. If Tuesday had 72 fruits, the weekly average would be too low. If Tuesday had 92 fruits, the weekly average would be 524 / 7, again not 75. Only 93 fruits gives the correct weekly average of 75. Common Pitfalls: A common mistake is to average the two averages directly, or to assume Tuesday also had 72 or 75 fruits. Another pitfall is forgetting to multiply the averages by the number of days to get totals. Always work with totals when combining or separating averages across different numbers of items or days. Final Answer: The number of fruits offered on Tuesday is 93.

Question 9

Nineteen persons go to a hotel for a combined dinner party. Thirteen of them spend Rs. 79 each on their dinner, and each of the remaining persons spends Rs. 4 more than the average expenditure of all 19 persons. What is the total amount of money spe…

Accepted Answer

Introduction / Context: This problem combines averages with a bit of algebra. Some people in a group spend a fixed amount, while the others spend a certain amount more than the average expenditure of the entire group. We need to use the definition of average and total expenditure to find the overall amount spent on the dinner. Given Data / Assumptions: Total number of persons at the dinner = 19. Thirteen persons each spend Rs. 79. The remaining 6 persons each spend Rs. 4 more than the average expenditure of all 19 persons. Let the average expenditure of all 19 persons be x rupees. All expenditures are per person for this one dinner. Concept / Approach: If the average expenditure per person is x, then: Total expenditure of all 19 persons = 19 * x. We can also express the total as the sum of the expenditures of the first 13 persons and the remaining 6 persons: 13 persons spend 13 * 79. 6 persons each spend (x + 4). Equating these two expressions for total expenditure gives an equation in x, which we can solve to find the average and then the total money spent. Step-by-Step Solution: Step 1: Let the average expenditure per person be x rupees. Then total expenditure = 19 * x. Step 2: Express total expenditure using the detailed information. Total expenditure = 13 * 79 + 6 * (x + 4). Step 3: Set up the equation. 19 * x = 13 * 79 + 6 * (x + 4). Step 4: Simplify the right hand side. 13 * 79 = 1027. 6 * (x + 4) = 6 * x + 24. So 19 * x = 1027 + 6 * x + 24 = 1051 + 6 * x. Step 5: Solve for x. 19 * x - 6 * x = 1051. 13 * x = 1051, so x = 1051 / 13 = 80.8461538 rupees (approximately). Step 6: Compute total expenditure. Total expenditure = 19 * x = 19 * 80.8461538 ≈ 1536.08 rupees. Verification / Alternative check: We can check by directly computing expenditure with this average. If the average is about Rs. 80.85, the 6 higher spending persons each pay about 84.85 rupees. Then: Total of 13 persons = 13 * 79 = 1027. Total of 6 persons ≈ 6 * 84.85 ≈ 509.1. Grand total ≈ 1027 + 509.1 ≈ 1536.1 rupees, which matches the earlier total within rounding error. Thus the total of about Rs. 1536.08 is consistent. Why Other Options Are Wrong: Values like Rs. 1534.00, Rs. 1628.40 and Rs. 1492.00 would correspond to different implied averages that do not satisfy the equation 19 * x = 13 * 79 + 6 * (x + 4). When these totals are substituted back, the per person average and the condition that six persons spent exactly Rs. 4 more than the average fail, so they are incorrect. Common Pitfalls: A typical mistake is to assume that the remaining 6 persons each pay Rs. 79 + 4 = Rs. 83 without considering that the extra amount is over the average x, not over Rs. 79. Another error is to forget that total expenditure must be 19 * x and instead try to compute average from partial groups directly. Always set up an algebraic equation equating total from the average with total from individual expenditures. Final Answer: The total money spent by all 19 persons is approximately Rs. 1536.08.

Question 10

Nine students of a class contribute money towards a charity fund. Seven of them give Rs. 50 each and the other two give, respectively, Rs. 50 and Rs. 90 more than the others. What is the average contribution (in rupees) of the nine students?

Accepted Answer

Introduction / Context: This question is a straightforward application of averages and total contributions in a small group. We know how much most of the students give and how much extra the remaining students contribute. From this, we can compute the total contribution and then the average per student. Given Data / Assumptions: Total number of students contributing = 9. Seven students each contribute Rs. 50. One of the remaining two contributes Rs. 50 more than the others, that is 50 + 50 = Rs. 100. The other contributes Rs. 90 more than the others, that is 50 + 90 = Rs. 140. We assume all contributions are in rupees and made once. Concept / Approach: The basic idea is: Average contribution = total contribution / number of students. So we first calculate the total contribution by summing the amounts from each category of students and then divide by 9 to obtain the average contribution per student. Step-by-Step Solution: Step 1: Compute total contribution of the 7 students who each give Rs. 50. Contribution of 7 students = 7 * 50 = Rs. 350. Step 2: Compute contribution of the student who gives Rs. 50 more than the others. That student gives = 50 + 50 = Rs. 100. Step 3: Compute contribution of the student who gives Rs. 90 more than the others. That student gives = 50 + 90 = Rs. 140. Step 4: Compute total contribution of all 9 students. Total = 350 + 100 + 140 = Rs. 590. Step 5: Compute the average contribution. Average = 590 / 9 ≈ 65.555..., which we round to Rs. 65.56. Verification / Alternative check: We can approximate to check reasonableness. If all 9 students had given Rs. 65 each, the total would be 9 * 65 = Rs. 585. Here the actual total is Rs. 590, which is slightly more, so the average should be slightly more than 65, around 65.5. This matches our exact value of approximately Rs. 65.56. Why Other Options Are Wrong: An average of Rs. 60 would give a total of 9 * 60 = Rs. 540, which is less than the known total of Rs. 590. An average of Rs. 70 would give a total of 630, which is too high. An average of Rs. 75 would give a total of 675, far above the actual sum. Only Rs. 65.56 corresponds correctly to the actual total of Rs. 590 and the given contributions. Common Pitfalls: A common error is misreading the phrase more than the others and adding 50 and 90 directly to 0 instead of to 50. Another mistake is to try to average the different contribution amounts directly without computing the total. Always sum contributions correctly and then divide by the total number of contributors to find the average. Final Answer: The average contribution of the nine students is approximately Rs. 65.56 per student.

Question 11

Seven years ago, the average age of A, B and C was 51 years. If at present A is 3 years older than B and B is 3 years older than C, then what are the present ages (in years) of A, B and C respectively?

Accepted Answer

Introduction / Context: This is a typical ages and averages problem. We are given the average age of three people at a past time and the age differences between them in the present. Using these facts, we need to determine their current ages. Such questions test the ability to translate age relationships and average information into algebraic equations. Given Data / Assumptions: Seven years ago, the average age of A, B and C was 51 years. At present, A is 3 years older than B. At present, B is 3 years older than C. We assume ages are in whole years, but the method works for any real ages. Concept / Approach: From the given average age some years ago, we can find the total of their ages at that time. We then add 7 years to each person to obtain present ages. In parallel, we express the present ages of A and B in terms of C using the age differences. Equating the total present age obtained from these two methods allows us to solve for the current ages. Step-by-Step Solution: Step 1: Compute total age seven years ago. Average age seven years ago = 51 years. Number of persons = 3, so total age then = 51 * 3 = 153 years. Step 2: Express present ages in terms of C. Let present age of C be c. Then present age of B = c + 3. Present age of A = (c + 3) + 3 = c + 6. Step 3: Relate present total age to past total age. Seven years have passed since the ages summed to 153. Each person has gained 7 years, so total gain = 3 * 7 = 21 years. So present total age = 153 + 21 = 174 years. Step 4: Form equation for present total in terms of c. Present total age = A + B + C = (c + 6) + (c + 3) + c = 3c + 9. Thus 3c + 9 = 174. Step 5: Solve for c and find all ages. 3c = 174 - 9 = 165, so c = 55. B = c + 3 = 55 + 3 = 58. A = c + 6 = 55 + 6 = 61. Verification / Alternative check: Seven years ago, their ages would have been A = 61 - 7 = 54, B = 58 - 7 = 51, C = 55 - 7 = 48. The average of 54, 51 and 48 is (54 + 51 + 48) / 3 = 153 / 3 = 51, matching the given information. The age gaps A minus B = 3 and B minus C = 3 are also satisfied. Hence the result is correct. Why Other Options Are Wrong: Option b and option d reverse the age order or give values that do not differ by exactly 3 years between consecutive persons. Option c lists ages in increasing order 55, 58 and 61, which does not match the requirement that A is the oldest. Moreover, if we test these other triplets seven years ago, the average is not 51 in any of those cases. Only 61, 58 and 55 satisfy all the given conditions. Common Pitfalls: Students sometimes subtract or add the 7 years incorrectly, or they mistakenly assume the average remains 51 at present. Another error is to assign the age differences backwards, for example making A younger than C. A disciplined approach is to define one person age as a variable, express others in terms of it, and then use the total from the average to solve the equations. Final Answer: The present ages of A, B and C are 61 years, 58 years and 55 years respectively.

Question 12

At a marriage party, 32% of the people are women, 54% are men and the remaining 196 are children. How many men are present at the party?

Accepted Answer

Introduction / Context: This question involves percentages and complements of percentages. We are given the percentage of women, the percentage of men, and the absolute number of children, and we need to determine the number of men present at the party. It is a typical problem of converting between percentages and absolute numbers. Given Data / Assumptions: Percentage of women at the party = 32%. Percentage of men at the party = 54%. The remaining people are children, whose number is 196. The total number of people at the party is denoted by N. Concept / Approach: We know that: Women = 32% of N. Men = 54% of N. Children = remaining percentage of N. The remaining percentage is 100% - (32% + 54%) = 14%. We also know that 14% of N equals 196 children. From this, we can find N and then compute 54% of N to get the number of men. Step-by-Step Solution: Step 1: Find the percentage of children. Children percentage = 100% - 32% - 54% = 14%. Step 2: Set up the relation between percentage and number of children. 14% of N = 196. Step 3: Solve for N. 14 / 100 * N = 196. N = 196 * 100 / 14. N = 196 * (100 / 14) = 196 * (50 / 7) = 28 * 50 = 1400. Step 4: Compute number of men, which is 54% of N. Men = 54% of 1400 = 54 / 100 * 1400. Men = 0.54 * 1400 = 756. Verification / Alternative check: We can also compute the number of women and children and check that totals match 1400. Women = 32% of 1400 = 0.32 * 1400 = 448. Children are given as 196. Sum of men, women and children = 756 + 448 + 196 = 1400, which equals the total number of people N, confirming the correctness of the calculation. Why Other Options Are Wrong: Option b (448) is actually the number of women, not men. Option c (332) and option d (324) do not correspond to 54% of 1400 and would give wrong totals when combined with the other groups. Only 756 satisfies the percentage condition and results in consistent totals. Common Pitfalls: A common mistake is to think that 196 is 32% or 54% of the total instead of the remainder. Another error is to miscalculate the total by adding percentages incorrectly or failing to convert percentages into fractions before solving. Always ensure that the sum of percentages equals 100% and that the complement percentage is correctly used to link the known count with the total. Final Answer: The number of men present at the party is 756.

Question 13

The average monthly salary of all workers in a workshop is Rs. 12,000. The average monthly salary of 7 technicians is Rs. 15,000 and the average salary of all the remaining workers is Rs. 9,000. What is the total number of workers in the workshop?

Accepted Answer

Introduction / Context: This problem involves combining averages from two subgroups of a workforce to recover the total number of workers. We know the overall average salary, the average salary of technicians, and the average salary of the remaining group. From this information we can determine how many workers there are in total. Given Data / Assumptions: Overall average salary of all workers = Rs. 12,000 per month. Average salary of 7 technicians = Rs. 15,000 per month. Average salary of the remaining workers = Rs. 9,000 per month. Let total number of workers (including the 7 technicians) be n. Concept / Approach: Let us denote: Total salary of all workers = 12,000 * n. Total salary of 7 technicians = 15,000 * 7. Number of remaining workers = n - 7, with total salary 9,000 * (n - 7). Since the total salary must be equal whether we compute it from the overall average or from the subgroup totals, we can form an equation and solve for n. Step-by-Step Solution: Step 1: Express total salary using the overall average. Total salary = 12,000 * n. Step 2: Express total salary as the sum of salaries of technicians and others. Total salary = 15,000 * 7 + 9,000 * (n - 7). Step 3: Form the equation. 12,000 * n = 15,000 * 7 + 9,000 * (n - 7). Step 4: Simplify the right hand side. 15,000 * 7 = 105,000. 9,000 * (n - 7) = 9,000 * n - 63,000. So 12,000 * n = 105,000 + 9,000 * n - 63,000 = 9,000 * n + 42,000. Step 5: Solve for n. 12,000 * n - 9,000 * n = 42,000. 3,000 * n = 42,000. n = 42,000 / 3,000 = 14. Verification / Alternative check: If there are 14 workers in total, then there are 7 technicians and 7 other workers. Total salary of technicians = 7 * 15,000 = 105,000. Total salary of other workers = 7 * 9,000 = 63,000. Combined total salary = 105,000 + 63,000 = 168,000. Average salary = 168,000 / 14 = 12,000, which matches the given overall average, confirming that 14 is correct. Why Other Options Are Wrong: For n = 12 or 13 or 15, when you compute the total salary using both methods, the averages do not match. For example, if n = 12, the total salary by subgroup calculation would be 15,000 * 7 + 9,000 * 5 = 105,000 + 45,000 = 150,000, giving an overall average of 150,000 / 12 = 12,500, not 12,000. Similar inconsistencies occur for n = 13 and n = 15. Only n = 14 is consistent with all the data. Common Pitfalls: Some students try to average 15,000 and 9,000 directly without accounting for different group sizes, which is incorrect. Others forget that the total salary is simply average times number of workers and struggle with unnecessary variables. The safest method is to write total salary in two ways, set them equal, and solve the resulting linear equation. Final Answer: The total number of workers in the workshop is 14.

Question 14

The average of 21 numbers is 12. The average of the first 11 numbers is 9 and the average of the last 11 numbers is 15. What is the value of the middle (11th) number in this ordered list?

Accepted Answer

Introduction / Context: This problem uses the idea of overlapping groups and averages. We know the average of all 21 numbers, the average of the first 11 numbers, and the average of the last 11 numbers. Since there is overlap at the middle number, we can use totals and subtraction to find its exact value. Given Data / Assumptions: Total count of numbers = 21. Average of all 21 numbers = 12. Average of the first 11 numbers = 9. Average of the last 11 numbers = 15. The middle number is the 11th number when they are ordered in sequence. Concept / Approach: Let us denote: Sum of all 21 numbers as S. Sum of first 11 numbers as S1. Sum of last 11 numbers as S2. The middle number is included in both S1 and S2. If we add S1 and S2, the middle number is counted twice. Using this, we can form a simple equation involving S, S1, S2 and the middle number x and then solve for x. Step-by-Step Solution: Step 1: Compute sum of all 21 numbers. S = 12 * 21 = 252. Step 2: Compute sum of the first 11 numbers. S1 = 9 * 11 = 99. Step 3: Compute sum of the last 11 numbers. S2 = 15 * 11 = 165. Step 4: Express relationship including the middle number x. When we add S1 and S2, all numbers except the middle one are counted once, but the middle number x is counted twice. Therefore S1 + S2 = S + x. Step 5: Substitute and solve for x. S1 + S2 = 99 + 165 = 264. We also have S = 252, so 264 = 252 + x. Thus x = 264 - 252 = 12. Verification / Alternative check: We can check by reconstructing the totals. If the middle number is 12, and S1 = 99, then the sum of the other 10 numbers in the first group is 99 - 12 = 87. In the last 11 numbers, S2 = 165, so the sum of the other 10 numbers in the last group is 165 - 12 = 153. Total sum of all 21 numbers is 87 + 12 + 153 = 252, and the average is 252 / 21 = 12, as given. So the answer is consistent. Why Other Options Are Wrong: If the middle number were 10, then S1 + S2 would equal S + 10, giving 99 + 165 = 252 + 10 and thus 264 = 262, which is impossible. Similar contradictions occur for 11 and 13. Only when the middle number is 12 does the relation S1 + S2 = S + x hold true. Common Pitfalls: Many students forget that the middle number is counted twice in S1 + S2 and may attempt to average 9 and 15 directly. Others confuse the count of numbers and mistakenly think there are only 20 numbers. The reliable method is to always convert given averages to sums and then carefully account for overlaps when groups share common elements. Final Answer: The value of the middle number is 12.

Question 15

There are 200 students in a school, of which seven-tenths are boys. How many girls are there in the school?

Accepted Answer

Introduction / Context: This is a basic ratio and fraction problem. We are told what fraction of the total students are boys and are asked to find how many students are girls. It tests the ability to convert fractional information about a group into actual numbers. Given Data / Assumptions: Total number of students in the school = 200. Fraction of students who are boys = 7/10. We assume all students are either boys or girls; there is no other category. Concept / Approach: If 7/10 of the students are boys, then the remaining 3/10 must be girls because the total fraction must add up to 1 (or 10/10). The number of boys is (7/10) of 200, and the number of girls is total students minus the number of boys or directly (3/10) of 200. Either method gives the same result. Step-by-Step Solution: Step 1: Compute the number of boys. Boys = (7 / 10) * 200 = 7 * 20 = 140. Step 2: Compute the number of girls from the total. Girls = total students - boys = 200 - 140 = 60. Alternative: Girls = (3 / 10) * 200 = 3 * 20 = 60. Verification / Alternative check: We can verify by checking that the sum of boys and girls matches the total. Boys = 140, girls = 60, so total = 140 + 60 = 200, which matches the given total number of students. Also, boys fraction = 140 / 200 = 7 / 10, and girls fraction = 60 / 200 = 3 / 10, which are complementary fractions adding up to 1, confirming correctness. Why Other Options Are Wrong: Option a (80) would give boys = 120 and girls = 80, so boys fraction would be 120 / 200 = 3/5, not 7/10. Option c (40) would give boys = 160 and boys fraction 160 / 200 = 4/5. Option d (120) would give boys = 80 and boys fraction 80 / 200 = 2/5. None of these matches the specified 7/10 fraction for boys. Only 60 yields the correct distribution. Common Pitfalls: One common mistake is to multiply the total by 3/10 instead of 7/10 when finding boys, or to misinterpret 7/10 as 7%. Another is forgetting that the remaining fraction automatically belongs to the other group. Always check that fractions for all groups add up to 1 and that the calculated numbers sum to the total. Final Answer: The number of girls in the school is 60.

Question 16

A batsman scores 95 runs in his 13th match and thereby increases his average runs per match by 4. What is his batting average (in runs per match) after the 13th match?

Accepted Answer

Introduction / Context: This is a typical cricket average problem. We are given how a player's average changes after one additional match and the score in that match. Using this information, we can find both the original average and the new average after the 13th match. Given Data / Assumptions: Number of matches before the latest innings = 12. The latest (13th) innings score = 95 runs. After this 13th innings, the batsman's average increases by 4 runs per match. We need to find the new average after 13 innings. Concept / Approach: Let the old average after 12 innings be a runs. Then: Total runs in first 12 innings = 12 * a. After the 13th innings, total runs = 12 * a + 95. New average after 13 innings = a + 4 (since it increases by 4). We use the definition of average: New average = (total runs after 13 innings) / 13. This gives an equation in a that we can solve. Step-by-Step Solution: Step 1: Let old average after 12 innings be a runs. Total runs after 12 innings = 12 * a. Step 2: Express total runs after the 13th innings. Total after 13 innings = 12 * a + 95. Step 3: Express new average in terms of a. New average = a + 4. By definition, new average = (12 * a + 95) / 13. Step 4: Form the equation and solve. (12 * a + 95) / 13 = a + 4. 12 * a + 95 = 13 * (a + 4) = 13 * a + 52. 12 * a + 95 = 13 * a + 52. 95 - 52 = 13 * a - 12 * a. 43 = a. Step 5: Compute the new average. New average = a + 4 = 43 + 4 = 47. Verification / Alternative check: If the old average was 43, total runs in 12 innings were 12 * 43 = 516. After scoring 95, total runs become 516 + 95 = 611. Now, 611 / 13 = 47 runs per innings, which is 4 more than 43. This matches the condition that the average increased by 4, confirming the result. Why Other Options Are Wrong: If the new average were 43, then the average would not have increased. If it were 45 or 49, solving backward would not give an integer old average and would not match the given 95 runs. Only 47 satisfies the equation relating old average, new average, and the additional score. Common Pitfalls: A common mistake is to treat 95 as the new average instead of a score, or to simply add 4 to 95. Another error is mixing up total runs and average. Always start by defining the original average as a variable, computing total runs, and then forming the equation with the new average after including the latest score. Final Answer: The batsman's average after the 13th match is 47 runs per match.

Question 17

In a class of 39 students there are 26 girls. The average weight of the girls is 42 kg, and the average weight of the entire class is 48 kg. What is the average weight (in kilograms) of the boys in the class?

Accepted Answer

Introduction / Context: This question is another example of finding the average of a subgroup when the averages of the full group and another subgroup are known. We need to find the average weight of boys given the number of boys and girls, the average weight of girls, and the average weight of the class. Given Data / Assumptions: Total number of students = 39. Number of girls = 26. Number of boys = 39 - 26 = 13. Average weight of girls = 42 kg. Average weight of the entire class = 48 kg. We assume all weights are accurately measured in kilograms. Concept / Approach: Let the average weight of the boys be b kg. Then: Total weight of class = 48 * 39. Total weight of girls = 42 * 26. Total weight of boys = b * 13. The total weight of the class is equal to the sum of total weights of girls and boys. Using this equality, we can solve for b. Step-by-Step Solution: Step 1: Compute total weight of the class. Total class weight = 48 * 39 = 1872 kg. Step 2: Compute total weight of girls. Total weight of girls = 42 * 26 = 1092 kg. Step 3: Compute total weight of boys. Total weight of boys = total class weight - total girls weight. Total weight of boys = 1872 - 1092 = 780 kg. Step 4: Compute average weight of boys. Number of boys = 13. Average weight of boys = 780 / 13 = 60 kg. Verification / Alternative check: Check the combined average: girls contribute 26 * 42 = 1092 kg, boys contribute 13 * 60 = 780 kg. Combined total = 1092 + 780 = 1872 kg. When divided by 39 students, 1872 / 39 = 48 kg, which matches the given class average. This confirms that the boys average of 60 kg is correct. Why Other Options Are Wrong: If the boys average were 54 kg, then boys total would be 13 * 54 = 702 kg, giving class total 1092 + 702 = 1794 kg and average 1794 / 39 ≈ 46, not 48 kg. For 66 kg, boys total becomes 858 kg, giving class average above 49 kg. For 62 kg, the average would again not match 48 kg. Only 60 kg yields the correct overall average. Common Pitfalls: Students may try to simply average 42 and 48 without considering the different numbers of boys and girls. Some may also miscalculate totals by forgetting to multiply by the number of students. Always remember that average = total / count, so when working with multiple groups, convert to totals first and then combine them carefully. Final Answer: The average weight of the boys in the class is 60 kilograms.

Question 18

A factory buys 7 machines: two of type A at Rs. 95,000 each, two of type B at Rs. 75,000 each, and the remaining machines of type C at Rs. 43,000 each. What is the average cost (in rupees) of these seven machines?

Accepted Answer

Introduction / Context: This problem requires computing the average cost of several machines bought at different prices. We know how many machines of each type were bought and their individual costs. From this we can obtain the total cost and then divide by the total number of machines to find the average cost per machine. Given Data / Assumptions: Number of machines of type A = 2, each costing Rs. 95,000. Number of machines of type B = 2, each costing Rs. 75,000. Total number of machines = 7, so machines of type C = 7 - 4 = 3. Each machine of type C costs Rs. 43,000. Concept / Approach: The average cost is defined as: Average cost per machine = total cost of all machines / total number of machines. So we first calculate the total cost contributed by each machine type, add them to get the grand total, and then divide by 7. Step-by-Step Solution: Step 1: Compute total cost of type A machines. Total cost of type A = 2 * 95,000 = Rs. 190,000. Step 2: Compute total cost of type B machines. Total cost of type B = 2 * 75,000 = Rs. 150,000. Step 3: Compute total cost of type C machines. Number of type C machines = 3. Total cost of type C = 3 * 43,000 = Rs. 129,000. Step 4: Compute grand total cost. Grand total = 190,000 + 150,000 + 129,000 = Rs. 469,000. Step 5: Compute average cost per machine. Average cost = 469,000 / 7 = Rs. 67,000. Verification / Alternative check: We can check the division: 67,000 * 7 = 469,000, which matches the total cost computed from individual machines. Also, the average cost lies between the highest price (Rs. 95,000) and the lowest price (Rs. 43,000), which is logically necessary, so the answer is reasonable. Why Other Options Are Wrong: An average of Rs. 71,000 would give a total of 7 * 71,000 = Rs. 497,000, which is higher than the actual total cost. An average of Rs. 75,000 gives an even higher total and is impossible given three machines cost only Rs. 43,000 each. An average of Rs. 65,500 gives a total of 458,500 rupees, which is too low. Only Rs. 67,000 reproduces the correct total of Rs. 469,000. Common Pitfalls: Some students mistakenly average the three price tags directly, ignoring the fact that there are different numbers of each machine type. Others may miscount the number of machines of type C or make arithmetic errors when adding large numbers. Always multiply each unit cost by its quantity, sum these totals, and divide by the total number of units to find the correct weighted average. Final Answer: The average cost of the seven machines is Rs. 67,000 per machine.

Question 19

The average of five numbers is 72. The first number is equal to one-eighth of the sum of the other four numbers. What is the value of this first number?

Accepted Answer

Introduction / Context: This question mixes the concept of averages with a special relationship between one number and the sum of the remaining numbers. We are told the overall average, and that the first number is one-eighth of the sum of the other four numbers. Using algebra, we can derive the value of the first number. Given Data / Assumptions: There are 5 numbers in total. The average of these 5 numbers is 72. The first number is one-eighth of the sum of the other four numbers. Let the first number be x. Concept / Approach: From the average, we can obtain the total sum of all five numbers. The special condition tells us how x relates to the total of the other four numbers. By expressing the sum of the other four numbers as total minus x, we can create an equation and solve for x. Step-by-Step Solution: Step 1: Compute total sum of the five numbers. Average = 72, number of numbers = 5. Total sum = 72 * 5 = 360. Step 2: Express the relationship using x. Let the first number be x. Sum of the other four numbers = 360 - x. Given that x is one-eighth of this sum: x = (1 / 8) * (360 - x). Step 3: Solve this equation for x. x = (360 - x) / 8. Multiply both sides by 8: 8 * x = 360 - x. 8x + x = 360. 9x = 360. x = 360 / 9 = 40. Verification / Alternative check: If the first number is 40, then the sum of the other four numbers is 360 - 40 = 320. One-eighth of 320 is 320 / 8 = 40, which matches x. So the relationship is satisfied. The average is (40 + sum of other four) / 5 = 360 / 5 = 72, which matches the given average. Hence x = 40 is correct. Why Other Options Are Wrong: If x = 60, then the sum of the other four numbers would be 360 - 60 = 300, and one-eighth of 300 is 37.5, not 60. For x = 26, one-eighth of the remaining sum does not match 26. For x = 80, the remaining sum would be 280 and one-eighth of 280 is 35, not 80. Only x = 40 satisfies both the total sum and the one-eighth relation simultaneously. Common Pitfalls: A common mistake is to misread the condition as the first number being one-eighth of the average or one-eighth of the total, rather than one-eighth of the sum of the other four. Another pitfall is not forming the equation correctly or forgetting to multiply out the fraction. Always translate verbal relationships into clear algebraic equations and solve them step by step. Final Answer: The first number is 40.

Question 20

The average weight of Shivshankar, Gopesh and Reena is 97 kg. The average weight of Shivshankar and Gopesh is 93 kg, and the average weight of Gopesh and Reena is 82 kg. What is the weight (in kilograms) of Gopesh?

Accepted Answer

Introduction / Context: This question involves determining an individual's weight when only averages of pairs and the trio are given. It is an example of solving a small system of linear equations that arises naturally from average relationships. Given Data / Assumptions: The average weight of Shivshankar (S), Gopesh (G) and Reena (R) is 97 kg. The average weight of S and G is 93 kg. The average weight of G and R is 82 kg. We need to find the weight of Gopesh, denoted G. Concept / Approach: We convert the averages into equations involving sums: (S + G + R) / 3 = 97, so S + G + R = 291. (S + G) / 2 = 93, so S + G = 186. (G + R) / 2 = 82, so G + R = 164. We now have three linear equations in three variables (S, G, R). Subtracting and combining these equations allows us to isolate G and find its value. Step-by-Step Solution: Step 1: Write equations from averages. From trio average: S + G + R = 3 * 97 = 291. From S and G average: S + G = 2 * 93 = 186. From G and R average: G + R = 2 * 82 = 164. Step 2: Use S + G = 186 to express S in terms of G. S = 186 - G. Step 3: Use G + R = 164 to express R in terms of G. R = 164 - G. Step 4: Substitute S and R into S + G + R = 291. (186 - G) + G + (164 - G) = 291. Simplify: 186 - G + G + 164 - G = 291. 350 - G = 291. Step 5: Solve for G. 350 - 291 = G. G = 59 kg. Verification / Alternative check: If G = 59 kg, then S + G = 186, so S = 127 kg. Also G + R = 164, so R = 164 - 59 = 105 kg. Check the trio average: (S + G + R) / 3 = (127 + 59 + 105) / 3 = 291 / 3 = 97 kg, which matches the given data. Also, average of S and G is (127 + 59) / 2 = 186 / 2 = 93, and average of G and R is (59 + 105) / 2 = 164 / 2 = 82, so all conditions are satisfied. Why Other Options Are Wrong: If G were 72, 56 or 63, the sums S + G and G + R derived from the given averages would not remain consistent with the total S + G + R = 291. Substituting any of these values for G produces contradictions in at least one of the equations, so they cannot be correct. Only G = 59 satisfies all three equations simultaneously. Common Pitfalls: A common error is to assume that the weights are symmetric and guess that G equals the overall average of 97, which is not supported by the pairwise averages. Others may misinterpret averages as requiring that each pair has similar weights. The systematic approach is to always convert averages to equations involving sums and then solve the resulting system carefully. Final Answer: The weight of Gopesh is 59 kg.

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