Question 1

Average of the first nine multiples of 3: Compute the mean of 3, 6, 9, ..., 27.

Accepted Answer

Introduction / Context: Multiples of a number form an arithmetic progression. The mean of an arithmetic progression equals the average of its first and last terms. Given Data / Assumptions: Sequence: 3, 6, 9, ..., 27 Count = 9 Concept / Approach: Average of an AP = (first + last) / 2. Here, first = 3 and last = 27. Step-by-Step Solution: Average = (3 + 27) / 2 = 30 / 2 = 15 Verification / Alternative check: The middle term (5th term) in an odd-length AP equals the mean. The 5th term here is 15, confirming the result. Why Other Options Are Wrong: 12.0, 12.5, 18.5, 13.5: do not match AP mean for 3 to 27 with equal spacing of 3. Common Pitfalls: Confusing the common difference with the average or miscounting terms. Final Answer: 15.0

Question 2

Find a missing result using averages: The average of five results is 46, and the average of the first four results is 45. What is the fifth result?

Accepted Answer

Introduction / Context: When a total average and a partial average are known, convert both to totals. The difference between these totals gives the missing entry directly. Given Data / Assumptions: Average of five = 46 Average of first four = 45 Concept / Approach: Compute total of all five and total of first four, then subtract to get the fifth value. Step-by-Step Solution: Total of five = 5 * 46 = 230Total of first four = 4 * 45 = 180Fifth result = 230 - 180 = 50 Verification / Alternative check: The five values must sum to 230. If the first four sum to 180, the last must be 50. Averages remain consistent. Why Other Options Are Wrong: 51, 49.5, 49, 52: do not satisfy the given averages when recomputed. Common Pitfalls: Arithmetic slips when multiplying averages or subtracting totals. Keep integers aligned. Final Answer: 50

Question 3

Combined average of two datasets: The average of 30 results is 20, and the average of another 20 results is 30. Find the average of all 50 results combined.

Accepted Answer

Introduction / Context: Combining datasets requires weighted averaging. Multiply each subgroup average by its count to get totals, then divide the sum by the combined count. Given Data / Assumptions: Group 1: 30 results, average 20 Group 2: 20 results, average 30 Concept / Approach: Overall average = (sum of group totals) / total number of results. Step-by-Step Solution: Total 1 = 30 * 20 = 600Total 2 = 20 * 30 = 600Combined total = 600 + 600 = 1200Overall average = 1200 / 50 = 24 Verification / Alternative check: The two groups contribute equally to the total. With equal totals but different counts, the overall mean lies between 20 and 30, specifically 24 given the 30 to 20 split. Why Other Options Are Wrong: 25, 27: incorrect weighting. 48, 50: misread as totals, not averages. Common Pitfalls: Doing an unweighted mean of 20 and 30 (which would be 25) without considering different counts. Final Answer: 24

Question 4

Mean of close decimal values: Compute the average of 0.64204, 0.64203, 0.64202, and 0.64201.

Accepted Answer

Introduction / Context: When numbers form a short arithmetic progression, the average equals the midpoint between first and last. This reduces calculation complexity and rounding errors. Given Data / Assumptions: Numbers: 0.64204, 0.64203, 0.64202, 0.64201 Count = 4 Concept / Approach: Average of an AP = (first + last) / 2. Using endpoints avoids repeated decimal addition. Step-by-Step Solution: Average = (0.64204 + 0.64201) / 2Average = 1.28405 / 2 = 0.642025 Verification / Alternative check: Add all numbers directly: total = 2.56810. Divide by 4 gives 0.642025, same result. Why Other Options Are Wrong: 0.64202, 0.64204, 0.642022, 0.642015: do not match the precise midpoint or full sum division. Common Pitfalls: Rounding too early or making carry mistakes when summing decimals. Using the midpoint is reliable here. Final Answer: 0.642025

Question 5

Three-person average weights: The average weight of A, B, and C is 45 kg. The average of A and B is 40 kg, and the average of B and C is 43 kg. Find the weight of B.

Accepted Answer

Introduction / Context: This is a system-of-equations average problem. Translating averages to totals creates linear equations that can be combined to isolate one person’s weight. Given Data / Assumptions: A + B + C average = 45 kg, so total 135 kg A + B average = 40 kg, so total 80 kg B + C average = 43 kg, so total 86 kg Concept / Approach: Add the two pair totals: (A + B) + (B + C) = A + 2B + C. Also note A + C = (A + B + C) - B. Substitute and solve for B. Step-by-Step Solution: A + B + C = 135A + B = 80, B + C = 86Add: A + 2B + C = 166But A + C = 135 - BSo 135 - B + 2B = 166 => B = 31 kg Verification / Alternative check: If B = 31, then A + C = 104. Pair sums: A + B = 80 gives A = 49; B + C = 86 gives C = 55. Check total: 49 + 31 + 55 = 135, average 45, consistent. Why Other Options Are Wrong: 17, 20, 26, 35 kg: do not satisfy all three average conditions simultaneously. Common Pitfalls: Mixing up which totals to add or subtract and forgetting that average times count equals total. Final Answer: 31 kg

Question 6

Admission changes average weight: The average of 19 students is 15 kg. After admitting one new student, the new average becomes 14.8 kg. Find the weight of the new student.

Accepted Answer

Introduction / Context: Average shift problems involving the addition of one value are solved by recomputing totals before and after the change. The difference equals the added value. Given Data / Assumptions: Initial: 19 students, average 15 kg After addition: 20 students, average 14.8 kg Concept / Approach: Compute initial total, compute new total, subtract to get the new student’s weight. Step-by-Step Solution: Initial total = 19 * 15 = 285 kgNew total = 20 * 14.8 = 296 kgWeight of new student = 296 - 285 = 11 kg Verification / Alternative check: With the new student at 11 kg, the total becomes 296 over 20 students, giving 14.8 kg average as stated. Why Other Options Are Wrong: 10.6, 10.8, 14.9, 12.5 kg: do not produce the new average when recalculated. Common Pitfalls: Using 19 in the second multiplication by mistake or misplacing decimals when multiplying 14.8 by 20. Final Answer: 11 kg

Question 7

Replacement effect on mean: The average of 8 persons increases by 2.5 kg when a 56 kg person is replaced by a new person. Find the weight of the new person.

Accepted Answer

Introduction / Context: Replacing one data point changes the total by an amount equal to (new value - old value). If the average rises by a known amount, multiply that by the count to get the change in total, then solve for the new value. Given Data / Assumptions: Count = 8 persons Old person weight = 56 kg Average increases by 2.5 kg Concept / Approach: Increase in total = increase in average * count. Then new weight = old weight + increase in total. Step-by-Step Solution: Increase in total = 2.5 * 8 = 20 kgNew person weight = 56 + 20 = 76 kg Verification / Alternative check: New total is old total + 20. Dividing by 8 gives the new average which is 2.5 kg higher, confirming the logic. Why Other Options Are Wrong: 66, 75, 78, 86 kg: do not produce the specified rise in mean across 8 persons. Common Pitfalls: Multiplying the increase by 1 instead of the group size, or subtracting from 56 instead of adding when the average rises. Final Answer: 76 kg

Question 8

Daily income relation and average: The average income of A, B, and C is Rs 60. B earns Rs 20 more than C, and A earns twice what C earns. Find the daily income of C.

Accepted Answer

Introduction / Context: Income relations can be set as variables and solved using the average. Translate each statement into algebra and solve systematically. Given Data / Assumptions: Average of A, B, C = Rs 60, so total Rs 180 B = C + Rs 20 A = 2 * C Concept / Approach: Let C = x. Then A = 2x and B = x + 20. Substitute into the total to solve for x. Step-by-Step Solution: Total = A + B + C = 2x + (x + 20) + x = 4x + 20Set 4x + 20 = 1804x = 160 => x = 40Therefore C = Rs 40 Verification / Alternative check: A = 80, B = 60, C = 40. Sum = 180, average = 60. Relations are satisfied. Why Other Options Are Wrong: Rs 70, 60, 35, 50: do not satisfy both the total and the linear relations. Common Pitfalls: Mistakes arise if B is misread as 20 percent more or if A is misinterpreted. The wording indicates absolute values, not percentages. Final Answer: Rs 40

Question 9

Average age drop after intake: An adult class has average age 40 years. When 12 new students with average age 32 join, the class average falls by 4 years to 36. What was the original class strength?

Accepted Answer

Introduction / Context: Average changes due to new members can be modeled with totals before and after. The new group average after addition lets us solve for the original size via algebra. Given Data / Assumptions: Original average = 40 years 12 join with average = 32 years New average = 36 years Concept / Approach: Let original size be n. Original total = 40n. New total = 40n + 12 * 32. New count = n + 12. Set new average to 36 and solve for n. Step-by-Step Solution: (40n + 384) / (n + 12) = 3640n + 384 = 36n + 4324n = 48 => n = 12 Verification / Alternative check: Original total 40 * 12 = 480. After adding 12 at 32, add 384 to get 864. New count 24. 864 / 24 = 36, correct. Why Other Options Are Wrong: 10, 11, 15, 18: do not satisfy the average equation. Common Pitfalls: Using 32 percent rather than 32 years, or neglecting that both numerator and denominator change when new members join. Final Answer: 12

Question 10

Class average with replacement: In a class of 20 boys, the overall average age decreases by 2 months when one boy aged 18 years is replaced by a new entrant. Determine the exact age of the new boy (express your answer in years and months, keeping al…

Accepted Answer

Introduction / Context: Average age problems often use the relation between total sum and average. A replacement that changes the average implies a precise change in the class total. We use years as the base unit and convert months where needed. Given Data / Assumptions: Number of boys = 20 One 18 year old leaves; a new boy joins Average decreases by 2 months = 2/12 year = 1/6 year Concept / Approach: The class total equals average * count. Let initial average be A years. New average is A - 1/6. Replacement changes total by (new_boy_age - 18). Step-by-Step Solution: Initial total = 20 * ANew total = 20 * (A - 1/6)But also New total = Initial total - 18 + xSo 20 * (A - 1/6) = 20 * A - 18 + x20A - 20*(1/6) = 20A - 18 + x-20/6 = -18 + xx = 18 - 20/6 = 18 - 10/3 = 44/3 years44/3 years = 14 years 8 months Verification / Alternative check: Decrease in total due to average drop = 20 * (1/6) = 10/3 years, which matches 18 - 14 2/3 = 10/3 years. Why Other Options Are Wrong: 15 years: Too high; would reduce total by only 3 years, not 10/3. 16 years 4 months: Higher still; change is smaller than required. 17 years 10 months: Nearly no change; incorrect. Common Pitfalls: Forgetting to convert months to a fraction of a year or mixing up which total increases or decreases. Final Answer: 14 years 8 months

Question 11

Family income average: A family has 4 earning members with an average monthly income of Rs 735. After one member passes away, the average for the remaining 3 becomes Rs 650. Find the monthly income of the deceased member.

Accepted Answer

Introduction / Context: This is a direct average to total conversion problem. Knowing the average and count allows recovery of totals and the missing value. Given Data / Assumptions: Initial earners = 4, average = Rs 735 After one leaves, earners = 3, average = Rs 650 Concept / Approach: Total = average * count. Income of the deceased = initial total - new total. Step-by-Step Solution: Initial total = 4 * 735 = 2940New total (3 members) = 3 * 650 = 1950Deceased income = 2940 - 1950 = 990 Verification / Alternative check: Average with all four: 2940/4 = 735. Removing 990 leaves 1950 for 3, which averages to 650. Checks out. Why Other Options Are Wrong: Rs 692.80, Rs 820, Rs 1385: Do not match the difference between totals computed from the given averages. Common Pitfalls: Confusing averages with totals and attempting to average averages directly without weighting by counts. Final Answer: Rs 990

Question 12

Teacher added to class: The average age of 24 students is 10 years. When the teacher is included, the average increases by 1 year. Determine the age of the teacher.

Accepted Answer

Introduction / Context: Adding one data point that raises the average lets us compute that data point by comparing totals before and after inclusion. Given Data / Assumptions: Students = 24, student average = 10 years With teacher, count = 25, new average = 11 years Concept / Approach: Total = average * count. Teacher age = new total - old total. Step-by-Step Solution: Old total = 24 * 10 = 240New total = 25 * 11 = 275Teacher age = 275 - 240 = 35 years Verification / Alternative check: Check average with teacher: (240 + 35)/25 = 275/25 = 11. Correct. Why Other Options Are Wrong: 25 or 30 years: Insufficient to raise the average by 1. 40 years: Would increase the average more than required for these counts. Common Pitfalls: Mistaking the direction of change, or forgetting that adding an older person increases the total appropriately. Final Answer: 35 years

Question 13

Monthly income from expenditures and savings: The average expenditure for the first 5 months is Rs 120 and for the next 7 months is Rs 130. If the man saves Rs 299 in the year, what is his average monthly income?

Accepted Answer

Introduction / Context: Average income can be deduced from total expenditure plus total savings across the year. The problem provides averages for two blocks of months and the total annual savings. Given Data / Assumptions: First 5 months average spend = Rs 120 Next 7 months average spend = Rs 130 Annual savings = Rs 299 Months in year considered = 12 Concept / Approach: Total income = total expenditure + total savings. Average monthly income = total income / 12. Step-by-Step Solution: Expenditure for first 5 months = 5 * 120 = 600Expenditure for next 7 months = 7 * 130 = 910Total annual expenditure = 600 + 910 = 1510Total income = 1510 + 299 = 1809Average monthly income = 1809 / 12 = 150.75 Verification / Alternative check: Average spend over 12 months = 1510/12. Adding average saving per month 299/12 yields 150.75. Same result. Why Other Options Are Wrong: Rs 150, Rs 160, Rs 170: Do not equal the exact computed average 150.75. Common Pitfalls: Using an approximate value or rounding too early instead of keeping the exact fraction for the final average. Final Answer: Rs 150.75

Question 14

Manager salary from averages: The average monthly salary of 20 workers is Rs 1900. When the manager is included, the average becomes Rs 2000 for 21 people. What is the manager's annual salary?

Accepted Answer

Introduction / Context: When an additional member changes the average, we recover the person's value by comparing totals before and after inclusion, then annualize by multiplying by 12. Given Data / Assumptions: Workers = 20, average = Rs 1900 With manager, count = 21, average = Rs 2000 Concept / Approach: Total = average * count. Monthly manager salary = new total - old total. Annual salary = monthly * 12. Step-by-Step Solution: Old total = 20 * 1900 = 38000New total = 21 * 2000 = 42000Manager monthly = 42000 - 38000 = 4000Manager annual = 4000 * 12 = 48000 Verification / Alternative check: Check new average: (38000 + 4000)/21 = 42000/21 = 2000. Correct. Why Other Options Are Wrong: Rs 24000, Rs 25200, Rs 45600: None equal the correct annual total of Rs 48000, hence 'None of these' is correct. Common Pitfalls: Forgetting to annualize or mixing monthly and annual figures. Final Answer: None of these

Question 15

Average adjustment after adding one value: The average of 6 observations is 12. After including a 7th observation, the new average decreases by 1. Find the value of the 7th observation.

Accepted Answer

Introduction / Context: This problem highlights how adding one item can decrease an average if the item is below the current average. We compute the exact value via totals. Given Data / Assumptions: Average of 6 = 12 Average of 7 after adding one = 11 Concept / Approach: Total = average * count. The new observation equals the difference between the new and old totals. Step-by-Step Solution: Old total = 6 * 12 = 72New total = 7 * 11 = 77Seventh observation = 77 - 72 = 5 Verification / Alternative check: New average check: (72 + 5)/7 = 77/7 = 11. Correct. Why Other Options Are Wrong: 1, 3, 6: These would produce totals 73, 75, or 78, which do not give an average of 11. Common Pitfalls: Assuming the new value equals the new average instead of using totals. Final Answer: 5

Question 16

Linked averages for overlapping triplets: Of four numbers a, b, c, d, the average of the first three is 15, the average of the last three is 16, and the last number d is 19. Find the first number a.

Accepted Answer

Introduction / Context: When averages of overlapping groups are known, sums can be equated to determine individual elements. Given Data / Assumptions: (a + b + c)/3 = 15 (b + c + d)/3 = 16 d = 19 Concept / Approach: Compute sums from averages, eliminate common parts, and solve for the unknown a. Step-by-Step Solution: a + b + c = 45b + c + d = 48Substitute d = 19 to get b + c = 29Therefore a = 45 - (b + c) = 45 - 29 = 16 Verification / Alternative check: Check last three: b + c + d = 29 + 19 = 48, average = 16. Works. Why Other Options Are Wrong: 15, 18, 19: They do not satisfy both average relations simultaneously. Common Pitfalls: Incorrectly averaging the two averages instead of using sums. Final Answer: 16

Question 17

Weekly temperature: The average of the first three days is 27 C and of the next three days is 29 C. If the average temperature for the entire 7 day week is 28.5 C, find the temperature on the last day.

Accepted Answer

Introduction / Context: Average over the whole equals total for the week divided by 7. Given two three day blocks, the seventh day is the remainder when those are removed from the total. Given Data / Assumptions: First 3 day average = 27 C Next 3 day average = 29 C Weekly average = 28.5 C Concept / Approach: Compute totals for each block and subtract from the weekly total. Step-by-Step Solution: Total week = 7 * 28.5 = 199.5First block total = 3 * 27 = 81Second block total = 3 * 29 = 87Seventh day = 199.5 - (81 + 87) = 31.5 C Verification / Alternative check: Average recomputed equals 28.5 C when all seven are used. Why Other Options Are Wrong: 10.5 C, 21 C, 42 C: Do not satisfy the computed remainder. Common Pitfalls: Using 6 days mistakenly for the weekly total or mixing Celsius symbols with arithmetic and rounding. Final Answer: 31.5 C

Question 18

Average speed for a round trip: A ship sails to a mark at 15 km/h and returns at 10 km/h over the same distance. What is the average speed for the whole journey?

Accepted Answer

Introduction / Context: Average speed for equal distance out and back uses the harmonic mean, not the arithmetic mean. Given Data / Assumptions: Outbound speed = 15 km/h Return speed = 10 km/h Concept / Approach: For equal distances, average speed = 2*x*y/(x + y). Step-by-Step Solution: Average = 2 * 15 * 10 / (15 + 10)= 300 / 25 = 12 km/h Verification / Alternative check: Pick a distance such as 30 km each way. Time = 30/15 + 30/10 = 2 + 3 = 5 h. Total distance = 60 km. 60/5 = 12 km/h. Why Other Options Are Wrong: 5 km/h and 25 km/h: Far from the harmonic mean. 12.5 km/h: Arithmetic mean trap, not correct for round trips. Common Pitfalls: Using the arithmetic mean of speeds instead of the harmonic mean for equal distances. Final Answer: 12 km/h

Question 19

Two block averages combined: A car consumes on average 110 litres of petrol per month for the first 7 months and 86 litres per month for the next 5 months. Find the average monthly consumption for the full year.

Accepted Answer

Introduction / Context: Weighted averages by month counts give the yearly mean consumption. Given Data / Assumptions: First 7 months average = 110 litres Next 5 months average = 86 litres Concept / Approach: Compute total litres in each block, add them, then divide by 12 months. Step-by-Step Solution: First block total = 7 * 110 = 770Second block total = 5 * 86 = 430Year total = 770 + 430 = 1200Average monthly = 1200 / 12 = 100 litres Verification / Alternative check: Proportional weighting by 7 and 5 months is correctly applied. Why Other Options Are Wrong: 96, 98, 102 litres: Do not equal the exact weighted result. Common Pitfalls: Taking a simple mean of 110 and 86 without weighting by month counts. Final Answer: 100 litres

Question 20

Batting average drop after a high score: A batsman has a certain average for 11 innings. In the 12th innings he scores 90 runs and his average decreases by 5. What is his average after 12 innings?

Accepted Answer

Introduction / Context: Averages in cricket can move in nonintuitive ways. Setting up an equation for totals before and after the nth innings resolves the value. Given Data / Assumptions: Average after 11 innings = A Score in 12th = 90 Average after 12 innings = A - 5 Concept / Approach: Relate totals: 11*A + 90 = 12*(A - 5). Solve for A, then compute A - 5. Step-by-Step Solution: 11*A + 90 = 12*A - 6090 + 60 = 12*A - 11*A150 = AAverage after 12 = A - 5 = 145 Verification / Alternative check: Total after 11 = 1650. After 12 with score 90 total = 1740. 1740/12 = 145. Works. Why Other Options Are Wrong: 127, 150, 217: Do not match the derived post 12th average. Common Pitfalls: Forgetting the average decreases despite a high single score due to a very high prior average. Final Answer: 145

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