Question 1

In a bus, the average weight of 45 passengers is 52 kg. Five passengers whose average weight is 48 kg get down at a stop, and five new passengers whose average weight is 54 kg board the bus at the same stop. What is the new average weight of all the…

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Introduction: This aptitude question tests the concept of average weight when some members of a group leave and new members join. Such problems are common in competitive exams and help you practice converting averages into totals and then back to a new average after a change in the group. Given Data / Assumptions: The original number of passengers in the bus = 45. Original average weight of passengers = 52 kg. Number of passengers who leave = 5, with average weight = 48 kg. Number of passengers who join = 5, with average weight = 54 kg. The total number of passengers after the change remains 45. Concept / Approach: The average of a group is defined as total weight divided by the number of persons. When some people leave and others join, it is always convenient to convert averages into total weights, adjust for the people leaving and joining, and then divide by the final number of people to get the new average. Step-by-Step Solution: Step 1: Compute the original total weight of all passengers. Original total weight = 45 * 52 = 2340 kg. Step 2: Compute the total weight of the 5 passengers who leave. Weight of passengers who leave = 5 * 48 = 240 kg. Step 3: Subtract this from the original total weight. Total after they leave = 2340 - 240 = 2100 kg. Step 4: Compute the total weight of the 5 new passengers. Weight of passengers who join = 5 * 54 = 270 kg. Step 5: Add this to the existing total. New total weight = 2100 + 270 = 2370 kg. Step 6: Find the new average weight. New average weight = 2370 / 45 = 52.666... kg which is approximately 52.66 kg. Verification / Alternative Check: You can also think of the net effect per passenger. The 5 who left were lighter than the original average by 4 kg each (52 - 48), and the 5 who joined were heavier than the original average by 2 kg each (54 - 52). So net change in total weight = 5 * (-4) + 5 * 2 = -20 + 10 = -10 kg. New total = 2340 - 10 = 2330 kg. But this reasoning mistakes the base average; the correct detailed computation above already confirms 2370 kg, so always prefer the full total method in exam settings to avoid conceptual slips. Why Other Options Are Wrong: 54.21 kg: This is higher than both the original average and the weight of most passengers, which does not match the calculated total. 51.07 kg: This is too low and would imply a large net reduction in total weight that does not follow from the given data. 53.45 kg: This is slightly higher than the correct average and would require a higher total weight than 2370 kg. 50.80 kg: This is clearly too low and inconsistent with the addition of heavier passengers. Common Pitfalls: Students often try to directly average 48 kg and 54 kg or average the averages of different groups without converting to totals. Another common mistake is to change the number of passengers while computing the new average, even though it remains 45 here. Always convert averages to totals first, adjust for people leaving or joining, and then divide by the final count. Final Answer: The new average weight of all passengers in the bus is 52.66 kg.

Question 2

A car owner buys petrol at Rs 7.50, Rs 8.00 and Rs 8.50 per litre for three successive years. If he spends Rs 4000 each year on petrol, what is the approximate average cost per litre of petrol over the three years?

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Introduction: This question checks your understanding of weighted average price when the same amount of money is spent at different rates over several periods. It is a very common quantitative aptitude idea used in profit and loss and data interpretation questions. Given Data / Assumptions: Petrol price in year 1 = Rs 7.50 per litre. Petrol price in year 2 = Rs 8.00 per litre. Petrol price in year 3 = Rs 8.50 per litre. Amount spent each year = Rs 4000. We need the overall average cost per litre over all three years. Concept / Approach: When the money spent is the same, but the price per litre is different, the quantity of petrol purchased each year is different. The correct method is: 1. Compute litres purchased each year. 2. Add all litres to get the total quantity. 3. Divide the total money spent over three years by the total quantity to get the weighted average price. Step-by-Step Solution: Step 1: Find litres bought in each year. Year 1: Litres = 4000 / 7.5 ≈ 533.33 litres. Year 2: Litres = 4000 / 8 = 500 litres. Year 3: Litres = 4000 / 8.5 ≈ 470.59 litres. Step 2: Find total litres of petrol. Total litres ≈ 533.33 + 500 + 470.59 ≈ 1503.92 litres. Step 3: Find total money spent. Total money spent in three years = 3 * 4000 = Rs 12000. Step 4: Compute average cost per litre. Average cost per litre ≈ 12000 / 1503.92 ≈ Rs 7.98 per litre. Verification / Alternative Check: Notice that all prices are between Rs 7.50 and Rs 8.50, and more litres are bought when petrol is cheaper (Rs 7.50). Therefore, the overall average must be slightly less than the simple average of the three prices. Simple average = (7.5 + 8 + 8.5) / 3 = 8.00. Since more quantity is purchased in the cheapest year, the weighted average should be slightly below 8, which matches Rs 7.98. Why Other Options Are Wrong: Rs 6.23 per litre: Far below all three given prices; impossible as an average of higher values. Rs 8.97 per litre: Higher than even the maximum price of Rs 8.50, so it cannot be an overall average. Rs 9.89 per litre: Also greater than all the individual prices, so it is clearly invalid. Rs 7.45 per litre: Below the minimum price of Rs 7.50 and inconsistent with the calculations. Common Pitfalls: A common mistake is to simply average the three rates (7.5, 8, 8.5) without considering that the number of litres bought each year is different. Another mistake is to average the three quantities of petrol instead of using the ratio of total money to total quantity. Always use total cost divided by total quantity for such questions. Final Answer: The approximate average cost per litre of petrol over the three years is Rs 7.98 per litre.

Question 3

Nineteen friends go to a restaurant for a combined weekend dinner party. Thirteen of them spend Rs 79 each on their dinner, and each of the remaining friends spends Rs 4 more than the average expenditure of all the 19 friends together. What is the t…

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Introduction: This problem uses the idea of averages where one part of the group spends a fixed amount and the remaining members spend an amount defined in terms of the overall average. It is a classic example of solving an equation where the average itself is unknown. Given Data / Assumptions: Total number of friends = 19. Number of friends in the first group = 13, each spending Rs 79. Number of remaining friends = 19 - 13 = 6. Each of these 6 friends spends Rs 4 more than the overall average expenditure of all 19 friends. We need the total money spent by all 19 friends. Concept / Approach: Let the average expenditure of all 19 friends be A rupees. Then: Total spending by 19 friends = 19 * A. Spending of the first 13 friends = 13 * 79. Spending of each of the remaining 6 friends = A + 4. We equate the sum of these two parts to 19 * A and solve for A, then find the total expenditure. Step-by-Step Solution: Step 1: Assume the overall average expenditure is A rupees. Step 2: Total expenditure of the 13 friends = 13 * 79 = 1027 rupees. Step 3: Total expenditure of the other 6 friends = 6 * (A + 4). Step 4: Total expenditure of all 19 friends = 19 * A. Step 5: Form the equation: 13 * 79 + 6 * (A + 4) = 19 * A. That is, 1027 + 6A + 24 = 19A. So, 1051 + 6A = 19A. 13A = 1051. A = 1051 / 13 ≈ 80.846 rupees. Step 6: Total money spent = 19 * A = 19 * (1051 / 13) = 19969 / 13 ≈ Rs 1536.07. Verification / Alternative Check: You can approximate mentally: since 13 people spend 79 each, their total is about 13 * 80 = 1040 (slightly less in reality). If the average is a bit above 80, total for 19 people should be slightly more than 19 * 80 = 1520, which is consistent with Rs 1536.07. Why Other Options Are Wrong: Rs 1398.96 and Rs 1457.09: These are too low compared to 19 people spending around Rs 80 on average. Rs 1662.35 and Rs 1601.50: These are too high and would imply an average much larger than the values derived from the equation. Common Pitfalls: Many learners incorrectly assume the 6 friends spend Rs 4 more than Rs 79, instead of Rs 4 more than the overall average A. Another pitfall is forgetting that the total expenditure must be 19 * A, not a simple sum of the known fixed amounts. Always express all unknown parts in terms of A, build a single equation, and then solve. Final Answer: The total amount of money spent by the 19 friends is approximately Rs 1536.07.

Question 4

In a hall, the average height of the boys is 180 cm and the average height of the girls is 150 cm. If the combined average height of the entire gathering is 165 cm and there are 78 boys present, how many girls are present in the hall?

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Introduction: This question deals with weighted averages of two groups, boys and girls, whose individual averages are known. You are asked to find the missing number of girls when the overall average height is given. Given Data / Assumptions: Average height of boys = 180 cm. Average height of girls = 150 cm. Combined average height of the gathering = 165 cm. Number of boys = 78. Number of girls = unknown (let this be g). Concept / Approach: We use the idea of weighted average. If one group has B members with average X and another group has G members with average Y, then the combined average is: Combined average = (B * X + G * Y) / (B + G). Here, B = 78, X = 180, Y = 150, and the combined average is 165. We form an equation in terms of G and solve it. Step-by-Step Solution: Step 1: Let the number of girls be g. Step 2: Total height of boys = 78 * 180. 78 * 180 = 14040 cm. Step 3: Total height of girls = g * 150. Step 4: Combined average is given as 165 cm, so (14040 + 150g) / (78 + g) = 165. Step 5: Cross multiply: 14040 + 150g = 165 * (78 + g). 14040 + 150g = 165 * 78 + 165g. 14040 + 150g = 12870 + 165g. 14040 - 12870 = 165g - 150g. 1170 = 15g. g = 1170 / 15 = 78. Verification / Alternative Check: If there are 78 boys and 78 girls, total persons = 156. Total height of boys = 14040 cm. Total height of girls = 78 * 150 = 11700 cm. Combined total = 14040 + 11700 = 25740 cm. Combined average = 25740 / 156 = 165 cm, which matches the given value, so the calculation is correct. Why Other Options Are Wrong: 56, 64, 87, 92: Substituting any of these values for g in the weighted average formula does not give a combined average of 165 cm. Either the average comes out greater than 165 or less than 165. Common Pitfalls: A frequent mistake is to assume the number of girls is simply proportional to the averages without forming the correct equation. Another error is to attempt to average the averages directly. Always use the formula for combined average with total sums: total of boys plus total of girls divided by the total number of people. Final Answer: The number of girls present in the hall is 78.

Question 5

The average of six numbers is 3.95. The average of two of these numbers is 3.40, and the average of another two of the numbers is 3.85. What is the average of the remaining two numbers?

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Introduction: This question is about breaking an overall average into parts. The average of all six numbers is known, and the averages of two separate pairs among them are given. You must find the average of the remaining two numbers. Given Data / Assumptions: Average of six numbers = 3.95. Average of one pair of numbers = 3.40. Average of another pair of numbers = 3.85. We need the average of the remaining two numbers. Concept / Approach: The idea is to translate averages into totals. For any group: Total = Average * Number of terms. We calculate: 1. Total of all six numbers. 2. Total of the first two numbers. 3. Total of the second pair of two numbers. Then subtract these from the total to get the sum of the remaining two numbers, and finally divide by 2 to get their average. Step-by-Step Solution: Step 1: Total of all six numbers = 6 * 3.95. 6 * 3.95 = 23.70. Step 2: Total of the first pair = 2 * 3.40 = 6.80. Step 3: Total of the second pair = 2 * 3.85 = 7.70. Step 4: Sum of the remaining two numbers = 23.70 - 6.80 - 7.70. Sum of remaining two numbers = 23.70 - 14.50 = 9.20. Step 5: Average of the remaining two numbers = 9.20 / 2 = 4.60. Verification / Alternative Check: You can quickly check: if two numbers average 3.40, their total is slightly less than 7; another two average 3.85, total slightly less than 8. The total of all six numbers is about 24, so the remaining pair must total about 9, giving an average a bit above 4, which matches 4.60. Why Other Options Are Wrong: 3.60: This would make the total of the last two too small, and the overall average would no longer be 3.95. 6.50 and 7.30: These are too high and would push the overall average far above 3.95. 5.10: This does not match the exact calculated sum of 9.20. Common Pitfalls: A common error is to average the three averages directly: (3.95 + 3.40 + 3.85) / 3, which is incorrect because the groups have different sizes. Always convert averages into totals, manipulate the totals, and then convert back to averages. Final Answer: The average of the remaining two numbers is 4.60.

Question 6

If the mean (average) of the three numbers a, b and c is M and it is given that ab + bc + ca = 0, then what is the mean of the three numbers a², b² and c²?

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Introduction: This question tests algebraic manipulation of means and symmetric expressions in three variables. It connects the mean of the original numbers a, b, c with the mean of their squares using a condition on ab + bc + ca. Given Data / Assumptions: Mean of a, b, c is M, so (a + b + c) / 3 = M. Therefore, a + b + c = 3M. It is given that ab + bc + ca = 0. We need the mean of a², b², c², that is (a² + b² + c²) / 3. Concept / Approach: A key algebraic identity is: (a + b + c)² = a² + b² + c² + 2(ab + bc + ca). We already know a + b + c and ab + bc + ca. Using this identity, we can express a² + b² + c² in terms of M, then divide by 3 to get the required mean. Step-by-Step Solution: Step 1: Use the given mean. a + b + c = 3M. Step 2: Square both sides of a + b + c = 3M. (a + b + c)² = (3M)² = 9M². Step 3: Expand the left-hand side using the identity. (a + b + c)² = a² + b² + c² + 2(ab + bc + ca). Step 4: Substitute ab + bc + ca = 0. So, 9M² = a² + b² + c² + 2 * 0. This simplifies to a² + b² + c² = 9M². Step 5: Find the mean of a², b², c². Mean of squares = (a² + b² + c²) / 3 = 9M² / 3 = 3M². Verification / Alternative Check: You can test with a simple example. Suppose a, b, c satisfy a + b + c = 3 and ab + bc + ca = 0. One such triple is (1, 1, 1) but that gives ab + bc + ca = 3, not 0. Instead, consider more advanced algebraic triples. However, the identity used is standard and algebraically sound, so the result 3M² is reliable. Why Other Options Are Wrong: M and M²: These ignore the effect of the squared terms and the given condition ab + bc + ca = 0. 2(M/3): This is dimensionally inconsistent with the squared quantity and does not follow from the identity. 3M: Again, this ignores the squared nature of the expression and is not derived from any correct manipulation. Common Pitfalls: A common error is to think that the mean of the squares is simply the square of the mean, which would give M², but this is only true in special cases. Another pitfall is forgetting to use the condition ab + bc + ca = 0, which is crucial to simplifying the expression. Always use known algebraic identities systematically. Final Answer: The mean of a², b² and c² is 3M².

Question 7

In an NGO, the daily average wages of 20 illiterate employees decrease from Rs 25 to Rs 10. As a result, the average daily salary of all employees, including both literate and illiterate workers, decreases by Rs 10 per day. How many educated (litera…

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Introduction: This question involves a change in average wages caused by a reduction in the wages of a subgroup of employees. You are asked to determine the number of educated employees from the information about how the overall average changes. Given Data / Assumptions: Number of illiterate employees = 20. Their daily wage decreases from Rs 25 to Rs 10. The overall average wage of all employees decreases by Rs 10. Number of literate employees = let this be x. Assume literate employees wages remain unchanged. Concept / Approach: The key idea is that a change in the average of the entire group is caused by a change in the total sum of wages, while the total number of workers stays the same. We: 1. Calculate the total decrease in wages of the 20 illiterate workers. 2. Relate that to the decrease in average wage of all employees. 3. Use this to find the total number of employees and then the number of educated workers. Step-by-Step Solution: Step 1: Change in wage per illiterate employee = 25 - 10 = Rs 15. Step 2: Total decrease in wages for 20 illiterate employees = 20 * 15 = Rs 300 per day. Step 3: Let the total number of employees (literate + illiterate) be N. Due to the wage cut, the average salary of all employees decreases by Rs 10 per day. Step 4: Total decrease in wages for all employees = N * 10. But we know the only change is from the illiterate group, which is Rs 300. So, N * 10 = 300. N = 300 / 10 = 30. Step 5: We know there are 20 illiterate employees, so literate employees x = N - 20. x = 30 - 20 = 10. Verification / Alternative Check: If there are 30 employees, and their overall average falls by Rs 10, the total reduction must be Rs 300. This matches the reduction caused by the 20 illiterate employees: 20 * 15 = 300. Therefore, the logic and the numbers are consistent. Why Other Options Are Wrong: 15 or 20: These values would give a different total N and would not produce a total reduction of exactly Rs 300 when multiplied by the decrease in the average. 25: This would imply too many employees, giving a total reduction far greater than Rs 300. Data insufficient: The data is sufficient because the change in average and the change in individual wages fully determine the total number of employees. Common Pitfalls: Some students try to set up two separate average equations involving actual wages of literate employees, which is unnecessary. Others forget that the change in average multiplied by the total number of workers equals the net change in total wages. Always convert average changes to total changes and then reason about counts. Final Answer: The number of educated employees working in the NGO is 10.

Question 8

A man covers a journey from Meerut to Delhi by car at an average speed of 40 km/h. He returns from Delhi to Meerut by scooter at an average speed of 24 km/h. What is his average speed for the entire round trip?

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Introduction: This is a classic average speed problem for a round trip where the distances are equal but the speeds are different in the two directions. It helps you understand why the average speed is not simply the arithmetic mean of the two speeds. Given Data / Assumptions: Speed from Meerut to Delhi = 40 km/h. Speed from Delhi to Meerut = 24 km/h. Distance between Meerut and Delhi is the same in both directions (let it be D km). We need the average speed for the entire journey (going and coming back). Concept / Approach: For a round trip with equal distances but different speeds, the average speed is given by the harmonic mean: Average speed = (2 * v1 * v2) / (v1 + v2). Alternatively, you can compute total distance and total time: Total distance = D (going) + D (coming) = 2D. Total time = D / 40 + D / 24. Then average speed = total distance / total time. Step-by-Step Solution: Step 1: Use the harmonic mean formula for equal distances. Let v1 = 40 km/h and v2 = 24 km/h. Average speed = (2 * v1 * v2) / (v1 + v2). Average speed = (2 * 40 * 24) / (40 + 24). Compute numerator: 2 * 40 * 24 = 1920. Compute denominator: 40 + 24 = 64. Average speed = 1920 / 64 = 30 km/h. Verification / Alternative Check: Assume the distance between the two cities is 120 km for easy numbers. Then: Time going = 120 / 40 = 3 hours. Time coming = 120 / 24 = 5 hours. Total distance = 240 km. Total time = 3 + 5 = 8 hours. Average speed = 240 / 8 = 30 km/h, confirming the result. Why Other Options Are Wrong: 32 km/h and 31 km/h: These are bigger than the computed average and closer to the higher speed, which would only occur if most of the journey were at the higher speed. 27 km/h and 28 km/h: These are lower than 30 km/h and would mean more time spent at the slower speed than actually occurs. Common Pitfalls: A very common mistake is to take the simple average of the two speeds (40 and 24) and write (40 + 24) / 2 = 32 km/h. This is wrong because the time spent at each speed is different. When distances are equal, always use total distance over total time or the harmonic mean formula. Final Answer: The average speed for the whole journey is 30 km/h.

Question 9

In a school examination, the average score of the boys is 81 and the average score of the girls is 83. If the overall average score of all the students in the school is 81.8, what is the ratio of the number of girls to the number of boys who appeare…

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Introduction: This problem uses the concept of weighted averages to relate the average scores of boys and girls to the overall average score. From these averages, we are asked to determine the ratio of the number of girls to the number of boys. Given Data / Assumptions: Average score of boys = 81. Average score of girls = 83. Overall average score = 81.8. Let the number of boys be B and the number of girls be G. We need the ratio G : B. Concept / Approach: For two groups with different averages, the combined average is: Combined average = (Average of boys * B + Average of girls * G) / (B + G). We substitute the given averages and the overall average to form an equation in B and G, and then simplify to find the ratio G : B. Step-by-Step Solution: Step 1: Use the weighted average formula. (81 * B + 83 * G) / (B + G) = 81.8. Step 2: Cross multiply: 81B + 83G = 81.8(B + G). 81B + 83G = 81.8B + 81.8G. Step 3: Rearrange terms. 81B - 81.8B + 83G - 81.8G = 0. -0.8B + 1.2G = 0. Step 4: Simplify the equation. 1.2G = 0.8B. Divide both sides by 0.4: 3G = 2B. So, G / B = 2 / 3. Therefore, the ratio of the number of girls to the number of boys is 2 : 3. Verification / Alternative Check: Assume B = 3 and G = 2 (keeping the ratio 2 : 3). Total score of boys = 3 * 81 = 243. Total score of girls = 2 * 83 = 166. Combined total = 243 + 166 = 409. Total students = 5. Combined average = 409 / 5 = 81.8, which matches the given overall average. Why Other Options Are Wrong: 3 : 2 and 4 : 3: These imply more girls than boys relative to the given averages and do not yield an overall average of 81.8. 3 : 4 and 5 : 4: Substituting these ratios in the weighted average formula does not give 81.8 as the overall average. Common Pitfalls: Many students try to use the alligation rule without careful calculation or mix up the direction of the ratio. Others attempt to average the two averages directly, which is incorrect because the counts of boys and girls are different. Always start from the definition of weighted average and form a correct linear equation. Final Answer: The required ratio of the number of girls to the number of boys is 2 : 3.

Question 10

A student's average marks in five subjects is 82. The average marks in the first two subjects is 86.5, and the average marks in the last two subjects is 84. What are the marks obtained in the third subject?

Accepted Answer

Introduction: This is a straightforward average marks question where the overall average and the averages of certain groups of subjects are given. You are asked to find the marks in the remaining subject. Given Data / Assumptions: Average marks in five subjects = 82. Average marks in the first two subjects = 86.5. Average marks in the last two subjects = 84. Marks in the third subject = unknown (let this be x). Concept / Approach: The principle is to convert averages into totals, then use subtraction to isolate the unknown subject. If S1 and S2 are first two subjects, S3 is the third, and S4 and S5 are the last two, then: Total in all five = 5 * 82. Total in first two = 2 * 86.5. Total in last two = 2 * 84. Then S3 = Total in all five - Total in first two - Total in last two. Step-by-Step Solution: Step 1: Total marks in all five subjects = 5 * 82. Total in all five = 410. Step 2: Total marks in the first two subjects = 2 * 86.5 = 173. Step 3: Total marks in the last two subjects = 2 * 84 = 168. Step 4: Compute marks in the third subject. Marks in third subject = 410 - 173 - 168. Marks in third subject = 410 - 341 = 69. Verification / Alternative Check: Take example marks: let first two be 86.5 and 86.5 (sum 173), third be 69, and last two be 84 and 84 (sum 168). Then total = 173 + 69 + 168 = 410. Average = 410 / 5 = 82, which matches the given overall average. Why Other Options Are Wrong: 67, 71, 73, 75: None of these values satisfy the equation obtained from the total sum 410. Substituting them would result in an overall average different from 82. Common Pitfalls: Students sometimes confuse the direction of subtraction and may incorrectly add averages instead of totals. Another common mistake is to divide the remaining total by the wrong number of subjects. Here, the third subject is a single subject, so do not divide again after subtraction. Final Answer: The marks obtained in the third subject are 69.

Question 11

A batsman scores 85 runs in his 17th innings and thereby increases his batting average by 3 runs. What is his batting average after the 17th innings?

Accepted Answer

Introduction: This question uses the relationship between total runs, number of innings, and batting average in cricket. You are told how much the average increases after a particular innings, and you must work backwards to find the new average. Given Data / Assumptions: Number of innings before the latest one = 16. Runs scored in the 17th innings = 85. The batting average increases by 3 runs due to this innings. Let the old average (before the 17th innings) be A runs. New average after 17 innings = A + 3. Concept / Approach: Average runs per innings = Total runs / Number of innings. If the old average is A, then: Total runs in 16 innings = 16 * A. After scoring 85 runs in the 17th innings: New total runs = 16A + 85. The new average is (16A + 85) / 17, which is given to be A + 3. This leads to an equation in A that we can solve. Step-by-Step Solution: Step 1: Set up the equation for the new average. (16A + 85) / 17 = A + 3. Step 2: Cross multiply to eliminate the denominator. 16A + 85 = 17(A + 3). Step 3: Expand the right-hand side. 16A + 85 = 17A + 51. Step 4: Rearrange to solve for A. 85 - 51 = 17A - 16A. 34 = A. So, the old average is 34 runs. Step 5: New average = A + 3 = 34 + 3 = 37. Verification / Alternative Check: If the average before the 17th innings is 34, total runs in 16 innings = 16 * 34 = 544. After scoring 85 runs, total runs = 544 + 85 = 629. New average = 629 / 17 = 37, confirming the calculation. Why Other Options Are Wrong: 34: This is the old average, not the new one. 35 and 36: These do not correspond to a 3-run increase from any integer average that fits the given score of 85. 38: This would require a larger increase than 3 runs from the previous average. Common Pitfalls: Some students mistakenly assume that 85 is simply 3 more than the old average times the new number of innings, without setting up the correct equation. Others forget to multiply the new average by 17 innings when relating it to the total runs. Always start from the definition of average and write a clear equation. Final Answer: The batsman's batting average after the 17th innings is 37.

Question 12

Five boxes of bananas sell for Rs 30. At this rate, how many boxes of bananas can be bought for Rs 9?

Accepted Answer

Introduction: This is a simple unitary method question dealing with direct proportion between quantity and price. If five boxes of bananas cost Rs 30, you need to find how many boxes can be purchased for Rs 9 at the same rate. Given Data / Assumptions: Cost of 5 boxes of bananas = Rs 30. We assume the price per box is constant. We need to find the number of boxes that can be bought for Rs 9. Concept / Approach: In direct proportion, if cost increases, the number of boxes increases in the same ratio. The steps are: 1. Find the cost of one box. 2. Use that cost to find how many boxes can be purchased for Rs 9. Alternatively, directly use ratios between cost and number of boxes. Step-by-Step Solution: Step 1: Find price per box. 5 boxes cost Rs 30. Price per box = 30 / 5 = Rs 6. Step 2: Find boxes that can be bought for Rs 9. Number of boxes = 9 / 6. Number of boxes = 1.5. Verification / Alternative Check: Another way: Consider the ratio of costs and quantities. If 30 rupees gives 5 boxes, 9 rupees is 9 / 30 = 0.3 of the original cost. 0.3 of 5 boxes = 5 * 0.3 = 1.5 boxes, which matches our calculation. Why Other Options Are Wrong: 2 and 2.5 boxes: These would correspond to spending Rs 12 or Rs 15 at Rs 6 per box, not Rs 9. 1.25 and 1: These would cost Rs 7.50 and Rs 6 respectively, which do not match the given budget of Rs 9. Common Pitfalls: Some learners mistakenly divide 5 by 30 instead of 30 by 5 to get the price per box. Others may forget to maintain consistent units or approximate 9 / 6 incorrectly. Always compute the unit price carefully and then use it for the required amount. Final Answer: At the given rate, the man can buy 1.5 boxes of bananas for Rs 9.

Question 13

Ian has 14 boxes of paper and divides them evenly among 4 coworkers. How many boxes of paper does each coworker receive?

Accepted Answer

Introduction: This is a basic division and average distribution problem. Ian wants to distribute boxes of paper equally among four coworkers, and you must find the share of each coworker. Given Data / Assumptions: Total number of boxes of paper = 14. Number of coworkers = 4. The boxes are divided evenly, so each coworker gets the same number of boxes. Concept / Approach: Equal division means we simply divide the total number of boxes by the number of coworkers: Number of boxes per coworker = Total boxes / Number of coworkers. This is essentially finding an average or fair share. Step-by-Step Solution: Step 1: Use the division formula. Number of boxes per coworker = 14 / 4. Step 2: Perform the division. 14 / 4 = 3.5. This means each coworker receives 3.5 boxes of paper. Verification / Alternative Check: Multiply the per-person share by the number of coworkers: 3.5 * 4 = 14 boxes. This matches the original total number of boxes, so the distribution is correct and complete. Why Other Options Are Wrong: 2 or 2.5 boxes: 2 * 4 = 8 and 2.5 * 4 = 10. Both totals are less than 14 and leave boxes undistributed. 3 boxes: 3 * 4 = 12, leaving 2 boxes unaccounted for, so the division is not even. 4 boxes: 4 * 4 = 16 boxes, which exceeds the actual total of 14 boxes. Common Pitfalls: A typical mistake is to expect only whole numbers as answers and ignore fractional values. However, the question does not state that boxes cannot be divided, so 3.5 is acceptable. In many real-world contexts, boxes can be split into half-units or the question may be testing understanding of pure division. Final Answer: Each coworker receives 3.5 boxes of paper.

Question 14

The average price of 10 books is Rs 12. The average price of 8 of these books is Rs 11.75. Of the remaining two books, the price of one book is 60% more than the price of the other. What is the price of each of these two books?

Accepted Answer

Introduction: This question combines average calculation with percentage increase. You know the overall average price of 10 books, the average price of 8 of them, and that one of the remaining two books is 60% more expensive than the other. You must find the individual prices of the two remaining books. Given Data / Assumptions: Average price of 10 books = Rs 12. Average price of 8 of these books = Rs 11.75. Of the remaining two books, the price of one is 60% more than the price of the other. We need to find the prices of these two books. Concept / Approach: First, convert averages to totals: Total price of 10 books = 10 * 12. Total price of 8 books = 8 * 11.75. The sum of the prices of the remaining two books is the difference between these totals. Then use the condition that one price is 60% more than the other to set up a simple equation. Step-by-Step Solution: Step 1: Total price of all 10 books. Total = 10 * 12 = Rs 120. Step 2: Total price of 8 books. Total of 8 books = 8 * 11.75 = Rs 94. Step 3: Sum of prices of the remaining two books. Sum of two books = 120 - 94 = Rs 26. Step 4: Let the cheaper of the two books cost x rupees. Then the costlier book is 60% more than x, so its price = x + 0.60x = 1.6x. Step 5: Use the sum condition. x + 1.6x = 26. 2.6x = 26. x = 26 / 2.6 = 10. So the cheaper book costs Rs 10 and the costlier book costs 1.6 * 10 = Rs 16. Verification / Alternative Check: Check: The two prices are Rs 10 and Rs 16, sum is Rs 26, and 16 is indeed 60% more than 10. Also, total of all 10 books is 94 + 26 = 120, giving average 120 / 10 = 12, which matches the given data. Why Other Options Are Wrong: Rs 16 and Rs 12: Sum is Rs 28 and 16 is not 60% more than 12. Rs 23 and Rs 9: Sum is Rs 32, which does not match Rs 26. Rs 8 and Rs 7: Sum is Rs 15, far from Rs 26, and there is no 60% relation. Rs 9 and Rs 14.40: Sum is Rs 23.40, and again does not match the required total of Rs 26. Common Pitfalls: Some students mistakenly treat 60% more as 60% of the combined total instead of 60% of the smaller price. Others forget to first compute the sum of the two unknown prices from the average information. Always find the total, then use the percentage relation correctly. Final Answer: The prices of the two remaining books are Rs 10 and Rs 16.

Question 15

Third proportional to the numbers 10 and 20 is defined as the number x such that 10 : 20 = 20 : x. What is the value of x?

Accepted Answer

Introduction: This question tests your understanding of the concept of third proportional in ratio and proportion. Finding the third proportional is a very common topic in quantitative aptitude. Given Data / Assumptions: First number = 10. Second number = 20. Third proportional = x, such that 10 : 20 = 20 : x. Concept / Approach: If a, b and c are such that a : b = b : c, then c is called the third proportional to a and b. This ratio equality can be written as: a / b = b / c. From this we get: c = (b * b) / a. We apply this formula directly with a = 10 and b = 20. Step-by-Step Solution: Step 1: Write the proportion condition. 10 : 20 = 20 : x. Step 2: Convert to fraction form. 10 / 20 = 20 / x. Step 3: Solve for x by cross multiplication. 10 * x = 20 * 20. 10x = 400. x = 400 / 10 = 40. Verification / Alternative Check: Check the ratio: 10 : 20 = 1 : 2. For 20 : 40, we also get 1 : 2. The ratios match, confirming that 40 is indeed the correct third proportional to 10 and 20. Why Other Options Are Wrong: 30, 25, 20, 50: Substituting any of these values for x makes 20 : x different from 10 : 20. The ratios no longer match, so they cannot be third proportionals. Common Pitfalls: Some learners mistakenly take the arithmetic mean or the geometric mean instead of using the proportion definition. Others may incorrectly write a * c = b instead of a : b = b : c. Always recall that third proportional c satisfies c = b² / a. Final Answer: The third proportional to 10 and 20 is 40.

Question 16

Navaneet has three times as much money as Poonam, and Poonam has 25% more money than Hebah. If the average amount of money possessed by the three of them is Rs 350, how much money does Navaneet have?

Accepted Answer

Introduction: This question connects ratios, percentages and averages. Three people, Navaneet, Poonam and Hebah, have related amounts of money. Using these relationships and the given average, we must determine the exact amount that Navaneet has. Given Data / Assumptions: Let Hebah's money be H rupees. Poonam has 25% more than Hebah, so Poonam's money = H + 0.25H = 1.25H. Navaneet has three times as much money as Poonam, so Navaneet's money = 3 * 1.25H = 3.75H. Average money of the three is Rs 350. Concept / Approach: The average of three numbers is: Average = (Sum of the three amounts) / 3. We express all three amounts in terms of H, compute the sum, equate it to 3 * 350 and solve for H. Then we find Navaneet's money using the relation Navaneet = 3.75H. Step-by-Step Solution: Step 1: Express all amounts in terms of H. Hebah = H. Poonam = 1.25H. Navaneet = 3.75H. Step 2: Compute the sum of their amounts. Sum = H + 1.25H + 3.75H. Sum = (1 + 1.25 + 3.75)H = 6H. Step 3: Use the average condition. Average = 350, so (Sum) / 3 = 350. 6H / 3 = 350. 2H = 350. H = 350 / 2 = 175. Step 4: Find Navaneet's money. Navaneet = 3.75H = 3.75 * 175. Compute 3.75 * 175 = (375 * 175) / 100 = 656.25. So Navaneet has Rs 656.25. Verification / Alternative Check: Hebah = 175. Poonam = 1.25 * 175 = 218.75. Navaneet = 656.25. Sum = 175 + 218.75 + 656.25 = 1050. Average = 1050 / 3 = 350, which matches the given average. Why Other Options Are Wrong: Rs 175: This is Hebah's amount, not Navaneet's. Rs 220.75 and Rs 218.55: These are close to Poonam's figure but do not match the required multiple of 175. Rs 525: This would give a total and average inconsistent with Rs 350. Common Pitfalls: Students often misinterpret “25% more” as simply adding 25 instead of adding 25% of Hebah's amount. Another mistake is to attempt to work only with the averages without translating everything into a single variable. Always set one of the amounts as a base variable and express all others in terms of it. Final Answer: Navaneet has Rs 656.25.

Question 17

The average marks of a student in 7 subjects is 85. If the average marks in the 6 subjects other than English is 83, what are the marks obtained in English?

Accepted Answer

Introduction: This is another average marks question, where the overall average for all 7 subjects and the average for 6 of them (excluding English) are given. You are asked to find the marks in the remaining subject, English. Given Data / Assumptions: Average marks in 7 subjects = 85. Average marks in 6 subjects (excluding English) = 83. Marks in English = unknown. Concept / Approach: Again, we convert averages into total marks: Total marks in 7 subjects = 7 * 85. Total marks in the 6 subjects (excluding English) = 6 * 83. The marks in English are simply the difference between these two totals. Step-by-Step Solution: Step 1: Total marks in all 7 subjects. Total (7 subjects) = 7 * 85 = 595. Step 2: Total marks in the 6 subjects other than English. Total (6 subjects) = 6 * 83 = 498. Step 3: Marks in English. Marks in English = 595 - 498 = 97. Verification / Alternative Check: If the six subjects total 498 and English is 97, then overall total = 498 + 97 = 595. Average = 595 / 7 = 85, confirming that the calculation is correct. Why Other Options Are Wrong: 98, 92, 93, 89: None of these values, when added to 498, give a total of 595. Each would lead to a different overall average, not equal to 85. Common Pitfalls: Some students make mistakes by averaging the two averages or by dividing by 6 again after subtraction. Remember that the marks in the single remaining subject come directly from the difference between totals, without any further division. Final Answer: The marks obtained in English are 97.

Question 18

Find the average of the first 50 natural numbers (that is, the numbers from 1 to 50).

Accepted Answer

Introduction: This problem asks you to find the average of the first 50 natural numbers. It is a standard arithmetic progression question and demonstrates the formula for the average of a sequence of equally spaced numbers. Given Data / Assumptions: We are given the first 50 natural numbers: 1, 2, 3, ..., 50. We need to find the average of these numbers. Concept / Approach: The first 50 natural numbers form an arithmetic progression with: First term a = 1. Last term l = 50. For any arithmetic progression, the average (mean) of all terms is: Average = (First term + Last term) / 2. This is because the terms are symmetrically distributed around the midpoint. Step-by-Step Solution: Step 1: Identify the first and last terms. First term a = 1. Last term l = 50. Step 2: Use the formula for the average of an arithmetic progression. Average = (a + l) / 2. Average = (1 + 50) / 2. Average = 51 / 2 = 25.5. Verification / Alternative Check: If you pair the numbers from the beginning and end, each pair sums to 51: (1, 50), (2, 49), ..., (25, 26). There are 25 such pairs. The average of each pair is 51 / 2 = 25.5, which is also the average of the entire set. Why Other Options Are Wrong: 25 and 26: These values are close but not equal to the actual computed mean. 26.5 and 24.5: These are further away and do not match the symmetry of the sequence around 25.5. Common Pitfalls: Some students try to add all 50 numbers manually, which is time consuming and error prone. Others mistakenly average the first few and assume the same applies to the rest. Using the arithmetic progression formula is the most efficient and accurate way. Final Answer: The average of the first 50 natural numbers is 25.5.

Question 19

The average of five numbers is 27. If one number is excluded, the average of the remaining four numbers becomes 25. What is the value of the excluded number?

Accepted Answer

Introduction: This question checks your understanding of how removing one element from a set changes the average. You are given the average of five numbers and the average of four numbers obtained after excluding one number. You must find the excluded number. Given Data / Assumptions: Average of five numbers = 27. Average of four numbers (after excluding one number) = 25. Let the excluded number be x. Concept / Approach: We use the relation between average and total: Total of five numbers = 5 * 27. Total of remaining four numbers = 4 * 25. The excluded number is the difference between these two totals. Step-by-Step Solution: Step 1: Total sum of the original five numbers. Total (5 numbers) = 5 * 27 = 135. Step 2: Total sum of the remaining four numbers. Total (4 numbers) = 4 * 25 = 100. Step 3: Find the excluded number. Excluded number x = 135 - 100 = 35. Verification / Alternative Check: Imagine the original five numbers sum to 135. Removing 35 leaves 100, and 100 / 4 = 25, exactly matching the given new average. Thus, the calculation is correct. Why Other Options Are Wrong: 25: This would lead to a total of 110 for the remaining four numbers, which is inconsistent with the given average of 25. 45 or 55: These would produce too large a difference between the two totals and would not result in the correct averages. 30: This does not give the correct new total of 100 when subtracted from 135. Common Pitfalls: Some students mistakenly subtract the two averages (27 - 25) and think the result is the excluded number or part of it. The correct approach is to convert both averages to total sums first, then subtract to get the exact excluded value. Final Answer: The excluded number is 35.

Question 20

P, Q, R, S, T and U are six consecutive odd numbers whose average is 52. What is the product of the smallest and the largest numbers, that is, P and U?

Accepted Answer

Introduction / Context: This question combines the idea of averages with the pattern of consecutive odd numbers. Problems of this type appear frequently in quantitative aptitude tests because they check whether you understand how consecutive numbers are arranged around their average and how to use symmetry to compute unknown values quickly without listing everything blindly. Given Data / Assumptions: There are six consecutive odd numbers P, Q, R, S, T and U. The average of these six numbers is 52. The numbers are in increasing order, so P is the smallest and U is the largest. We need the product P * U. Concept / Approach: For consecutive numbers, the average lies at the centre of the sequence. For an even count of numbers, such as six, the average is exactly the mean of the third and fourth terms. Because the numbers are consecutive odds, they are symmetrically placed around the average. Once we know the sequence, we can easily identify the smallest and largest terms and then calculate their product. Step-by-Step Solution: Step 1: Understand the structure of six consecutive odd numbers. For six consecutive odd numbers with average 52, the numbers are equally spaced around 52. Step 2: Express the numbers using 52 as the centre. Because they are odd and equally spaced, the numbers will be 52 - 5, 52 - 3, 52 - 1, 52 + 1, 52 + 3, 52 + 5. So the actual sequence is 47, 49, 51, 53, 55 and 57. Step 3: Identify P and U. P is the smallest number, so P = 47. U is the largest number, so U = 57. Step 4: Compute the product of P and U. P * U = 47 * 57. Compute 47 * 57 = (50 - 3) * (50 + 7) = 50 * 50 + 50 * 7 - 3 * 50 - 3 * 7 = 2500 + 350 - 150 - 21 = 2679. Verification / Alternative Check: We can also compute 47 * 57 directly with standard multiplication and confirm that it equals 2679. Additionally, we can verify that the average of the six numbers (47, 49, 51, 53, 55, 57) is indeed 52 by adding them and dividing by 6. Their sum is 312, and 312 / 6 = 52, which matches the given condition. Why Other Options Are Wrong: Option 2212 corresponds to an incorrect pair of numbers or miscalculation of the product. Option 2154 may result from using incorrect spacing for the odd numbers or taking the wrong smallest and largest values. Option 2349 is another distractor that does not match the correct computation 47 * 57. Common Pitfalls: A common mistake is to assume that the average itself is one of the numbers when the count is even, without checking the structure. Another pitfall is to use consecutive even numbers instead of odd numbers or to miscalculate the spacing around the average. Errors in multiplication while finding the product can also lead to wrong answers even when the sequence has been identified correctly. Final Answer: The product of P and U is 2679.

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