Question 1

After one member of a five-member club is replaced by a new member, it is found that the average age of the five members is the same as it was 3 years ago. What is the difference (in years) between the age of the member who left and the age of the n…

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Introduction / Context: This is a classic average-age puzzle. There are five members in a club. After 3 years, one original member leaves and a new one joins. Surprisingly, the average age of the five members now is the same as it was 3 years ago. You must determine the difference between the ages of the old member who left and the new member who joined, using the idea of total age and average. Given Data / Assumptions: Number of members = 5. Three years pass between the two comparisons of average age. After 3 years, one old member leaves and a new member joins. The average age of the 5 members now equals the average age of the original 5 members 3 years ago. We need the difference between the age of the replaced member and the new member. Concept / Approach: Let the average age of the 5 members 3 years ago be A. Then the total age 3 years ago is 5A. After 3 years, if nobody left, each of the 5 members would be 3 years older, so the total age would increase by 5 * 3 = 15 years. However, one member leaves and a new member joins, and the resulting average returns to A. By carefully tracking the total ages, we can find the difference between the ages of the leaving and entering members. Step-by-Step Solution: Step 1: Let the average age 3 years ago be A years for the 5 members. Step 2: Total age 3 years ago = 5A. Step 3: After 3 years, if no one left, each person would gain 3 years, so total increase = 5 * 3 = 15 years. Thus, total age just before replacement = 5A + 15. Step 4: Let the age 3 years ago of the member who will later leave be a years. Then just before leaving, his age is a + 3. Step 5: After replacement, total age = (5A + 15) - (a + 3) + N, where N is the age of the new member at that time. Step 6: The new average age is given to be the same as 3 years ago, i.e. A. So, [(5A + 15) - (a + 3) + N] / 5 = A. Step 7: Multiply both sides by 5: (5A + 15) - (a + 3) + N = 5A. Simplify left side: 5A + 15 - a - 3 + N = 5A → 5A + 12 - a + N = 5A. Step 8: Cancel 5A on both sides to get 12 - a + N = 0 → N - a = -12. Thus, a - N = 12. The replaced member (old one) is 12 years older than the new member. Verification / Alternative check: The key result is that (old age now) - (new member age) = 12 years. This difference ensures that the extra 15 years gained by aging of the original group is offset in such a way that the total age comes back to 5A. Why Other Options Are Wrong: Options B (13), C (14), and D (15) do not follow from the balance of total ages when the averages are kept the same. Option E (9) also cannot satisfy the equation derived from the average condition. Common Pitfalls: Students sometimes incorrectly assume that the difference is 3 * number of members (i.e., 15) without doing the algebra. Forgetting that the average is compared across a 3-year gap is another common source of confusion. Final Answer: The difference between the ages of the replaced member and the new member is 12 years.

Question 2

In a class, one pupil's marks were wrongly entered as 83 instead of 63. Because of this single error, the average marks for the whole class increased by 0.5 marks. How many pupils are there in the class?

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Introduction / Context: This question involves a classic average error scenario. One student's marks were recorded incorrectly, and this mistake caused the class average to change. By comparing the original correct total and the incorrect total implied by the faulty entry, we can find the number of pupils in the class using a simple algebraic equation. Given Data / Assumptions: One pupil's correct marks = 63. Marks were wrongly entered as 83. Difference between incorrect and correct marks = 83 - 63 = 20. This error increased the average marks of the class by 0.5 marks. Let the number of pupils in the class be n. Concept / Approach: If the average increases by 0.5 marks for n pupils, then the total increase in marks must be: increase in total marks = n * 0.5. But we also know that the incorrect entry added 20 extra marks to the total. By equating these two expressions for the increase in total marks, we can solve for n. Step-by-Step Solution: Step 1: Extra marks added due to error = 83 - 63 = 20. Step 2: Because of this, the average for the whole class increased by 0.5 marks. Step 3: Let the number of pupils be n. Step 4: Increase in total marks due to rise in average = n * 0.5. Step 5: This increase must equal the extra 20 marks from the wrong entry. Therefore, n * 0.5 = 20. Step 6: So n = 20 / 0.5 = 20 * 2 = 40. Verification / Alternative check: If n = 40, then an increase of 0.5 in average means total marks increased by 40 * 0.5 = 20, which matches the extra marks due to the wrong entry. Thus, the calculation is consistent and confirms that n = 40. Why Other Options Are Wrong: If n were 45, the increase in total would be 45 * 0.5 = 22.5, which does not match the extra 20 marks. Similarly, for n = 39, the increase would be 19.5; for n = 37, it would be 18.5; for n = 50, it would be 25. None of these equal 20. Common Pitfalls: A frequent mistake is to divide 20 by 0.5 incorrectly (for example, getting 10 instead of 40). Another error is to forget to connect increase in average across the entire class with the total increase in marks. Final Answer: The number of pupils in the class is 40.

Question 3

The mean of 50 observations was calculated as 36. Later, it was discovered that one observation had been recorded as 23 instead of the correct value 48. What is the corrected mean of the 50 observations?

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Introduction / Context: This question tests your understanding of how a single incorrect observation affects the calculated mean and how to adjust the mean once the correct value is known. Instead of recomputing everything from scratch, we use the difference between the wrong value and the correct value to update the total and thus the mean efficiently. Given Data / Assumptions: Number of observations = 50. Initially computed mean = 36. One observation was wrongly taken as 23. The correct value of that observation is 48. We need to find the corrected mean. Concept / Approach: The mean is defined as: mean = total sum / number of observations. We first compute the total sum based on the incorrect mean. Then, we adjust the total by adding the difference between the correct value and the wrong value. Finally, we divide the corrected total by the same number of observations (50) to obtain the corrected mean. Step-by-Step Solution: Step 1: Incorrect total sum based on mean 36: Incorrect total = 50 * 36 = 1800. Step 2: The wrong value recorded was 23; the correct value is 48. Difference = correct value - wrong value = 48 - 23 = 25. Step 3: Corrected total sum = incorrect total + difference. Corrected total = 1800 + 25 = 1825. Step 4: Corrected mean = corrected total / number of observations. Corrected mean = 1825 / 50. Step 5: Compute 1825 / 50 = 36.5. Verification / Alternative check: If the single value had originally been 48 instead of 23, the total would indeed increase by 25, so the mean would rise by 25 / 50 = 0.5. So the corrected mean is 36 + 0.5 = 36.5, which matches our detailed calculation. Why Other Options Are Wrong: Option A (35) and Option C (40) both differ significantly from 36.5 and do not correspond to adding the correct adjustment of 25 to the total sum. Option D (42) and Option E (37) similarly do not reflect the small 0.5 increase in the mean that results from the extra 25 marks spread over 50 observations. Common Pitfalls: Students might subtract instead of add the difference, leading to 1800 - 25, which is incorrect. Another frequent mistake is to divide the difference 25 by the wrong number of observations, such as 49 instead of 50. Final Answer: The corrected mean of the 50 observations is 36.5.

Question 4

The average of four numbers is 60. The first number is equal to one-fourth of the sum of the other three numbers. What is the value of the first number?

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Introduction / Context: This question involves both the concept of average and a special relationship among four numbers. You are told that their average is 60 and that the first number is one-fourth of the sum of the remaining three. Using these two conditions, you must find the value of the first number by setting up simple algebraic equations. Given Data / Assumptions: There are four numbers: let them be a, b, c and d. The average of the four numbers is 60. The first number a is one-fourth of the sum of the last three numbers, i.e. a = (b + c + d) / 4. We need to find a. Concept / Approach: From the average, we find the total sum of all four numbers. Then we translate the condition relating a to b, c and d into an equation. Combining the two equations allows us to solve for a directly. The key step is to express the sum of the last three numbers in terms of a and the total. Step-by-Step Solution: Step 1: Average of four numbers = 60. So total sum S = 4 * 60 = 240. Step 2: Given that a = (b + c + d) / 4. This implies b + c + d = 4a. Step 3: But we know that a + b + c + d = 240. Substitute b + c + d = 4a into this: a + (4a) = 240 → 5a = 240. Step 4: Solve for a: a = 240 / 5 = 48. Verification / Alternative check: If a = 48, then b + c + d = 4a = 4 * 48 = 192. Total sum = a + b + c + d = 48 + 192 = 240. Average = 240 / 4 = 60, which matches the given average. Also, a is indeed one-fourth of the sum of the last three numbers, since (b + c + d) / 4 = 192 / 4 = 48 = a. Why Other Options Are Wrong: Option A (17), B (29), C (36), and E (60) do not satisfy both conditions simultaneously. For example, if a were 60, then b + c + d = 4 * 60 = 240, making total 300 and the average 75, not 60. Common Pitfalls: Some students mistakenly assume that a is directly equal to the average, which is not given. Others misinterpret the condition and write a = 4(b + c + d) instead of (b + c + d) / 4. Final Answer: The first number is 48.

Question 5

Ten years ago, the average age of a family of 4 members was 24 years. Since then, two children have been born in the family, and the age difference between the two children is 2 years. If the present average age of the family (all 6 members) is stil…

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Introduction / Context: This problem is about average age in a family over time. Ten years ago there were 4 members, and their average age was 24 years. Over the next 10 years, two children were born, with a 2-year age difference between them. Interestingly, the current average age remains the same. Using this information, we determine the current age of the youngest child by focusing on total age changes over time. Given Data / Assumptions: Number of family members 10 years ago = 4. Average age 10 years ago = 24 years. Present number of family members = 6 (original 4 plus 2 children). Present average age of all 6 members = 24 years. Age difference between the two children = 2 years. We need to find the present age of the younger child. Concept / Approach: Average age times number of persons gives total age. Over 10 years, each of the original 4 members ages by 10 years, adding a fixed total of 4 * 10 years to their combined age. The two children contribute additional age. Because the average stays the same, we can equate the total present age to a fixed value and then compute how much age belongs to the children. From there, we use the age difference to find each child's age. Step-by-Step Solution: Step 1: Ten years ago, total age of 4 members = 4 * 24 = 96 years. Step 2: After 10 years, each of the original 4 members gains 10 years. So increase for these 4 = 4 * 10 = 40 years. Total present age of original 4 members = 96 + 40 = 136 years. Step 3: Currently there are 6 family members, and the present average age is still 24. Therefore, total present age of all 6 members = 6 * 24 = 144 years. Step 4: Combined age of the two children now = total present age - present age of original 4. Children's total age = 144 - 136 = 8 years. Step 5: Let the present age of the younger child be x years. Then the older child is x + 2 years (2-year age difference). Step 6: Sum of their ages = x + (x + 2) = 2x + 2. We know 2x + 2 = 8, so 2x = 6, x = 3. Verification / Alternative check: If the younger child is 3, the older is 5. Their total age is 3 + 5 = 8, which matches the previous calculation. Thus, the total present age of the family is 136 + 8 = 144, and the average is 144 / 6 = 24, as required. Why Other Options Are Wrong: If the younger child were 2, the older would be 4, total = 6, which is too small compared to the required 8. If the younger child were 1 or 4 or 5, the sums would be 1 + 3 = 4, 4 + 6 = 10, 5 + 7 = 12 respectively, none of which equal 8. Common Pitfalls: Forgetting that the average remained constant and not using it to compute current total age can lead to unnecessary algebra. Another common mistake is mixing up years ago with present ages, which can shift totals incorrectly. Final Answer: The present age of the youngest child is 3 years.

Question 6

Eight out of nine persons each spend Rs 30 on their meals. The ninth person spends Rs 20 more than the average expenditure of all the nine persons. What is the total amount of money spent by all nine persons together?

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Introduction / Context: This word problem mixes average expenditure with a condition involving one person spending more than that average. You are told that eight people spend the same amount, and the ninth person spends Rs 20 more than the overall average. The task is to determine the total expenditure of all nine people by setting up and solving an equation involving the average. Given Data / Assumptions: Number of persons = 9. Eight persons each spend Rs 30. Ninth person spends (average expenditure of all nine + Rs 20). Let the average expenditure be A rupees. We need to find the total expenditure of all nine persons. Concept / Approach: Let A be the average spending per person. Then the ninth person spends A + 20 rupees. The total spending is thus: 8 * 30 + (A + 20). But the total can also be written as 9A because the average is A over 9 people. Equating these two expressions for total expenditure allows us to solve for A and then compute the total amount. Step-by-Step Solution: Step 1: Let average expenditure per person = A rupees. Step 2: Expenditure of 8 persons = 8 * 30 = Rs 240. Step 3: Expenditure of 9th person = A + 20. Step 4: Total expenditure = 240 + (A + 20) = 260 + A. Step 5: But by definition of average, total expenditure also equals 9A. So, 260 + A = 9A. Step 6: Rearranging: 260 = 9A - A = 8A. Step 7: Therefore, A = 260 / 8 = 32.5 rupees. Step 8: Total expenditure = 9A = 9 * 32.5 = 292.5 rupees. Verification / Alternative check: Check the ninth person's expenditure: A + 20 = 32.5 + 20 = 52.5 rupees. Total expenditure = 8 * 30 + 52.5 = 240 + 52.5 = 292.5 rupees. Average = 292.5 / 9 = 32.5 rupees, which matches A and satisfies the condition that the ninth person spends Rs 20 more than the average. Why Other Options Are Wrong: Options B (Rs. 297.50), C (Rs. 298) and D (Rs. 298.50) correspond to averages different from Rs 32.5 and do not satisfy the relationship that the ninth person spends A + 20. Option E (Rs. 300) would give an average of 300 / 9 ≈ 33.33, so the ninth person would need to spend approximately Rs 53.33, not Rs 20 more than 33.33 exactly. Common Pitfalls: Some students treat Rs 20 as added to one of the known 30 rupee amounts instead of to the average, which changes the equation incorrectly. Others forget to express the total as 9A and instead equate the wrong expressions. Final Answer: The total money spent by all nine persons is Rs. 292.50.

Question 7

In an examination, a student's average mark is 63 per paper. If he had obtained 20 more marks in Geography and 2 more marks in History, his average mark per paper would have become 65. How many papers did he take in the examination?

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Introduction / Context: This average-based problem involves a student who has a certain average mark per paper. When we hypothetically add extra marks to two specific subjects (Geography and History), his average would increase. The key idea is to understand how changes in total marks relate to changes in average marks, allowing us to find the number of papers in the exam. Given Data / Assumptions: Original average mark = 63 marks per paper. He took n papers (we need to find n). If he had scored 20 more marks in Geography and 2 more marks in History, his average would become 65 marks per paper. Total additional marks in that hypothetical scenario = 20 + 2 = 22 marks. Concept / Approach: Let n be the number of papers. The original total marks are 63 * n. If we increase his marks by 22 in total, the new total would be 63 * n + 22. We are told that this new total, when divided by n, would give a new average of 65 marks. This gives one linear equation in n, which we can solve easily. Step-by-Step Solution: Step 1: Let the number of papers be n. Step 2: Original total marks = 63 * n. Step 3: Additional marks (20 in Geography + 2 in History) = 22 marks. Step 4: Hypothetical new total marks = 63 * n + 22. Step 5: New average is given to be 65 marks per paper. Therefore, (63 * n + 22) / n = 65. Step 6: Multiply both sides by n: 63n + 22 = 65n. Step 7: Rearrange: 22 = 65n - 63n = 2n. Step 8: So n = 22 / 2 = 11. Verification / Alternative check: If n = 11, original total marks = 63 * 11 = 693. Hypothetical new total = 693 + 22 = 715. New average = 715 / 11 = 65 marks per paper, which matches the condition. Why Other Options Are Wrong: If n were 8, 9, 10, or 12, the equation (63n + 22) / n = 65 would not hold. For example, if n = 10, new average = (630 + 22) / 10 = 652 / 10 = 65.2, not exactly 65. Common Pitfalls: Some students mistakenly divide the extra 22 marks directly by the difference in averages (2) and stop, not realizing that this effectively already assumes the correct approach; however, they may misinterpret the result as additional papers instead of the total number of papers. Others forget that the average change applies across all papers, not just the two with extra marks. Final Answer: The student appeared in 11 papers in the examination.

Question 8

The average of 11 numbers is 10.9. If the average of the first six numbers is 10.5 and the average of the last six numbers is 11.4, then what is the value of the middle number in this list?

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Introduction / Context:This question checks a very common aptitude concept involving averages, overlapping groups, and how a single middle value affects two different partial averages. Such problems are popular in competitive exams because they require candidates to combine the average formula with a little logical thinking about how sums are counted more than once. Given Data / Assumptions: There are 11 numbers in total. The average of all 11 numbers is 10.9. The average of the first 6 numbers is 10.5. The average of the last 6 numbers is 11.4. The middle number is counted in both the first 6 and the last 6 numbers because of overlap. Concept / Approach:The key concepts are: average = sum / number of items and the idea that when two overlapping groups share a common element, that element is counted twice if we add the sums of those groups. We can use this to form a simple equation and solve for the middle number. This technique is widely used in problems about consecutive terms, marks of students, and daily temperature lists. Step-by-Step Solution:Let the 11 numbers be N1, N2, ..., N11 with N6 as the middle number.Total sum of all 11 numbers = average * count = 10.9 * 11 = 119.9.Sum of first 6 numbers = 10.5 * 6 = 63.Sum of last 6 numbers = 11.4 * 6 = 68.4.When we add the sums of the first 6 and the last 6, the middle number N6 is included twice.So, 63 + 68.4 = (sum of all 11 numbers) + N6.Therefore, 131.4 = 119.9 + N6.N6 = 131.4 - 119.9 = 11.5. Verification / Alternative check:We can quickly check: suppose the total sum is 119.9 and the middle number is 11.5. Then the sum of the other 10 numbers is 119.9 - 11.5 = 108.4. The first 5 and last 5 plus the middle should still match the given averages. The calculations we did are consistent and follow the standard overlapping averages method, so the answer is reliable. Why Other Options Are Wrong:Values like 9.5 or 10.5 would be too small and would reduce the combined overlap sum, making the derived averages incorrect. Numbers like 12.5 and 13.5 are too large and would make the sums of the first six or last six inconsistent with 10.5 and 11.4 respectively. Only 11.5 keeps all three averages valid at the same time. Common Pitfalls:A frequent mistake is to treat the two sets of six numbers as disjoint and forget that the middle number is common to both sets. Another common error is to average the two averages directly, which does not respect the overlap structure. Careless multiplication or subtraction can also lead to wrong answers even with the correct method. Writing the equation for sums clearly usually avoids these errors. Final Answer:The value of the middle number is 11.5.

Question 9

The average age of a husband and his wife was 23 years at the time of their marriage. After five years they have a one-year-old child. What is the current average age of the family of three members?

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Introduction / Context:This problem tests your understanding of average age and how averages change when time passes and new members join a group. Questions involving families, children, and changes in average age are a staple of aptitude exams, because they check whether you can convert statements about averages into total sums of ages and then back into a new average. Given Data / Assumptions: Initially, only the husband and wife are considered. Their average age at marriage is 23 years. After five years, both spouses are 5 years older. They now have a one-year-old child. We must find the present average age of the three family members. Concept / Approach:The key idea is that average = total sum of ages / number of persons. When time passes, every existing person's age increases by the same amount, so the total sum increases accordingly. When a new member is added, the total sum gets that person's age added and the number of persons increases by one. We work through totals instead of trying to manipulate averages directly. Step-by-Step Solution:At marriage, there are 2 people with average age 23 years.Total age then = 23 * 2 = 46 years.After five years, each of the two adults becomes 5 years older.Increase in total age due to time = 2 * 5 = 10 years.New total age of husband and wife = 46 + 10 = 56 years.They now have a child aged 1 year.Total age of all three family members = 56 + 1 = 57 years.Number of family members now = 3.Average age now = total age / number of members = 57 / 3 = 19 years. Verification / Alternative check:As a quick check, imagine the current ages. Let the husband and wife both have ages that average 28 years after five years, because 23 + 5 = 28. So together they total 56 years, and with a one-year-old child the total is 57 years. Dividing 57 by 3 again gives 19. This confirms that the calculation and reasoning are consistent. Why Other Options Are Wrong:Options like 25 or 23 years ignore the effect of adding the child or treat the situation as if only two people were still being averaged. Options 18 and 21 years are just approximate guesses without a proper sum-based calculation. Only 19 years fits the exact arithmetic derived from the given information. Common Pitfalls:Students often forget that the number of individuals changes from two to three and still divide by two. Others simply add 5 years to 23 and assume that is the new average without accounting for the child's much lower age. Always convert averages into totals, adjust for time and new members, and then convert back to an average. Final Answer:The current average age of the family is 19 years.

Question 10

The captain of a cricket team of 11 members is 26 years old and the wicket keeper is 29 years old. If the ages of these two players are excluded, the average age of the remaining 9 players is one year less than the average age of the whole team. Wha…

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Introduction / Context:This question involves average age in a sports team, a very common pattern in aptitude tests. The twist is that the removal of two members changes the average of the remaining players in a known way. The goal is to convert the information about averages into algebraic equations involving total ages, and then solve for the unknown team average. Given Data / Assumptions: Total number of team members = 11. Age of the captain = 26 years. Age of the wicket keeper = 29 years. Average age of remaining 9 players is one year less than the average age of the whole team. We need the average age of all 11 players. Concept / Approach:When you know an average, you can always write total = average * number of items. Here, let the unknown team average be A. Then the total age of all 11 players is 11A. We know the ages of two specific players, so we can subtract them to get the total age of the remaining 9 players. Using the fact that the remaining players have an average of A - 1, we can form an equation and solve for A. Step-by-Step Solution:Let the average age of the whole team be A years.Total age of all 11 players = 11 * A.Ages of captain and wicket keeper together = 26 + 29 = 55 years.Total age of the remaining 9 players = 11A - 55.Average age of these 9 players is given as (A - 1) years.So, (11A - 55) / 9 = A - 1.Multiply both sides by 9: 11A - 55 = 9A - 9.Bring like terms together: 11A - 9A = -9 + 55, so 2A = 46.Therefore, A = 23 years. Verification / Alternative check:If the average is 23, total age of 11 players = 11 * 23 = 253. After removing the captain and wicket keeper, remaining total = 253 - 55 = 198. Average of 9 players = 198 / 9 = 22, which is exactly one year less than 23. This matches the statement in the problem, so the solution is correct. Why Other Options Are Wrong:At 20 or 21 years, the numbers do not satisfy the relationship that removing two players reduces the average by exactly one year. At 25 years, the remaining players would not average 24 years. Only 23 years gives a consistent solution for all conditions given in the question. Common Pitfalls:Common errors include mixing up the number of players (using 10 instead of 9 for the remaining group), or assuming the difference in ages of the removed players directly affects the change in average. The correct method is always to set up total sums using the average formula and then solve the resulting linear equation carefully. Final Answer:The average age of the entire team is 23 years.

Question 11

The average temperature of a town on the first four days of a certain month was 58 degrees. The average for the second, third, fourth and fifth days was 60 degrees. If the temperatures on the first and fifth days were in the ratio 7 : 8, then what w…

Accepted Answer

Introduction / Context:This average temperature problem checks your ability to work with overlapping averages and to use a given ratio between two values. The structure is similar to many exam questions where the averages of two overlapping groups are known and one or more individual values have to be determined, often with the help of a ratio condition. Given Data / Assumptions: Average temperature of days 1, 2, 3 and 4 is 58 degrees. Average temperature of days 2, 3, 4 and 5 is 60 degrees. Temperature on day 1 and day 5 are in the ratio 7 : 8. We need to find the temperature on the fifth day. Concept / Approach:We convert averages to total sums, use the overlap of days 2, 3 and 4, and express the temperatures on day 1 and day 5 using a ratio. The key idea is that sums of overlapping groups differ only by the values at the ends. Using the ratio, we can express one temperature in terms of the other and then solve the equations to find the actual values. Step-by-Step Solution:Let the temperatures on days 1, 2, 3, 4, 5 be T1, T2, T3, T4, T5.Given average for days 1 to 4 is 58, so T1 + T2 + T3 + T4 = 58 * 4 = 232.Given average for days 2 to 5 is 60, so T2 + T3 + T4 + T5 = 60 * 4 = 240.Subtract the first equation from the second: (T2 + T3 + T4 + T5) - (T1 + T2 + T3 + T4) = 240 - 232.This simplifies to T5 - T1 = 8.We are also told T1 : T5 = 7 : 8, so we can write T1 = 7k and T5 = 8k for some k.Then T5 - T1 = 8k - 7k = k, but we already know this difference is 8.So k = 8, therefore T1 = 7 * 8 = 56 and T5 = 8 * 8 = 64.Hence the temperature on the fifth day is 64 degrees. Verification / Alternative check:We can quickly verify by computing sums again. If T1 = 56 and T5 = 64, then T5 - T1 = 8, matching the overlap difference. Using T1 + T2 + T3 + T4 = 232, we get T2 + T3 + T4 = 176. Then T2 + T3 + T4 + T5 = 176 + 64 = 240, whose average is 240 / 4 = 60. Everything is consistent. Why Other Options Are Wrong:Values like 62, 65 or 66 degrees do not satisfy both the overlap condition T5 - T1 = 8 and the 7 : 8 ratio simultaneously. If you plug those values into the equations, you will get inconsistent or impossible values for T1. Only 64 degrees works with both given constraints. Common Pitfalls:A typical mistake is to ignore the ratio and assume T1 and T5 are equally spaced around some average, or to average 58 and 60 directly. Another error is to mis-handle the subtraction of equations, leading to T1 - T5 instead of T5 - T1. Writing the two sum equations and subtracting carefully is the safest way to avoid such mistakes. Final Answer:The temperature on the fifth day was 64 degrees.

Question 12

The average monthly salary of all the employees in a firm is Rs 60. Among them, 12 officers receive an average salary of Rs 400 per month and the remaining employees receive an average salary of Rs 56 per month. Find the total number of employees wo…

Accepted Answer

Introduction / Context:This is a classic weighted average salary problem where a group is divided into two subgroups with different averages. The overall average is known, and the average of each subgroup is known. The question asks for the total number of employees, which requires setting up an equation based on total salary paid to all employees. Given Data / Assumptions: The average salary of all employees is Rs 60 per month. There are 12 officers. The average salary of the 12 officers is Rs 400 per month. The remaining employees (non-officers) have an average salary of Rs 56 per month. We must find the total number of employees in the firm. Concept / Approach:We use the idea of weighted averages: overall average = (total salary of all employees) / (total number of employees). If the total number of employees is N, then the total salary can be expressed in terms of N and the given subgroup averages. By equating this to the expression derived from the overall average, we get a linear equation in N that we can solve. Step-by-Step Solution:Let the total number of employees be N.Overall average salary = Rs 60, so total salary of all employees = 60 * N.We know there are 12 officers with an average salary of Rs 400.Total salary of officers = 12 * 400 = Rs 4800.The remaining employees are (N - 12) in number, each earning on average Rs 56.Total salary of remaining employees = (N - 12) * 56.Total salary of all employees can also be written as 4800 + 56 * (N - 12).Equate both expressions for total salary: 60N = 4800 + 56(N - 12).Expand the right side: 60N = 4800 + 56N - 672.So 60N = 56N + 4128, hence 4N = 4128.Therefore, N = 4128 / 4 = 1032 employees. Verification / Alternative check:If there are 1032 employees, 12 are officers and 1020 are non-officers. Salary of officers = 12 * 400 = Rs 4800. Salary of non-officers = 1020 * 56 = Rs 57120. Total salary = 4800 + 57120 = Rs 61920. Average salary = 61920 / 1032 = Rs 60, exactly as given. Hence, the calculation is correct. Why Other Options Are Wrong:If you try 1030, 1060 or any other option, the recomputed overall average will differ from Rs 60. For example, a smaller N would make the overall average higher than 60 because the officers get a much larger salary. A larger N would pull the overall average closer to Rs 56. Only N = 1032 satisfies the weighted average condition. Common Pitfalls:Common mistakes include failing to subtract 12 when calculating the number of non-officers, or trying to average 400 and 56 directly without using proper weights. Always express total salary in two ways (overall and by groups) and then equate them to derive a correct equation. Final Answer:The total number of employees in the firm is 1032.

Question 13

The average salary of all workers in a workshop is Rs 8000 per month. The average salary of 7 technicians is Rs 12000 per month and the average salary of the remaining workers is Rs 6000 per month. What is the total number of workers in the workshop?

Accepted Answer

Introduction / Context:This question is another example of a weighted average problem with two subgroups: technicians and other workers. You are given the average salary of each subgroup and the overall average salary. Your job is to determine how many workers there are in total. This type of question is very common in quantitative aptitude tests involving mixtures, groups, and weighted averages. Given Data / Assumptions: The average salary of all workers is Rs 8000 per month. There are 7 technicians. The average salary of the 7 technicians is Rs 12000 per month. The average salary of the remaining workers is Rs 6000 per month. We must find the total number of workers in the workshop. Concept / Approach:We again use the weighted average idea. If the total number of workers is N, then the overall total salary is 8000 * N. The total salary can also be expressed as the sum of the total salary of technicians plus the total salary of the remaining workers. Equating these two forms allows us to solve for N. Step-by-Step Solution:Let the total number of workers be N.Overall average salary is Rs 8000, so total salary = 8000 * N.There are 7 technicians with an average salary of Rs 12000.Total salary of technicians = 7 * 12000 = Rs 84000.The remaining workers are (N - 7) in number, with average salary Rs 6000.Total salary of remaining workers = (N - 7) * 6000.Total salary of all workers can also be written as 84000 + 6000 * (N - 7).Equate both expressions: 8000N = 84000 + 6000(N - 7).Expand: 8000N = 84000 + 6000N - 42000.Simplify: 8000N = 6000N + 42000, so 2000N = 42000.Therefore, N = 42000 / 2000 = 21 workers. Verification / Alternative check:If there are 21 workers, 7 are technicians and 14 are other workers. Salary of technicians = 7 * 12000 = Rs 84000. Salary of remaining 14 workers = 14 * 6000 = Rs 84000. Total salary = 84000 + 84000 = Rs 168000. Average salary = 168000 / 21 = Rs 8000, which matches the given overall average. Why Other Options Are Wrong:If N were 20 or 24, the overall average would not be Rs 8000. Too few workers or too many workers would skew the weighted average. For example, with fewer low-salary workers, the average would be closer to Rs 12000, and with more, closer to Rs 6000. Only 21 produces the correct balance to maintain an overall average of Rs 8000. Common Pitfalls:Some learners mistakenly average 12000 and 6000 to get 9000, not noticing that the group sizes are different. Others forget to subtract 7 when determining how many workers are not technicians. Always use total salary expressions and equate them according to the weighted average formula. Final Answer:The total number of workers in the workshop is 21.

Question 14

A team of 8 persons takes part in a shooting competition. The best marksman scored 85 points. If he had scored 92 points instead, the average score of the team would have been 84 points. What was the total number of points actually scored by the tea…

Accepted Answer

Introduction / Context:This question deals with averages and shows how changing one person's score affects the overall average of a group. It is a standard type of aptitude problem where you must think in terms of total points rather than only averages. Here, the score of the best marksman is hypothetically increased, and we are told how the team average would change, from which we can work backwards. Given Data / Assumptions: There are 8 persons in the shooting team. The best marksman actually scored 85 points. If he had scored 92 points, the average score of the team would have been 84 points. We must find the actual total points scored by the team. Concept / Approach:The key idea is that average = total points / number of shooters. We do not know the actual average, but we know a hypothetical scenario: if one shooter scored 7 points more (92 instead of 85), then the team average would be 84. From that hypothetical average, we can find the hypothetical total, subtract the extra 7 points, and obtain the real total. Step-by-Step Solution:Number of shooters = 8.If the best marksman scored 92 points, the average of the team would be 84.Hypothetical total points = 84 * 8 = 672.Currently, he actually scores 85 points, which is 7 points less than 92.So the actual total is 672 - 7 = 665 points.Therefore, the total number of points that the team actually scored is 665. Verification / Alternative check:In the hypothetical case, extra points given to the best marksman are 7. That extra 7 points spread over 8 members increases the average by 7 / 8 = 0.875. If the final average is 84 in the hypothetical case, then the actual average must be 84 - 0.875 = 83.125. Multiplying 83.125 by 8 again gives 665 total points. This confirms that the earlier calculation is consistent. Why Other Options Are Wrong:Any total less than 665, such as 657 or 658, would correspond to a lower actual average and not give an exact increase of 7 points when moving to the hypothetical scenario. Similarly, a total like 678 or 672 does not match the description, since the hypothetical total was already calculated as 672. Only 665 correctly accounts for the 7 point difference between the real and hypothetical scores. Common Pitfalls:Some students wrongly assume that the current average is 84 or try to adjust the average directly without considering total points. Another typical mistake is to divide the 7 points by something other than 8 or to add rather than subtract it from the hypothetical total. Always remember to use total points = average * number of members and then account for the change step by step. Final Answer:The team actually scored a total of 665 points.

Question 15

The arithmetic mean (average) of seventy-five numbers is 35. If each of these numbers is increased by 5, then what will be the new mean of the resulting numbers?

Accepted Answer

Introduction / Context:This problem tests a very basic but powerful property of averages. When every number in a data set is increased or decreased by the same amount, the average also increases or decreases by that same amount. Recognizing this saves time and avoids unnecessary calculations in exams. Given Data / Assumptions: There are 75 numbers in total. The current arithmetic mean of these 75 numbers is 35. Each number is increased by 5. We must find the new arithmetic mean after the increase. Concept / Approach:The average of a set of numbers is defined as the total sum divided by the count of numbers. If each number in the set is increased by a constant value k, the total sum increases by k times the number of elements, but the count itself does not change. Therefore, the average also increases by exactly k. In this question, k is +5, so the new average is simply the old average plus 5. Step-by-Step Solution:Number of values = 75.Old mean = 35.Total sum of the original 75 numbers = 35 * 75.Each number is increased by 5, so the total increase in sum = 5 * 75.New total sum = old total sum + 5 * 75.New mean = (old total sum + 5 * 75) / 75.This simplifies to (35 * 75) / 75 + (5 * 75) / 75 = 35 + 5 = 40.Hence the new mean is 40. Verification / Alternative check:Instead of using sums, we can apply the direct property: adding the same constant to every term shifts the mean by that constant. Since each number increases by 5, the average must also increase by 5. Starting from 35, we get the new mean as 35 + 5 = 40. Both the formal and intuitive reasoning give the same result. Why Other Options Are Wrong:Options 30 and 35 would mean that the average has decreased or remained unchanged, which is impossible when all numbers increase by 5. Options 50 and 60 represent much larger increases in the mean and do not correspond to the constant increase of only 5 per number. Only 40 is consistent with the basic property of averages in this situation. Common Pitfalls:Some students try to recalculate the total sum explicitly, which is unnecessary and time consuming, but not wrong. Others mistakenly add 5 only once to the average of 35/75 or confuse the number of observations. Remember, when every value increases by k, the mean also increases by k, regardless of how many values there are. Final Answer:The new arithmetic mean of the numbers is 40.

Question 16

A factory employs 600 men and 400 women. The average wage of all 1000 workers together is Rs 25.50 per day. Each woman earns Rs 5 less per day than each man. What are the daily wages of a man and of a woman?

Accepted Answer

Introduction / Context:This question combines average wage, total workers, and a fixed difference between the wages of two groups. It is a standard application of the weighted average concept and linear equations. You must express the total wage paid to all workers in terms of the unknown man's wage and then use the given overall average to solve for that unknown. Given Data / Assumptions: Total workers = 600 men + 400 women = 1000 workers. Average wage of all workers together = Rs 25.50 per day. Each woman earns Rs 5 less per day than each man. We need to find the daily wage of a man and that of a woman. Concept / Approach:Let the daily wage of a man be m rupees. Then the daily wage of a woman is (m - 5) rupees. Using the average wage information, we know that the total daily wage bill equals (average wage) * (number of workers) = 25.5 * 1000. We also know that this total wage is the sum of wages paid to men plus the sum of wages paid to women. This gives an equation in terms of m that we can solve directly. Step-by-Step Solution:Let the daily wage of each man be m rupees.Then the daily wage of each woman is m - 5 rupees.Total daily wage paid to all workers = average wage * number of workers = 25.5 * 1000 = Rs 25500.Total wage paid to 600 men = 600 * m.Total wage paid to 400 women = 400 * (m - 5).So, 600m + 400(m - 5) = 25500.Simplify: 600m + 400m - 2000 = 25500, so 1000m - 2000 = 25500.Therefore, 1000m = 27500, and m = 27.5.Thus, the daily wage of a man is Rs 27.50 and that of a woman is m - 5 = 22.5 rupees. Verification / Alternative check:Check using totals: wage of a man = Rs 27.50, wage of a woman = Rs 22.50. Total wage to men = 600 * 27.5 = Rs 16500. Total wage to women = 400 * 22.5 = Rs 9000. Combined total = 16500 + 9000 = Rs 25500. Average wage = 25500 / 1000 = Rs 25.50, which matches the given average. Therefore, the solution is consistent. Why Other Options Are Wrong:Option a reverses the relationship and gives a woman more pay than a man, contradicting the statement that a woman earns Rs 5 less. Options c and d also do not preserve the fixed difference of Rs 5 in the correct direction or fail to produce an overall average of Rs 25.50 when weighted by 600 men and 400 women. Option e is arbitrarily large and does not satisfy the total wage and average constraints. Only option b matches all the conditions. Common Pitfalls:Errors can come from mixing up the direction of the Rs 5 difference or forgetting that there are 1000 workers, not 600 or 400 only. Some students also mistakenly average the two group wages without weighting them by their respective counts. Always set up the total wage equation carefully and solve step by step. Final Answer:The daily wage of a man is Rs 27.50 and the daily wage of a woman is Rs 22.50.

Question 17

The average age of all the students in a class is 15.8 years. The average age of the boys is 16.4 years and the average age of the girls is 15.4 years. What is the ratio of the number of boys to the number of girls in the class?

Accepted Answer

Introduction / Context:This is a classic mixture and average problem. Instead of liquids or salaries, we are dealing with ages of boys and girls. The overall class average lies between the average of boys and the average of girls. Using the concept of weighted averages and a simple equation, we can find the ratio of the number of boys to the number of girls. Given Data / Assumptions: The overall average age of all students = 15.8 years. The average age of boys = 16.4 years. The average age of girls = 15.4 years. We are asked to find the ratio of the number of boys to the number of girls. Concept / Approach:Let the number of boys be B and the number of girls be G. Then the total age of boys is 16.4B and the total age of girls is 15.4G. The overall average gives a relationship: (16.4B + 15.4G) / (B + G) = 15.8. This equation can be solved for the ratio B : G. Conceptually, the overall average is closer to the girls' average because there are slightly more girls than boys, which we will see in the final ratio. Step-by-Step Solution:Let the number of boys be B and the number of girls be G.Total age of boys = 16.4 * B.Total age of girls = 15.4 * G.Overall average age is 15.8, so (16.4B + 15.4G) / (B + G) = 15.8.Multiply both sides by (B + G): 16.4B + 15.4G = 15.8B + 15.8G.Rearrange terms: 16.4B - 15.8B = 15.8G - 15.4G.This gives 0.6B = 0.4G.So B / G = 0.4 / 0.6 = 2 / 3.Thus, the ratio of boys to girls is 2 : 3. Verification / Alternative check:We can test with small numbers in this ratio. Suppose B = 2 and G = 3. Then total boys' age = 2 * 16.4 = 32.8. Total girls' age = 3 * 15.4 = 46.2. Combined total age = 32.8 + 46.2 = 79. Average age = 79 / 5 = 15.8, which perfectly matches the given overall average. That confirms the correctness of the ratio 2 : 3. Why Other Options Are Wrong:Ratios such as 1 : 2 or 3 : 4 do not satisfy the equation derived from the averages. If we plug them in, the recalculated overall average will not be 15.8. The ratio must balance the contributions of the 16.4 and 15.4 ages to reach 15.8. Only 2 : 3 achieves that exact balance. Common Pitfalls:Students sometimes mis-handle decimals or try to use an alligation diagram but mix up which difference corresponds to boys or girls. Another error is to cross-multiply incorrectly or forget to divide both sides properly when solving 0.6B = 0.4G. Writing the equation clearly and simplifying step by step prevents these mistakes. Final Answer:The ratio of the number of boys to the number of girls is 2 : 3.

Question 18

The average temperature for Monday, Tuesday, Wednesday and Thursday was 48 degrees. The average for Tuesday, Wednesday, Thursday and Friday was 46 degrees. If the temperature on Monday was 42 degrees, what was the temperature on Friday?

Accepted Answer

Introduction / Context:This problem is another example of overlapping averages used to find an unknown value. It involves daily temperatures across two overlapping groups of four days. The first group includes Monday and the second group includes Friday. Using the difference between the two group sums and the known Monday temperature, we can determine Friday's temperature. Given Data / Assumptions: Average temperature for Monday, Tuesday, Wednesday, Thursday = 48 degrees. Average temperature for Tuesday, Wednesday, Thursday, Friday = 46 degrees. Temperature on Monday = 42 degrees. We must find the temperature on Friday. Concept / Approach:The approach is to convert averages to total sums over four days each, then subtract to find the difference between Friday and Monday. The key relationship is: (sum of second group) - (sum of first group) = Friday temperature minus Monday temperature, because the middle three days are common to both groups and cancel out. Step-by-Step Solution:Let the temperatures on Monday to Friday be M, T, W, Th, F.Average for Monday to Thursday is 48, so M + T + W + Th = 48 * 4 = 192.Average for Tuesday to Friday is 46, so T + W + Th + F = 46 * 4 = 184.Subtract the first equation from the second: (T + W + Th + F) - (M + T + W + Th) = 184 - 192.This simplifies to F - M = -8.Given M (Monday) = 42 degrees, so F - 42 = -8.Hence F = 42 - 8 = 34 degrees.Therefore, the temperature on Friday was 34 degrees. Verification / Alternative check:Check the sums explicitly. If Monday is 42 and Friday is 34, then F - M = 34 - 42 = -8, matching our equation. Let the sum of Tuesday, Wednesday and Thursday be S. Then M + S = 192 gives S = 192 - 42 = 150. Also, S + F must equal 184, and 150 + 34 = 184. Both conditions are satisfied, so the answer is consistent. Why Other Options Are Wrong:If you choose 36, 38 or 40 degrees for Friday, the difference F - M would not equal -8 and you would not obtain the given averages. Each incorrect value results in inconsistent sums for the two groups of days. Only Friday = 34 degrees keeps both averages correct. Common Pitfalls:Some exam takers attempt to average 48 and 46 directly or forget that the middle three days cancel when subtracting. Others accidentally compute M - F instead of F - M, which flips the sign and leads to a wrong answer. Carefully writing the equations and subtracting them in the correct order avoids these errors. Final Answer:The temperature on Friday was 34 degrees.

Question 19

A student was asked to find the arithmetic mean of the numbers 3, 11, 7, 9, 15, 13, 8, 19, 17, 21, 14 and x. He found that the mean of these twelve numbers is 12. What should be the value of x?

Accepted Answer

Introduction / Context:This question illustrates how to recover a missing number when the average of the entire set is known. Such problems appear frequently in aptitude tests, and the key is to work with total sums rather than trying to adjust averages directly. Once we know the total sum needed for a given mean, we can subtract the sum of known numbers to find the unknown value. Given Data / Assumptions: The list of numbers is 3, 11, 7, 9, 15, 13, 8, 19, 17, 21, 14 and x. The arithmetic mean of these 12 numbers is 12. We need to find the value of x that makes this mean correct. Concept / Approach:If the mean of 12 numbers is 12, then the total sum of those 12 numbers must be 12 * 12. We first compute the sum of the 11 known numbers. The difference between the required total sum and this known sum must equal x. This is a simple but very useful application of the definition of arithmetic mean. Step-by-Step Solution:We have 12 numbers in total, so required total sum = mean * count = 12 * 12 = 144.Now, find the sum of the known 11 numbers: 3 + 11 + 7 + 9 + 15 + 13 + 8 + 19 + 17 + 21 + 14.Compute this sum stepwise: 3 + 11 = 14, 14 + 7 = 21, 21 + 9 = 30, 30 + 15 = 45, 45 + 13 = 58, 58 + 8 = 66, 66 + 19 = 85, 85 + 17 = 102, 102 + 21 = 123, 123 + 14 = 137.So the sum of the 11 known numbers is 137.Since total required sum is 144, x must satisfy 137 + x = 144.Therefore, x = 144 - 137 = 7. Verification / Alternative check:Create the full set including x = 7: 3, 11, 7, 9, 15, 13, 8, 19, 17, 21, 14, 7. Their sum is 137 + 7 = 144. Divide by 12 to get the mean: 144 / 12 = 12. This matches the given mean, confirming that the calculation is correct. Why Other Options Are Wrong:If x were 6, 8, 9 or 10, the total sum would become 143, 145, 146 or 147 respectively, giving means of 143 / 12, 145 / 12, 146 / 12 or 147 / 12, none of which equals exactly 12. Only x = 7 produces the exact mean of 12. Common Pitfalls:Students sometimes average the known numbers first and then adjust that average, which can cause confusion. Others miscalculate the sum of the 11 numbers or forget that there are 12 numbers when multiplying the mean. Carefully computing the total required sum and subtracting the known sum is the fastest and most reliable method. Final Answer:The value that should replace x is 7.

Question 20

A library has on average 510 visitors on Sundays and 240 visitors on each of the other days. In a 30-day month that begins on a Sunday, what is the average number of visitors per day during the month?

Accepted Answer

Introduction / Context:This problem combines averages with a calendar structure. The number of Sundays in the month affects how many days have the higher visitor count. This is a typical exam question that checks whether you can count the number of specific weekdays in a month and then use total visitors to compute an overall average. Given Data / Assumptions: The average number of visitors on Sundays is 510. The average number of visitors on each non-Sunday is 240. The month has 30 days and starts on a Sunday. We need to find the average number of visitors per day for the entire month. Concept / Approach:Since the month starts on a Sunday and has 30 days, we must determine how many Sundays occur in that period. Once we know how many Sundays and non-Sundays there are, we can compute the total visitors in the month by multiplying the appropriate average for each type of day. Finally, we divide the total visitors by 30 to get the average per day. Step-by-Step Solution:A 30-day month that begins on Sunday has Sundays on days 1, 8, 15, 22 and 29.So there are 5 Sundays in the month.Number of non-Sunday days = 30 - 5 = 25.Average visitors on a Sunday = 510, so total Sunday visitors = 5 * 510 = 2550.Average visitors on a non-Sunday = 240, so total non-Sunday visitors = 25 * 240.Compute 25 * 240 = 6000.Total visitors in the month = 2550 + 6000 = 8550.Average visitors per day for the month = total visitors / total days = 8550 / 30 = 285. Verification / Alternative check:We can sanity check by noticing that the overall average should lie between 240 and 510. Since Sundays (with 510 visitors) are fewer than non-Sundays (with 240 visitors), the overall average should be much closer to 240 than to 510. The value 285 fits this expectation and matches the exact calculation. Why Other Options Are Wrong:Values like 290, 297, 305 or 275 are not equal to 8550 / 30. If you recompute total visitors using those averages, the implied total would not match 8550. Only 285 gives the correct total number of visitors and respects the tallies for both Sundays and other days. Common Pitfalls:Students sometimes miscount the number of Sundays, especially if they forget that the pattern repeats every 7 days. Others mistakenly assume 4 Sundays in every month regardless of the starting day. Errors can also occur when multiplying or adding the totals. To avoid mistakes, always list out the Sundays by date and then confirm that the total days add up correctly. Final Answer:The average number of visitors per day during the month is 285.

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