Question 1

A cricket player's average over 10 innings is 32 runs. How many runs must he score in the 11th innings to raise his average by 4 runs (to 36)?

Accepted Answer

Introduction / Context: Average = total / count. To raise an average after one more data point, compute the target new total and subtract the current total to find the required score. Given Data / Assumptions: Current average over 10 innings = 32 ⇒ current total = 10 * 32 = 320. Desired new average over 11 innings = 36 ⇒ target total = 11 * 36 = 396. Concept / Approach: Required runs in 11th = target total − current total. Step-by-Step Solution: Required = 396 − 320 = 76 Verification / Alternative check: After scoring 76, new total = 396; 396/11 = 36, as desired. Why Other Options Are Wrong: 79, 85, 87 yield averages of 36.18..., 36.64..., 36.91..., not exactly 36. Final Answer: 76

Question 2

The average of 20 numbers is 0. At most, how many of these numbers can be greater than 0?

Accepted Answer

Introduction / Context: An average of zero means the sum of all numbers is zero. We are asked for the maximum possible count of positive numbers under this constraint, with no restriction (other than arithmetic) on the negative numbers. This is a classic extremal question about sums. Given Data / Assumptions: There are 20 real numbers (no further restrictions). Their sum must be 0 (average 0). Concept / Approach: To maximize the count of positive numbers while keeping the sum at 0, we can take 19 of them to be any positive values and choose the remaining one number negative with magnitude equal to the sum of the 19 positives. This construction satisfies the sum constraint and uses as many positive entries as possible. Step-by-Step Solution: Let x1, x2, ..., x19 > 0 be arbitrarily chosen.Let x20 = −(x1 + x2 + ... + x19), which is ≤ 0.Then total sum = (x1 + ... + x19) + x20 = 0 ⇒ average = 0.Thus, we can have 19 positive numbers at most. Verification / Alternative check: If all 20 were positive, the sum could not be zero. Hence 20 is impossible; 19 is attainable by construction, so it is the maximum. Why Other Options Are Wrong: 0 or 1 are not maxima; much larger positive counts are possible. -1 is meaningless as a count of elements. Final Answer: none of these (the correct maximum is 19).

Question 3

A grocer records monthly sales of Rs. 6435, Rs. 6927, Rs. 6855, Rs. 7230 and Rs. 6562 for five consecutive months. What sales amount must he achieve in the sixth month so that his average monthly sales for all six months is Rs. 6500?

Accepted Answer

Introduction / Context: This aptitude question tests the concept of arithmetic mean applied to monthly sales. A grocer has sales figures for five months and wants to know what sales are required in the sixth month to reach a target average of Rs. 6500. Such questions are very common in banking, SSC and placement exams because they check the ability to reverse engineer an unknown value from a required average. Given Data / Assumptions: - Sales for the first five months: Rs. 6435, Rs. 6927, Rs. 6855, Rs. 7230 and Rs. 6562. - Total number of months to be considered: 6. - Required average monthly sale for all six months: Rs. 6500. - Let the sales in the sixth month be x rupees. Concept / Approach: The arithmetic mean or average of a set of values is defined as total sum of the values divided by the number of values. When the desired average and some of the values are known, we can find the missing value by first computing the required total sum and then subtracting the known sum. Here we will compute the total sales required for six months and then deduce the sixth month sales from it. Step-by-Step Solution: Step 1: Required total sales for six months = required average * number of months = 6500 * 6 = Rs. 39000.Step 2: Compute the sum of the known five months: 6435 + 6927 + 6855 + 7230 + 6562 = Rs. 34009.Step 3: Let the required sales in the sixth month be x. Then total sales for six months = 34009 + x.Step 4: Equate this to the required total sales: 34009 + x = 39000.Step 5: Solve for x: x = 39000 - 34009 = Rs. 4991. Verification / Alternative Check: If the grocer sells Rs. 4991 in the sixth month, the total of all six months becomes 34009 + 4991 = Rs. 39000. Dividing by 6 gives 39000 / 6 = Rs. 6500, which exactly matches the required average. This confirms that the computed value is correct and consistent with the definition of average. Why Other Options Are Wrong: Rs. 5467, Rs. 5987 and Rs. 6453 all produce a total that is greater than or less than Rs. 39000, giving an average different from Rs. 6500. For example, if the sixth month sales were Rs. 6453, the total would be 34009 + 6453 = 40462 and the average would be 40462 / 6 which is greater than Rs. 6500. Therefore these values do not satisfy the required average condition. Common Pitfalls: A common mistake is to average the given five months first and then try to adjust by intuition instead of using the formula. Another frequent error is to divide by 5 instead of 6 when calculating the required total or to forget that the target average applies to all six months together. Some learners also accidentally add 6500 instead of multiplying 6500 by 6. Careful use of the average formula avoids these issues. Final Answer: The grocer must have sales of Rs. 4991 in the sixth month to obtain an average monthly sale of Rs. 6500.

Question 4

The average weight of 8 persons increases by 2.5 kg when a new person replaces one of them who weighs 65 kg. What is the weight of the new person?

Accepted Answer

Introduction / Context: This question tests the concept of average and how it changes when one value in a group is replaced by another. It appears frequently in aptitude tests for bank PO, SSC, and campus placements. Understanding how the average shifts allows us to find the weight of the new person who joins the group of eight and causes a known increase in the average weight. Given Data / Assumptions: - Number of persons in the group = 8. - Weight of the person who leaves the group = 65 kg. - The new person joins in place of this person. - The average weight of the 8 persons increases by 2.5 kg after the replacement. - We need to find the weight of the new person in kilograms. Concept / Approach: When the average of a group changes due to replacing one member, the net change in total weight can be computed by multiplying the change in average by the number of members. That net change must be equal to the difference between the weight of the new person and the weight of the person who left. Using this relation, we can find the weight of the new person directly without first calculating the original average. Step-by-Step Solution: Step 1: Let the original average weight of the 8 persons be A kg.Step 2: Original total weight of the group = 8 * A.Step 3: After replacement, the new average becomes A + 2.5 kg, so the new total weight = 8 * (A + 2.5).Step 4: The new total weight is also equal to original total weight minus 65 kg plus weight of the new person, say W: 8 * A - 65 + W.Step 5: Equate both expressions: 8 * (A + 2.5) = 8 * A - 65 + W.Step 6: Expand left side: 8 * A + 8 * 2.5 = 8 * A - 65 + W.Step 7: Simplify: 8 * 2.5 = -65 + W, so 20 = W - 65.Step 8: Therefore W = 65 + 20 = 85 kg. Verification / Alternative Check: The increase in average is 2.5 kg across 8 persons, so the net increase in total weight is 2.5 * 8 = 20 kg. This net increase must come from the new person being heavier than the old one. Since the old person weighed 65 kg, the new person must weigh 65 + 20 = 85 kg. This quick method confirms our detailed algebraic solution. Why Other Options Are Wrong: A weight of 70 kg or 75 kg would increase the total weight by only 5 kg or 10 kg, which would increase the average by less than 2.5 kg. A weight of 80 kg would increase the total by 15 kg, giving an average increase of 15 / 8 which is less than 2.5 kg. Only 85 kg produces a total increase of 20 kg and therefore the correct average increase of 2.5 kg. Common Pitfalls: Many students mistakenly divide the increase in total weight by 7 instead of 8 or they incorrectly add 2.5 kg directly to 65 kg without considering the number of persons involved. Another mistake is to assume that the total number of persons changes, whereas in this problem the group size remains 8 because one person leaves and another person joins. Keeping track of the group size and using the relation change in total = change in average * number of terms avoids such errors. Final Answer: The weight of the new person is 85 kg.

Question 5

The average weight of a class of 24 students is 35 kg. When the teacher is included, the average becomes 35.4 kg (an increase of 400 g). What is the weight of the teacher?

Accepted Answer

Introduction / Context: This problem involves the concept of average weight when an additional person is added to a group. We are given the average weight of 24 students and the new average when the teacher joins them. From this information, we must determine the teacher weight. This type of question trains students to work comfortably with averages and total sums. Given Data / Assumptions: - Number of students = 24. - Average weight of students alone = 35 kg. - When the teacher is included, total persons = 25. - New average weight = 35.4 kg (which is 35 kg + 0.4 kg, that is 400 g increase). - We need to find the teacher weight in kilograms. Concept / Approach: The idea is that average multiplied by number of persons gives the total weight. We first compute the total weight of the 24 students. Then we compute the total combined weight of 24 students plus teacher using the new average. The difference between these two totals must be exactly the teacher weight. This is a direct application of average = total sum / number of terms. Step-by-Step Solution: Step 1: Total weight of the 24 students = 24 * 35 = 840 kg.Step 2: After adding the teacher, the number of persons is 25 and the new average is 35.4 kg.Step 3: Total weight of 24 students plus teacher = 25 * 35.4.Step 4: Compute this total: 25 * 35.4 = 885 kg.Step 5: Teacher weight = combined total weight minus weight of students = 885 - 840 = 45 kg. Verification / Alternative Check: We can confirm by checking the new average. If the teacher weighs 45 kg, the total weight is 840 + 45 = 885 kg for 25 persons. The average is then 885 / 25 = 35.4 kg, which matches the given new average. This confirms that 45 kg is correct. Why Other Options Are Wrong: If the teacher weighed 46 kg, the total would be 886 kg and the average would be 886 / 25 = 35.44 kg, not 35.4 kg. For 47 kg, the average would be even higher, and for 48 kg the deviation is larger. Any weight other than 45 kg fails to give the exact new average of 35.4 kg specified in the question statement. Common Pitfalls: A common error is to add 0.4 kg directly to 35 kg and assume that the teacher weight is 35.4 kg, which is incorrect because that value is an average, not an individual weight. Another mistake is to forget to change the number of persons from 24 to 25 after including the teacher. Students should always translate averages into total sums and then work backward to find missing values. Final Answer: The weight of the teacher is 45 kg.

Question 6

The mean temperature from Monday to Wednesday is 37 degrees Celsius and the mean temperature from Tuesday to Thursday is 34 degrees Celsius. If the temperature on Thursday is four fifths of the temperature on Monday, what is the temperature on Thurs…

Accepted Answer

Introduction / Context: This problem involves average temperatures over overlapping groups of days and a ratio relation between two specific days. It is a classic example of how averages and simple algebra combine to solve practical questions about weather data. Students need to translate average conditions into equations for the individual day temperatures. Given Data / Assumptions: - Average temperature for Monday, Tuesday and Wednesday = 37 degrees Celsius. - Average temperature for Tuesday, Wednesday and Thursday = 34 degrees Celsius. - Temperature on Thursday is four fifths of the temperature on Monday. - We need to find the temperature on Thursday in degrees Celsius. Concept / Approach: Let the temperatures on Monday, Tuesday, Wednesday and Thursday be M, T, W and Th. The given averages can be converted to equations involving sums of these variables. Subtracting these equations allows us to relate Monday and Thursday directly. The ratio condition Th = (4 / 5) * M gives a second equation. Solving these two equations together yields the actual temperatures. Step-by-Step Solution: Step 1: From the first average, (M + T + W) / 3 = 37 so M + T + W = 3 * 37 = 111.Step 2: From the second average, (T + W + Th) / 3 = 34 so T + W + Th = 3 * 34 = 102.Step 3: Subtract the second equation from the first: (M + T + W) - (T + W + Th) = 111 - 102.Step 4: This simplifies to M - Th = 9.Step 5: We are also given Th = (4 / 5) * M.Step 6: Substitute into M - Th = 9 to get M - (4 / 5) * M = 9.Step 7: This becomes (1 / 5) * M = 9, so M = 9 * 5 = 45 degrees Celsius.Step 8: Then Th = (4 / 5) * 45 = 36 degrees Celsius. Verification / Alternative Check: We can check quickly. If Monday is 45 and Thursday is 36, compute sums: M + T + W = 111 and T + W + Th = 102 from the equations. Their difference gives M - Th = 9 which matches 45 - 36. Also Th is four fifths of M because (4 / 5) * 45 = 36. Thus all conditions are satisfied, confirming the answer. Why Other Options Are Wrong: Values such as 38, 39 or 40 degrees Celsius do not satisfy both the difference equation M - Th = 9 and the ratio Th = (4 / 5) * M at the same time. If we assume any of those as Thursday temperature and back calculate Monday, one or both conditions fail. Only 36 degrees Celsius fits all given relationships. Common Pitfalls: Some students confuse which average corresponds to which group of days, leading to incorrect equations. Others may misapply the ratio, using 5 / 4 instead of 4 / 5, or they forget to multiply the averages by the number of days to get total sums. The safe method is always to convert each average into a sum equation, subtract them carefully, and then use the given ratio to solve for the unknowns. Final Answer: The temperature on Thursday is 36 degrees Celsius.

Question 7

The marks of a pupil were wrongly entered as 83 instead of 63, and due to this mistake the average marks of the class increased by 0.5. How many pupils are there in the class?

Accepted Answer

Introduction / Context: This problem focuses on how a single incorrect entry can affect the average marks of a class. The marks of one pupil were entered 20 marks higher than the actual marks, resulting in an increase in the class average. We need to determine the class strength from this information. Such questions help students understand the sensitivity of averages to individual data points. Given Data / Assumptions: - Correct marks of the pupil should be 63. - Wrongly entered marks are 83. - The wrong entry is 20 marks higher than the correct value. - Due to this mistake, the average marks of the class increased by 0.5 mark. - Let the number of pupils in the class be n. Concept / Approach: The key idea is that a change in one value by a certain amount changes the total sum by the same amount. When average changes, the total change in sum is equal to change in average multiplied by the number of pupils. Here, the total increase in the sum due to the mistake is 20 marks. This must be equal to the increase in average (0.5) multiplied by the number of pupils n, which allows us to solve for n. Step-by-Step Solution: Step 1: Extra marks added due to wrong entry = 83 - 63 = 20 marks.Step 2: Let the number of pupils be n.Step 3: Increase in average due to mistake = 0.5 mark.Step 4: Increase in total marks = increase in average * number of pupils = 0.5 * n.Step 5: This increase must equal the extra 20 marks, so 0.5 * n = 20.Step 6: Solve for n: n = 20 / 0.5 = 40. Verification / Alternative Check: We can cross check by imagining the correct scenario. If n = 40, then an increase of 0.5 mark in the average means the total marks rise by 40 * 0.5 = 20 marks. This matches exactly the extra marks due to entering 83 instead of 63. Therefore, the class must consist of 40 pupils for this change in average to occur. Why Other Options Are Wrong: If there were 45 pupils, then an increase of 0.5 in the average would mean a total increase of 22.5 marks, which does not match the actual extra 20 marks. Similarly, 39 pupils would give total increase of 19.5, and 37 pupils would give total increase of 18.5 marks. Only 40 pupils produce exactly 20 extra marks in total, consistent with the incorrect entry. Common Pitfalls: Students sometimes set up the equation incorrectly by dividing 20 by the original average or by using 20 / n = 0.5 instead of 0.5 * n = 20. Another common mistake is to think that the average increases for only one student instead of all students. The correct viewpoint is that the class average is over the entire group, so any change in a single entry affects the sum for all pupils together. Final Answer: The class has 40 pupils.

Question 8

The mean of 50 observations is 36. Later it is found that one observation was recorded as 23 instead of 48. What is the corrected mean of the 50 observations?

Accepted Answer

Introduction / Context: This aptitude question checks your understanding of how a single incorrect value affects the mean of a data set. You are given the original mean of 50 observations and told that one value was recorded wrongly. You must recalculate the corrected mean after fixing that erroneous observation. This type of problem is common in exams for bank jobs, SSC and campus placements because it combines ideas of averages and error correction. Given Data / Assumptions: - Number of observations = 50. - Original mean (with error) = 36. - Wrongly taken value = 23. - Correct value should be = 48. - We need to find the new corrected mean of all 50 observations after replacing 23 by 48. Concept / Approach: The mean of n observations is defined as total sum divided by n. If one value in the set is changed, the total sum changes by the difference between the new and the old value. Here, instead of recomputing everything from scratch, we can adjust the original total sum by adding the correction difference and then divide again by 50 to obtain the corrected mean. Step-by-Step Solution: Step 1: Original total sum with error = mean * number of observations = 36 * 50 = 1800.Step 2: Because 23 was recorded instead of 48, the total is short by 48 - 23 = 25.Step 3: Corrected total sum = 1800 + 25 = 1825.Step 4: Corrected mean = corrected total sum / number of observations = 1825 / 50.Step 5: 1825 / 50 = 36.5. Verification / Alternative Check: You can also think directly in terms of the effect on the mean. The total has increased by 25 while the number of observations is still 50. So the mean should increase by 25 / 50 = 0.5. The original mean was 36, so the corrected mean must be 36 + 0.5 = 36.5. This fast reasoning matches the detailed calculation and confirms that 36.5 is correct. Why Other Options Are Wrong: A mean of 35 would be smaller than the original 36, which is impossible because the correction increased the total. Means of 40 and 42 are much too high and would require a much larger increase in total sum than the given difference of 25. Therefore those values do not agree with the actual change in the data set. Common Pitfalls: Students sometimes subtract 25 instead of adding it, or they divide the difference by 49 or 51 instead of 50. Another mistake is to try to adjust the mean directly without understanding that it is the total sum that changes first. Always start from total = mean * number of observations, correct the total, and then divide by the same number of observations. Final Answer: The corrected mean of the 50 observations is 36.5.

Question 9

Four numbers have an average of 60. The first number is one fourth of the sum of the last three numbers. What is the value of the first number?

Accepted Answer

Introduction / Context: This question tests your understanding of averages and simple algebraic relationships between numbers. You are told that the average of four numbers is known, and that the first number is one fourth of the sum of the remaining three. Using these conditions, you must determine the value of the first number. Such problems reinforce the link between averages, totals and linear equations. Given Data / Assumptions: - There are four numbers: let them be a, b, c and d. - Average of the four numbers = 60. - Therefore, total sum of the four numbers = 4 * 60 = 240. - The first number a is one fourth of the sum of the last three numbers b, c and d. - So a = (1 / 4) * (b + c + d). Concept / Approach: From the average, we know the total sum of all four numbers. The condition relating a to the sum of b, c, and d can be combined with this total to form an equation involving a only. After that, solving the equation gives us the exact value of the first number. This is a straightforward application of algebra with averages. Step-by-Step Solution: Step 1: Total sum of all four numbers = a + b + c + d = 240.Step 2: From the given condition, a = (1 / 4) * (b + c + d).Step 3: Note that b + c + d = 240 - a.Step 4: Substitute into the relation a = (1 / 4) * (b + c + d) to get a = (1 / 4) * (240 - a).Step 5: Multiply both sides by 4: 4a = 240 - a.Step 6: Bring terms together: 4a + a = 240, so 5a = 240.Step 7: Therefore a = 240 / 5 = 48. Verification / Alternative Check: If a = 48, then b + c + d = 240 - 48 = 192. One fourth of 192 is 48, which matches a. Thus the condition is exactly satisfied. Also, the average becomes (48 + 192) / 4 = 240 / 4 = 60, which matches the given average. So 48 is consistent with all information in the question. Why Other Options Are Wrong: Values like 17, 29 or 36 do not satisfy the equation a = (1 / 4) * (240 - a). Substituting any of those options leads to a mismatch between left-hand side and right-hand side. Therefore, they cannot be the correct first number for the given average and relationship. Common Pitfalls: Learners sometimes confuse the phrase one fourth of the sum of the last three with one fourth of each of the last three. Another error is to assume that the first number is simply 60, the same as the average. Writing down the equations carefully and using the total sum 240 avoids these misunderstandings. Final Answer: The first number is 48.

Question 10

Out of 9 persons, 8 persons spend Rs. 30 each for their meals. The ninth person spends Rs. 20 more than the average expenditure of all 9 persons. What is the total money spent by all of them?

Accepted Answer

Introduction / Context: This question is a classic average and total expenditure problem. You know the expenditure of eight persons and are given a special condition about how much the ninth person spends relative to the overall average. Using this information, you must work backwards to find the average expenditure and then the total money spent by all nine persons. This type of problem is frequently asked in aptitude tests. Given Data / Assumptions: - Number of persons = 9. - 8 persons each spend Rs. 30. - The ninth person spends Rs. 20 more than the average expenditure of all nine persons. - Let the average expenditure of all nine persons be x rupees. - We need the total expenditure by all nine persons. Concept / Approach: The key is to express everything in terms of the unknown average x. The total expenditure is the sum of the expenditures of the eight persons plus that of the ninth person. This total, divided by 9, must equal x. Forming an equation from this relation allows us to solve for x and then compute the total expenditure easily. Step-by-Step Solution: Step 1: Let the average expenditure of all nine persons be x rupees.Step 2: Total expenditure of 8 persons who each spend Rs. 30 is 8 * 30 = Rs. 240.Step 3: The ninth person spends x + 20 rupees.Step 4: Total expenditure of all nine persons = 240 + (x + 20) = 260 + x.Step 5: By definition of average, total expenditure = 9 * x.Step 6: Therefore, 9x = 260 + x.Step 7: Simplify: 9x - x = 260, so 8x = 260.Step 8: Hence x = 260 / 8 = 32.5 rupees.Step 9: Total expenditure = 9 * x = 9 * 32.5 = Rs. 292.50. Verification / Alternative Check: Check the condition for the ninth person. Average expenditure is Rs. 32.50, so the ninth person spends 32.50 + 20 = Rs. 52.50. Sum of expenditures of 8 persons is 8 * 30 = Rs. 240. Adding the ninth person gives 240 + 52.50 = Rs. 292.50. Dividing 292.50 by 9 again gives 32.50, confirming that the average is consistent. Why Other Options Are Wrong: Totals such as Rs. 297.50, Rs. 298 or Rs. 298.50 would give averages that are not equal to x when you apply the given condition about the ninth person spending x + 20. Only Rs. 292.50 ensures that the ninth person expenditure and the overall average agree with the problem statement. Common Pitfalls: Many students treat 20 as extra over Rs. 30 instead of extra over the average, which leads to a wrong equation. Others forget to include x inside the total on both sides. Always read the phrase carefully: more than the average expenditure of all nine persons, and then set up the equation accordingly. Final Answer: The total money spent by all nine persons is Rs. 292.50.

Question 11

The average of 11 numbers is 10.9. The average of the first six numbers is 10.5 and the average of the last six numbers is 11.4. What is the value of the middle (6th) number?

Accepted Answer

Introduction / Context: This question involves overlapping averages and the idea of a middle term in a sequence of numbers. You are given the average of all 11 numbers, the average of the first six, and the average of the last six. From these overlapping pieces of information, you must determine the middle number, which is common to both groups. This type of question is useful for building confidence in handling sums and averages of partially overlapping sets. Given Data / Assumptions: - There are 11 numbers in total. - Average of all 11 numbers = 10.9. - Average of the first six numbers = 10.5. - Average of the last six numbers = 11.4. - The middle number is the 6th number, which belongs to both the first six and the last six. Concept / Approach: Let the 11 numbers be a1, a2, ..., a11. The 6th number is a6. The first six numbers are a1 to a6 and the last six numbers are a6 to a11. If we add the sums of the first six and the last six numbers, the middle term a6 is counted twice, while the others are counted once. We can relate this combined sum to the sum of all 11 numbers and then solve for a6. Step-by-Step Solution: Step 1: Total sum of all 11 numbers = 11 * 10.9 = 119.9.Step 2: Sum of the first six numbers = 6 * 10.5 = 63.Step 3: Sum of the last six numbers = 6 * 11.4 = 68.4.Step 4: Let the middle number (6th number) be m.Step 5: The sum of first six plus last six equals (sum of all 11 numbers) + m, because a6 is counted twice.Step 6: Therefore, 63 + 68.4 = 119.9 + m.Step 7: Left side = 131.4, so 131.4 = 119.9 + m.Step 8: Hence m = 131.4 - 119.9 = 11.5. Verification / Alternative Check: The combined sum of the first six and last six numbers is 63 + 68.4 = 131.4. If the middle number is 11.5, then the sum of all 11 numbers becomes 131.4 - 11.5 = 119.9. Dividing 119.9 by 11 yields 10.9, which matches the given overall average. This consistency confirms that the middle number must be 11.5. Why Other Options Are Wrong: Values like 10.5, 12.5, or 13.5 do not make the total sum match 119.9 when you apply the overlapping average logic. Substituting any of these values in place of 11.5 leads to a mismatch between the computed overall average and the given 10.9. Common Pitfalls: Some learners mistakenly think the middle number is simply the average of 10.5 and 11.4, or they miscount how many times the middle term appears in the combined sum of the first six and last six numbers. Always remember that the overlapping element is counted twice in such combined sums, and set up the equation based on that fact. Final Answer: The middle (6th) number is 11.5.

Question 12

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?

Accepted Answer

Introduction / Context: This is a straightforward average age problem involving a family. You know the average age of a couple at the time of marriage, and a few years later they have a child. You are asked to compute the new average age of the family. Such questions are designed to build comfort with age related word problems and average calculations. Given Data / Assumptions: - At the time of marriage, the average age of husband and wife was 23 years. - Therefore, at marriage, total age of husband and wife together = 2 * 23 = 46 years. - After 5 years, both husband and wife are 5 years older. - After 5 years, they have a one year old child. - We need to find the current average age of the family of three members. Concept / Approach: Age problems use the idea that everyone's age increases by the same number of years over any time period. The total age of the couple at marriage is known, so after 5 years, the combined age simply increases by 10 years, because there are 2 persons. Then we add the age of the child to obtain the family total, and divide by the number of family members to get the average age. Step-by-Step Solution: Step 1: Total age of husband and wife at marriage = 46 years.Step 2: After 5 years, each of them is 5 years older, so total increase in combined age = 2 * 5 = 10 years.Step 3: Combined age of husband and wife after 5 years = 46 + 10 = 56 years.Step 4: At this time, they have a one year old child, so child age = 1 year.Step 5: Total age of the family now = 56 + 1 = 57 years.Step 6: Number of family members = 3.Step 7: Average age of the family = total age / number of members = 57 / 3 = 19 years. Verification / Alternative Check: To verify, think of possible ages. Suppose at marriage the husband was 25 and the wife was 21; that gives an average of 23. After 5 years, they would be 30 and 26, and with a one year old child, the ages would be 30, 26 and 1. The total is 57 and the average is 19, which matches the calculated value. Any specific numbers that respect the original average lead to the same final average. Why Other Options Are Wrong: An average of 23 years would mean the total age now is 69, which is too large compared to the original total of 46 plus only 10 additional years for the couple and 1 year for the child. Values 18 or 25 also do not match the correct total of 57 when multiplied by 3. Common Pitfalls: Some students forget that the total age of the couple increases by twice the number of years because there are two adults. Others mistakenly take the average of 23 and 1 directly. Carefully tracking totals over time and then dividing by the number of persons avoids these mistakes. Final Answer: The current average age of the family is 19 years.

Question 13

The captain of a cricket team of 11 members is 26 years old and the wicket keeper is 3 years older than the captain. If the ages of these two players are excluded, the average age of the remaining players is one year less than the average age of the…

Accepted Answer

Introduction / Context: This age based question uses the idea of removing two members from a group and seeing how the average changes. We know the ages of the captain and the wicket keeper, and we are told that when they are removed, the average age of the remaining nine players is one year less than the overall team average. From this, we have to find the average age of the whole cricket team of 11 players. Given Data / Assumptions: - Total number of team members = 11. - Age of the captain = 26 years. - Age of the wicket keeper = 26 + 3 = 29 years. - If these two players are excluded, 9 players remain. - The average age of the 9 remaining players is 1 year less than the average age of the whole team. - Let the average age of the whole team be A years. Concept / Approach: The average age of the whole team, multiplied by 11, gives the total age of all members. When we remove the captain and the wicket keeper, we subtract their ages from this total to get the sum of ages of the remaining nine players. Dividing that new total by 9 gives their average, which is given as A - 1. Setting up this equation allows us to solve for A. Step-by-Step Solution: Step 1: Total age of all 11 players = 11 * A.Step 2: Combined age of captain and wicket keeper = 26 + 29 = 55 years.Step 3: Total age of the remaining 9 players = 11 * A - 55.Step 4: Average age of the remaining 9 players is given as A - 1.Step 5: Therefore, (11 * A - 55) / 9 = A - 1.Step 6: Multiply both sides by 9 to remove the denominator: 11A - 55 = 9A - 9.Step 7: Rearrange terms: 11A - 9A = 55 - 9, so 2A = 46.Step 8: Hence A = 46 / 2 = 23 years. Verification / Alternative Check: If the average age of the team is 23 years, total age of 11 players is 11 * 23 = 253 years. Removing the captain and wicket keeper removes 55 years, leaving 253 - 55 = 198 years for 9 players. Their average becomes 198 / 9 = 22 years, which is indeed one year less than 23. This confirms that our value for A is consistent with the given condition. Why Other Options Are Wrong: If the team average were 20, 21 or 25 years, the resulting average of the remaining 9 players would not be exactly one year less than the overall average when the given ages of 26 and 29 are subtracted. Only an average of 23 years satisfies the equation and the described situation. Common Pitfalls: Some learners mistakenly subtract 2 from the average instead of 1, or they forget that the total age reduces by the sum of both removed players. Others divide by 11 instead of 9 after removal. To avoid such errors, always compute totals first, then use the correct number of players in each average. Final Answer: The average age of the entire cricket team is 23 years.

Question 14

The average temperature of a town during the first four days of a month is 58 degrees. The average for the second, third, fourth and fifth days is 60 degrees. If the temperatures on the first and fifth days are in the ratio 7 : 8, what is the temper…

Accepted Answer

Introduction / Context: This temperature problem combines average calculations with a ratio relation. The averages are given for two overlapping sets of four days each. In addition, you know the ratio of temperatures on the first and fifth days. Using this information, you need to determine the temperature on the fifth day. It is a typical exam style question that checks algebraic manipulation and understanding of averages. Given Data / Assumptions: - Let the temperatures on days 1, 2, 3, 4 and 5 be T1, T2, T3, T4 and T5 respectively. - Average for days 1 to 4 is 58 degrees. - Average for days 2 to 5 is 60 degrees. - Temperatures T1 and T5 are in the ratio 7 : 8, so T1 : T5 = 7 : 8. - We need to find T5, the temperature on the fifth day. Concept / Approach: The average of a group of days can be written as the sum of those day temperatures divided by the number of days. The given averages give two equations for the sums of T1 to T4 and T2 to T5. Subtracting these equations relates T1 and T5. The ratio condition between T1 and T5 provides a second relation. Solving these two simple equations gives the values of T1 and T5. Step-by-Step Solution: Step 1: From the first average, (T1 + T2 + T3 + T4) / 4 = 58, so T1 + T2 + T3 + T4 = 4 * 58 = 232.Step 2: From the second average, (T2 + T3 + T4 + T5) / 4 = 60, so T2 + T3 + T4 + T5 = 4 * 60 = 240.Step 3: Subtract the first sum from the second: (T2 + T3 + T4 + T5) - (T1 + T2 + T3 + T4) = 240 - 232.Step 4: This simplifies to T5 - T1 = 8.Step 5: From the ratio T1 : T5 = 7 : 8, we have T5 = (8 / 7) * T1.Step 6: Substitute into T5 - T1 = 8 to get (8 / 7) * T1 - T1 = 8.Step 7: Factor T1: T1 * (8 / 7 - 1) = 8, so T1 * (1 / 7) = 8.Step 8: Therefore T1 = 8 * 7 = 56 degrees.Step 9: Then T5 = (8 / 7) * 56 = 64 degrees. Verification / Alternative Check: We can quickly verify: T5 - T1 = 64 - 56 = 8, which matches the difference derived from the averages. Also, T1 : T5 = 56 : 64 simplifies to 7 : 8, which agrees with the given ratio. So T5 is consistent with all conditions in the problem. Why Other Options Are Wrong: Temperatures such as 62, 65 or 66 degrees fail to satisfy both relations simultaneously. For each of those values, either the difference T5 - T1 does not equal 8 when combined with the ratio, or the ratio T1 : T5 is not exactly 7 : 8. Only 64 degrees fits both the difference and the ratio conditions. Common Pitfalls: Many learners mistakenly divide by 5 instead of 4 when converting averages to sums, or they misinterpret the ratio and use T5 : T1 = 7 : 8 instead of T1 : T5. Another common error is not subtracting the equations correctly, which leads to an incorrect relation between T1 and T5. Careful algebra avoids these problems. Final Answer: The temperature on the fifth day is 64 degrees.

Question 15

The average monthly salary of all the employees of a firm is Rs. 60. Out of these employees, 12 officers receive an average salary of Rs. 400 per month and the rest receive an average of Rs. 56 per month. How many employees are there in the firm?

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Introduction / Context: This salary based question illustrates how a combined average arises from two subgroups with different averages. You know the overall average salary for all employees, the number and average salary of officers, and the average salary of the remaining employees. From this information, you must determine the total number of employees. This type of mixture and average problem is very common in quantitative aptitude tests. Given Data / Assumptions: - Let the total number of employees be N. - Overall average salary of all employees = Rs. 60 per month. - Number of officers = 12. - Average salary of officers = Rs. 400 per month. - Average salary of remaining employees = Rs. 56 per month. - We need to find N. Concept / Approach: The overall average salary comes from the total salary paid to all employees divided by the total number of employees. The total salary can be expressed as the sum of the salary paid to officers plus the salary paid to the remaining employees. By writing this total in two different ways and equating them, we can solve for the total number of employees N. Step-by-Step Solution: Step 1: Total salary paid to all employees = overall average * total number of employees = 60 * N.Step 2: Salary paid to officers = number of officers * average officer salary = 12 * 400 = Rs. 4800.Step 3: Number of remaining employees = N - 12.Step 4: Total salary paid to remaining employees = (N - 12) * 56.Step 5: Total salary for all employees can also be written as 4800 + 56 * (N - 12).Step 6: Equate both expressions for total salary: 60 * N = 4800 + 56 * (N - 12).Step 7: Expand the right side: 60N = 4800 + 56N - 672.Step 8: Simplify: 60N = 56N + 4128, so 60N - 56N = 4128.Step 9: This gives 4N = 4128, so N = 4128 / 4 = 1032. Verification / Alternative Check: If N = 1032, then remaining employees = 1032 - 12 = 1020. Total salary for officers = 4800. Total salary for others = 1020 * 56 = 57120. Combined total salary = 4800 + 57120 = 61920. Dividing this by 1032 gives 61920 / 1032 = 60, which matches the given overall average salary. So N = 1032 is consistent. Why Other Options Are Wrong: Values like 1030, 1060 or 1062 do not satisfy the equation 60N = 4800 + 56(N - 12). Substituting any of these into the equation leads to a mismatch between the total salary based on the overall average and the sum of subgroup salaries. Common Pitfalls: Students sometimes forget to subtract 12 officers when computing the number of remaining employees, or they mistakenly average the salaries instead of working with totals. The safest path is always to convert every average into a total amount of salary and then form a simple linear equation in N. Final Answer: The firm has 1032 employees.

Question 16

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

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Introduction / Context: This is another mixture and average problem involving salaries in a workshop. The overall average salary is known, along with the number and average salary of technicians and the average salary of the remaining workers. Your task is to find the total number of workers. Such questions are helpful in understanding how subgroups with different averages combine to form an overall average. Given Data / Assumptions: - Let the total number of workers be N. - Average salary of all workers = Rs. 8000. - Number of technicians = 7. - Average salary of technicians = Rs. 12000. - Average salary of the remaining workers = Rs. 6000. - We have to find N. Concept / Approach: The overall total salary equals the sum of the salaries of technicians and the salaries of all the other workers. The overall average multiplied by N gives the total salary in one expression. Another expression for the total salary comes from adding the subgroup totals. Equating these two expressions allows us to compute N. Step-by-Step Solution: Step 1: Total salary of all workers = 8000 * N.Step 2: Total salary paid to 7 technicians = 7 * 12000 = Rs. 84000.Step 3: Number of remaining workers = N - 7.Step 4: Total salary paid to remaining workers = (N - 7) * 6000.Step 5: Total salary can also be written as 84000 + 6000 * (N - 7).Step 6: Equate the two expressions for total salary: 8000N = 84000 + 6000(N - 7).Step 7: Expand the right side: 8000N = 84000 + 6000N - 42000.Step 8: Simplify: 8000N = 6000N + 42000.Step 9: Subtract 6000N: 2000N = 42000.Step 10: N = 42000 / 2000 = 21. Verification / Alternative Check: If N = 21, then remaining workers = 21 - 7 = 14. Salary for technicians = 84000. Salary for remaining workers = 14 * 6000 = 84000. Total salary = 84000 + 84000 = 168000. Average salary = 168000 / 21 = 8000, which matches the given overall average. This confirms that N = 21 is correct. Why Other Options Are Wrong: If N were 20, 23 or 24, the total salary computed from subgroup averages would not equal 8000 times N. A quick substitution shows that the equation 8000N = 84000 + 6000(N - 7) is not satisfied for those values, so they cannot be the total number of workers. Common Pitfalls: Some students mistakenly apply a simple average formula like (12000 + 6000) / 2, ignoring the different group sizes. Others forget to subtract 7 when finding the number of non technicians. The correct technique is always to use totals: overall total salary equals the sum of subgroup totals. Final Answer: The total number of workers in the workshop is 21.

Question 17

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

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Introduction / Context: This question tests understanding of how changing a single score affects the average of the whole team. You know the score of the best marksman and how the average would change if he had scored more points. From this hypothetical situation, you must find the actual total score of the team. This is a common type of average and what if scenario in aptitude exams. Given Data / Assumptions: - Number of team members = 8. - Best marksman's actual score = 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 with the original scores. Concept / Approach: The key idea is that increasing one player's score from 85 to 92 increases the team total by 7 points. The hypothetical average of 84 gives us the hypothetical total score, from which we can subtract the 7 extra points to recover the actual total. This method avoids needing to know each individual score and instead uses only total and average relations. Step-by-Step Solution: Step 1: If the best marksman had scored 92 points, the average score of the 8 members would have been 84.Step 2: Hypothetical total score with 92 points = 84 * 8 = 672 points.Step 3: The difference between the hypothetical and actual score of the best marksman is 92 - 85 = 7 points.Step 4: So the hypothetical total is 7 points more than the actual total.Step 5: Actual total score of the team = 672 - 7 = 665 points. Verification / Alternative Check: If the actual total is 665 and one player has 85, then the sum of the other seven players' scores is 665 - 85 = 580. If that same player had scored 92, the new team total would be 580 + 92 = 672, and the average would be 672 / 8 = 84, matching the given hypothetical situation. This confirms that 665 is the correct total score. Why Other Options Are Wrong: Totals such as 657, 658 or 678 do not yield an average of 84 when 7 extra points are added to the best marksman's score. Only 665 satisfies the condition that increasing a single player's score by 7 yields a team total of 672, which corresponds to an average of 84. Common Pitfalls: A common mistake is to set up an equation with an unknown average first, which is not necessary here. Another error is to subtract instead of add when translating between actual and hypothetical totals. Remember that changing one score by a certain amount changes the total by the same amount. Work carefully with these differences to avoid sign errors. Final Answer: The actual total number of points scored by the team is 665.

Question 18

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

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Introduction / Context: This wages problem combines overall average income with a fixed difference between the wages of men and women. You know the counts of men and women, the average wage of all workers together, and that each woman receives Rs. 5 less than each man. The task is to determine the individual daily wages of a man and a woman. This is a typical linear equation and weighted average problem. Given Data / Assumptions: - Number of men = 600. - Number of women = 400. - Total workers = 600 + 400 = 1000. - Average wage for all workers = Rs. 25.50 per day. - Each woman receives Rs. 5 less than each man. - Let the daily wage of a man be M rupees. - Then the daily wage of a woman is M - 5 rupees. Concept / Approach: The total wage paid per day to all workers equals the overall average multiplied by the total number of workers. This same total can be expressed as wages paid to men plus wages paid to women. Writing the total in these two ways yields an equation in M that can be solved easily. Once M is found, the woman's wage is M - 5. Step-by-Step Solution: Step 1: Total wage per day for all workers based on the overall average = 25.50 * 1000 = Rs. 25500.Step 2: Total wage per day paid to men = 600 * M.Step 3: Total wage per day paid to women = 400 * (M - 5).Step 4: Combined total wage = 600M + 400(M - 5).Step 5: This combined total must equal Rs. 25500, so 600M + 400(M - 5) = 25500.Step 6: Expand the equation: 600M + 400M - 2000 = 25500.Step 7: Simplify: 1000M - 2000 = 25500.Step 8: Add 2000 to both sides: 1000M = 27500.Step 9: Therefore M = 27500 / 1000 = Rs. 27.50.Step 10: Wage of a woman = M - 5 = 27.50 - 5 = Rs. 22.50. Verification / Alternative Check: Check totals with these wages. Men: 600 * 27.50 = 16500. Women: 400 * 22.50 = 9000. Combined total = 16500 + 9000 = 25500. Dividing 25500 by 1000 workers gives an average of Rs. 25.50, which matches the given overall average. So the values are consistent with all data in the problem. Why Other Options Are Wrong: The other options either have the woman earning more than the man or do not preserve a difference of exactly Rs. 5. Substituting those pairs into the total wage calculation does not give a total of Rs. 25500, so they do not produce the required overall average of Rs. 25.50. Common Pitfalls: A typical mistake is to reverse the Rs. 5 difference, assuming that men earn less than women, or to forget to multiply by the correct number of workers in each group. Another error is to take a simple average of the two unknown wages instead of using weighted totals. Always base your calculation on total wage, not on a naive average of subgroups. Final Answer: The daily wage of a man is Rs. 27.50 and the daily wage of a woman is Rs. 22.50.

Question 19

The average temperature for Monday, Tuesday, Wednesday and Thursday is 48 degrees. The average temperature for Tuesday, Wednesday, Thursday and Friday is 52 degrees. If the temperature on Monday is 42 degrees, what is the temperature on Friday?

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Introduction / Context: This temperature problem uses overlapping averages over consecutive days of the week. You know the average from Monday to Thursday, and from Tuesday to Friday, and you also know the temperature on Monday. From this information, you must compute the temperature on Friday. It is a standard example of using sums from averages for consecutive groups. Given Data / Assumptions: - Let the temperatures on Monday, Tuesday, Wednesday, Thursday and Friday be M, T, W, Th and F respectively. - Average for Monday to Thursday = 48 degrees. - Average for Tuesday to Friday = 52 degrees. - Temperature on Monday M = 42 degrees. - We need to find the temperature on Friday F. Concept / Approach: The average of four days is the sum of their temperatures divided by 4. So each given average can be turned into an equation for the sum of those four days. Subtracting these equations allows you to isolate the difference between Friday and Monday temperatures. Since Monday's temperature is known, you can then solve for Friday's temperature directly. Step-by-Step Solution: Step 1: From the first average, (M + T + W + Th) / 4 = 48.Step 2: Multiply by 4: M + T + W + Th = 4 * 48 = 192.Step 3: From the second average, (T + W + Th + F) / 4 = 52.Step 4: Multiply by 4: T + W + Th + F = 4 * 52 = 208.Step 5: Subtract the first sum from the second: (T + W + Th + F) - (M + T + W + Th) = 208 - 192.Step 6: This simplifies to F - M = 16.Step 7: Given M = 42, substitute to get F - 42 = 16.Step 8: Therefore F = 42 + 16 = 58 degrees. Verification / Alternative Check: If Friday's temperature is 58, the sum Tuesday to Friday is T + W + Th + 58 = 208. Since M + T + W + Th = 192 and M = 42, we have T + W + Th = 150. Then T + W + Th + 58 = 150 + 58 = 208, matching the second sum. So the averages and individual values are consistent, confirming F = 58. Why Other Options Are Wrong: Values like 56, 52 or 50 do not satisfy the equation F - M = 16 when M = 42. Substituting any of these into the sum equations would not produce the correct second average of 52 degrees. Only 58 degrees satisfies all conditions of the problem. Common Pitfalls: Some students incorrectly average 48 and 52 directly or forget to multiply by 4 when finding the sums. Others make mistakes during subtraction of the equations. To avoid such errors, always convert averages to sums first, then subtract carefully to isolate the unknown temperature. Final Answer: The temperature on Friday is 58 degrees.

Question 20

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

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Introduction / Context: This problem involves calculating the overall average number of visitors to a library in a month, given different daily averages for Sundays and other days. You are told that the month has 30 days and begins on a Sunday. Using this information, you must find the average number of visitors per day for the entire month. It is a simple but important example of weighted averages over different types of days. Given Data / Assumptions: - Average visitors on Sundays = 510 per day. - Average visitors on other days (non Sundays) = 240 per day. - The month has 30 days and starts on a Sunday. - We need to determine how many Sundays occur in such a month and then compute the overall daily average number of visitors. Concept / Approach: If a month of 30 days starts on a Sunday, the days of the month cover four complete weeks (28 days) plus 2 extra days. In every full week there is exactly one Sunday, so there are 4 Sundays in the first 28 days. The extra two days are Sunday and Monday, giving one more Sunday. Hence the total number of Sundays is 5 and the remaining 25 days are non Sundays. After finding the total visitors on Sundays and on other days, we divide the combined total by 30 to get the required average. Step-by-Step Solution: Step 1: Count Sundays in the month. In 4 complete weeks there are 4 Sundays. The remaining 2 days start with Sunday, so the extra days are Sunday and Monday, giving 1 more Sunday.Step 2: Total number of Sundays = 4 + 1 = 5.Step 3: Total number of non Sundays = 30 - 5 = 25.Step 4: Total visitors on Sundays = 5 * 510 = 2550.Step 5: Total visitors on other days = 25 * 240 = 6000.Step 6: Combined total visitors in the month = 2550 + 6000 = 8550.Step 7: Average visitors per day for the 30 day month = 8550 / 30 = 285. Verification / Alternative Check: To check, note that if every day had 240 visitors, total visitors would be 240 * 30 = 7200. On Sundays, the library actually receives 510 visitors, which is 270 more than 240. There are 5 Sundays, so the extra visitors contribute 5 * 270 = 1350 additional visitors. Hence total visitors = 7200 + 1350 = 8550. Dividing by 30 again gives 285, confirming the result. Why Other Options Are Wrong: Values like 290, 297 or 305 correspond to totals that are either too high or too low compared to the sum of 2550 Sunday visitors and 6000 other day visitors. Only an average of 285 visitors per day reproduces the correct total of 8550 visitors for the month. Common Pitfalls: Some learners miscount the number of Sundays in a 30 day month that begins on a Sunday, mistakenly taking 4 or 6 Sundays. Others forget to multiply the averages by the number of days of each type before taking the overall average. Carefully counting Sundays and using total visitors for each category avoids these mistakes. Final Answer: The average number of visitors per day in the month is 285.

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