Question 1

In the first 32 overs of a cricket match, the run rate was 7.2 runs/over. What is the required run rate in the remaining 18 overs to reach the target of 297 runs?

Accepted Answer

3.7

Question 2

The mean height of 25 boys in a class is 150 cm, and the mean height of 35 girls in the same class is 145 cm. The combined mean height of 60 students in the class is ___ (app)

Accepted Answer

147

Question 3

Of the 5 subjects, the average marks of Rakesh in first 3 Subjects was 79 and the average marks in the last 3 subjects was 86. If his marks in the third subject was 80, then what was his average marks in all the 5 subjects?

Accepted Answer

83

Question 4

The mode of the data 12, 1, 10, 1, 9, 3, 4, 9, 7, 9 is

Accepted Answer

9

Question 5

In a class of 17 students, the average marks were first calculated as 71, but it was later found that one student's mark had been entered as 65 instead of 56 and another student's mark as 24 instead of 50. After correcting both entries, what is the …

Accepted Answer

Introduction / Context: This question tests the concept of correcting an average when some of the individual data values were entered incorrectly. Such problems are very common in competitive exams because they mix arithmetic mean, totals, and error analysis in one compact question. Understanding how to update the total when a few marks are wrongly recorded helps you handle many real classroom and survey situations accurately. Given Data / Assumptions: Number of students = 17. Originally calculated average marks = 71. One mark was entered as 65 instead of 56. Another mark was entered as 24 instead of 50. We assume all other marks are correct. Concept / Approach: The key idea is that average = total sum / number of observations. If some values are wrong, the original total is also wrong. To correct the average, we first compute the old total using the wrong average, then remove the wrong entries, add the correct ones, and finally divide the new total by the same number of students to get the corrected average. Step-by-Step Solution: Step 1: Original total marks (with mistakes) = 71 * 17.Step 2: 71 * 17 = 1207, so original total = 1207.Step 3: Replace 65 with 56 and 24 with 50. Net correction = (56 - 65) + (50 - 24).Step 4: (56 - 65) = -9 and (50 - 24) = 26. Net correction = -9 + 26 = 17.Step 5: Correct total marks = 1207 + 17 = 1224.Step 6: Correct average = 1224 / 17 = 72. Verification / Alternative check: Another way is to think directly in terms of totals. If two entries are adjusted by a net increase of 17 marks, the average increase per student must be 17 / 17 = 1 mark. Since the original average was 71, the new average becomes 71 + 1 = 72. This matches our previous calculation, so the result is consistent. Why Other Options Are Wrong: Option 70 would correspond to a net negative correction, which contradicts the given changes. Option 71 would mean no overall correction, which is not possible because the two corrections do not cancel each other. Option 73 would require a larger positive change in the total than what actually occurs. Option 69 is even further away and does not align with any reasonable miscalculation of the corrections. Common Pitfalls: Forgetting to multiply average by the number of students to get the total. Mixing up which mark is the correct one and which mark is the wrong one. Adding both differences instead of considering the correct sign (one change reduces the total, the other increases it). Final Answer: The corrected average marks of the 17 students is 72.

Question 6

A cricketer has an average of 99 runs over 99 innings. How many runs must he score in his 100th innings so that his overall batting average becomes exactly 100 runs after 100 innings?

Accepted Answer

Introduction / Context: This question uses the basic concept of average runs in cricket. It checks whether you can translate the verbal condition about the current average and the desired average into equations involving totals and then work backwards to find the required score in the next innings. Such problems help in strengthening the link between arithmetic mean and cumulative totals over time. Given Data / Assumptions: Number of innings already played = 99. Current average runs = 99 runs per innings. Target average after 100 innings = 100 runs per innings. We assume all previous scores are fixed and only the 100th innings can change. Concept / Approach: The average is defined as total runs divided by number of innings. To reach a new average, we first compute the total runs required for that new average over 100 innings. We also compute the current total runs from the existing average over 99 innings. The difference between the required total and the current total is exactly the number of runs the batsman needs to score in the 100th innings. Step-by-Step Solution: Step 1: Current total runs = 99 innings * 99 runs per innings = 99 * 99 = 9801.Step 2: To average 100 runs over 100 innings, total required runs = 100 * 100 = 10000.Step 3: Runs needed in the 100th innings = 10000 - 9801.Step 4: 10000 - 9801 = 199.Step 5: Therefore, the batsman must score 199 runs in his 100th innings. Verification / Alternative check: If he scores 199 runs, the new total becomes 9801 + 199 = 10000. Dividing 10000 by 100 innings gives an average of exactly 100 runs, which matches the target average. This confirms that our calculation is accurate and that no arithmetic step has been missed. Why Other Options Are Wrong: Option 100 would lead to a total of 9901 runs, giving an average of 99.01, which is still below 100. Option 99 would keep the average almost unchanged. Option 101 would only give a total of 9902 runs. Option 150 gives a total of 9951 runs, which results in an average of 99.51, still less than 100. Common Pitfalls: Thinking that an increase of 1 in the average requires only 1 extra run, without considering the number of innings. Forgetting to convert averages into totals before doing any subtraction. Making mistakes in squaring 99 or multiplying 100 by 100. Final Answer: The batsman must score 199 runs in his 100th innings to raise his average to 100.

Question 7

In a class of 65 students, there are 39 girls whose average weight is 60 kg. The average weight of the whole class is 64 kg. What is the average weight of the boys in the class?

Accepted Answer

Introduction / Context: This question tests weighted average concepts where there are two groups within a larger group, each with its own average. You are given the overall average as well as the average of one subgroup and asked to compute the average of the remaining subgroup. This is a classic mixture and alligation style situation, useful in aptitude tests and real life when combining data from different segments. Given Data / Assumptions: Total students in the class = 65. Number of girls = 39. Average weight of girls = 60 kg. Average weight of entire class = 64 kg. Number of boys = 65 - 39 = 26. Concept / Approach: The approach is to translate the average information into totals. The total weight of all students is overall average multiplied by total students. The total weight of the girls is girls average multiplied by number of girls. Subtracting the girls total from the class total gives the total weight of boys. Dividing this by the number of boys yields the average weight of the boys. Step-by-Step Solution: Step 1: Total class weight = 64 * 65 = 4160 kg.Step 2: Total weight of girls = 60 * 39 = 2340 kg.Step 3: Total weight of boys = 4160 - 2340 = 1820 kg.Step 4: Number of boys = 26.Step 5: Average weight of boys = 1820 / 26 = 70 kg. Verification / Alternative check: We can verify by reconstructing the overall average. Suppose each boy weighs on average 70 kg, so total weight of boys is 26 * 70 = 1820 kg. Adding the girls total of 2340 kg gives 4160 kg. Dividing 4160 by 65 again yields 64 kg, confirming consistency with the given overall average. Why Other Options Are Wrong: Option 69 or 68 would reduce the boys total and hence reduce the overall average below 64 kg. Option 66 is even smaller and clearly cannot produce an overall average greater than the given girls average of 60 kg with the given proportions. Option 64 would mean boys and girls have similar weights, which contradicts the requirement of an overall average of 64 with the girls fixed at 60 kg. Common Pitfalls: Using a simple average of 60 and 64 instead of a weighted average based on actual counts. Incorrectly computing the number of boys or making arithmetic errors in multiplication. Trying to guess the answer without checking against the overall average condition. Final Answer: The average weight of the boys in the class is 70 kg.

Question 8

The sum of a non zero number and ten times its reciprocal is 7. What is the value of the number?

Accepted Answer

Introduction / Context: This problem tests your ability to translate a verbal statement involving a number and its reciprocal into an algebraic equation. Once the equation is set up, you must solve a quadratic to find possible values of the number. Questions of this type are common in algebra and aptitude exams, and they reinforce the relationship between a quantity and its reciprocal. Given Data / Assumptions: The number is non zero. Let the number be x. The expression given is x plus ten times its reciprocal. x + 10/x = 7. Concept / Approach: The general method is to clear the denominator by multiplying both sides of the equation by x, which produces a quadratic equation. Then we factor the quadratic and identify the valid roots. Both roots are algebraically valid, but only the root that appears in the option list can be chosen as the correct multiple choice answer. Step-by-Step Solution: Step 1: Start with x + 10/x = 7.Step 2: Multiply both sides by x to remove the denominator: x^2 + 10 = 7x.Step 3: Rearrange to standard quadratic form: x^2 - 7x + 10 = 0.Step 4: Factor the quadratic: x^2 - 7x + 10 = (x - 5)(x - 2).Step 5: Set each factor equal to zero: x - 5 = 0 or x - 2 = 0, so x = 5 or x = 2. Verification / Alternative check: Check x = 5: 5 + 10/5 = 5 + 2 = 7, which satisfies the condition. Check x = 2: 2 + 10/2 = 2 + 5 = 7, which also satisfies the condition. However, in the given answer choices only 5 appears. Therefore, 5 is the correct option to select in this multiple choice setting. Why Other Options Are Wrong: Option 4 gives 4 + 10/4 = 4 + 2.5 = 6.5, not 7. Option 3 gives 3 + 10/3, which is greater than 6 and less than 7, so it does not match exactly. Option 6 yields 6 + 10/6, which is more than 7. Option 2 is algebraically correct but is not listed as a choice; in the context of this MCQ, the only matching available choice is 5. Common Pitfalls: Forgetting to multiply every term by x when clearing the denominator. Making sign errors when rearranging the quadratic. Not checking which algebraic root actually appears among the given options. Final Answer: The required number, matching the given options, is 5.

Question 9

The average age of 80 boys in a class is 15 years. The average age of a group of 15 boys is 16 years and the average age of another group of 25 boys is 14 years. What is the average age of the remaining boys in the class?

Accepted Answer

Introduction / Context: This problem is a classic example of weighted averages and group averages. You are given the overall average of a large group and the averages of two subgroups, and you must determine the average of the remaining subgroup. This type of question is common in aptitude tests to assess your ability to combine and separate averages correctly. Given Data / Assumptions: Total number of boys = 80. Overall average age = 15 years. First subgroup: 15 boys with average age = 16 years. Second subgroup: 25 boys with average age = 14 years. Remaining boys = 80 - (15 + 25) = 40 boys. Concept / Approach: The main idea is that average = total sum / number of items. We first find the total age of all 80 boys using the overall average. Then we compute the total age of the two known subgroups from their averages. Subtracting these totals from the overall total gives the combined age of the remaining boys, which then yields their average. Step-by-Step Solution: Step 1: Total age of all 80 boys = 80 * 15 = 1200 years.Step 2: Total age of first 15 boys = 15 * 16 = 240 years.Step 3: Total age of second 25 boys = 25 * 14 = 350 years.Step 4: Combined age of these 40 boys = 240 + 350 = 590 years.Step 5: Age of remaining 40 boys = 1200 - 590 = 610 years.Step 6: Average age of remaining 40 boys = 610 / 40 = 15.25 years. Verification / Alternative check: We can quickly verify by recombining. The remaining group has total age 610 years, so adding 610 + 590 gives 1200, which matches the original total. Dividing 1200 by 80 again yields 15 years as the overall average, confirming that the computed subgroup average is consistent. Why Other Options Are Wrong: Option 14 years would drag the overall average below 15. Option 14.75 years is too low to keep the overall average at 15 when combined with the other subgroups. Option 15 years would give a different total age, not matching 1200. Option “Cannot be determined” is incorrect because sufficient data are given to compute a unique answer. Common Pitfalls: Using a simple average of 16 and 14 instead of applying correct weights based on group sizes. Forgetting that there are 40 remaining boys, not 80 or 60. Arithmetic errors while multiplying or subtracting totals. Final Answer: The average age of the remaining boys is 15.25 years.

Question 10

What is the average of the first seven positive multiples of 7 (that is, 7, 14, 21, 28, 35, 42 and 49)?

Accepted Answer

Introduction / Context: This question checks your understanding of averages of equally spaced numbers, specifically an arithmetic progression. The first seven positive multiples of 7 form a simple arithmetic sequence with constant difference. Recognizing this pattern allows you to compute the average efficiently without adding every term individually, a useful shortcut for timed exams. Given Data / Assumptions: The sequence is 7, 14, 21, 28, 35, 42 and 49. Number of terms = 7. This is an arithmetic progression with first term 7 and last term 49. Common difference = 7. Concept / Approach: For any arithmetic progression, the average (arithmetic mean) of all terms is simply equal to the mean of the first and last term. That is, average = (first term + last term) / 2. This is because the terms are symmetrically spaced around the middle value, and each pair equidistant from the ends sums to the same constant. Step-by-Step Solution: Step 1: Identify first term a = 7 and last term l = 49.Step 2: Use the property of arithmetic progressions: average = (a + l) / 2.Step 3: Compute (7 + 49) / 2 = 56 / 2.Step 4: 56 / 2 = 28.Step 5: Therefore, the average of the first seven multiples of 7 is 28. Verification / Alternative check: We can also directly sum the seven terms and divide by 7. Sum = 7 + 14 + 21 + 28 + 35 + 42 + 49 = 196. Then average = 196 / 7 = 28. Both methods give the same result, confirming the calculation is correct. Why Other Options Are Wrong: Option 7 is just the first term, not the average. Option 14 is the second term, and option 21 is the fourth term, both below the true central value. Option 35 is above the true average and would shift the balance of total too high. Only 28 matches both the arithmetic progression property and the explicit computation. Common Pitfalls: Assuming the average is equal to the middle term without confirming the sequence is an arithmetic progression. Making arithmetic errors while adding several terms under exam pressure. Confusing the concept of multiples with that of powers. Final Answer: The average of the first seven multiples of 7 is 28.

Question 11

In a class there are 28 boys and some girls. The average score of the boys in a test is 12.5, while the average score of the girls is 14.5. If the overall average score of all students is 13.1, how many students are there in the class in total?

Accepted Answer

Introduction / Context: This problem involves combining averages from two subgroups (boys and girls) to obtain the overall class average and then working backwards to deduce the unknown group size. It is a practical example of weighted averages, where each group contributes according to its size and mean score. Such questions are standard in quantitative aptitude tests. Given Data / Assumptions: Number of boys = 28. Average score of boys = 12.5. Average score of girls = 14.5. Overall average score of the class = 13.1. Number of girls is unknown; let it be g. Concept / Approach: The key is to convert each average into a total score and then set up an equation. The total score of boys is fixed, the total score of girls depends on g, and their sum must equal the overall average multiplied by the total number of students (28 + g). Solving this equation gives g, and so we can find the total students. Step-by-Step Solution: Step 1: Total score of boys = 28 * 12.5 = 350.Step 2: Total score of girls = 14.5 * g.Step 3: Total number of students = 28 + g, so overall total score = 13.1 * (28 + g).Step 4: Set up equation: 350 + 14.5g = 13.1(28 + g).Step 5: Compute right side: 13.1 * 28 = 366.8, so 350 + 14.5g = 366.8 + 13.1g.Step 6: Simplify: 14.5g - 13.1g = 366.8 - 350, so 1.4g = 16.8, hence g = 12.Step 7: Total students = 28 + 12 = 40. Verification / Alternative check: Check totals: Boys total = 350, girls total = 12 * 14.5 = 174, combined total = 524. Average = 524 / 40 = 13.1, which matches the given overall average, confirming that 40 is correct. Why Other Options Are Wrong: Options 42, 44, or 38 would correspond to different values of g and would break the relationship between subgroup averages and the overall average. Substituting those totals back into the equation would not produce an overall average of 13.1. Option 36 similarly fails to satisfy the weighted average condition. Common Pitfalls: Using a simple unweighted average of 12.5 and 14.5 instead of considering the number of boys and girls. Arithmetic mistakes when distributing 13.1 or simplifying the equation. Forgetting that total students are 28 + g, not just g. Final Answer: The total number of students in the class is 40.

Question 12

Two students appeared for an examination. One of them scored 21 marks more than the other, and his marks were 80 percent of the sum of both their marks. What are the marks obtained by the two students?

Accepted Answer

Introduction / Context: This algebra problem involves forming and solving simultaneous equations from a word description. The relationship between the two students scores includes both a difference (21 marks) and a percentage condition (80 percent of their sum). Translating this correctly into equations is crucial in many exam situations and real problems involving comparisons and percentages. Given Data / Assumptions: Let the higher mark be x and the lower mark be y. One student scored 21 marks more than the other, so x = y + 21. The higher score x is 80 percent of the sum of both scores. So x = 0.8(x + y). Concept / Approach: We use algebraic substitution. Starting from the 80 percent relation, we derive an equation in x and y. Then we combine this with the simple difference relation x = y + 21. Solving the system gives unique values for both x and y, which can be checked against the options. Step-by-Step Solution: Step 1: From x = 0.8(x + y), expand the right side to get x = 0.8x + 0.8y.Step 2: Rearrange: x - 0.8x = 0.8y, so 0.2x = 0.8y.Step 3: Divide both sides by 0.2 to get x = 4y.Step 4: From the difference relation, x = y + 21.Step 5: Substitute x = 4y into x = y + 21 to get 4y = y + 21.Step 6: Solve 4y - y = 21, giving 3y = 21, so y = 7 and x = 28. Verification / Alternative check: Check the conditions. The difference is 28 - 7 = 21, which agrees with the problem statement. The sum is 28 + 7 = 35, and 80 percent of 35 is 0.8 * 35 = 28, which equals the higher score. Both conditions are satisfied, confirming the solution is consistent. Why Other Options Are Wrong: Pairs like 88 and 67 or 98 and 77 have a difference of 21 but do not satisfy the 80 percent condition when checking x = 0.8(x + y). The pair 89 and 68 does not even maintain the stated difference exactly. The pair 49 and 28 has a difference of 21 but would not make the higher score equal to 80 percent of the sum of the two scores. Common Pitfalls: Misreading the 80 percent condition as the higher score being 80 percent of the lower score. Forgetting to use both relationships and trying to guess from one condition only. Arithmetic errors in handling decimal coefficients like 0.8 and 0.2. Final Answer: The two students scored 28 and 7 marks respectively.

Question 13

In a class there are 9 boys and some girls. The average score of the boys in a test is 12, while the average score of the girls is 14. If the overall average score of all the students is 13.1, how many students are there in the class?

Accepted Answer

Introduction / Context: This is another weighted average problem similar to earlier ones, but now with different group sizes. The question tests your ability to handle averages when one group size is known and the other is unknown, yet the overall average is given. Solving such questions correctly is important in exam scenarios involving statistics and arithmetic mean. Given Data / Assumptions: Number of boys = 9. Average score of boys = 12. Average score of girls = 14. Overall average score of the class = 13.1. Number of girls is unknown, let it be g. Concept / Approach: Again we convert averages into totals. The total score for boys is fixed. The total score for girls is 14g. The overall total must equal 13.1 times the total number of students, which is 9 + g. Setting these expressions equal allows us to solve for g and thus determine the total class size. Step-by-Step Solution: Step 1: Total score of boys = 9 * 12 = 108.Step 2: Total score of girls = 14 * g.Step 3: Total number of students = 9 + g.Step 4: Overall total score = 13.1 * (9 + g).Step 5: Write equation: 108 + 14g = 13.1(9 + g).Step 6: Compute right side: 13.1 * 9 = 117.9, so 108 + 14g = 117.9 + 13.1g.Step 7: Subtract 13.1g from both sides: 0.9g = 9.9, hence g = 11.Step 8: Total students = 9 + 11 = 20. Verification / Alternative check: Check totals: Girls total = 11 * 14 = 154. Combined total = 108 + 154 = 262. Average = 262 / 20 = 13.1, which matches the given overall average. This confirms that the computed number of students is correct. Why Other Options Are Wrong: Options 21, 22, or 19 would require different numbers of girls and would change the weighted average, failing to preserve the overall mean of 13.1. Option 18 is too small a total and cannot produce the given average with the specified subgroup averages. Only 20 yields a consistent solution. Common Pitfalls: Using a simple average of 12 and 14 instead of using the correct weighted relation. Errors while distributing 13.1 or when simplifying decimal coefficients. Assuming equal numbers of boys and girls, which is not given in the problem. Final Answer: The total number of students in the class is 20.

Question 14

The average of X, Y and Z is 24. Also, the ratio X : Y is 2 : 3 and X + Y = 60. What is the value of X − Z?

Accepted Answer

Introduction / Context: This problem combines the idea of averages with ratios and simple algebra. You are given the average of three numbers, the ratio of two of them, and the sum of those two numbers. From this information, you must find the difference between one number and the third. Such questions strengthen skills in manipulating equations and ratios in quantitative aptitude exams. Given Data / Assumptions: Average of X, Y and Z is 24. Therefore, X + Y + Z = 3 * 24 = 72. Ratio X : Y = 2 : 3. X + Y = 60. We are asked to find X − Z. Concept / Approach: First, we use the ratio X : Y = 2 : 3 with the information X + Y = 60 to find the individual values of X and Y. Then, using the total X + Y + Z = 72, we determine Z. Once X and Z are known, computing X − Z is straightforward. Step-by-Step Solution: Step 1: From X : Y = 2 : 3 and X + Y = 60, we can write X = 2k and Y = 3k.Step 2: Then X + Y = 2k + 3k = 5k = 60.Step 3: Solve for k: 5k = 60 gives k = 12.Step 4: Therefore, X = 2k = 24 and Y = 3k = 36.Step 5: From X + Y + Z = 72, we get 24 + 36 + Z = 72 so Z = 72 − 60 = 12.Step 6: Now compute X − Z = 24 − 12 = 12. Verification / Alternative check: Check averages: (24 + 36 + 12) / 3 = 72 / 3 = 24, which matches the given average. Check ratio: 24 : 36 simplifies to 2 : 3, matching the given ratio. All conditions are satisfied, confirming the internal consistency of the solution. Why Other Options Are Wrong: Options 9, 10 or 11 would correspond to different values of Z or X that would not satisfy the average or the ratio conditions simultaneously. Option 8 is even further away and would clearly break the total sum of 72 or the ratio of 2 : 3. Only 12 fits all of the given relationships. Common Pitfalls: Forgetting to multiply the average by 3 to get the total X + Y + Z. Misapplying the ratio and not using X + Y = 60 correctly. Arithmetic mistakes when solving for k and then for Z. Final Answer: The value of X − Z is 12.

Question 15

The average age of 5 members of a family is 24 years. The youngest member is 8 years old. What was the average age of the family at the time of the youngest member's birth?

Accepted Answer

Introduction / Context: This problem involves backward reasoning with averages and ages. You are given the present average age of a family, the current age of the youngest member, and you must find the family average at the time of that child's birth. Age problems of this type appear frequently in aptitude tests and require careful thinking about how ages change over time. Given Data / Assumptions: Number of family members now = 5. Current average age = 24 years. Youngest member's current age = 8 years. At the time of the youngest member's birth, there were only 4 family members. The ages of those 4 members were each 8 years less than their current ages. Concept / Approach: The idea is to first find the current total age of all 5 members. Then subtract the youngest member's age to get the combined age of the other four. To find their total age 8 years ago, we subtract 8 years from each of these four members, effectively subtracting 4 * 8 from that total. This gives the total age of the family at the time of birth, and dividing by 4 yields the required average. Step-by-Step Solution: Step 1: Current total age of 5 members = 5 * 24 = 120 years.Step 2: Current age of youngest member = 8 years.Step 3: Combined current age of the other 4 members = 120 − 8 = 112 years.Step 4: 8 years ago, each of these 4 members was 8 years younger, so total reduction in their combined ages = 4 * 8 = 32 years.Step 5: Total age of the family at the time of the youngest member's birth = 112 − 32 = 80 years.Step 6: Average age then = 80 / 4 = 20 years. Verification / Alternative check: We can move forward from the time of birth. If the average age of 4 members then was 20, total = 80. After 8 years, each gains 8 years, so total gained = 4 * 8 = 32 and new total becomes 112. Adding the youngest member now aged 8 gives 120, which divided by 5 gives the current average of 24. This confirms that our reasoning is correct. Why Other Options Are Wrong: An average of 16 or 18 years would not grow to give the correct current totals when moved forward by 8 years and combined with the child. An average of 24 or 30 years at birth is not realistic since people were younger at that time and would produce incorrect current averages. Common Pitfalls: Forgetting that the youngest member did not exist at the time of birth and mistakenly dividing by 5 instead of 4. Subtracting 8 years only once instead of for each of the 4 older members. Confusing current averages with past averages without adjusting ages. Final Answer: The average age of the family at the time of the youngest member's birth was 20 years.

Question 16

The average age of 120 members of a society is 60.7 years. After 30 new members join, the average age of all 150 members becomes 56.3 years. What is the average age of the 30 newly joined members?

Accepted Answer

Introduction / Context: This question focuses on how the average of a group changes when new members join. It uses the idea of total sums behind averages and asks you to deduce the average of the newly added members from the change in overall average. This is a common pattern in quantitative aptitude tests involving populations, salaries or age distributions. Given Data / Assumptions: Original number of members = 120. Original average age = 60.7 years. New members added = 30. Total members after addition = 150. New average age = 56.3 years. We assume only the new members changed the composition. Concept / Approach: The average age multiplied by the number of members gives the total age. We first compute the total age of the original 120 members. Then we compute the total age of all 150 members after the new members join. The difference between these totals is the combined age of the 30 new members, and dividing by 30 gives their average age. Step-by-Step Solution: Step 1: Original total age = 120 * 60.7 = 7284 years.Step 2: New total age after all 150 members = 150 * 56.3 = 8445 years.Step 3: Combined age of 30 new members = 8445 − 7284 = 1161 years.Step 4: Average age of the 30 new members = 1161 / 30.Step 5: 1161 / 30 = 38.7 years. Verification / Alternative check: We can reason qualitatively: the overall average age decreased when new members joined, which implies that their average age must be lower than 60.7. The calculated value 38.7 is indeed lower. If we reconstruct the total using 30 members of average age 38.7 and 120 members maintaining their original total, we obtain 8445 total age and 56.3 average again, confirming the calculation. Why Other Options Are Wrong: An average such as 36.5 or 37.2 would not reduce the total age enough to bring the overall average down to 56.3. A value like 38.3 would be close but not exact and would slightly change the final average. A value like 39.5 would not reduce the overall average sufficiently. Only 38.7 makes both totals consistent. Common Pitfalls: Not converting averages into totals before subtracting. Attempting to use simple differences between averages without considering group sizes. Arithmetic mistakes when multiplying decimals by large numbers. Final Answer: The average age of the newly joined 30 members is 38.7 years.

Question 17

A company starts with 8000 workers. The number of workers increases by 5 percent at the end of the first year, by 10 percent at the end of the second year, and by 20 percent at the end of the third year. How many workers does the company have in the…

Accepted Answer

Introduction / Context: This problem tests understanding of successive percentage increases, which is essentially an application of compound growth. Instead of a single percentage applied once, different rates are applied at the end of each year. Recognizing that each year's increase is calculated on the latest value is essential when dealing with populations, investments, salaries and other growing quantities. Given Data / Assumptions: Initial number of workers = 8000. Increase at end of first year = 5 percent. Increase at end of second year = 10 percent. Increase at end of third year = 20 percent. We are asked for the number of workers at the start of the fourth year, after all three increases. Concept / Approach: For successive percentage changes, we multiply the starting value by each growth factor in sequence. A 5 percent increase corresponds to multiplying by 1.05, a 10 percent increase to multiplying by 1.10, and a 20 percent increase to multiplying by 1.20. The final number is obtained by 8000 * 1.05 * 1.10 * 1.20. Step-by-Step Solution: Step 1: After first year, workers = 8000 * 1.05 = 8400.Step 2: After second year, workers = 8400 * 1.10 = 9240.Step 3: After third year, workers = 9240 * 1.20 = 11088.Step 4: Thus, at the beginning of the fourth year the company has 11088 workers. Verification / Alternative check: We can do the combined multiplication in one step: 8000 * 1.05 * 1.10 * 1.20 = 8000 * (1.05 * 1.10 * 1.20) = 8000 * 1.386 = 11088. This matches the stepwise calculation, confirming consistency. Both methods rely on the idea that each yearly percentage increase applies to the most up to date number of workers. Why Other Options Are Wrong: Option 10188 or 10080 would correspond to using simple addition of percentages or incorrect rounding rather than compounding. Option 11008 is close but does not match the correct compound product of the three factors. Option 11808 is too high and would require larger percentage increases than those given. Common Pitfalls: Adding the percentages first (5 + 10 + 20) and applying a single 35 percent increase to 8000, which is incorrect. Applying each percentage change to the original 8000 instead of the progressively updated worker count. Rounding intermediate results too aggressively and causing noticeable errors in the final value. Final Answer: The number of workers in the fourth year is 11088.

Question 18

The average of 11 numbers is 7. If every number in the list is doubled, what will be the new average of these 11 numbers?

Accepted Answer

Introduction / Context: This question checks your understanding of how the average behaves under simple linear transformations of the data, such as doubling every value. It illustrates that if each number in a list is multiplied by a constant, then the average is also multiplied by that constant. This is a fundamental property of averages that simplifies many calculations. Given Data / Assumptions: There are 11 numbers. Their original average is 7. Each number is doubled, that is, multiplied by 2. We need to find the new average after doubling. Concept / Approach: Average is defined as total sum divided by number of items. If every number is multiplied by a constant factor k, the total sum is also multiplied by k while the number of items stays the same. Therefore, the new average is simply k times the original average. Here, k = 2. Step-by-Step Solution: Step 1: Original average = 7, number of numbers = 11.Step 2: Original total sum = 7 * 11 = 77.Step 3: After doubling each number, every term is multiplied by 2, so new total sum = 2 * 77 = 154.Step 4: The number of items remains 11.Step 5: New average = 154 / 11 = 14. Verification / Alternative check: Instead of computing totals, we can directly apply the rule that multiplying each value by 2 multiplies the average by 2. Hence new average = 2 * 7 = 14. This simple rule gives the same result, confirming that our understanding of the effect of scaling on averages is correct. Why Other Options Are Wrong: Option 7 is just the original average and would be correct only if the numbers were unchanged. Option 3.5 would correspond to halving the numbers, not doubling them. Options 10.5 and 21 correspond to multiplying the average by factors of 1.5 or 3 respectively, which do not match the given operation. Common Pitfalls: Thinking the average remains unchanged regardless of operations on all data points. Confusing the effect of adding a constant with multiplying by a constant. Attempting to recompute each hypothetical new number instead of using the scaling property of averages. Final Answer: The new average of the 11 numbers is 14.

Question 19

In a cricket match, the average runs scored by 7 players is 53. The runs scored by 6 of the players are 121, 40, 26, 56, 37 and 48. How many runs did the seventh player score?

Accepted Answer

Introduction / Context: This question is a straightforward application of the definition of average. You know the average runs of all 7 players and the individual scores of 6 of them. To find the missing seventh score, you reconstruct the total runs from the average and subtract the known scores. This type of question reinforces the idea that average carries complete information about the total when the count is known. Given Data / Assumptions: Number of players = 7. Average runs of all players = 53. Runs of 6 players: 121, 40, 26, 56, 37 and 48. Score of the seventh player is unknown. Concept / Approach: Average = total runs / number of players. Therefore total runs = average * number of players. Once total runs are known, we subtract the sum of the 6 known scores to get the missing score of the seventh player. Step-by-Step Solution: Step 1: Total runs scored by all 7 players = 53 * 7 = 371.Step 2: Sum of known 6 scores = 121 + 40 + 26 + 56 + 37 + 48.Step 3: Compute the sum: 121 + 40 = 161, 161 + 26 = 187, 187 + 56 = 243, 243 + 37 = 280, 280 + 48 = 328.Step 4: Runs scored by the seventh player = 371 − 328 = 43. Verification / Alternative check: We can verify by recomputing the average. New list of scores includes 43 as the seventh value. Total runs = 328 + 43 = 371. Average = 371 / 7 = 53, which matches the original average. Therefore, the calculation is correct. Why Other Options Are Wrong: Options 26, 37, 48 or 53 would produce totals that do not give an average of 53 when combined with the other six scores. They either overshoot or undershoot the required total of 371 runs, so the average would be different from the stated value. Common Pitfalls: Miscalculating the sum of the known scores. Using division incorrectly instead of reconstructing the total from the average. Guessing the missing score from the pattern of existing numbers rather than using the formula. Final Answer: The seventh player scored 43 runs.

Question 20

The average age of 24 students in a class is 12 years. It is later found that one student's age was taken as 14 years instead of 8 years. What is the correct average age of the class?

Accepted Answer

Introduction / Context: This question is about correcting an average when an individual value was misreported. You are given the average based on incorrect data and the nature of the mistake. The task is to adjust the total accordingly and compute the new corrected average. Such questions reflect real world data correction scenarios in classrooms or surveys. Given Data / Assumptions: Number of students = 24. Original average age (with error) = 12 years. One student's age was recorded as 14 years instead of the correct 8 years. We assume all other ages were recorded correctly. Concept / Approach: We start by computing the total age based on the incorrect average. Then we adjust the total to correct the misrecorded age by subtracting the wrong value and adding the correct value. Finally, we divide the corrected total by the original number of students to obtain the corrected average. Step-by-Step Solution: Step 1: Incorrect total age based on original average = 24 * 12 = 288.Step 2: The wrong entry was 14, but it should have been 8.Step 3: Net correction needed = correct age − wrong age = 8 − 14 = −6.Step 4: Correct total age = 288 − 6 = 282.Step 5: Correct average age = 282 / 24.Step 6: 282 divided by 24 is 11.75 years. Verification / Alternative check: We can see that the incorrect total was 6 years too high because one student's age was overstated by 6 years. Reducing the total by 6 and dividing by 24 reduces the average by 6 / 24 = 0.25 years. Thus, the corrected average = 12 − 0.25 = 11.75 years, which matches our computation. Why Other Options Are Wrong: Option 11.25 would represent a decrease of 0.75 from 12, which is too much. Option 11.5 corresponds to a correction of 12 years in total, not 6. Option 12.25 increases the average, which contradicts the fact that the misrecorded age was higher than the true age. Option 12 is simply the uncorrected average. Common Pitfalls: Adding the correction instead of subtracting when the wrong age is larger than the correct age. Forgetting to recalculate the total before dividing. Making arithmetic errors when dividing 282 by 24. Final Answer: The corrected average age of the class is 11.75 years.

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