Question 1

In a cricket match the average of 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 7th player score?

Accepted Answer

Introduction / Context: This is a straightforward average and missing term problem. The total runs can be found from the average and number of players, and subtracting the known contributions reveals the unknown score of the seventh player. Given Data / Assumptions: - Total number of players considered = 7. - Average runs scored by these 7 players = 53. - Runs of 6 players: 121, 40, 26, 56, 37 and 48. - We must find the runs scored by the 7th player. Concept / Approach: Average runs = total runs / number of players. Hence total runs for 7 players are 7 times 53. The sum of the given 6 scores is subtracted from this total to obtain the remaining score. This is a classic technique for determining a missing value from an average. Step-by-Step Solution: Step 1: Total runs of 7 players = 7 * 53 = 371. Step 2: Sum of known runs = 121 + 40 + 26 + 56 + 37 + 48. Step 3: 121 + 40 = 161, 161 + 26 = 187, 187 + 56 = 243, 243 + 37 = 280, 280 + 48 = 328. Step 4: Total of known runs is 328. Step 5: Runs by 7th player = total runs - known runs = 371 - 328. Step 6: 371 - 328 = 43. Verification / Alternative check: Add the found value back and check the average. New total = 328 + 43 = 371. Average = 371 / 7 = 53, which matches the original average given in the question. This confirms that 43 is correct. Why Other Options Are Wrong: - 26, 37 and 48 are already existing scores of other players. - 50 is a plausible cricket score but does not give the correct overall average when added. Common Pitfalls: Some candidates try to average the known runs first or guess a middle value. Others may make arithmetic slips while adding the six scores. It is safer to group additions carefully or write them down step by step before subtracting from the total based on the average. Final Answer: The 7th player scored 43 runs.

Question 2

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

Accepted Answer

Introduction / Context: This question is about correcting an average when one data value was recorded incorrectly. By understanding how a single correction alters the total sum, we can quickly recalculate the true average for the entire class. Given Data / Assumptions: - Number of students = 24. - Recorded average age = 12 years. - One student age was recorded as 14 years instead of the actual age of 8 years. - Only this single error is present in the data. Concept / Approach: Average equals total sum divided by number of students. From the recorded average, we know the incorrect total sum. We adjust this total by replacing the incorrect value with the correct one; that is, we subtract the wrong value and add the correct value. Dividing the corrected total by 24 gives the correct average age. Step-by-Step Solution: Step 1: Recorded total age = 24 * 12 = 288 years. Step 2: The age 14 was used but the actual age should be 8. Step 3: Correct total age = recorded total - wrong value + correct value. Step 4: So, correct total age = 288 - 14 + 8. Step 5: 288 - 14 = 274, and 274 + 8 = 282 years. Step 6: Correct average age = 282 / 24. Step 7: 282 / 24 = 11.75 years. Verification / Alternative check: Observe that the recorded age was 6 years higher than it should have been. Thus the recorded total is 6 years too high. If we subtract 6 from 288, we again get 282, which leads to the same correct average 11.75. This provides a quick mental check on the calculation. Why Other Options Are Wrong: - 11.25, 11.5 and 12.25 correspond to incorrect adjustments to the total sum. - 12 is the original incorrect average and ignores the correction. Common Pitfalls: Learners sometimes add the difference instead of subtracting it, or they divide the correction by the wrong number. A reliable method is to always start from total = average * number of items, then carefully adjust that total based on the direction and size of the error. Final Answer: The correct average age of the class is 11.75 years.

Question 3

A bowler concedes runs in 9 cricket matches. The average runs conceded in the first 5 matches is 45 and the average runs conceded in the remaining 4 matches is 15.75. What is the average number of runs conceded per match over all 9 matches?

Accepted Answer

Introduction / Context: This question is about combining averages from two groups of matches to find a single overall average. The bowler has very different performances in two sets of matches, and we need the combined average runs conceded across all matches. Given Data / Assumptions: - Number of matches in first group = 5. - Average runs conceded in first 5 matches = 45. - Number of matches in second group = 4. - Average runs conceded in last 4 matches = 15.75. - Total matches played = 9. Concept / Approach: To find the combined average, we first find the total runs conceded in each group using average * number of matches. Adding these totals gives total runs for all 9 matches. Finally we divide this overall total by 9 to get the combined average. This is a typical weighted average situation where more matches weigh more heavily. Step-by-Step Solution: Step 1: Total runs in the first 5 matches = 5 * 45 = 225. Step 2: Total runs in the last 4 matches = 4 * 15.75. Step 3: 4 * 15.75 = 63. Step 4: Total runs in all 9 matches = 225 + 63 = 288. Step 5: Total matches = 5 + 4 = 9. Step 6: Combined average runs conceded = 288 / 9 = 32. Verification / Alternative check: Note that the average 32 is between 15.75 and 45, closer to 45 because there are more matches in the first group. This behaviour matches how weighted averages work, so the answer is reasonable. Rechecking the arithmetic 225 + 63 = 288 and 288 / 9 = 32 confirms the calculation. Why Other Options Are Wrong: - 15 and 35 are either too low or too high compared with the weighted average. - 53.5 is larger than even the first group average of 45, so it cannot be correct. - 28 results from incorrect summation or division. Common Pitfalls: A frequent error is to simply average 45 and 15.75 without weighting, which ignores that they cover different numbers of matches. Another mistake is to divide the total runs by 2 instead of 9. Always multiply averages by their counts to recover totals, then compute a new overall average correctly. Final Answer: The average runs conceded over 9 matches is 32.

Question 4

A canteen requires 56 kg of rice for seven days. Assuming consumption is uniform, what quantity of rice will be required for the combined months of April and May?

Accepted Answer

Introduction / Context: This is a simple proportionality problem where daily consumption is constant. Once the daily requirement is known, we can scale it to any number of days, here the total days in April and May together. Given Data / Assumptions: - Rice required for 7 days = 56 kg. - Consumption is assumed uniform per day. - We consider April and May together with 30 and 31 days respectively. - Total days in April and May combined = 61. Concept / Approach: If 56 kg is used in 7 days, the daily consumption is 56 / 7 kg. That daily requirement is then multiplied by the total number of days, 61, to get the total rice required. This uses the direct proportion idea that more days mean more rice in linear fashion when daily usage is fixed. Step-by-Step Solution: Step 1: Daily rice consumption = 56 / 7 kg. Step 2: 56 / 7 = 8 kg per day. Step 3: Number of days in April = 30, in May = 31. Step 4: Total days for April and May = 30 + 31 = 61. Step 5: Total rice needed = daily consumption * number of days. Step 6: So total required = 8 * 61 = 488 kg. Verification / Alternative check: As a quick check, for 56 days the requirement would be 8 * 56 = 448 kg. For 5 extra days, we need 5 * 8 = 40 kg more. Adding gives 448 + 40 = 488 kg, which matches the earlier calculation and confirms the result. Why Other Options Are Wrong: - 468, 498, 508 and 512 kg all correspond to incorrect multiplication or an incorrect day count. - Any answer not equal to 8 * 61 is mathematically inconsistent with the given daily rate. Common Pitfalls: Some learners forget that April has 30 days and May has 31 days, or they miscalculate 56 / 7. Others directly scale 56 by an incorrect factor without finding daily usage. Being precise with day counts and intermediate steps prevents such errors. Final Answer: The canteen requires 488 kg of rice for April and May together.

Question 5

Due to a price hike of 20%, 4 kg less sugar is available for Rs 120 than before. What was the initial price per kg of sugar before the increase?

Accepted Answer

Introduction / Context: This is a percentage and unit cost question involving a fixed amount of money and a change in price. When price increases, the same amount of money buys less quantity. The decrease in quantity, along with the percentage change, allows us to find the original price per kilogram. Given Data / Assumptions: - Original price of sugar = Rs p per kg. - Price increases by 20%, so new price = 1.2p per kg. - Budget for sugar = Rs 120. - Due to the increase, 4 kg less sugar is obtained for Rs 120 than before. Concept / Approach: Initially, quantity bought for Rs 120 is 120 / p kg. After the price hike, quantity is 120 / (1.2p) kg. The difference between these two quantities is given as 4 kg. This leads to an equation in p, which we can solve to find the original price per kilogram. Step-by-Step Solution: Step 1: Original quantity for Rs 120 = 120 / p kg. Step 2: New price after 20% increase = 1.2p. Step 3: New quantity for Rs 120 = 120 / (1.2p) kg. Step 4: Simplify 120 / (1.2p) = 100 / p. Step 5: According to question, original quantity minus new quantity = 4. Step 6: So, 120 / p - 100 / p = 4. Step 7: This gives 20 / p = 4, so p = 20 / 4 = 5. Verification / Alternative check: At original price Rs 5 per kg, Rs 120 buys 120 / 5 = 24 kg. After 20% increase, new price = 1.2 * 5 = Rs 6 per kg. Now Rs 120 buys 120 / 6 = 20 kg. The reduction is 24 - 20 = 4 kg, which matches the question exactly. So the answer is consistent. Why Other Options Are Wrong: - Rs 4, Rs 4.5, Rs 5.5 and Rs 6 per kg do not produce a difference of exactly 4 kg when recalculated. - In each wrong case, either the percentage increase or the quantity difference fails to match the given data. Common Pitfalls: A common mistake is to apply the percentage increase directly to the quantity instead of the price. Others forget to convert 20% to a factor of 1.2 or mishandle the fraction simplification from 120 / (1.2p) to 100 / p. Keeping track of units and operations on price versus quantity is essential. Final Answer: The initial price of sugar was Rs 5 per kg.

Question 6

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

Accepted Answer

Introduction / Context: This question uses overlapping averages over consecutive days. By converting each average into a total for four days and using the known temperature for Monday, we can deduce the temperature for Friday. Given Data / Assumptions: - Average temperature for Monday to Thursday = 48 degrees. - Average temperature for Tuesday to Friday = 46 degrees. - Temperature on Monday = 42 degrees. - Temperatures are measured in the same consistent unit (degrees). Concept / Approach: Average of four days equals the sum of those four daily temperatures divided by 4. We first find the total temperature for Monday through Thursday, then subtract the Monday temperature to get the sum for Tuesday through Thursday. We also find the total for Tuesday through Friday. Equating the Tuesday to Thursday part from both expressions allows us to solve for the unknown Friday temperature. Step-by-Step Solution: Step 1: Let M, T, W, Th and F be the temperatures from Monday to Friday. Step 2: (M + T + W + Th) / 4 = 48, so M + T + W + Th = 192. Step 3: (T + W + Th + F) / 4 = 46, so T + W + Th + F = 184. Step 4: Given M = 42. Step 5: From Step 2, T + W + Th = 192 - M = 192 - 42 = 150. Step 6: Substitute into Step 3: 150 + F = 184. Step 7: Therefore F = 184 - 150 = 34 degrees. Verification / Alternative check: Check consistency with averages. If Monday is 42 and the total Monday to Thursday is 192, then the average is 192 / 4 = 48, as stated. For Tuesday to Friday, the total is 150 + 34 = 184 and the average is 184 / 4 = 46, also matching. So the solution is correct. Why Other Options Are Wrong: - Values 36, 38, 40 and 42 would change the total for Tuesday to Friday and would not give an average of 46. - Only 34 satisfies both overlapping average conditions at the same time. Common Pitfalls: Some learners try to average 48 and 46 or guess a middle value for Friday. Others forget that Monday appears only in the first average and Friday only in the second. Writing equations for sums makes the relationship clear and avoids such confusion. Final Answer: The temperature on Friday was 34 degrees.

Question 7

What is the average of the first 93 natural numbers?

Accepted Answer

Introduction / Context: This question tests knowledge of the properties of consecutive natural numbers and the standard formula for their average. The first n natural numbers form an arithmetic progression, so their average is easy to compute using endpoints. Given Data / Assumptions: - We consider the first 93 natural numbers: 1, 2, 3, ..., 93. - They form an arithmetic progression with common difference 1. - We must find the average of these 93 numbers. Concept / Approach: For any arithmetic progression with first term a and last term l, the average (mean) of all terms is equal to (a + l) / 2. This is because the sequence is symmetric around the middle value. The first n natural numbers fit this pattern with a = 1 and l = n, so the average is (1 + n) / 2. Step-by-Step Solution: Step 1: Identify first term a = 1. Step 2: Identify last term l = 93. Step 3: Recognize that the numbers 1 to 93 form an arithmetic progression. Step 4: Use the formula for the average of an arithmetic progression: average = (a + l) / 2. Step 5: Substitute values: average = (1 + 93) / 2. Step 6: 1 + 93 = 94, and 94 / 2 = 47. Verification / Alternative check: Note that the middle of 1 and 93 is 47, and the sequence is symmetric: 1 pairs with 93, 2 with 92, and so on, each pair averaging to 47. Since every such pair has average 47, the whole set must also have average 47, confirming the formula based result. Why Other Options Are Wrong: - 45, 46, 49 and 50 do not match the midpoint between 1 and 93 and therefore cannot equal the average of this symmetric sequence. Common Pitfalls: Some learners try to sum all numbers up to 93 and then divide, which is time consuming and error prone. Others mistakenly think that the average equals the number of terms, 93, or half of 93. Remember the simple midpoint formula (first + last) / 2 for any consecutive sequence. Final Answer: The average of the first 93 natural numbers is 47.

Question 8

The average marks in Science for a class of 20 students is 68. Later it is found that the marks of two students were misread as 48 and 65 instead of the correct marks 72 and 61 respectively. What is the correct average marks of the class?

Accepted Answer

Introduction / Context: This problem involves correcting an average when two entries were misread. You must adjust the total marks based on the difference between the wrong and correct values, and then recompute the class average. Given Data / Assumptions: - Number of students = 20. - Recorded average marks = 68. - Two marks were recorded as 48 and 65. - The correct marks should be 72 and 61 respectively. - Only these two entries are in error. Concept / Approach: From the recorded average we find the inaccurate total marks. We then adjust this total by removing the wrong values and adding the correct ones. The net change in the total divided by the number of students will show how much the average shifts. Finally, we compute the corrected average by dividing the corrected total by 20. Step-by-Step Solution: Step 1: Recorded total marks = 20 * 68 = 1360. Step 2: Sum of wrong entries = 48 + 65 = 113. Step 3: Sum of correct entries = 72 + 61 = 133. Step 4: Correct total marks = recorded total - wrong sum + correct sum. Step 5: Correct total = 1360 - 113 + 133. Step 6: 1360 - 113 = 1247; 1247 + 133 = 1380. Step 7: Correct average = 1380 / 20 = 69. Verification / Alternative check: Note that the net change in the total is 133 - 113 = 20 marks. Spreading an extra 20 marks over 20 students increases the average by 1 mark. Thus the corrected average is 68 + 1 = 69, matching the detailed computation above. This provides a quick mental verification. Why Other Options Are Wrong: - 68.5 and 69.5 correspond to spreading an incorrect adjustment over the class. - 70 assumes a total increase of 40 marks rather than 20. - 67.5 would imply a fall in average, which contradicts the fact that the corrected marks are higher overall. Common Pitfalls: Learners often forget to subtract the misread values before adding the correct ones, or they adjust the average directly without considering the number of students. Always work through the total marks step by step to avoid these errors. Final Answer: The correct class average in Science is 69 marks.

Question 9

A group of boys has an average weight of 36 kg. One boy weighing 42 kg leaves the group and another boy weighing 30 kg joins the group. If the new average weight becomes 35.7 kg, how many boys are there in the group?

Accepted Answer

Introduction / Context: This question is a classic average and replacement problem. Removing a heavier boy and adding a lighter one decreases the average weight slightly. From the size of this decrease we can compute how many boys are in the group. Given Data / Assumptions: - Initial average weight of the group = 36 kg. - One boy of weight 42 kg leaves. - A new boy of weight 30 kg joins. - New average weight of the group = 35.7 kg. - Let the initial number of boys be n. Concept / Approach: Initially, total weight equals 36n. After the change, the total weight becomes 36n minus 42 plus 30. The number of boys remains n, so the new average is (36n - 12) / n, which is given as 35.7. Solving this equation for n gives the size of the group. This approach directly uses the definition of average as total divided by count. Step-by-Step Solution: Step 1: Initial total weight = 36n. Step 2: When the 42 kg boy leaves, total becomes 36n - 42. Step 3: A 30 kg boy joins, so new total weight = 36n - 42 + 30 = 36n - 12. Step 4: New average weight = (36n - 12) / n. Step 5: Given new average = 35.7, so (36n - 12) / n = 35.7. Step 6: Multiply both sides by n: 36n - 12 = 35.7n. Step 7: Rearrange: 36n - 35.7n = 12, so 0.3n = 12 and n = 12 / 0.3 = 40. Verification / Alternative check: For n = 40, initial total weight = 36 * 40 = 1440 kg. Removing 42 kg gives 1398 kg. Adding 30 kg yields 1428 kg. New average = 1428 / 40 = 35.7 kg, which matches the question. This confirms that the calculation is correct. Why Other Options Are Wrong: - 30, 32, 36 and 56 do not satisfy the equation (36n - 12) / n = 35.7. - Substituting any of these values leads to either a different new average or a non integer group effect. Common Pitfalls: Some candidates replace the average directly without forming an equation, or incorrectly subtract both 42 and 30 from the total. Others forget that the number of boys remains unchanged. Always express the new situation in terms of the same variable n and then solve systematically. Final Answer: There are 40 boys in the group.

Question 10

Three numbers a, b and c satisfy the equations a + b = 5, b + c = 7.5 and c + a = 8.5. What is the average value of a, b and c?

Accepted Answer

Introduction / Context: This algebra question uses pairwise sums of three numbers to find their average. Rather than solving individually for a, b and c, you can use symmetry and summation to find the total directly and then compute the average. Given Data / Assumptions: - a + b = 5. - b + c = 7.5. - c + a = 8.5. - All three are real numbers. - We must find (a + b + c) / 3. Concept / Approach: By adding the three given equations, each variable a, b and c appears exactly twice in the total sum. Thus the left side becomes 2(a + b + c), and the right side becomes the sum of the three pairwise sums. This allows us to find a + b + c directly without solving for each variable separately. Dividing by 3 gives the required average. Step-by-Step Solution: Step 1: Add the three equations: (a + b) + (b + c) + (c + a). Step 2: Left side simplifies to 2a + 2b + 2c = 2(a + b + c). Step 3: Right side equals 5 + 7.5 + 8.5. Step 4: Compute right side: 5 + 7.5 = 12.5, 12.5 + 8.5 = 21. Step 5: So 2(a + b + c) = 21. Step 6: Therefore a + b + c = 21 / 2 = 10.5. Step 7: Average of a, b and c = (a + b + c) / 3 = 10.5 / 3 = 3.5. Verification / Alternative check: Optionally, you can solve for individual values. From a + b = 5 and a + c = 8.5, subtract to get (a + c) - (a + b) = 3.5, so c - b = 3.5. Combine with b + c = 7.5 to solve and you will again obtain a, b, c that sum to 10.5. This confirms that the average 3.5 is consistent. Why Other Options Are Wrong: - 1.5, 2.5, 3 and 4.5 do not satisfy the derived condition that a + b + c = 10.5. - Any average other than 3.5 would give a different total and violate one or more of the given equations. Common Pitfalls: Some learners attempt to guess values or solve three equations for three unknowns in a longer way. Others misadd the constants on the right side. Recognizing the symmetry and adding equations is the fastest and cleanest approach here. Final Answer: The average of a, b and c is 3.5.

Question 11

Seven friends spend Rs 14 each on a project. An eighth friend spends Rs 21 more than the average expenditure of all eight friends. What is the total amount of money, in rupees, spent by all of them together?

Accepted Answer

Introduction / Context: This question involves an unknown average where one of the contributors spends an amount defined relative to that average. We must set up an equation in terms of the average and then use it to compute the total expenditure. Given Data / Assumptions: - Seven friends each spend Rs 14. - There are 8 friends in total. - The eighth friend spends Rs 21 more than the average expenditure of all eight friends. - Let the average expenditure of all eight friends be A rupees. Concept / Approach: Total expenditure equals number of friends multiplied by the average, that is 8A. It also equals the sum of the amounts spent by the seven friends plus the amount spent by the eighth friend, which is A + 21. Equating these two expressions for the total gives a linear equation for A. Once A is found, the overall total can be calculated as 8A. Step-by-Step Solution: Step 1: Total spent by seven friends = 7 * 14 = 98 rupees. Step 2: Let average expenditure of all eight be A rupees. Step 3: Eighth friend spends A + 21 rupees. Step 4: Total expenditure T can be written as T = 98 + (A + 21) = 119 + A. Step 5: By definition of average, T is also equal to 8A. Step 6: Set 8A = 119 + A. Step 7: Rearrange: 8A - A = 119, so 7A = 119 and A = 17. Step 8: Total expenditure T = 8A = 8 * 17 = 136 rupees. Verification / Alternative check: Check the eighth friend expenditure. Eighth friend spends A + 21 = 17 + 21 = 38 rupees. Then total from eight friends is 7 * 14 + 38 = 98 + 38 = 136. Average = 136 / 8 = 17, which matches A. This confirms internal consistency. Why Other Options Are Wrong: - 133, 141, 147 and 155 lead to different implied averages that do not satisfy the condition that the eighth friend spends 21 more than that average. Common Pitfalls: Some learners treat the extra 21 rupees as added to 14 instead of to the average, or forget to include the eighth friend in the total count for the average. Others attempt guesswork instead of forming an equation. A clear definition of A and equating totals is the safest method. Final Answer: The total amount of money spent by the eight friends is Rs 136.

Question 12

In a class the ratio of the number of boys to the number of girls is 2 : 3. The average weight of the boys is 18 kg and the average weight of the girls is 21 kg. What is the average weight, in kg, of all the students together?

Accepted Answer

Introduction / Context: This question tests understanding of weighted averages when group sizes differ. The class has boys and girls in a fixed ratio, and their separate average weights are given. We must find the combined average weight of all students. Given Data / Assumptions: - Ratio of boys : girls = 2 : 3. - Average weight of boys = 18 kg. - Average weight of girls = 21 kg. - We assume all students are counted once and weights are in kilograms. Concept / Approach: Choose convenient absolute numbers consistent with the ratio, for example 2 boys and 3 girls. Their total weight can be computed using the given averages. The overall average weight is then total weight divided by total number of students. Because averages depend only on relative ratios, the result will be the same for any scaled group sizes with ratio 2 : 3. Step-by-Step Solution: Step 1: Assume there are 2 boys and 3 girls. Step 2: Total weight of boys = 2 * 18 = 36 kg. Step 3: Total weight of girls = 3 * 21 = 63 kg. Step 4: Total weight of all students = 36 + 63 = 99 kg. Step 5: Total number of students = 2 + 3 = 5. Step 6: Average weight = total weight / total students = 99 / 5 kg. Step 7: 99 / 5 = 19.8 kg if expressed as a decimal. Verification / Alternative check: Alternatively, use the weighted average formula directly. Overall average = (2 * 18 + 3 * 21) / (2 + 3). This equals (36 + 63) / 5 = 99 / 5, perfectly matching the earlier calculation. This confirms that the chosen counts respect the ratio and yield the correct combined average. Why Other Options Are Wrong: - 101/5, 96/5 and 109/6 represent different overall totals or weights and do not match the correct weighted sum. - 21 kg would be the average only if all students had the same average as the girls, ignoring the lighter boys. Common Pitfalls: Many learners simply average 18 and 21 without weighting, which ignores the fact that there are more girls than boys. Others misuse the ratio and pick wrong counts. Always multiply each group average by its relative group size to form the correct weighted sum before dividing by the total count. Final Answer: The average weight of all students together is 99/5 kg.

Question 13

The average of 8 consecutive natural numbers is 38.5. What is the largest of these 8 numbers?

Accepted Answer

Introduction / Context: This problem uses properties of consecutive natural numbers and their average. When numbers are consecutive, the average lies exactly in the middle of the first and last terms, which helps us reconstruct the sequence from the average. Given Data / Assumptions: - There are 8 consecutive natural numbers. - Their average is 38.5. - We must determine the largest of the 8 numbers. Concept / Approach: Let the first number be x, then the sequence is x, x + 1, x + 2, ..., x + 7. For any arithmetic progression, the average equals the mean of first and last terms. Thus average = (x + x + 7) / 2 = x + 3.5. Setting this equal to 38.5 allows us to solve for x and then find x + 7, the largest number in the list. Step-by-Step Solution: Step 1: Represent the 8 numbers as x, x + 1, x + 2, x + 3, x + 4, x + 5, x + 6, x + 7. Step 2: The first term is x and the last term is x + 7. Step 3: Average of an arithmetic progression = (first + last) / 2. Step 4: So average = (x + (x + 7)) / 2 = (2x + 7) / 2 = x + 3.5. Step 5: Given average is 38.5, so x + 3.5 = 38.5. Step 6: Solve: x = 38.5 - 3.5 = 35. Step 7: Largest number = x + 7 = 35 + 7 = 42. Verification / Alternative check: List the numbers explicitly: 35, 36, 37, 38, 39, 40, 41 and 42. Their sum is 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 = 308. Average = 308 / 8 = 38.5, which matches the given value and confirms that 42 is indeed the largest number. Why Other Options Are Wrong: - 39, 40 and 41 are smaller than the correct largest term for this sequence. - 45 is too large; including such a number would shift the average above 38.5. Common Pitfalls: Learners sometimes mistakenly treat 38.5 as one of the terms or misplace the largest term by one or two positions. Remember that for an even number of consecutive terms, the average lies between the two middle terms, not necessarily equal to one of them. Using the first and last term relation is more reliable. Final Answer: The largest of the 8 consecutive natural numbers is 42.

Question 14

In a school, the mean marks of 65 students in Division A of Class X is 54, for 30 students in Division B is 50 and for 55 students in Division C is 48. What is the combined mean marks of all the students of the three divisions of Class X?

Accepted Answer

Introduction / Context: This aptitude question tests the concept of weighted mean, where different groups have different sizes and different averages. Instead of simply averaging the three given means, you must correctly account for the number of students in each division to find the overall mean for the full class. Given Data / Assumptions: Division A of Class X has 65 students with mean marks 54. Division B has 30 students with mean marks 50. Division C has 55 students with mean marks 48. We need the combined mean marks of all students in the three divisions together. Concept / Approach: When several groups have different sizes, the overall mean is a weighted mean. We first convert each group mean to a total marks figure by multiplying mean by number of students. Then we add all these totals to get the grand total. Finally, we divide the grand total by the total number of students in all divisions. Taking a simple average of the three means would ignore the different group sizes and give a wrong answer. Step-by-Step Solution: Step 1: Total marks of Division A = 65 * 54 = 3510. Step 2: Total marks of Division B = 30 * 50 = 1500. Step 3: Total marks of Division C = 55 * 48 = 2640. Step 4: Grand total marks of all students = 3510 + 1500 + 2640 = 7650. Step 5: Total number of students = 65 + 30 + 55 = 150. Step 6: Overall mean marks = 7650 / 150 = 51. Verification / Alternative check: As a quick reasonableness check, note that Division A has the highest mean and also a relatively large strength. Division C has the lowest mean but not the largest strength. Therefore the overall mean must lie between the extremes 48 and 54, and it should be a little closer to 54 because Division A has many students. The value 51 fits this intuition well. Why Other Options Are Wrong: Option 50.3 and option 49.6 are too close to the lowest group mean of 48 and ignore the strong pull of Division A with mean 54. Option 52.4 is too high and would require several more students scoring near 54 or above, which is not supported by the given data. Only 51 matches the correct weighted mean calculation. Common Pitfalls: A very common mistake is to simply average 54, 50 and 48 without considering the different numbers of students. That simple average is (54 + 50 + 48) / 3 = 50.67, which does not match any given choice and is incorrect. Always remember that when group sizes differ, total marks and total students must be used to compute the overall mean. Final Answer: The combined mean marks of all three divisions of Class X is 51.

Question 15

Among four students sitting in a row, the average age of the last three students is 20 years and the average age of the first three students is 21 years. If the age of the first student is 26 years, what is the age in years of the last student?

Accepted Answer

Introduction / Context: This problem uses averages of overlapping groups to find an individual value. Four students sit in a row, and we know the average age of the first three and the last three. Using these overlapping averages and the age of the first student, we can determine the age of the last student at the end of the row. Given Data / Assumptions: There are four students in a row, call them A, B, C and D in order. The average age of the last three students B, C and D is 20 years. The average age of the first three students A, B and C is 21 years. The age of the first student A is 26 years. We must find the age of the last student D. Concept / Approach: When we know averages of overlapping groups, we can convert them to equations using sums. From the average of the last three students we get B + C + D. From the average of the first three students we get A + B + C. Substituting the given value of A lets us find B + C, which can then be used to find D. This is a standard technique for handling overlapping averages in aptitude questions. Step-by-Step Solution: Step 1: From the last three students, (B + C + D) / 3 = 20, so B + C + D = 60. Step 2: From the first three students, (A + B + C) / 3 = 21, so A + B + C = 63. Step 3: The age of A is 26, so B + C = 63 - 26 = 37. Step 4: Substitute B + C = 37 into B + C + D = 60 to get 37 + D = 60. Step 5: Therefore D = 60 - 37 = 23. Verification / Alternative check: Check that these ages are consistent. We do not know the exact values of B and C, but we know B + C is 37. So A + B + C = 26 + 37 = 63, whose average is 63 / 3 = 21. Also B + C + D = 37 + 23 = 60, whose average is 60 / 3 = 20. Both match the original conditions, confirming the value of D as 23 years. Why Other Options Are Wrong: If D were 37, then B + C would have to be 23 to keep B + C + D = 60, which would make A + B + C less than 63 and the first average smaller than 21. If D were 24 or 29, similar checks break one of the two given average conditions. Only D = 23 satisfies both equations at the same time. Common Pitfalls: Learners sometimes try to treat the averages as simple independent values and forget to convert them to sums. Others mistakenly assume that B and C must have the same age. The safe method is to set up equations using sums, substitute known values step by step, and solve for the unknown age carefully. Final Answer: The age of the last student is 23 years.

Question 16

Of three numbers whose average is 40, the first number is one third of the sum of the other two numbers. What is the value of the first number?

Accepted Answer

Introduction / Context: This question combines averages with a simple algebraic relationship between three numbers. You are given the average of the numbers and a condition that connects the first number with the sum of the other two. The task is to translate the words into equations and then solve for the unknown first number. Given Data / Assumptions: There are three numbers, call them x, y and z. Their average is 40. The first number x is one third of the sum of the other two numbers, so x is equal to one third of (y + z). We need to find the value of the first number x. Concept / Approach: The average of three numbers is given by (x + y + z) / 3. We turn the average information into an equation for the sum x + y + z. The second condition links x directly to y + z. By substituting this relationship into the total sum, we reduce the problem to a single variable equation in x and solve it using basic algebra. Step-by-Step Solution: Step 1: From the average, (x + y + z) / 3 = 40, so x + y + z = 40 * 3 = 120. Step 2: From the condition, x is one third of the sum of the other two numbers, so x = (y + z) / 3. Step 3: Let S = y + z. Then x = S / 3 and x + y + z = x + S = 120. Step 4: Substitute x = S / 3 into x + S = 120 to get S / 3 + S = 120. Step 5: Combine terms: S / 3 + S = S / 3 + 3S / 3 = 4S / 3, so 4S / 3 = 120. Step 6: Multiply both sides by 3 to get 4S = 360, so S = 360 / 4 = 90. Step 7: Now x = S / 3 = 90 / 3 = 30. Verification / Alternative check: Take x = 30. Then y + z must be 90 to satisfy x = one third of (y + z). The total x + y + z is 30 + 90 = 120, and the average is 120 / 3 = 40, which matches the given information. Thus the value 30 is fully consistent with both conditions. Why Other Options Are Wrong: If x were 20, then y + z would be 60, giving a total of 80 and an average well below 40. If x were 50, then y + z would be 150, giving a total of 200 and an average far above 40. If x were 25, the equations would also fail to produce the given average of 40. Only x = 30 satisfies both the average condition and the one third relationship. Common Pitfalls: Some learners confuse one third of the sum with the sum being three times the first number and write incorrect equations. Others forget to multiply the average by 3 to get the total. Writing the equations carefully and introducing a helper symbol like S for y + z makes the algebra easier to manage. Final Answer: The first number is 30.

Question 17

The average of 14 numbers is 32. If the average of the last 5 numbers is 26, what is the average of the remaining 9 numbers?

Accepted Answer

Introduction / Context: This question focuses on how averages change when a subset of numbers is separated out. You know the overall average for 14 numbers and a separate average for the last 5 numbers. Using these, you must compute the average of the remaining 9 numbers. Given Data / Assumptions: There are 14 numbers in total. Their overall average is 32. The average of the last 5 numbers is 26. We are asked to find the average of the first 9 numbers or equivalently the remaining 9 numbers. Concept / Approach: The key idea is that average multiplied by count gives the total sum. We first find the total of all 14 numbers from the overall average. Then we find the total of the last 5 numbers from their average. By subtracting the total of these 5 numbers from the grand total, we obtain the total of the remaining 9 numbers. Finally, we divide that result by 9 to get their average. Step-by-Step Solution: Step 1: Total of all 14 numbers = 14 * 32. Step 2: Compute 14 * 32 = 448. Step 3: Total of the last 5 numbers = 5 * 26 = 130. Step 4: Total of the remaining 9 numbers = 448 - 130 = 318. Step 5: Average of the remaining 9 numbers = 318 / 9. Step 6: Compute 318 / 9 = 35.333..., which is written as 35.33 to two decimal places. Verification / Alternative check: If the 9 earlier numbers have an average slightly above 35, and the 5 later numbers have an average of 26, it is reasonable that the overall average of all 14 numbers is between these two values at 32. The calculation 448 = 318 + 130 also checks out, so the arithmetic is consistent. Why Other Options Are Wrong: An average of 41.33 would be too high, pulling the overall average above 32 when combined with the last 5 numbers at 26. Options 27.5 and 44.5 are even more extreme and do not produce the correct grand total when back calculated. Only 35.33 leads to a total of 318 for the 9 numbers, which correctly combines with 130 for the last 5 to give 448 overall. Common Pitfalls: Many learners mistakenly average 32 and 26 directly, which has no mathematical justification here because the two groups have different sizes. Others forget to multiply by the number of terms when converting averages to totals. Always work with totals when combining or separating groups of numbers with known averages. Final Answer: The average of the remaining 9 numbers is 35.33.

Question 18

The average weight of Shubha, Govinda and Reshma is 65 kg. If the average weight of Shubha and Govinda is 63 kg and the average weight of Govinda and Reshma is 70 kg, what is the weight of Govinda in kilograms?

Accepted Answer

Introduction / Context: This question uses averages of small groups to determine the individual weight of one person. You are given averages for three persons together and for two different pairs among them. By translating these averages into equations, you can isolate and find the weight of Govinda. Given Data / Assumptions: Three people are involved: Shubha, Govinda and Reshma, with weights S, G and R in kilograms. The average weight of all three is 65 kg. The average weight of Shubha and Govinda is 63 kg. The average weight of Govinda and Reshma is 70 kg. We want the value of Govinda's weight G in kilograms. Concept / Approach: We convert each average into a sum equation. From the three person average, we get S + G + R. From each two person average we get S + G and G + R. Using these three equations, we can first find R and then find G, or directly manipulate them to isolate G. This is a common technique in average problems involving three unknowns and three equations. Step-by-Step Solution: Step 1: From the three person average, (S + G + R) / 3 = 65, so S + G + R = 195. Step 2: From the average of Shubha and Govinda, (S + G) / 2 = 63, so S + G = 126. Step 3: From the average of Govinda and Reshma, (G + R) / 2 = 70, so G + R = 140. Step 4: Use S + G + R = 195 and S + G = 126 to get R = 195 - 126 = 69. Step 5: Use G + R = 140 and R = 69 to get G = 140 - 69 = 71. Verification / Alternative check: Check all three conditions with S, G and R. From S + G = 126 and G = 71, we get S = 55. Now S + G + R = 55 + 71 + 69 = 195, and 195 / 3 = 65, matching the given three person average. Also, G + R = 71 + 69 = 140, and 140 / 2 = 70, matching the two person average. This confirms that Govinda's weight is 71 kg. Why Other Options Are Wrong: If Govinda weighed 41, 88 or 46 kilograms, the pair averages 63 and 70 could not both hold at the same time. Substituting any of those values into S + G = 126 or G + R = 140 would lead to contradictions when checking the three person total of 195. Only G = 71 satisfies all three equations simultaneously. Common Pitfalls: A frequent mistake is to misinterpret the phrase average of two people and treat 63 or 70 as totals instead of averages. Another issue is mixing up who is in each pair. Always convert averages to totals by multiplying by the number of people, write clear equations and then solve step by step. Final Answer: Govinda weighs 71 kilograms.

Question 19

The average of 100 items is found to be 30. During the calculation two items were wrongly taken as 32 and 12 instead of 23 and 11. What is the correct average of the 100 items?

Accepted Answer

Introduction / Context: This question involves correcting an average when two observations have been entered incorrectly. Such data correction problems are very common in aptitude and data interpretation sections. The key is to adjust the total sum using the correct and incorrect values and then recompute the average. Given Data / Assumptions: The reported average of 100 items is 30. Two items were mistakenly taken as 32 and 12. The correct values for those two items are 23 and 11. We need to find the correct average of all 100 items. Concept / Approach: Average equals total sum divided by number of items. The reported average is based on an incorrect total. We first find this incorrect total by multiplying average by number of items. Then we remove the wrong values and add the correct values to get the corrected total. Finally, we divide the corrected total by 100 again to obtain the correct average. Step-by-Step Solution: Step 1: Incorrect total sum = reported average * number of items = 30 * 100 = 3000. Step 2: The wrong values used were 32 and 12, whose total is 32 + 12 = 44. Step 3: The correct values are 23 and 11, whose total is 23 + 11 = 34. Step 4: To correct the total, subtract the wrong total and add the correct total: corrected total = 3000 - 44 + 34. Step 5: Compute 3000 - 44 = 2956, then 2956 + 34 = 2990. Step 6: Correct average = corrected total / number of items = 2990 / 100 = 29.9. Verification / Alternative check: Notice that the net effect of the error on the total is 44 - 34 = 10 extra marks added by mistake. Therefore, the corrected total should be 3000 - 10 = 2990, which matches our detailed calculation. Dividing by 100 gives 29.9, so the answer is consistent by two different checks. Why Other Options Are Wrong: Option 29.8 would correspond to a total of 2980, which is 10 less than the correct total. Option 29 would require a total of 2900, implying an error of 100 marks. Option 29.5 would imply a total of 2950. None of these match the precise net correction of 10 marks from the given wrong and correct values. Common Pitfalls: Some learners mistakenly average the wrong and correct values or adjust the average directly by the difference divided by number of changed items rather than the total number of observations. The safe method is always to work with the total sum, apply the correction, and then recompute the average with the full sample size. Final Answer: The correct average of the 100 items is 29.9.

Question 20

The average revenues of 11 consecutive years of a company is Rs 69 lakhs. The average revenue of the first 6 years is Rs 64 lakhs and the average revenue of the last 6 years is Rs 76 lakhs. What is the revenue for the 6th year?

Accepted Answer

Introduction / Context: This problem deals with average revenues over overlapping periods of years. You are told the average for 11 consecutive years, as well as the averages for the first 6 and the last 6 years. Because the 6th year is counted in both of these six year averages, we can use this overlap to find the revenue for that particular year. Given Data / Assumptions: There are 11 consecutive years of revenue figures. The overall average revenue for the 11 years is Rs 69 lakhs. The average revenue for the first 6 years is Rs 64 lakhs. The average revenue for the last 6 years is Rs 76 lakhs. Year 6 is common to both the first 6 and the last 6 years. We need to find the revenue in the 6th year. Concept / Approach: We convert each average into a total revenue for that period. Let the revenues in years 1 to 11 be Y1, Y2, ..., Y11. Then we express the total of years 1 to 6 and the total of years 6 to 11 in terms of these symbols. Because both range sums include Y6, we can add or subtract equations to isolate Y6. We also use the total for all 11 years from the overall average. Step-by-Step Solution: Step 1: Total revenue for all 11 years = 69 * 11 = 759 lakhs. Step 2: Total revenue for the first 6 years = 64 * 6 = 384 lakhs. Step 3: Total revenue for the last 6 years (years 6 to 11) = 76 * 6 = 456 lakhs. Step 4: Let Y6 be the revenue in year 6. Then sum of years 1 to 5 is 384 - Y6 and sum of years 7 to 11 is 456 - Y6. Step 5: Total revenue for all 11 years is (sum of years 1 to 5) + Y6 + (sum of years 7 to 11) = (384 - Y6) + Y6 + (456 - Y6) = 840 - Y6. Step 6: Set this equal to the total 759: 840 - Y6 = 759, so Y6 = 840 - 759 = 81. Verification / Alternative check: If the 6th year revenue is Rs 81 lakhs, then the first 6 years total 384, so the first 5 years sum to 384 - 81 = 303 lakhs. The last 6 years total 456, so years 7 to 11 sum to 456 - 81 = 375 lakhs. Adding 303 + 81 + 375 gives 759, and 759 / 11 is 69, which matches the overall average, confirming the result. Why Other Options Are Wrong: Values such as 83, 79 or 77 lakhs would lead to totals for the first and last sets of years that do not match 384 and 456 simultaneously when combined into 11 years. Substituting any of these alternatives for Y6 would break the equation 840 - Y6 = 759, so they cannot be correct. Common Pitfalls: A frequent mistake is to think that the 6th year revenue is simply the difference between two averages or to misidentify which years are included in each range. Another pitfall is not converting averages to totals and trying to reason only with averages. Always write out total revenues and use clear algebra when years overlap. Final Answer: The revenue in the 6th year is Rs 81 lakhs.

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