Question 1

There were 35 students in a hostel. If the number of the students is increased by 7, then the expenses of the mess increase by ₹ 42 per day, while the average expenditure per head diminishes by ₹ 1. The original expenditure of the mess per day was

Accepted Answer

Concept & Logic When a group expands, total expenditure changes by a fixed addition, while the new average becomes the old average minus the reduction amount. Use linear equations tracking the original average expenditure per head. Step-by-Step Solution • Let the original average expenditure per student per day be ₹ x. • Original daily total expenditure for 35 students = 35x. • New number of students = 35 + 7 = 42. • New average expenditure per head = ₹ (x - 1). • New daily total expenditure for 42 students = 42(x - 1). • Given that the total daily expenditure increased by ₹ 42: 42(x - 1) - 35x = 42 • Solve the linear equation: 42x - 42 - 35x = 42 7x = 84 x = 12 • Find the original daily expenditure of the mess: Original Expenditure = 35x = 35 × 12 = ₹ 420 Exam Strategy & Shortcut Option Elimination Method: The original daily expenditure must be a multiple of 35 because Total = 35 × x. • 400 is not divisible by 35. • 420 divided by 35 gives exactly 12 (an integer value for initial head cost). Let's check: New total = 420 + 42 = 462. New per head cost = 462 / 42 = 11, which is exactly 1 less than 12. Option (b) matches perfectly! Common Pitfall Be cautious when structuring the difference equation. Ensure you subtract the original cost from the new cost (42(x-1) - 35x = 42) rather than accidentally swapping them, which results in negative numbers. Final Answer Therefore, the correct answer is ₹ 420.

Question 2

Out of three numbers, the first is twice the second and is half of the third. If the average of the three numbers is 56, then difference of first and third numbers is

Accepted Answer

Concept & Logic Establish a common ratio between the numbers to find their total sum in terms of parts. Use the given average to find the actual value of one part, then calculate the required difference directly from the ratio parts. Step-by-Step Solution • Let the three numbers be A, B, and C. • Based on the conditions: A = 2B \implies B = (A/2) A = (C/2) \implies C = 2A • To avoid fractions, let's assume a base variable for B. • Let Second number B = x. • First number A = 2x. • Third number C = 2A = 2(2x) = 4x. • The sum of the numbers is 2x + x + 4x = 7x. • The average of the three numbers is given as 56. Average = \frac{Sum}{3} = 56 \implies Sum = 56 × 3 = 168 • Equate the sum expressions: 7x = 168 x = 24 • The First number A = 2(24) = 48. • The Third number C = 4(24) = 96. • Difference between the third and first number = 96 - 48 = 48. Exam Strategy & Shortcut Write the ratio directly: A:B:C = 2:1:4. The average of these parts is (2+1+4/3) = (7/3) units. We know (7/3) units = 56. Therefore, 1 unit = 56 × (3/7) = 24. The question asks for the difference between the first and third numbers (4 - 2 = 2 units). Difference = 2 × 24 = 48. This ratio method skips solving for every individual number. Common Pitfall Misinterpreting the phrasing "and is half of the third". Students sometimes mistakenly think the *second* number is half of the third (B = C/2), instead of the *first* number (A = C/2). Always link the conjunction "and" back to the primary subject of the sentence. Final Answer Therefore, the correct answer is 48.

Question 3

Of the three numbers, the first is twice the second and the second is twice the third. The average of the reciprocal of the numbers is $\frac{7}{72}$. The numbers are :

Accepted Answer

Concept & Logic When a question provides the average of reciprocals along with multiple-choice options, it is often significantly faster to test the options rather than solving the algebraic equation, provided the relations are simple. Step-by-Step Solution • Let the third number be x. • The second number is 2x. • The first number is 2(2x) = 4x. • The numbers are 4x, 2x, and x. • Their reciprocals are (1/4x), (1/2x), and (1/x). • Find the sum of these reciprocals. The common denominator is 4x: Sum = (1/4x) + (2/4x) + (4/4x) = (7/4x) • The average of the 3 reciprocals is: Average = (\frac{7/4x)}{3} = (7/12x) • We are given that the average is (7/72). Equate them: (7/12x) = (7/72) 12x = 72 \implies x = 6 • Substitute x = 6 back into our expressions for the numbers: Third = 6 Second = 2(6) = 12 First = 4(6) = 24 • The numbers are 24, 12, 6. Exam Strategy & Shortcut Option Elimination: Notice that all given options already follow the "first is twice the second, second is twice the third" rule (4:2:1 ratio). Just test the reciprocals of Option (c) 24, 12, 6. Reciprocals: (1/24), (1/12), (1/6). Sum = (1 + 2 + 4/24) = (7/24). Average = (7/24) \div 3 = (7/72). It matches perfectly. You can do this mental arithmetic in 10 seconds. Common Pitfall A classic error is forgetting to divide the sum of the reciprocals by 3 to find the average, mistakenly equating the sum (7/4x) directly to (7/72), yielding x = 18, which leads to the incorrect option (d) 72, 36, 18 (or 36, 18, 9 if further miscalculated). Final Answer Therefore, the correct answer is 24, 12, 6.

Question 4

In the first $10$ overs of a cricket game, the run rate was only $3.2$. What should be the run rate in the remaining $40$ overs to reach the target of $282$ runs?

Accepted Answer

Concept & Strategy The total runs scored in any segment of a cricket match is the product of the number of overs and the run rate. To find the required run rate, we must determine the remaining runs needed and divide it by the remaining overs. Total Runs = Overs × Run Rate Step-by-Step Solution • Given: Target score = 282 runs. First 10 overs run rate = 3.2. Remaining overs = 40. • Calculate the runs scored in the first 10 overs: 10 × 3.2 = 32 runs. • Determine the remaining runs required to reach the target: 282 - 32 = 250 runs. • Calculate the required run rate for the remaining 40 overs: 250 / 40 = 25 / 4 = 6.25. Exam Strategy & Shortcut Instead of writing out full equations, you can calculate this mentally. Mentally shift the decimal: 10 × 3.2 = 32. Subtract 32 from 282 in your head to get 250. The fraction 250/40 simplifies to 25/4. Since 24/4 = 6 and 1/4 = 0.25, the sum is 6.25. Common Pitfall A frequent mistake is dividing the total target (282) by the total overs (50) to get the overall run rate, and then trying to subtract the initial run rate (3.2), which mathematically does not yield the correct required run rate. Always work with absolute runs, not averages of averages. Final Answer Therefore, the correct answer is 6.25.

Question 5

In a one-day cricket match the captain of one of the teams scored $30$ runs more than the average runs scored by the remaining six batsmen of that team who batted in the match. If the total runs scored by all the batsmen of that team were $310$, how…

Accepted Answer

Concept & Logic The total score of a team is the sum of the runs scored by the captain and the runs scored by the remaining batsmen. By setting a variable for the average of the remaining batsmen, we can form a linear equation based on the total team score. Total Runs = Captain's Runs + Remaining Batsmen's Runs Step-by-Step Solution • Let the average runs scored by the remaining 6 batsmen be x. • The total runs scored by these 6 batsmen = 6x. • The captain scored 30 runs more than their average, so the captain's score = x + 30. • The total runs scored by all 7 batsmen (captain + 6 others) is given as 310. • Set up the equation: (x + 30) + 6x = 310. • Simplify and solve for x: 7x + 30 = 310 \implies 7x = 280 \implies x = 40. • The captain's score is x + 30 = 40 + 30 = 70. Exam Strategy & Shortcut Subtract the captain's "extra" 30 runs from the team total immediately: 310 - 30 = 280. This 280 represents the runs if the captain had scored exactly the average of the other 6 batsmen. Divide this adjusted total by the total number of batsmen (7): 280 / 7 = 40. This is the average x. Add back the 30 to find the captain's score: 40 + 30 = 70. Common Pitfall A common error is confusing the total number of players. Students might divide by 11 (a full cricket team) instead of the 7 batsmen who actually batted as stated by the constraints of the problem. Always read the specific headcount given. Final Answer Therefore, the correct answer is 70.

Question 6

In a group of $120$ people, one-fifth are men, one-fourth are women and the rest children. The average age of women is five-sixth of the average age of men. Average age of children is one-fourth of the average age of men. If average age of men is $6…

Accepted Answer

Concept & Strategy To find the average age of a mixed group, we use the weighted average formula. We need to determine the exact number of people in each subgroup (men, women, children) and their respective average ages, then calculate the total age sum and divide by the total population. Weighted Average = (n_1A_1 + n_2A_2 + n_3A_3/n_1 + n_2 + n_3) Step-by-Step Solution • Given: Total people = 120. Average age of men = 60 years. • Calculate population sizes: - Men = 1/5 of 120 = 24. - Women = 1/4 of 120 = 30. - Children = Rest = 120 - (24 + 30) = 120 - 54 = 66. • Calculate group average ages based on the men's average (60): - Women = (5/6) × 60 = 50 years. - Children = (1/4) × 60 = 15 years. • Calculate the total sum of all ages: - Total men's age = 24 × 60 = 1440. - Total women's age = 30 × 50 = 1500. - Total children's age = 66 × 15 = 990. • Total age of the group = 1440 + 1500 + 990 = 3930. • Calculate group average = 3930 / 120. Simplify by dropping the zero: 393 / 12 = 131 / 4 = 32.75. Exam Strategy & Shortcut Instead of large multiplications, you can use the ratio of the population to simplify calculations. The ratio of Men : Women : Children is 24 : 30 : 66 \implies 4 : 5 : 11. The total parts are 20. Weighted average = ((4 × 60) + (5 × 50) + (11 × 15)/20) = (240 + 250 + 165/20) = (655/20) = 32.75. This keeps numbers smaller and saves time. Common Pitfall A frequent trap is calculating the simple unweighted average of the three ages: (60 + 50 + 15) / 3 = 125 / 3 \approx 41.6. This completely ignores the vastly different population sizes (there are nearly three times as many children as men). Always use a weighted average. Final Answer Therefore, the correct answer is 32.75.

Question 7

Visitors to a show were charged ₹ $15$ each on the first day, ₹ $7.50$ each on the second day and ₹ $2.50$ each on the third day. The attendance on the three days was in the ratio $2 : 5 : 13$. The average charge per person for the whole show was

Accepted Answer

Concept & Formula This problem requires finding the weighted average. When exact numbers aren't given but ratios are, you can use the ratio components directly as the "weights" or assumed number of visitors, because the common multiplier will cancel out in the fraction. Weighted Average = (w_1x_1 + w_2x_2 + w_3x_3/w_1 + w_2 + w_3) Step-by-Step Solution • Given charges for Day 1, Day 2, and Day 3: ₹ 15, ₹ 7.50, and ₹ 2.50. • Given attendance ratio: 2 : 5 : 13. Let the attendance be 2x, 5x, and 13x. • Total number of visitors over three days = 2x + 5x + 13x = 20x. • Calculate total revenue collected: - Day 1: 2x × 15 = 30x - Day 2: 5x × 7.50 = 37.5x - Day 3: 13x × 2.50 = 32.5x • Total revenue = 30x + 37.5x + 32.5x = 100x. • Average charge per person = \frac{Total Revenue}{Total Visitors} = (100x/20x) = 5. Exam Strategy & Shortcut Ignore the variable x completely in exams. Assume exactly 2, 5, and 13 people visited. Calculate the total money collected: (2 × 15) + (5 × 7.5) + (13 × 2.5) = 30 + 37.5 + 32.5 = 100. Divide by the total number of people (2 + 5 + 13 = 20). So, 100 / 20 = 5. Common Pitfall A common error is just taking the average of the three ticket prices: (15 + 7.5 + 2.5) / 3 = 25 / 3 = 8.33. This is conceptually wrong because it assumes an equal number of people visited each day, totally disregarding the 2:5:13 attendance ratio. Final Answer Therefore, the correct answer is ₹ 5.

Question 8

The mean of $25$ observations was found to be $78.4$. But later on it was found that $96$ was misread as $69$. The correct mean is

Accepted Answer

Concept & Formula: When a value is incorrectly read while calculating a mean, we can find the correct mean by adjusting the total sum. Instead of recalculating the entire sum from scratch, we can simply find the net difference caused by the error and distribute it evenly across all observations. Correct Mean = Old Mean + \frac{Correct Value - Incorrect Value}{Total Observations} Step-by-Step Solution: • Given: • Total observations (n) = 25 • Incorrect mean = 78.4 • Correct value = 96 • Incorrect value = 69 • Calculation: 1. Determine the net error in the total sum. The sum was recorded lower than it should have been because 96 was read as 69. 2. Net increase required in sum = 96 - 69 = 27. 3. This increase of 27 needs to be distributed equally among the 25 observations. 4. Increase in average = (27/25) = (108/100) = 1.08. 5. Correct Mean = 78.4 + 1.08 = 79.48. Exam Strategy & Shortcut: The Deviation Method: Never multiply 25 × 78.4 in an exam! Calculate the difference between the correct and incorrect values (+27). Divide this difference by the total number of items (25) to find the change in the average (+1.08). Add this to the old average. Common Pitfall: Students often waste precious time calculating the total sum (25 × 78.4 = 1960), subtracting 69, adding 96, and then dividing by 25 again. While this works, it is highly prone to arithmetic errors and slows you down significantly. Final Answer: Therefore, the correct answer is 79.48.

Question 9

The average of $10$ numbers is $40.2$. Later it is found that two numbers have been wrongly added. The first is $18$ greater than the actual number and the second number added is $13$ instead of $33$. Find the correct average.

Accepted Answer

Concept & Strategy: To correct an average when items are incorrectly recorded, we determine the net deviation (the total difference between the correct values and the incorrect values) and divide it by the total number of items. Change in Average = \frac{Net Deviation}{Total Numbers} Step-by-Step Solution: • Given: • Total numbers (n) = 10 • Initial Average = 40.2 • Error 1: The wrong number is 18 greater than the actual number. • Error 2: The wrong number added was 13, but the actual number was 33. • Calculation: 1. Analyze the first error: The recorded number is 18 too high. Therefore, to correct it, we must subtract 18. Deviation = -18. 2. Analyze the second error: The recorded number was 13, but it should be 33. The recorded sum is too low. To correct it, we must add 20 (since 33 - 13 = 20). Deviation = +20. 3. Calculate the net deviation: -18 + 20 = +2. 4. Calculate the change in the average: (+2/10) = +0.2. 5. Calculate the new average: 40.2 + 0.2 = 40.4. Exam Strategy & Shortcut: Mental Math: Interpret the wording carefully. "First is 18 greater" means a -18 correction. "13 instead of 33" means a +20 correction. Net correction is +2. Divided by 10 numbers is +0.2. Original average 40.2 + 0.2 = 40.4. Solved in 5 seconds without writing anything down. Common Pitfall: Misinterpreting the phrasing "The first is 18 greater than the actual number". Students sometimes add 18 instead of subtracting it. Always map out whether the *correction* requires an addition or subtraction to the total. Final Answer: Therefore, the correct answer is 40.4.

Question 10

The average age of seven boys sitting in a row facing North is $26$ years. If the average age of the first three boys is $19$ years and the average age of the last three boys is $32$ years, what is the age of the boy who is sitting in the middle of …

Accepted Answer

Concept & Strategy: When given the average of an entire group and the averages of non-overlapping subgroups that exclude exactly one item, the value of that excluded item can be found by subtracting the sum of the subgroups from the total sum. Middle Item = Total Sum - (Sum of Group 1 + Sum of Group 2) Step-by-Step Solution: • Given: • Total boys = 7, Average age = 26 years. • First 3 boys, Average age = 19 years. • Last 3 boys, Average age = 32 years. • Calculation: 1. Calculate the total age of all 7 boys: 7 × 26 = 182 years. 2. Calculate the total age of the first 3 boys: 3 × 19 = 57 years. 3. Calculate the total age of the last 3 boys: 3 × 32 = 96 years. 4. Notice that the first 3 boys and the last 3 boys cover 6 distinct boys. The only one left out is the 4th boy (the middle one). 5. Sum of the 6 boys = 57 + 96 = 153 years. 6. Age of the middle boy = Total Age - Sum of the 6 boys. 7. Age = 182 - 153 = 29 years. Exam Strategy & Shortcut: Deviation Method: Compare the subgroup averages to the main average (26). First 3 boys: deviation is 19 - 26 = -7 per boy. Total deviation = 3 × (-7) = -21. Last 3 boys: deviation is 32 - 26 = +6 per boy. Total deviation = 3 × (+6) = +18. Net deviation of the 6 boys = -21 + 18 = -3. In order to balance the overall average back to zero deviation, the middle boy must be +3 relative to the average. Middle boy = 26 + 3 = 29. This involves minimal calculation! Common Pitfall: Getting bogged down by the "facing North" detail. This is extraneous information meant to confuse you into thinking this is a seating arrangement/logical reasoning puzzle rather than a pure mathematics question. Final Answer: Therefore, the correct answer is 29 years.

Question 11

The average of five numbers is $306.4$. The average of the first two numbers is $431$ and the average of the last two numbers is $214.5$. What is the third number?

Accepted Answer

Concept & Strategy: This follows the standard subgroup subtraction principle. By finding the sum of all components and subtracting the parts we know, we can isolate the unknown central item. Missing Number = Total Sum of All - (Sum of First Part + Sum of Last Part) Step-by-Step Solution: • Given: • Total numbers = 5, Overall average = 306.4. • First 2 numbers, Average = 431. • Last 2 numbers, Average = 214.5. • Calculation: 1. Calculate the total sum of all 5 numbers: 5 × 306.4. • 5 × 300 = 1500 • 5 × 6.4 = 32 • Total sum = 1532. 2. Calculate the total sum of the first 2 numbers: 2 × 431 = 862. 3. Calculate the total sum of the last 2 numbers: 2 × 214.5 = 429. 4. The 3rd number is the only number not included in the first two or the last two. 5. Third number = Total Sum - (Sum of First 2 + Sum of Last 2) 6. Third number = 1532 - (862 + 429) = 1532 - 1291 = 241. 7. Looking at the given options: (a) 52, (b) 108, (c) 321, (d) Cannot be determined. The answer 241 is not present. Exam Strategy & Shortcut: Unit Digit Check: Let's check the unit digits of the sums. Total sum (5 × 306.4) ends in a 0 (since 5 × .4 = 2.0, technically the integer part ends in 2). Let's be precise: 1532 (ends in 2). First two sum (2 × 431) ends in 2. Last two sum (2 × 214.5) ends in 9 (since 214.5 × 2 = 429). Sum of parts ends in 2 + 9 = 11 \rightarrow unit digit 1. Third number unit digit = 2 - 1 = 1. None of the numerical options (52, 108, 321) logically fit if you quickly check approximations (1500 - 1300 = 200). 321 has the right unit digit but is too large. Hence, "None of these". Common Pitfall: Selecting "Cannot be determined" simply because the answer isn't among the numerical options. If a precise unique value *can* be calculated (in this case, 241), the answer is "None of these", not "Cannot be determined". Final Answer: Therefore, the correct answer is None of these.

Question 12

The average of 11 numbers is 10.9. If the average of the first six numbers is 10.5 and that of the last six numbers is 11.4, then the middle number is :

Accepted Answer

Concept & Strategy: When two subgroups overlap by exactly one item (e.g., first 6 and last 6 out of 11 total items), the overlapping item is counted twice when you add the sums of the subgroups. To find the overlapping item, subtract the true total sum from the combined sum of the subgroups. Overlapping Item = (Sum of Group 1 + Sum of Group 2) - Total Sum Step-by-Step Solution: • Given: • Total numbers (n) = 11, Average = 10.9. • First 6 numbers, Average = 10.5. • Last 6 numbers, Average = 11.4. • Calculation: 1. Find the total sum of all 11 numbers: 11 × 10.9 = 119.9. 2. Find the sum of the first 6 numbers: 6 × 10.5 = 63.0. 3. Find the sum of the last 6 numbers: 6 × 11.4 = 68.4. 4. Add the sums of the two groups: 63.0 + 68.4 = 131.4. 5. This combined sum (131.4) contains all 11 numbers, but the 6th number is counted twice. 6. Middle number = Combined Sum - True Total Sum = 131.4 - 119.9 = 11.5. Exam Strategy & Shortcut: Deviation Method: Compare subgroup averages to the overall average (10.9). First 6: Deviation is 10.5 - 10.9 = -0.4. Total deviation = 6 × (-0.4) = -2.4. Last 6: Deviation is 11.4 - 10.9 = +0.5. Total deviation = 6 × (+0.5) = +3.0. Net deviation = -2.4 + 3.0 = +0.6. In an overlapping sequence, you simply add the net deviation to the overall average to find the overlapping number. Middle number = 10.9 + 0.6 = 11.5. Common Pitfall: Confusing the overlapping formula with the excluded item formula. If subgroups overlap (first 6 + last 6 = 12 > 11), you subtract the total sum from the subgroup sum. If there is a gap (first 5 + last 5 = 10 < 11), you subtract the subgroup sum from the total sum. Final Answer: Therefore, the correct answer is 11.5.

Question 13

The average temperature for the first four days of a week is 40.2°C and that of the last four days is 41.3°C. If the average temperature for the whole week is 40.6°C, then the temperature on the fourth day is

Accepted Answer

Concept & Strategy: A week has 7 days. The "first four days" and the "last four days" sum up to 8 days. This means the 4th day is counted in both groups. We can find the temperature of this overlapping day by finding the sum of both groups and subtracting the sum of the entire week. Overlapping Day = (Sum of Group 1 + Sum of Group 2) - Sum of Whole Step-by-Step Solution: • Given: • Total days in a week = 7, Average temp = 40.6^\circC. • First 4 days, Average temp = 40.2^\circC. • Last 4 days, Average temp = 41.3^\circC. • Calculation: 1. Calculate total sum of temperatures for the whole week (7 days): 7 × 40.6 = 284.2^\circC. 2. Calculate sum of temperatures for the first 4 days: 4 × 40.2 = 160.8^\circC. 3. Calculate sum of temperatures for the last 4 days: 4 × 41.3 = 165.2^\circC. 4. Summing the two groups gives the total for all 7 days plus the 4th day counted a second time: 160.8 + 165.2 = 326.0^\circC. 5. Temperature on 4th day = 326.0 - 284.2 = 41.8^\circC. Exam Strategy & Shortcut: Deviation Method: Check the difference from the weekly average (40.6^\circC). First 4 days deviation: 40.2 - 40.6 = -0.4^\circC. Total = 4 × (-0.4) = -1.6. Last 4 days deviation: 41.3 - 40.6 = +0.7^\circC. Total = 4 × (+0.7) = +2.8. Net deviation = -1.6 + 2.8 = +1.2. Because the 4th day overlaps, we add this net deviation directly to the overall average: 40.6 + 1.2 = 41.8^\circC. This avoids multiplying decimals like 7 × 40.6. Common Pitfall: Simple arithmetic mistakes when multiplying decimals. Using the deviation method completely bypasses the need to calculate 284.2 and 326, which are prime spots for calculation errors under time pressure. Final Answer: Therefore, the correct answer is 41.8°C.

Question 14

The average of the ages of Sumit, Krishna and Rishabh is 43 and the average of the ages of Sumit, Rishabh and Rohit is 49. If Rohit is 54 years old, what is Krishna's age?

Accepted Answer

Concept & Strategy: When given the averages of two groups that share members, setting up algebraic equations for their sums allows you to eliminate the shared members via subtraction, revealing the relationship between the unshared members. Difference in Sums = Difference between Unshared Elements Step-by-Step Solution: • Given: • Average of Sumit (S), Krishna (K), and Rishabh (Ri) = 43. • Average of Sumit (S), Rishabh (Ri), and Rohit (Ro) = 49. • Rohit (Ro) = 54 years. • Calculation: 1. Find the sum of the ages of the first group: S + K + Ri = 3 × 43 = 129 (Equation 1) 2. Find the sum of the ages of the second group: S + Ri + Ro = 3 × 49 = 147 (Equation 2) 3. Subtract Equation 1 from Equation 2 to eliminate the overlapping variables (S and Ri): (S + Ri + Ro) - (S + K + Ri) = 147 - 129 Ro - K = 18 4. This tells us Rohit is 18 years older than Krishna. 5. Substitute Rohit's given age (54) into the equation: 54 - K = 18 K = 54 - 18 = 36 Exam Strategy & Shortcut: Average Difference Multiplier: Instead of calculating full sums, observe the shift in the average. When Krishna is swapped out for Rohit, the average of the 3 people increases from 43 to 49. Increase in average = 6 years. Since there are 3 people, the total sum must have increased by 3 × 6 = 18 years. This means the new person (Rohit) must be exactly 18 years older than the old person (Krishna). Rohit is 54. Therefore, Krishna is 54 - 18 = 36. This skips multiplying 43 × 3 and 49 × 3 completely! Common Pitfall: Getting overwhelmed by the multiple names and trying to find the individual ages of Sumit and Rishabh. You do not need to know their individual ages because they cancel each other out in the equations. Final Answer: Therefore, the correct answer is 36 years.

Question 15

The mean temperature of Monday to Wednesday was 37°C and of Tuesday to Thursday was 34°C. If the temperature on Thursday was $\frac{4}{5}$ that of Monday, the temperature on Thursday was

Accepted Answer

Concept & Logic: When dealing with overlapping sequence averages (like Mon-Wed and Tue-Thu), the days in the middle cancel out when you find the difference between their sums. This leaves a direct relationship between the first item of the first sequence and the last item of the second sequence. Difference in Sums = First Item - Last Item Step-by-Step Solution: • Given: • Mean of Mon, Tue, Wed = 37°C • Mean of Tue, Wed, Thu = 34°C • Thursday (Th) = (4/5) × Monday (M) • Calculation: 1. Find the total temperature from Monday to Wednesday: M + T + W = 3 × 37 = 111 2. Find the total temperature from Tuesday to Thursday: T + W + Th = 3 × 34 = 102 3. Subtract the second equation from the first to eliminate the overlapping days (T and W): (M + T + W) - (T + W + Th) = 111 - 102 M - Th = 9 4. Substitute the given ratio (Th = (4/5)M, which means M = (5/4)Th) into the equation: (5/4)Th - Th = 9 (1/4)Th = 9 5. Solve for Thursday's temperature: Th = 9 × 4 = 36 Exam Strategy & Shortcut: Direct Difference Multiplier: You don't need to calculate 111 and 102. Just look at the difference in the averages (37°C - 34°C = 3°C). Since there are 3 days in each group, the total difference between the unshared days (Monday and Thursday) is 3 × 3 = 9°C. Since Th = (4/5)M, the ratio of M : Th is 5 : 4. The difference in ratio units is 5 - 4 = 1 unit. 1 unit = 9°C. Thursday is 4 units. So, 4 × 9 = 36°C. Solved in 10 seconds! Common Pitfall: Calculating the value for Monday (5 × 9 = 45°C) instead of Thursday, and then either panicking because 45 isn't in the options or blindly guessing. Always double-check which specific variable the question is asking for at the very end. Final Answer: Therefore, the correct answer is 36°C.

Question 16

The average weight of $A, B$ and $C$ is 45 kg. If the average weight of $A$ and $B$ be 40 kg and that of $B$ and $C$ be 43 kg, then the weight of $B$ is:

Accepted Answer

Concept & Logic: When you have the total sum of a complete group, and the sums of two subgroups that overlap on exactly one member, you can find the overlapping member by adding the two subgroups and subtracting the total group. Overlapping Member = (Sum of Group 1 + Sum of Group 2) - Total Sum Step-by-Step Solution: • Given: • Average of A, B, C = 45 kg • Average of A, B = 40 kg • Average of B, C = 43 kg • Calculation: 1. Find the total weight of all three people (A, B, C): A + B + C = 3 × 45 = 135 kg 2. Find the total weight of the first subgroup (A, B): A + B = 2 × 40 = 80 kg 3. Find the total weight of the second subgroup (B, C): B + C = 2 × 43 = 86 kg 4. Add the two subgroups together. Notice this counts B twice: (A + B) + (B + C) = A + 2B + C 80 + 86 = 166 kg 5. Subtract the true total sum from this combined sum to isolate B: (A + 2B + C) - (A + B + C) = 166 - 135 = 31 kg Exam Strategy & Shortcut: Net Deviation Method: Compare the subgroup averages to the main group average of 45 kg. Subgroup 1 (A, B): Deviation is -5 kg per person. Total deviation = 2 × (-5) = -10 kg. Subgroup 2 (B, C): Deviation is -2 kg per person. Total deviation = 2 × (-2) = -4 kg. Total combined deviation = -10 + (-4) = -14 kg. Because B is the overlapping element tying these averages together, B must be exactly this total deviation away from the main average to balance the scales. Weight of B = 45 - 14 = 31 kg. Common Pitfall: A frequent mistake is attempting to solve for A and C individually before finding B. While possible (e.g., finding C by subtracting A+B from the total), it adds unnecessary steps. Combining the subgroups and subtracting the total is the most direct mathematical route. Final Answer: Therefore, the correct answer is 31 kg.

Question 17

The average monthly income of $P$ and $Q$ is ₹ 5050. The average monthly income of $Q$ and $R$ is ₹ 6250 and the average monthly income of $P$ and $R$ is ₹ 5200. The monthly income of $P$ is

Accepted Answer

Concept & Formula: When you are given the averages of overlapping pairs, you can find the total sum of all entities by adding the sums of the pairs and dividing by 2. Sum = Average × Number of Entities Step-by-Step Solution: • Sum of P and Q: Since their average is ₹ 5050, P + Q = 5050 × 2 = 10100 • Sum of Q and R: Since their average is ₹ 6250, Q + R = 6250 × 2 = 12500 • Sum of P and R: Since their average is ₹ 5200, P + R = 5200 × 2 = 10400 • Find the Total Sum (P + Q + R): Add all three equations together: (P + Q) + (Q + R) + (P + R) = 10100 + 12500 + 10400 2(P + Q + R) = 33000 P + Q + R = 16500 • Calculate P's Income: Subtract the sum of (Q + R) from the total sum: P = (P + Q + R) - (Q + R) P = 16500 - 12500 = 4000 Exam Strategy & Shortcut: Instead of writing out full variables, simply add the three given averages and subtract the average of the pair that *does not* include the person you are looking for. P's income = Sum of averages containing P - Average not containing P Wait, the formula for this specific shortcut is: P = (Average of P+Q) + (Average of P+R) - (Average of Q+R) P = 5050 + 5200 - 6250 = 10250 - 6250 = 4000. This is incredibly fast! Common Pitfall: Students often add the totals together (33000) and forget to divide by 2, erroneously thinking 33000 is the sum of P, Q, and R. Always remember that each person is counted twice when adding overlapping pairs. Final Answer: Therefore, the correct answer is ₹ 4000.

Question 18

Average age of ten persons learning yoga is 32 years. When the age of their instructor is added, the average age becomes 34 years. The age of their instructor is

Accepted Answer

Concept & Strategy: When a new person joins a group and the average increases, the new person's value must be greater than the old average. Their age covers the original average PLUS the total increase distributed among all members (including themselves). Total = Average × Number of Entities Step-by-Step Solution: • Calculate the initial total age: For 10 persons with an average of 32 years: Sum_{initial} = 10 × 32 = 320 years • Calculate the new total age: When the instructor joins, there are 11 people. The new average is 34 years: Sum_{new} = 11 × 34 = 374 years • Find the instructor's age: Subtract the initial total from the new total: Instructor = 374 - 320 = 54 years Exam Strategy & Shortcut: Use the "Deviation Method" for a lightning-fast mental calculation. The instructor brings enough age to be exactly the old average (32), plus enough extra age to increase all 11 people's averages by 2 years (34 - 32). Age = Old Average + (Total New Members × Increase in Average) Age = 32 + (11 × 2) = 32 + 22 = 54 Common Pitfall: Multiplying the increase in average by the *old* number of members (10) instead of the *new* number of members (11). The instructor also needs those 2 extra years to bring their own base age of 32 up to 34. Final Answer: Therefore, the correct answer is 54 years.

Question 19

The average weight of a class of 24 students is 35 kg. If the weight of the teacher be included, the average rises by 400 g. The weight of the teacher is

Accepted Answer

Concept & Strategy: When a new person joins and raises the average, their weight is equal to the old average plus the total weight increase distributed among all members (including themselves). Make sure all units are consistent (convert grams to kilograms). Step-by-Step Solution: • Convert units: The increase is 400 g. Let's convert this to kg: 400 g = 0.4 kg • Calculate initial total weight: For 24 students with an average of 35 kg: Sum_{initial} = 24 × 35 = 840 kg • Calculate new total weight: Including the teacher makes 25 people. The new average is 35 + 0.4 = 35.4 kg: Sum_{new} = 25 × 35.4 = 885 kg • Find the teacher's weight: Teacher = 885 - 840 = 45 kg Exam Strategy & Shortcut: Avoid calculating large multiplications like 24 × 35. Use the deviation shortcut. The teacher brings a base weight of 35 kg (the old average), plus enough extra weight to raise the average of all 25 people by 0.4 kg. Weight = Old Average + (Total New Members × Increase) Weight = 35 + (25 × 0.4) Weight = 35 + 10 = 45 kg Common Pitfall: Forgetting to convert 400 grams into kilograms before calculating. Adding 400 directly to 35 will lead to bizarre and incorrect results. Always normalize your units first. Final Answer: Therefore, the correct answer is 45 kg.

Question 20

The average age of the mother and her six children is 12 years which is reduced by 5 years if the age of the mother is excluded. How old is the mother?

Accepted Answer

Concept & Strategy: When a member leaves and the average drops dramatically, that member held a disproportionately high value. The departing member's value is the new average plus the total decrease they "took away" from the rest of the group. Total = Average × Number of Entities Step-by-Step Solution: • Calculate initial total age: The group consists of the mother + 6 children = 7 people. Sum_{initial} = 7 × 12 = 84 years • Calculate new total age (without mother): There are now 6 children left. The average is reduced by 5 years (12 - 5 = 7 years). Sum_{new} = 6 × 7 = 42 years • Find the mother's age: Subtract the children's total from the entire group's total: Mother = 84 - 42 = 42 years Exam Strategy & Shortcut: Use the deviation method to bypass calculating totals. When the mother leaves, she takes her share of the old average (12 years). Her departure also causes the remaining 6 children to lose 5 years each from their average. She was providing those extra years. Age = Old Average + (Remaining Members × Decrease in Average) Age = 12 + (6 × 5) Age = 12 + 30 = 42 years Common Pitfall: Miscounting the initial number of people. Students often read "six children" and use 6 as the total initial group size instead of 7 (6 children + 1 mother). Always count the specific entities making up the average. Final Answer: Therefore, the correct answer is 42 years.

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