Q: A computer printed 176400 lines in a given day. If the printer was in operation for 7 hours during the day, how many lines did it print per minute?

Concept & Formula: This is a basic rate problem. To find the rate per minute, you must divide the total work done by the total time taken in minutes. Rate = \frac{Total Lines}{Total Minutes} Step-by-Step Solution: • Given: Total lines = 176400. Total time = 7 hours. • Conversion: First, convert hours into minutes. • Total minutes = 7 × 60 = 420 minutes. • Calculation: Divide the total lines by the total minutes. • Rate = (176400/420) • Cancel out the zeros: (17640/42) • Simplify by dividing by 7: (2520/6) • Simplify by dividing by 6: 420. Exam Strategy & Shortcut: Look at the numbers: 1764 is a multiple of 7. (1764/7) = 252. Then you just need to divide by 60 (for minutes). (25200/60) = 420. Handling the hours division first keeps the numbers smaller. Common Pitfall: Failing to convert hours to minutes. A student might do (176400/7) = 25200 and then get confused when it doesn't match the options, or worse, make a secondary math error trying to force an answer. Always align the units first. Final Answer: Therefore, the correct answer is 420.

Q: In a test, a candidate secured 336 marks out of maximum marks $x$. If the maximum marks $x$ had been converted into 400 marks, he would have secured 192 marks. What was the maximum marks of the test?

Concept & Formula: This problem requires setting up a direct proportion. The ratio of the marks obtained to the maximum marks remains constant, even if the maximum marks scale is converted. \frac{Obtained Marks_1}{Maximum Marks_1} = \frac{Obtained Marks_2}{Maximum Marks_2} Step-by-Step Solution: • Given: • Initial state: Obtained = 336, Maximum = x. • Converted state: Obtained = 192, Maximum = 400. • Equation: Set up the proportion: • (336/x) = (192/400) • Calculation: • Simplify the right side fraction first: (192/400) divides cleanly by 8 to give (24/50), and dividing by 2 gives (12/25). • Now, (336/x) = (12/25) • Cross-multiply to solve for x: 12x = 336 × 25 • x = (336 × 25/12) • Divide 336 by 12: 336 / 12 = 28. • x = 28 × 25 • x = 700 Exam Strategy & Shortcut: Recognize the multiplier scale. The new base is 400. The ratio is (192/400). Observe that 192 is slightly less than half of 400. Thus, 336 must be slightly less than half of x. This means x is slightly more than double 336 (which is 672). The only logical option nearby is 700. Alternatively, multiplying 28 × 25 is quickly done as 28 × (100/4) = 7 × 100 = 700. Common Pitfall: Getting bogged down in massive cross-multiplication without simplifying the fractions first. Doing 336 × 400 = 134400 and then trying to divide by 192 by hand wastes precious exam time and increases the likelihood of arithmetic errors. Always simplify fractions first. Final Answer: Therefore, the correct answer is 700.

Q: The marks scored in an examination are converted from 50 to 10 for the purpose of internal assessment. The highest marks were 47 and the lowest were 14. The difference between the maximum and the minimum internal assessment scores is

Concept & Formula: When a set of values is scaled by a constant factor, the difference between any two values in the set is also scaled by that exact same factor. Scaled Difference = Original Difference × Scale Factor Step-by-Step Solution: • Scale Factor: The base marks are converting from 50 to 10. • Scale Factor = (10/50) = (1/5). This means every score is divided by 5. • Original Difference: Calculate the difference between the highest and lowest original marks. • Original Maximum = 47, Original Minimum = 14. • Original Difference = 47 - 14 = 33. • Calculation: Apply the scale factor to the original difference. • Scaled Difference = 33 × (1/5) = (33/5) • Scaled Difference = 6.6 Exam Strategy & Shortcut: Instead of converting both the maximum and the minimum scores individually ((47/5) = 9.4 and (14/5) = 2.8) and then subtracting them (9.4 - 2.8 = 6.6), it is vastly faster to subtract the raw scores first (33) and divide by the scale factor once ((33/5) = 6.6). Common Pitfall: Converting the individual numbers first. While mathematically correct, dealing with decimals like 9.4 and 2.8 under time pressure invites subtraction errors. Stick to integers as long as possible before dividing. Final Answer: Therefore, the correct answer is 6.6.

Q: In an examination, a student scores 4 marks for every correct answer and loses 1 mark for every wrong answer. A student attempted all the 200 questions and scored in all 200 marks. The number of questions he answered correctly was

Concept & Strategy: This is a classic penalty scoring problem. It is best solved using the "Deviation Method" (or assumed total method), which calculates the total possible score if all questions were correct, and then divides the "loss" in marks by the penalty per wrong answer to find the number of incorrect answers. Wrong Answers = \frac{Maximum Possible Score - Actual Score}{Marks per Correct Answer + Penalty per Wrong Answer} Step-by-Step Solution: • Assume all are correct: If the student answered all 200 questions correctly, the maximum possible score would be 200 × 4 = 800 marks. • Calculate the total loss: The student actually scored 200 marks. The marks lost due to wrong answers = 800 - 200 = 600 marks. • Determine loss per wrong answer: Every wrong answer costs the student 4 marks (the marks they *would* have gotten) PLUS 1 mark (the negative penalty). Total loss per wrong answer = 4 + 1 = 5 marks. • Find number of wrong answers: Total wrong answers = (600/5) = 120. • Find number of correct answers: Total questions minus wrong answers = 200 - 120 = 80. Exam Strategy & Shortcut: Using the algebraic method (4c - 1(200 - c) = 200) works, but the assumed total method is much faster mentally. You can simply think: "Max score is 800. He lost 600. Each mistake costs 5 points. 600 / 5 = 120 wrong. So 80 right." This eliminates algebra entirely. Common Pitfall: The most common mistake is assuming the loss per wrong answer is just the penalty mark (1). This leads to calculating the wrong answers as 600 / 1 = 600, which makes no sense. You must remember that getting a question wrong also means forfeiting the positive marks it carried. Final Answer: Therefore, the correct answer is 80.

Q: There are some cows, bulls and 45 hens in a group. One caretaker looks after 15 animals. The number of bulls is twice the number of cows. If the number of heads is less than the total number of feet by 186 (including the caretakers), how many careta…

Concept & Logic: This is an advanced algebraic puzzle requiring you to construct equations based on physical attributes (heads and feet) across multiple categories (animals and humans), while maintaining relational constraints (bulls to cows, caretakers to animals). Total Feet - Total Heads = 186 Step-by-Step Solution: • Define Variables: Let the number of cows be c. • Bulls (b) = 2c. • Hens (h) = 45. • Total Animals (A): A = c + 2c + 45 = 3c + 45. • Caretakers (k): One caretaker for every 15 animals. • k = (3c + 45/15) = (c/5) + 3. • Total Heads (H): Every creature has 1 head. • H = Animals + Caretakers = (3c + 45) + ((c/5) + 3) = 3.2c + 48. • Total Feet (F): Cows and bulls have 4 feet, hens and caretakers have 2. • F = 4(c) + 4(2c) + 2(45) + 2((c/5) + 3) • F = 4c + 8c + 90 + 0.4c + 6 = 12.4c + 96. • Apply the Given Condition: Feet minus Heads = 186. • (12.4c + 96) - (3.2c + 48) = 186 • 9.2c + 48 = 186 • 9.2c = 138 • c = (138/9.2) = 15. • Final Calculation: Number of cows is 15. We need the number of caretakers (k). • k = (c/5) + 3 = (15/5) + 3 = 3 + 3 = 6. Exam Strategy & Shortcut: Notice the caretaker fraction: k = (c/5) + 3. For caretakers to be a whole number (you can't have a fraction of a person), the number of cows (c) *must* be a multiple of 5 (e.g., 5, 10, 15, 20). If you test c=15, then k = 6. Check the total feet and heads with these numbers to verify it equals 186. Guessing and checking logically is often faster than solving decimals algebraically. Common Pitfall: Forgetting to include the caretakers in the head and feet count. The problem explicitly states "(including the caretakers)", but under pressure, students often calculate heads and feet only for the animals, leading to a completely unsolvable equation. Final Answer: Therefore, the correct answer is 6.

Q: Mr. Shah decided to walk down the escalator of a tube station. He found that if he walks down 26 steps, he requires 30 seconds to reach the bottom. However, if he steps down 34 stairs he would only require 18 seconds to get to the bottom. If the t…

Concept & Strategy: The total number of steps on an escalator is constant. The total steps is equal to the steps walked by the person plus the steps the escalator moves on its own during the time it took the person to reach the bottom. Total Steps = Steps Walked + (Escalator Speed × Time) Step-by-Step Solution: • Set up the variables: Let N be the total number of steps, and e be the speed of the escalator in steps per second. • Formulate Equation 1 (Case 1): He walks 26 steps in 30 seconds. • N = 26 + 30e • Formulate Equation 2 (Case 2): He walks 34 steps in 18 seconds. • N = 34 + 18e • Equate and Solve: Since N is the same in both cases: • 26 + 30e = 34 + 18e • 30e - 18e = 34 - 26 • 12e = 8 • e = (8/12) = (2/3) steps per second. • Calculate Total Steps (N): Substitute e back into the first equation. • N = 26 + 30 × ((2/3)) • N = 26 + 20 = 46. Exam Strategy & Shortcut: Observe the differences directly. Difference in steps walked = 34 - 26 = 8 steps. Difference in time taken = 30 - 18 = 12 seconds. This means the escalator covers those 8 steps in 12 seconds, giving an escalator speed of (8/12) or (2/3) steps per second. In the first scenario (30 seconds), the escalator contributes 30 × ((2/3)) = 20 steps. Total steps = 26 (walked) + 20 (escalator) = 46. Common Pitfall: Assuming that speed equals \frac{Distance}{Time} for the person alone (e.g., trying to calculate 26/30). You must account for the fact that the escalator acts as a moving frame of reference, adding its own speed to the person's descent. Final Answer: Therefore, the correct answer is 46.

Q: The taxi charges in a city comprise of a fixed charge, together with the charge of the distance covered. For a journey of 16 km, the charges paid are ₹ 156 and for a journey of 24 km, the charges paid are ₹ 204. What will a person have to pay for …

Concept & Formula: This problem represents a system of linear equations. The total cost is a function of a constant (y-intercept) and a variable rate (slope). Total Cost = Fixed Charge + (Distance × Rate per km) Step-by-Step Solution: • Set up the variables: Let the fixed charge be F and the rate per km be R. • Create Equation 1: For 16 km, the cost is 156. • F + 16R = 156 • Create Equation 2: For 24 km, the cost is 204. • F + 24R = 204 • Solve for R (Rate): Subtract Equation 1 from Equation 2 to eliminate F. • (F + 24R) - (F + 16R) = 204 - 156 • 8R = 48 • R = 6 (The rate is ₹ 6 per km). • Solve for F (Fixed Charge): Substitute R=6 back into Equation 1. • F + 16(6) = 156 • F + 96 = 156 • F = 60 (The fixed charge is ₹ 60). • Calculate cost for 30 km: * Total Cost = 60 + 30(6) = 60 + 180 = 240. Exam Strategy & Shortcut: Skip solving for the fixed charge entirely to save time. You know the cost increases by ₹ 48 for an extra 8 km (24km - 16km). Therefore, the rate is 48/8 = ₹ 6 per km. You need the cost for 30 km. 30 km is exactly 6 km more than the 24 km journey. Just add the cost of the extra 6 km to the 24 km fare: 204 + (6 × 6) = 204 + 36 = 240. Common Pitfall: A common trap is assuming direct proportionality. A student might divide 156 / 16 = 9.75 and then multiply by 30 to get 292.5. This completely ignores the fixed charge component of the fare. Final Answer: Therefore, the correct answer is ₹ 240.

Q: $M$ men agree to purchase a gift for ₹ $D$. If three men drop out how much more will each have to contribute towards the purchase of the gift?

Concept & Strategy: This is an algebraic fraction problem. The "extra" contribution required is simply the difference between the new per-person cost and the original per-person cost. Extra Contribution = New Share - Old Share Step-by-Step Solution: • Calculate the Original Share: The total cost is D and the original number of men is M. • Original Share per man = (D/M). • Calculate the New Share: 3 men drop out, so the new number of men is M - 3. The total cost D remains unchanged. • New Share per man = (D/M - 3). • Calculate the Difference: Subtract the old share from the new share to find the increase. • Increase = (D/M - 3) - (D/M). • Simplify the Expression: Find a common denominator, which is M(M - 3). • Multiply the first fraction by (M/M): (DM/M(M - 3)) • Multiply the second fraction by (M - 3/M - 3): (D(M - 3)/M(M - 3)) • Combine numerators: (DM - D(M - 3)/M(M - 3)) • Expand the numerator: DM - DM + 3D = 3D. • Expand the denominator: M^2 - 3M. • Final expression: (3D/M^2 - 3M). Exam Strategy & Shortcut: When variables confuse you under pressure, plug in simple numbers. Let D = 20 (Total cost) and M = 5 (Total men). Original share = 20 / 5 = ₹ 4. If 3 drop out, new men = 2. New share = 20 / 2 = ₹ 10. Increase = 10 - 4 = ₹ 6. Now, plug D=20, M=5 into the options and look for the one that equals 6: (a) 20 / 2 = 10 (Incorrect) (b) 100 / 3 = 33.3 (Incorrect) (c) 5 / 17 (Incorrect) (d) (3 × 20) / (25 - 15) = 60 / 10 = 6. (Match!) Common Pitfall: Subtracting the denominators incorrectly or selecting option (a), which is just the *new* total share, not the *increase* (the "how much more" asked in the question). Always pay attention to whether the question asks for the final state or the difference between states. Final Answer: Therefore, the correct answer is (3D/M^2 - 3M).

Q: Some students planned a picnic. The budget for food was ₹ 500. But, 5 of them failed to go and thus the cost of food for each member increased by ₹ 5. How many students attended the picnic?

Concept & Logic: This problem uses the classic inverse relationship between the number of contributors and the cost per head when the total budget is fixed. Cost per Head = \frac{Total Budget}{Number of People} Step-by-Step Solution: • Given: Total food budget = ₹ 500. Dropouts = 5 students. Cost increase per head = ₹ 5. • Calculation: Let the originally planned number of students be x. • Original cost per head = (500/x) • Number of students who actually attended = x - 5 • New cost per head = (500/x - 5) • The difference between the new cost and the original cost is ₹ 5: (500/x - 5) - (500/x) = 5 • Divide by 5 to simplify: (100/x - 5) - (100/x) = 1 • Multiply through by x(x - 5): 100x - 100(x - 5) = x^2 - 5x 100x - 100x + 500 = x^2 - 5x x^2 - 5x - 500 = 0 • Factorize (factors of -500 adding to -5 are -25 and +20): (x - 25)(x + 20) = 0 • Since the number of students must be positive, x = 25 (original plan). • The question asks for the number of students who attended: 25 - 5 = 20 Exam Strategy & Shortcut: Option Testing: Note that the question asks for the number who *attended*. If attended = 20 (Option b), then planned = 25. Check costs: 500 / 25 = 20 per head originally. 500 / 20 = 25 per head now. Difference = 5. Matches instantly! Common Pitfall: The most common trap is solving the quadratic equation to get x = 25, and immediately marking 25 (Option c) without reading the final sentence. Always double-check what the question is asking—in this case, those who *attended* (x - 5). Final Answer: Therefore, the correct answer is 20.

Question 1

A man has 1044 candles. After burning, he can make a new candle from 9 stubs left behind. Find the maximum number of candles that can be made.

Accepted Answer

Concept & Logic: This problem requires an iterative calculation. Creating a new candle from stubs allows you to burn that newly created candle, which then leaves behind another stub. You must repeat the division process until the remaining stubs are fewer than the required amount (9) to make a new candle. Step-by-Step Solution: • Initial Step: The man burns 1044 candles. This leaves 1044 stubs. • First Remake: Number of new candles made = (1044/9) = 116. • He burns these 116 candles. This leaves 116 stubs. • Second Remake: Number of new candles made = (116/9) = 12 new candles, with 8 stubs left over. • He burns these 12 candles, leaving 12 stubs. • Total stubs available now = 12 (new) + 8 (left over) = 20 stubs. • Third Remake: Number of new candles made = (20/9) = 2 new candles, with 2 stubs left over. • He burns these 2 candles, leaving 2 stubs. Total stubs = 2 + 2 = 4. • Since 4 < 9, no more candles can be made. • Total new candles: 116 + 12 + 2 = 130. Exam Strategy & Shortcut: Follow the chain of division carefully and maintain a running tally of remainders. Don't discard remainders until the very end, as combining them with new stubs often yields enough material for additional rounds. Common Pitfall: The most common mistake is stopping after the first division. Students calculate (1044/9) = 116 and select option (a). You must remember that newly created candles also leave stubs when burned! Final Answer: Therefore, the correct answer is 130.

Question 2

A computer printed 176400 lines in a given day. If the printer was in operation for 7 hours during the day, how many lines did it print per minute?

Accepted Answer

Concept & Formula: This is a basic rate problem. To find the rate per minute, you must divide the total work done by the total time taken in minutes. Rate = \frac{Total Lines}{Total Minutes} Step-by-Step Solution: • Given: Total lines = 176400. Total time = 7 hours. • Conversion: First, convert hours into minutes. • Total minutes = 7 × 60 = 420 minutes. • Calculation: Divide the total lines by the total minutes. • Rate = (176400/420) • Cancel out the zeros: (17640/42) • Simplify by dividing by 7: (2520/6) • Simplify by dividing by 6: 420. Exam Strategy & Shortcut: Look at the numbers: 1764 is a multiple of 7. (1764/7) = 252. Then you just need to divide by 60 (for minutes). (25200/60) = 420. Handling the hours division first keeps the numbers smaller. Common Pitfall: Failing to convert hours to minutes. A student might do (176400/7) = 25200 and then get confused when it doesn't match the options, or worse, make a secondary math error trying to force an answer. Always align the units first. Final Answer: Therefore, the correct answer is 420.

Question 3

In a test, a candidate secured 336 marks out of maximum marks $x$. If the maximum marks $x$ had been converted into 400 marks, he would have secured 192 marks. What was the maximum marks of the test?

Accepted Answer

Concept & Formula: This problem requires setting up a direct proportion. The ratio of the marks obtained to the maximum marks remains constant, even if the maximum marks scale is converted. \frac{Obtained Marks_1}{Maximum Marks_1} = \frac{Obtained Marks_2}{Maximum Marks_2} Step-by-Step Solution: • Given: • Initial state: Obtained = 336, Maximum = x. • Converted state: Obtained = 192, Maximum = 400. • Equation: Set up the proportion: • (336/x) = (192/400) • Calculation: • Simplify the right side fraction first: (192/400) divides cleanly by 8 to give (24/50), and dividing by 2 gives (12/25). • Now, (336/x) = (12/25) • Cross-multiply to solve for x: 12x = 336 × 25 • x = (336 × 25/12) • Divide 336 by 12: 336 / 12 = 28. • x = 28 × 25 • x = 700 Exam Strategy & Shortcut: Recognize the multiplier scale. The new base is 400. The ratio is (192/400). Observe that 192 is slightly less than half of 400. Thus, 336 must be slightly less than half of x. This means x is slightly more than double 336 (which is 672). The only logical option nearby is 700. Alternatively, multiplying 28 × 25 is quickly done as 28 × (100/4) = 7 × 100 = 700. Common Pitfall: Getting bogged down in massive cross-multiplication without simplifying the fractions first. Doing 336 × 400 = 134400 and then trying to divide by 192 by hand wastes precious exam time and increases the likelihood of arithmetic errors. Always simplify fractions first. Final Answer: Therefore, the correct answer is 700.

Question 4

The marks scored in an examination are converted from 50 to 10 for the purpose of internal assessment. The highest marks were 47 and the lowest were 14. The difference between the maximum and the minimum internal assessment scores is

Accepted Answer

Concept & Formula: When a set of values is scaled by a constant factor, the difference between any two values in the set is also scaled by that exact same factor. Scaled Difference = Original Difference × Scale Factor Step-by-Step Solution: • Scale Factor: The base marks are converting from 50 to 10. • Scale Factor = (10/50) = (1/5). This means every score is divided by 5. • Original Difference: Calculate the difference between the highest and lowest original marks. • Original Maximum = 47, Original Minimum = 14. • Original Difference = 47 - 14 = 33. • Calculation: Apply the scale factor to the original difference. • Scaled Difference = 33 × (1/5) = (33/5) • Scaled Difference = 6.6 Exam Strategy & Shortcut: Instead of converting both the maximum and the minimum scores individually ((47/5) = 9.4 and (14/5) = 2.8) and then subtracting them (9.4 - 2.8 = 6.6), it is vastly faster to subtract the raw scores first (33) and divide by the scale factor once ((33/5) = 6.6). Common Pitfall: Converting the individual numbers first. While mathematically correct, dealing with decimals like 9.4 and 2.8 under time pressure invites subtraction errors. Stick to integers as long as possible before dividing. Final Answer: Therefore, the correct answer is 6.6.

Question 5

In an examination, a student scores 4 marks for every correct answer and loses 1 mark for every wrong answer. A student attempted all the 200 questions and scored in all 200 marks. The number of questions he answered correctly was

Accepted Answer

Concept & Strategy: This is a classic penalty scoring problem. It is best solved using the "Deviation Method" (or assumed total method), which calculates the total possible score if all questions were correct, and then divides the "loss" in marks by the penalty per wrong answer to find the number of incorrect answers. Wrong Answers = \frac{Maximum Possible Score - Actual Score}{Marks per Correct Answer + Penalty per Wrong Answer} Step-by-Step Solution: • Assume all are correct: If the student answered all 200 questions correctly, the maximum possible score would be 200 × 4 = 800 marks. • Calculate the total loss: The student actually scored 200 marks. The marks lost due to wrong answers = 800 - 200 = 600 marks. • Determine loss per wrong answer: Every wrong answer costs the student 4 marks (the marks they *would* have gotten) PLUS 1 mark (the negative penalty). Total loss per wrong answer = 4 + 1 = 5 marks. • Find number of wrong answers: Total wrong answers = (600/5) = 120. • Find number of correct answers: Total questions minus wrong answers = 200 - 120 = 80. Exam Strategy & Shortcut: Using the algebraic method (4c - 1(200 - c) = 200) works, but the assumed total method is much faster mentally. You can simply think: "Max score is 800. He lost 600. Each mistake costs 5 points. 600 / 5 = 120 wrong. So 80 right." This eliminates algebra entirely. Common Pitfall: The most common mistake is assuming the loss per wrong answer is just the penalty mark (1). This leads to calculating the wrong answers as 600 / 1 = 600, which makes no sense. You must remember that getting a question wrong also means forfeiting the positive marks it carried. Final Answer: Therefore, the correct answer is 80.

Question 6

There are some cows, bulls and 45 hens in a group. One caretaker looks after 15 animals. The number of bulls is twice the number of cows. If the number of heads is less than the total number of feet by 186 (including the caretakers), how many careta…

Accepted Answer

Concept & Logic: This is an advanced algebraic puzzle requiring you to construct equations based on physical attributes (heads and feet) across multiple categories (animals and humans), while maintaining relational constraints (bulls to cows, caretakers to animals). Total Feet - Total Heads = 186 Step-by-Step Solution: • Define Variables: Let the number of cows be c. • Bulls (b) = 2c. • Hens (h) = 45. • Total Animals (A): A = c + 2c + 45 = 3c + 45. • Caretakers (k): One caretaker for every 15 animals. • k = (3c + 45/15) = (c/5) + 3. • Total Heads (H): Every creature has 1 head. • H = Animals + Caretakers = (3c + 45) + ((c/5) + 3) = 3.2c + 48. • Total Feet (F): Cows and bulls have 4 feet, hens and caretakers have 2. • F = 4(c) + 4(2c) + 2(45) + 2((c/5) + 3) • F = 4c + 8c + 90 + 0.4c + 6 = 12.4c + 96. • Apply the Given Condition: Feet minus Heads = 186. • (12.4c + 96) - (3.2c + 48) = 186 • 9.2c + 48 = 186 • 9.2c = 138 • c = (138/9.2) = 15. • Final Calculation: Number of cows is 15. We need the number of caretakers (k). • k = (c/5) + 3 = (15/5) + 3 = 3 + 3 = 6. Exam Strategy & Shortcut: Notice the caretaker fraction: k = (c/5) + 3. For caretakers to be a whole number (you can't have a fraction of a person), the number of cows (c) *must* be a multiple of 5 (e.g., 5, 10, 15, 20). If you test c=15, then k = 6. Check the total feet and heads with these numbers to verify it equals 186. Guessing and checking logically is often faster than solving decimals algebraically. Common Pitfall: Forgetting to include the caretakers in the head and feet count. The problem explicitly states "(including the caretakers)", but under pressure, students often calculate heads and feet only for the animals, leading to a completely unsolvable equation. Final Answer: Therefore, the correct answer is 6.

Question 7

Mr. Shah decided to walk down the escalator of a tube station. He found that if he walks down 26 steps, he requires 30 seconds to reach the bottom. However, if he steps down 34 stairs he would only require 18 seconds to get to the bottom.   If the t…

Accepted Answer

Concept & Strategy: The total number of steps on an escalator is constant. The total steps is equal to the steps walked by the person plus the steps the escalator moves on its own during the time it took the person to reach the bottom. Total Steps = Steps Walked + (Escalator Speed × Time) Step-by-Step Solution: • Set up the variables: Let N be the total number of steps, and e be the speed of the escalator in steps per second. • Formulate Equation 1 (Case 1): He walks 26 steps in 30 seconds. • N = 26 + 30e • Formulate Equation 2 (Case 2): He walks 34 steps in 18 seconds. • N = 34 + 18e • Equate and Solve: Since N is the same in both cases: • 26 + 30e = 34 + 18e • 30e - 18e = 34 - 26 • 12e = 8 • e = (8/12) = (2/3) steps per second. • Calculate Total Steps (N): Substitute e back into the first equation. • N = 26 + 30 × ((2/3)) • N = 26 + 20 = 46. Exam Strategy & Shortcut: Observe the differences directly. Difference in steps walked = 34 - 26 = 8 steps. Difference in time taken = 30 - 18 = 12 seconds. This means the escalator covers those 8 steps in 12 seconds, giving an escalator speed of (8/12) or (2/3) steps per second. In the first scenario (30 seconds), the escalator contributes 30 × ((2/3)) = 20 steps. Total steps = 26 (walked) + 20 (escalator) = 46. Common Pitfall: Assuming that speed equals \frac{Distance}{Time} for the person alone (e.g., trying to calculate 26/30). You must account for the fact that the escalator acts as a moving frame of reference, adding its own speed to the person's descent. Final Answer: Therefore, the correct answer is 46.

Question 8

The taxi charges in a city comprise of a fixed charge, together with the charge of the distance covered. For a journey of 16 km, the charges paid are ₹ 156 and for a journey of 24 km, the charges paid are ₹ 204.   What will a person have to pay for …

Accepted Answer

Concept & Formula: This problem represents a system of linear equations. The total cost is a function of a constant (y-intercept) and a variable rate (slope). Total Cost = Fixed Charge + (Distance × Rate per km) Step-by-Step Solution: • Set up the variables: Let the fixed charge be F and the rate per km be R. • Create Equation 1: For 16 km, the cost is 156. • F + 16R = 156 • Create Equation 2: For 24 km, the cost is 204. • F + 24R = 204 • Solve for R (Rate): Subtract Equation 1 from Equation 2 to eliminate F. • (F + 24R) - (F + 16R) = 204 - 156 • 8R = 48 • R = 6 (The rate is ₹ 6 per km). • Solve for F (Fixed Charge): Substitute R=6 back into Equation 1. • F + 16(6) = 156 • F + 96 = 156 • F = 60 (The fixed charge is ₹ 60). • Calculate cost for 30 km: * Total Cost = 60 + 30(6) = 60 + 180 = 240. Exam Strategy & Shortcut: Skip solving for the fixed charge entirely to save time. You know the cost increases by ₹ 48 for an extra 8 km (24km - 16km). Therefore, the rate is 48/8 = ₹ 6 per km. You need the cost for 30 km. 30 km is exactly 6 km more than the 24 km journey. Just add the cost of the extra 6 km to the 24 km fare: 204 + (6 × 6) = 204 + 36 = 240. Common Pitfall: A common trap is assuming direct proportionality. A student might divide 156 / 16 = 9.75 and then multiply by 30 to get 292.5. This completely ignores the fixed charge component of the fare. Final Answer: Therefore, the correct answer is ₹ 240.

Question 9

$M$ men agree to purchase a gift for ₹ $D$. If three men drop out how much more will each have to contribute towards the purchase of the gift?

Accepted Answer

Concept & Strategy: This is an algebraic fraction problem. The "extra" contribution required is simply the difference between the new per-person cost and the original per-person cost. Extra Contribution = New Share - Old Share Step-by-Step Solution: • Calculate the Original Share: The total cost is D and the original number of men is M. • Original Share per man = (D/M). • Calculate the New Share: 3 men drop out, so the new number of men is M - 3. The total cost D remains unchanged. • New Share per man = (D/M - 3). • Calculate the Difference: Subtract the old share from the new share to find the increase. • Increase = (D/M - 3) - (D/M). • Simplify the Expression: Find a common denominator, which is M(M - 3). • Multiply the first fraction by (M/M): (DM/M(M - 3)) • Multiply the second fraction by (M - 3/M - 3): (D(M - 3)/M(M - 3)) • Combine numerators: (DM - D(M - 3)/M(M - 3)) • Expand the numerator: DM - DM + 3D = 3D. • Expand the denominator: M^2 - 3M. • Final expression: (3D/M^2 - 3M). Exam Strategy & Shortcut: When variables confuse you under pressure, plug in simple numbers. Let D = 20 (Total cost) and M = 5 (Total men). Original share = 20 / 5 = ₹ 4. If 3 drop out, new men = 2. New share = 20 / 2 = ₹ 10. Increase = 10 - 4 = ₹ 6. Now, plug D=20, M=5 into the options and look for the one that equals 6: (a) 20 / 2 = 10 (Incorrect) (b) 100 / 3 = 33.3 (Incorrect) (c) 5 / 17 (Incorrect) (d) (3 × 20) / (25 - 15) = 60 / 10 = 6. (Match!) Common Pitfall: Subtracting the denominators incorrectly or selecting option (a), which is just the *new* total share, not the *increase* (the "how much more" asked in the question). Always pay attention to whether the question asks for the final state or the difference between states. Final Answer: Therefore, the correct answer is (3D/M^2 - 3M).

Question 10

Some students planned a picnic. The budget for food was ₹ 500. But, 5 of them failed to go and thus the cost of food for each member increased by ₹ 5.  How many students attended the picnic?

Accepted Answer

Concept & Logic: This problem uses the classic inverse relationship between the number of contributors and the cost per head when the total budget is fixed. Cost per Head = \frac{Total Budget}{Number of People} Step-by-Step Solution: • Given: Total food budget = ₹ 500. Dropouts = 5 students. Cost increase per head = ₹ 5. • Calculation: Let the originally planned number of students be x. • Original cost per head = (500/x) • Number of students who actually attended = x - 5 • New cost per head = (500/x - 5) • The difference between the new cost and the original cost is ₹ 5: (500/x - 5) - (500/x) = 5 • Divide by 5 to simplify: (100/x - 5) - (100/x) = 1 • Multiply through by x(x - 5): 100x - 100(x - 5) = x^2 - 5x 100x - 100x + 500 = x^2 - 5x x^2 - 5x - 500 = 0 • Factorize (factors of -500 adding to -5 are -25 and +20): (x - 25)(x + 20) = 0 • Since the number of students must be positive, x = 25 (original plan). • The question asks for the number of students who attended: 25 - 5 = 20 Exam Strategy & Shortcut: Option Testing: Note that the question asks for the number who *attended*. If attended = 20 (Option b), then planned = 25. Check costs: 500 / 25 = 20 per head originally. 500 / 20 = 25 per head now. Difference = 5. Matches instantly! Common Pitfall: The most common trap is solving the quadratic equation to get x = 25, and immediately marking 25 (Option c) without reading the final sentence. Always double-check what the question is asking—in this case, those who *attended* (x - 5). Final Answer: Therefore, the correct answer is 20.

Question 11

A total of 324 coins of 20 paise and 25 paise make a sum of ₹ 71.  The number of 25-paise coins is

Accepted Answer

Concept & Logic: This is a classic denomination mixture problem that can be solved by setting up simultaneous equations or using a strategic assumption method. We establish an algebraic relationship between the count of the coins and their total monetary value. Step-by-Step Solution: • Given: Total number of coins = 324. Total monetary value = ₹ 71 = 7100 paise. The denominations are 20p and 25p. • Calculation: Let the number of 25-paise coins be x. • Because there are 324 coins in total, the number of 20-paise coins must be (324 - x). • Set up the total value equation in paise to avoid decimals: 25x + 20(324 - x) = 7100 25x + 6480 - 20x = 7100 • Isolate x: 5x = 7100 - 6480 5x = 620 x = (620/5) = 124 Exam Strategy & Shortcut: The "Assume All" Shortcut: Assume ALL 324 coins were of the lower denomination (20-paise). Total value would logically be 324 × 20 = 6480 paise (₹ 64.80). However, the actual value given is 7100 paise (₹ 71.00). The shortfall in our assumption is 7100 - 6480 = 620 paise. Every time we replace a 20p coin with a 25p coin, the total value increases by 5p. Therefore, the number of 25p coins needed to make up the 620p shortfall is 620 / 5 = 124. This avoids algebraic equations entirely. Common Pitfall: Mixing up units between Rupees and Paise. It is always safer and cleaner to convert everything into Paise to avoid decimals during the calculation, which inherently prevents arithmetic errors. Final Answer: Therefore, the correct answer is 124.

Question 12

A bag contains three types of coins — 1-rupee coins, 50 p-coins and 25 p-coins totalling 175 coins.  If the total value of the coins of each kind be the same, the total amount in the bag is

Accepted Answer

Concept & Logic: This problem introduces an inverse relationship constraint: instead of the *number* of coins being equal, the *total value* contributed by each denomination is equal. If the value is constant, lower denominations require proportionally more coins to reach that same value. Step-by-Step Solution: • Given: Total coins = 175. Denominations = ₹ 1, 50p (₹ 0.50), 25p (₹ 0.25). The total monetary value contributed by each type is identical. • Calculation: Let the total value contributed by each specific denomination be ₹ V. • Because each of the 3 types contributes ₹ V, the total amount in the bag is 3V. Our goal is to find V. • Number of ₹ 1 coins needed to make ₹ V = (V/1) = V • Number of 50p coins needed to make ₹ V = (V/0.50) = 2V • Number of 25p coins needed to make ₹ V = (V/0.25) = 4V • The total number of coins is the sum of these quantities: V + 2V + 4V = 175 7V = 175 V = 25 • Each denomination contributes ₹ 25 to the bag. • Total amount in the bag = 3 × V = 3 × 25 = 75. Exam Strategy & Shortcut: Since the value of each coin type is the same, the ratio of the *number of coins* is simply the inverse of their *face values*. Face values ratio: 1 : 1/2 : 1/4. Coin quantities ratio: 1 : 2 : 4. Total ratio parts = 1 + 2 + 4 = 7 parts. 7 parts = 175 coins \implies 1 part = 25 coins. Since the "1 part" corresponds to the ₹ 1 coins, there are 25 ₹ 1 coins, which are worth ₹ 25. Since all 3 types have equal value, total amount = 25 × 3 = 75. Common Pitfall: Calculating V = 25 and immediately looking for 25 in the options. Remember that V is the value of just *one* denomination type. You must multiply by 3 to get the total amount in the bag. Final Answer: Therefore, the correct answer is ₹ 75.

Question 13

These questions are based on the following information:  85 children went to an amusement park where they could ride on merry-go-round, roller coaster and Ferris wheel. It was known that 20 of them took all the three rides and 55 of them have taken …

Accepted Answer

Concept & Logic: Following up on the previous calculation, the total number of children is the sum of children who took 0, 1, 2, and 3 rides. Once we have the subsets, finding the remainder is straightforward addition and subtraction. Step-by-Step Solution: • Given: Total children = 85. • From previous analysis, we know the breakdown of rides: • Children taking exactly 3 rides: n(3) = 20 • Children taking exactly 2 rides: n(2) = 35 • Children taking exactly 1 ride: n(1) = 15 • Calculation: The sum of all categories must equal the total number of children. n(0) + n(1) + n(2) + n(3) = Total Children n(0) + 15 + 35 + 20 = 85 • Sum the known values: n(0) + 70 = 85 • Isolate n(0): n(0) = 85 - 70 = 15 • Thus, 15 children did not try any of the rides. Exam Strategy & Shortcut: If you have already calculated that n(1) = 15 from the previous question, this step takes 5 seconds. Total children = 85. Active children = (Exactly 1) + (At least 2) = 15 + 55 = 70. Inactive children = 85 - 70 = 15. Common Pitfall: Accidentally double counting the children who took 3 rides when trying to sum the groups. Remember that the "55" figure for "at least two rides" *already includes* the 20 who took three rides. Final Answer: Therefore, the correct answer is 15.

Question 14

In a class of 50 students, 25 take Bengali, 16 take Hindi, 12 students take no language.  How many take both Bengali and Hindi?

Accepted Answer

Concept & Logic: This is a fundamental 2-set Venn diagram problem. We use the Principle of Inclusion-Exclusion. First, find the "active" population (those who take at least one language), and then use the formula: n(A \cup B) = n(A) + n(B) - n(A \cap B). Step-by-Step Solution: • Given: * Total students = 50. • Students taking no language = 12. • n(Bengali) = 25. • n(Hindi) = 16. • Calculation: First, determine how many students take *at least* one language. Active Students = Total - None n(B \cup H) = 50 - 12 = 38 • Now apply the set formula to find the intersection (those taking both): n(B \cup H) = n(B) + n(H) - n(B \cap H) 38 = 25 + 16 - n(B \cap H) 38 = 41 - n(B \cap H) • Solve for the intersection: n(B \cap H) = 41 - 38 = 3 Exam Strategy & Shortcut: Add the individual subject totals and subtract the number of active students. Active students = 50 - 12 = 38. Sum of subjects = 25 + 16 = 41. The overlap (both) = Sum of subjects - Active students = 41 - 38 = 3. You can do this entirely in your head without writing a single formula. Common Pitfall: Forgetting to subtract the 12 students who take no language from the total 50. If you mistakenly use 50 as n(A \cup B), you will get a negative number (41 - 50), which should immediately signal an error. Final Answer: Therefore, the correct answer is 3.

Question 15

In a group of 52 persons, 16 drink tea but not coffee and 33 drink tea.  How many drink coffee but not tea?

Accepted Answer

Concept & Logic: This problem requires breaking down two sets into three distinct zones: Set A only, Set B only, and Both. By definition, the total group (Union) is the sum of these three non-overlapping zones: (A only) + (B only) + (Both). Step-by-Step Solution: • Given: Total group size (assuming everyone drinks at least one), n(T \cup C) = 52 Tea only, n(T - C) = 16 Total Tea, n(T) = 33 • Calculation: * First, let's find how many people drink *both*. We know the total Tea drinkers consist of "Tea only" plus "Both". n(T) = n(T - C) + n(T \cap C) 33 = 16 + n(T \cap C) n(T \cap C) = 33 - 16 = 17 • Now, we want to find "Coffee only", which is n(C - T). We know the total group is 52. Total = (Tea only) + (Both) + (Coffee only) 52 = 16 + 17 + n(C - T) 52 = 33 + n(C - T) • Solve for Coffee only: n(C - T) = 52 - 33 = 19 Exam Strategy & Shortcut: Notice that the total Tea circle n(T) perfectly encompasses both "Tea only" and "Both". Therefore, if you subtract the entire Tea circle from the total population, you are left exclusively with "Coffee only". Coffee Only = Total Population - Total Tea Drinkers Coffee Only = 52 - 33 = 19 You do not even need the "16 drink tea but not coffee" data point—it is entirely redundant filler! Common Pitfall: Getting distracted by the redundant "16 drink tea but not coffee" and spending time calculating the intersection (17), only to forget what the final goal is and mistakenly selecting 17 (Option c) as the final answer. Always keep your eye on the specific metric requested. Final Answer: Therefore, the correct answer is 19.

Question 16

In a town with population of 4000, 3000 people are egg eaters, 2000 meat eaters and 1500 eat both eggs and meat.  How many are pure vegetarians?

Accepted Answer

Concept & Logic: This is a fundamental Venn diagram problem. "Pure vegetarians" implies people who eat *neither* eggs nor meat. We must find the total number of non-vegetarians (the union of egg and meat eaters) and subtract it from the total population. Step-by-Step Solution: • Given: Total Population = 4000 Egg eaters, n(E) = 3000 Meat eaters, n(M) = 2000 Both, n(E \cap M) = 1500 • Calculation: * Find the total number of people who eat eggs, meat, or both using the inclusion-exclusion principle: n(E \cup M) = n(E) + n(M) - n(E \cap M) n(E \cup M) = 3000 + 2000 - 1500 n(E \cup M) = 5000 - 1500 = 3500 • This means 3500 people in the town are non-vegetarians (they consume animal products). • To find the pure vegetarians, subtract this number from the total town population: Vegetarians = Total Population - n(E \cup M) Vegetarians = 4000 - 3500 = 500 Exam Strategy & Shortcut: Add the single categories together: 3000 + 2000 = 5000. Since the town only has 4000 people, the "excess" must represent the overlap (people counted twice). Wait, the overlap is given as 1500. So, distinct eaters = 5000 - 1500 = 3500. Remaining population = 4000 - 3500 = 500. Common Pitfall: The most frequent error is simply subtracting the two single categories from the total: 4000 - 3000 - 2000 = -1000. This fails because you are heavily double-subtracting the 1500 people who do both. Always establish the precise union before subtracting from the master total. Final Answer: Therefore, the correct answer is 500.

Question 17

In a group of players in a college, 20 are in the basketball team, 24 in the hockey team and 27 in the cricket team. If 12 play hockey and basketball, 10 play cricket and basketball, 14 play hockey and cricket and 7 play all the three games, then th…

Accepted Answer

Concept & Logic: This problem requires the three-set version of the Inclusion-Exclusion Principle. When you add the totals of three overlapping circles, you double-count the pairwise intersections, and triple-count the center intersection. The formula corrects for this by adding the singles, subtracting the doubles, and re-adding the triple overlap. Step-by-Step Solution: • Given: Basketball, n(B) = 20 Hockey, n(H) = 24 Cricket, n(C) = 27 Hockey & Basketball, n(H \cap B) = 12 Cricket & Basketball, n(C \cap B) = 10 Hockey & Cricket, n(H \cap C) = 14 All three, n(B \cap H \cap C) = 7 • Calculation: * Use the grand formula for the union of three sets: n(B \cup H \cup C) = n(B) + n(H) + n(C) - n(H \cap B) - n(C \cap B) - n(H \cap C) + n(B \cap H \cap C) • Substitute the provided numbers into the equation: Total Players = 20 + 24 + 27 - 12 - 10 - 14 + 7 • Group the positive and negative numbers for easier mental math: Positives = 20 + 24 + 27 + 7 = 78 Negatives = 12 + 10 + 14 = 36 • Subtract the totals: Total Players = 78 - 36 = 42 Exam Strategy & Shortcut: You can calculate this rapidly by treating the positive and negative pairs together to keep numbers small. (20 - 12 = 8) (24 - 10 = 14) (27 - 14 = 13) Add these up with the center overlap: 8 + 14 + 13 + 7 = 42. This minimizes the risk of arithmetic errors associated with summing large strings of numbers. Common Pitfall: The most dangerous pitfall is forgetting to *add back* the 7 players who play all three games at the end of the formula. Because they were subtracted three times (in each of the pairwise intersections), they have been entirely removed from the total count and must be reinstated. Final Answer: Therefore, the correct answer is 42.

Question 18

Directions: These questions are based on the following information: Children in a class play only one or two or all of the three games - badminton, football and cricket. 5 children play only cricket, 8 children play only football and 7 children play…

Accepted Answer

Concept & Strategy: The key insight for solving 3-set Venn diagram problems is distinguishing between "playing exactly two games" and "playing two games." When asked for children who play "football as well as cricket," it includes everyone in the intersection of the Football and Cricket sets, regardless of whether they also play badminton. n(F \cap C) = n(Only F and C) + n(All three) Step-by-Step Solution: • Given: • Children playing *only* cricket and football = 4 • Children playing *all three* games = 2 • Calculation: • The phrase "football as well as cricket" translates to the entire overlapping region between the football and cricket circles. • This region is composed of two distinct parts: those who play *only* those two, and those who play *all three*. • Add the two values together: 4 + 2 = 6. Exam Strategy & Shortcut: Instead of drawing a complete Venn diagram, scan the text specifically for the intersection requested. Look for the "only [Game A] and [Game B]" value, then immediately add the "all three" value. Common Pitfall: The most common mistake is assuming "football as well as cricket" means *only* football and cricket, leading students to select 4. Always remember that "A and B" inherently includes "A, B, and C" unless the word "only" is explicitly used. Final Answer: Therefore, the correct answer is 6.

Question 19

Directions: These questions are based on the following information: Children in a class play only one or two or all of the three games - badminton, football and cricket. 5 children play only cricket, 8 children play only football and 7 children play…

Accepted Answer

Concept & Strategy: When calculating the intersection of two specific sets, you must combine the region where *only* those two sets intersect with the central region where *all three* sets intersect. n(B \cap C) = n(Only B and C) + n(All three) Step-by-Step Solution: • Given: • Children playing *only* badminton and cricket = 4 • Children playing *all three* games = 2 • Calculation: • The question asks for "badminton as well as cricket" without the restrictive word "only". • This means we must include everyone who plays these two games, even if they also happen to play football. • Add the dual-sport group to the tri-sport group: 4 + 2 = 6. Exam Strategy & Shortcut: Identify the two games in question (Badminton and Cricket). Find the "only two games" value for this pair (4), and automatically add the universal "all three" value (2) to it. Common Pitfall: Forgetting to add the center intersection (2) is the most frequent mistake. Students see "only two games - badminton and cricket = 4" and immediately choose 4. The absence of the word "only" in the question itself changes everything. Final Answer: Therefore, the correct answer is 6.

Question 20

In the cinema set of a movie, 125 mechanical aliens were created. Some of these aliens had peculiar features. 40 had two noses, 30 had three legs, 20 had four ears, 10 had two noses and three legs, 12 had three legs and four ears, 5 had two noses an…

Accepted Answer

Concept & Formula: To find the number of elements with *none* of the features, we first calculate the total number of elements that have *at least one* feature using the standard 3-set Union formula, and then subtract that from the universal total. n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C) Step-by-Step Solution: • Given: • Universal Total = 125 • Noses (N) = 40, Legs (L) = 30, Ears (E) = 20 • N \cap L = 10, L \cap E = 12, N \cap E = 5 • N \cap L \cap E = 3 • Calculation: • First, plug the given values directly into the union formula to find how many aliens have at least one unusual feature: • Total with features = 40 + 30 + 20 - 10 - 12 - 5 + 3 • Total with features = 90 - 27 + 3 = 66 • Now, subtract the aliens with features from the total created aliens to find those without any: • None = Total - (Total with features) • None = 125 - 66 = 59 • Since 59 is not present in options (a), (b), or (c), the answer is "None of these". Exam Strategy & Shortcut: For standard formula applications, group all the positive terms and negative terms separately for faster mental math: (40+30+20+3) - (10+12+5) = 93 - 27 = 66. Then compute 125 - 66 = 59. Avoid drawing Venn diagrams when the data strictly maps to the union formula components. Common Pitfall: A frequent error is assuming that "10 had two noses and three legs" means *only* those two features. In set theory, unless the word "only" is used, the intersection includes those who might have the third feature as well. The standard formula handles this overlap perfectly. Final Answer: Therefore, the correct answer is None of these.

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