Question 1

If the third day of a month is a Tuesday, which of the following will be the fourth day before the 27th day of that same month?

Accepted Answer

Introduction / Context: This question is another calendar based reasoning problem. A specific day of the week is given for the third day of the month, and we are asked to find the day of the week that is four days before the 27th of the same month. It tests understanding of how days of the week repeat every seven days and how to move forward or backward through the calendar using modular arithmetic. Given Data / Assumptions: The 3rd day of the month is a Tuesday. We must determine the day four days before the 27th day of the month. The month is continuous without interruptions and follows the standard 7 day weekly cycle. Concept / Approach: First determine the day of the week for the 27th. To do that, calculate how many days separate the 3rd and the 27th, and then find the remainder when this difference is divided by 7. That remainder tells us how many net steps forward to move from Tuesday. Once the day for the 27th is known, move four days backward to find the required day, again using the 7 day cycle. Step-by-Step Solution: Step 1: The 3rd is Tuesday. Step 2: The 27th is 27 - 3 = 24 days after the 3rd. Step 3: Compute 24 mod 7. Since 21 is a multiple of 7, the remainder is 3. Step 4: Move 3 days forward from Tuesday: Tuesday (start), Wednesday (1), Thursday (2), Friday (3). Thus, the 27th is a Friday. Step 5: We need the day four days before the 27th. Count backward four days from Friday. Step 6: One day before Friday is Thursday, two days before is Wednesday, three days before is Tuesday, and four days before is Monday. Step 7: Therefore, the fourth day before the 27th is Monday. Verification / Alternative check: Instead of first finding the day of the 27th, we can go directly from the 3rd to the 23rd, which is four days before the 27th. The difference between the 3rd and the 23rd is 20 days. Now 20 mod 7 is 6, and moving 6 days forward from Tuesday lands on Monday. This shorter route gives the same answer and confirms that the day four days before the 27th is Monday. Why Other Options Are Wrong: Tuesday, Wednesday and Sunday do not match the correct backward counting from Friday. They correspond to other offsets that do not equal four days before the 27th. Common Pitfalls: Students often miscount the number of days between dates or forget to use remainders modulo 7, which can lead to going too far forward or backward. Another mistake is confusing counting of days inclusively versus exclusively. Carefully calculating the difference in dates and then using mod 7 arithmetic helps keep the process accurate. Final Answer: The fourth day before the 27th of the month is Monday.

Question 2

If the day before yesterday was Thursday, then on which day will Sunday fall relative to today?

Accepted Answer

Introduction / Context: This is a basic day relationship problem that asks you to relate the current day to other days of the week based on a simple statement. The information given connects today to the day before yesterday, and from this we must infer the position of Sunday relative to today (whether it is today, tomorrow, or another day). Given Data / Assumptions: The day before yesterday was Thursday. We must determine when Sunday occurs with respect to today. The weekly cycle is the standard Sunday to Saturday cycle repeating every 7 days. Concept / Approach: If the day before yesterday is known, we can find yesterday by moving one day forward, and today by moving one more day forward. Once today is identified, we can determine the relationship of Sunday to today by comparing positions in the weekly cycle. We need to express today in terms of a known weekday and then count forward to Sunday. Step-by-Step Solution: Step 1: Let us denote the current day as X. Step 2: The day before yesterday is two days before today. The statement says that this day is Thursday. Step 3: If X is today, then X - 2 days is Thursday. Step 4: Move forward from Thursday: one day after Thursday is Friday (yesterday), and two days after Thursday is Saturday (today). Step 5: Therefore, today is Saturday. Step 6: Sunday is the day immediately after Saturday. Step 7: So Sunday is tomorrow relative to today. Verification / Alternative check: We can verify by listing: Thursday (day before yesterday), Friday (yesterday), Saturday (today), Sunday (tomorrow). This matches the given condition and clearly shows Sunday is one day after Saturday, that is, on the next day after today. Why Other Options Are Wrong: Sunday cannot be today, because today has been identified as Saturday. Sunday is not the day after tomorrow in this scenario, because that would make today Friday, which contradicts the starting statement. The option none is not correct because a definite relationship exists, and Sunday is specifically tomorrow. Common Pitfalls: Some learners mistakenly shift the days in the wrong direction or misinterpret which day is "before yesterday". Remember that the day before yesterday is two days prior to today. Drawing a small timeline with three consecutive days marked can help avoid confusion about the relative positions of yesterday, today and tomorrow. Final Answer: Sunday will be tomorrow.

Question 3

How many days are there from 26th January 2008 to 15th May 2008, if both the starting day and the ending day are included in the count?

Accepted Answer

Introduction / Context: This question asks for the number of calendar days between two specific dates in the year 2008, with the important detail that both the starting date and the ending date are to be included. Since 2008 is a leap year, February has 29 days, and this must be taken into account. The problem tests careful counting of days across multiple months and correct handling of inclusive ranges. Given Data / Assumptions: Start date: 26th January 2008. End date: 15th May 2008. Both 26th January and 15th May are included in the count. Year 2008 is a leap year, so February has 29 days. Concept / Approach: The total number of days in an inclusive interval from date A to date B can be found by summing days in each month segment between the two dates, and adjusting for partial months at the beginning and end. For each complete month between the two endpoints, we add the full month length; for partial months, we count only the relevant days. Finally, we ensure that both the start and end dates are included in the total. Step-by-Step Solution: Step 1: In January 2008, count days from 26th to 31st inclusive. That gives 31 - 26 + 1 = 6 days. Step 2: February 2008 is in a leap year, so it has 29 days. All 29 are between the two dates, so we add 29. Step 3: March has 31 days. All 31 days fall within the interval, so add 31. Step 4: April has 30 days. All 30 days fall inside the range, so add 30. Step 5: In May, we count from 1st May to 15th May inclusive, which is 15 days. Step 6: Now sum all partial and full month contributions: 6 (January) + 29 (February) + 31 (March) + 30 (April) + 15 (May). Step 7: Compute 6 + 29 = 35; 35 + 31 = 66; 66 + 30 = 96; 96 + 15 = 111. Step 8: Therefore, there are 111 days from 26th January 2008 to 15th May 2008 inclusive. Verification / Alternative check: An equivalent method is to compute the serial day number of each date within the year and subtract. The day number of 26th January is 26. For 15th May, add full months before May: January 31, February 29, March 31, April 30, then plus 15 days of May. This sums to 31 + 29 + 31 + 30 + 15 = 136. Now the inclusive count between day 26 and day 136 is 136 - 26 + 1 = 111, which matches the previous calculation. Why Other Options Are Wrong: 110, 112 and 113 would arise only if one of the months is miscounted, if February is treated as having 28 instead of 29 days, or if the inclusion of both endpoints is mishandled. Common Pitfalls: Students often forget that 2008 is a leap year and mistakenly use 28 days for February. Others may misapply inclusive counting and either omit the start day or the end day. Breaking the period month by month and using the simple formula last day minus first day plus one for partial months helps ensure accurate counting. Final Answer: There are 111 days between 26th January 2008 and 15th May 2008, including both dates.

Question 4

The digits in the number 25673948 are arranged in ascending order from left to right. In this new arrangement, what is the sum of the digit that is fourth from the right and the digit that is third from the left?

Accepted Answer

Introduction / Context: This problem combines basic ordering of digits and positional reasoning within a number. We are asked to rearrange the digits of a given number in ascending order and then identify two particular digits according to their positions from the left and from the right. Finally, we must compute the sum of these two digits. It tests careful manipulation of digit sequences and position counting. Given Data / Assumptions: Original number: 25673948. Digits must be rearranged in ascending order from left to right. In the new arrangement, we must find: The digit that is third from the left. The digit that is fourth from the right. The required answer is the sum of these two digits. Concept / Approach: First, list all digits of the original number without changing them, then sort them in ascending order. After obtaining the sorted sequence, position counting can be done easily: from the left, positions are counted 1, 2, 3 and so on; from the right, positions are counted starting from the units place. Once the correct digits are identified, adding them gives the final result. Step-by-Step Solution: Step 1: Write down the digits of 25673948: 2, 5, 6, 7, 3, 9, 4, 8. Step 2: Arrange these digits in ascending order: 2, 3, 4, 5, 6, 7, 8, 9. Step 3: In the new arrangement, the digit third from the left is the third element in this list. Step 4: The list from the left is: position 1 = 2, position 2 = 3, position 3 = 4, position 4 = 5, position 5 = 6, position 6 = 7, position 7 = 8, position 8 = 9. So the digit third from the left is 4. Step 5: Now count from the right. The rightmost digit (units place) is 9, second from the right is 8, third from the right is 7, and fourth from the right is 6. Step 6: Therefore, the digit fourth from the right is 6. Step 7: The required sum is 4 + 6 = 10. Verification / Alternative check: We can also mark positions explicitly under the ascending sequence 2 3 4 5 6 7 8 9. Label positions from the left as 1 through 8 and from the right as 1 through 8 in reverse order. This clearly shows that the third from the left is 4 and the fourth from the right is 6. Adding them again confirms the total of 10. Why Other Options Are Wrong: Sums of 9, 8 or 6 arise only if one of the positional digits is misidentified, for example by confusing the third from the left with the fourth, or miscounting from the right. Common Pitfalls: Errors often occur when students miscount positions from the right, especially in an 8 digit sequence, or forget to fully reorder the digits before counting. Another pitfall is to overlook repeated digits (though in this number all digits are distinct). Sorting the digits first and then carefully counting positions from both ends helps avoid these issues. Final Answer: The required sum is 10.

Question 5

Using only the second, fourth and seventh digits of the number 739142658, each used exactly once, it is possible to form a number that is the perfect square of a two digit odd number. In that case, which of the following is the second digit of that …

Accepted Answer

Introduction / Context: This problem combines digit selection with knowledge of perfect squares and basic number properties. We are told to use three specific digits from a larger number, form a new number using each digit exactly once, and require that this new number be the square of a two digit odd number. The question then asks for the second digit of that two digit odd number, and we must check whether any of the listed options match or whether the correct digit is not among them. Given Data / Assumptions: Original number: 739142658. Digits are labelled from left to right as positions 1 to 9. We may only use the second, fourth and seventh digits, each exactly once, to form a new number. The new number must be a perfect square of a two digit odd number. We then need the second digit (units place) of that two digit odd number. Concept / Approach: First, identify the second, fourth and seventh digits in the given nine digit number. Then list all permutations of these three digits to form possible three digit numbers. Next, check which of these candidates are perfect squares. A perfect square must equal n * n for some integer n. If one of the candidate numbers is a perfect square of a two digit odd number, we find that two digit number and record its second digit. Finally, we compare that digit with the provided options and decide whether it is present or if the correct response is that none of the listed digits matches. Step-by-Step Solution: Step 1: Write the digits of 739142658 with positions: 1:7, 2:3, 3:9, 4:1, 5:4, 6:2, 7:6, 8:5, 9:8. Step 2: The second digit is 3, the fourth digit is 1, and the seventh digit is 6. Step 3: We must form a number using digits 3, 1 and 6, each used exactly once. The possible three digit numbers are: 136, 163, 316, 361, 613 and 631. Step 4: Check which of these are perfect squares. 361 is equal to 19 * 19, so 361 is a perfect square. Step 5: 19 is a two digit odd number, so 361 satisfies all the given conditions. Step 6: The two digit odd number whose square is 361 is 19. The second digit of 19 is 9. Step 7: Now compare the digit 9 with the options given: 4, 7, 3 and none. Step 8: The digit 9 does not appear among 4, 7 or 3, so the correct choice is that none of the listed digits matches the required second digit. Verification / Alternative check: We can verify by checking other candidate squares in the three digit range. None of 136, 163, 316, 613 or 631 is a perfect square of an integer, while 361 is clearly 19^2. Therefore there is exactly one suitable three digit number that fits all constraints, and it leads uniquely to the two digit odd number 19 and its second digit 9. Since 9 is absent from the explicit options, the answer must be none of these. Why Other Options Are Wrong: Digits 4, 7 and 3 are not the second digit of the required two digit odd number 19. They may appear in the original nine digit number or be among the selected digits, but that does not make them correct for the question asked. Common Pitfalls: A common mistake is to assume that the second digit must be one of the digits used to form the square, which is not required by the problem statement. Another error is to overlook 361 as a well known perfect square and assume no such square exists, leading to an incorrect conclusion. Systematically checking all permutations and knowing basic square values such as 19^2 = 361 helps to avoid these pitfalls. Final Answer: The correct second digit is 9, which is not among the listed digits, so the answer is none.

Question 6

In a non-leap year, if the first day of the year was Monday, then which day of the week was the last day of that year?

Accepted Answer

Introduction / Context: In calendar based reasoning questions, exams often test whether you understand how days of the week repeat over fixed intervals. A non leap year has 365 days, which is not exactly a multiple of seven, so the starting and ending days of the year are closely related. This question asks you to connect the first day of a non leap year with its last day using simple modular arithmetic on days of the week. Given Data / Assumptions: - The year under consideration is explicitly stated to be a non leap year. - A non leap year has 365 days. - The first day of this year is Monday. - The week cycle repeats every 7 days. Concept / Approach: The key concept is that days of the week repeat after every block of seven days. If the total number of days in a period is of the form 7k, the starting and ending days are the same. For a non leap year with 365 days, we look at 365 in terms of complete weeks plus the remainder. Once we know how many extra days are left beyond full weeks, we shift the first day forward by that remainder to get the last day of the year. Step-by-Step Solution: Step 1: Note that a non leap year has 365 days. Step 2: Express 365 as full weeks plus remainder: 365 = 52 * 7 + 1. Step 3: The term 52 * 7 represents 52 complete weeks, which do not change the day of the week pattern. Step 4: The remainder is 1 day, which means the last day of the year is exactly one weekday step ahead of the first day. Step 5: If the first day is Monday, shifting forward by 1 normally gives Tuesday. However, here we must be careful about counting positions of days. Step 6: Day 1 of the year is Monday. Day 8 is also Monday, and so on. Day 365 is 364 days after day 1, which is 52 full weeks (364 days). Therefore the 365th day falls on the same weekday as the first day. Step 7: Hence the last day of the year is again Monday. Verification / Alternative check: One quick verification is to consider the simple idea that 364 is exactly 52 * 7. If you start from Monday and move forward by 364 days, you complete 52 full weeks and land on Monday again. Day 1 is Monday, day 365 is 364 days later, so day 365 is also Monday. This confirms that the first and last days of a non leap year are the same weekday. Therefore the answer Monday is fully consistent with the arithmetic of weeks and days. Why Other Options Are Wrong: Friday: This would be correct only if the remainder created a shift of four days, which is not the case here. Saturday: This would correspond to a remainder of five days between first and last day, which does not match 365 days. Sunday: This would be appropriate if the effective shift produced a six day difference, which again does not match the calculation. Common Pitfalls: Many learners mistakenly assume that because 365 divided by 7 leaves a remainder of 1, the last day must be one day ahead of the first, and they forget to relate the remainder to the position of the last day. The correct way is to compare day 1 and day 365 by looking at the difference of 364 days, not 365. Another common confusion is mixing up leap years and non leap years, which changes the total days and the remainder. Being precise about which day is counted as day 1 avoids these errors. Final Answer: Thus, in a non leap year where the first day is Monday, the last day of the year is also Monday.

Question 7

In a class of 31 students, Manoj and Satya are ranked seventh and eleventh from the top. What are their respective ranks from the bottom of the class?

Accepted Answer

Introduction / Context: Ranking and ordering problems are very common in reasoning and aptitude examinations. They test your ability to relate positions from the top and from the bottom within a fixed total. In this question, you are given two ranks from the top in a class of 31 students and you must find the corresponding ranks from the bottom for both students. Given Data / Assumptions: - Total number of students in the class = 31. - Manoj is ranked seventh from the top. - Satya is ranked eleventh from the top. - There is exactly one student at each rank, with no shared positions. Concept / Approach: The core relationship between a rank from the top and a rank from the bottom in a group of N students is bottom rank = N - top rank + 1. This works because you count how many students are after a given student plus the student himself. Applying this formula separately to Manoj and Satya gives their positions from the opposite end. Once this direct formula is clear, such questions become straightforward calculations. Step-by-Step Solution: Step 1: Note that N, the total number of students, is 31. Step 2: Manoj is seventh from the top, so his top rank T1 = 7. Step 3: Use the relation for Manoj: bottom rank B1 = N - T1 + 1 = 31 - 7 + 1 = 25. Step 4: Therefore, Manoj is twenty fifth from the bottom. Step 5: Satya is eleventh from the top, so his top rank T2 = 11. Step 6: Use the same relation for Satya: bottom rank B2 = N - T2 + 1 = 31 - 11 + 1 = 21. Step 7: Therefore, Satya is twenty first from the bottom. Step 8: Combine the results: Manoj and Satya are respectively 25th and 21st from the bottom. Verification / Alternative check: A quick check is to imagine the positions in reverse. If Manoj is 25th from the bottom, then there are 24 students below him and 6 above him. That makes 6 + 1 + 24 = 31 students, and his top rank is 7, which matches the given data. Similarly, for Satya at 21st from the bottom, there are 20 students below and 10 above, so 10 + 1 + 20 = 31 and his top rank is 11. This confirms that our conversion between top and bottom ranks is consistent. Why Other Options Are Wrong: 26th and 22nd: These values come from subtracting one incorrectly or misapplying the formula. 20th and 24th: This option reverses and miscalculates the positions, not matching either student correctly. 24th and 20th: Both ranks are off by one for each student and do not satisfy the conversion back to the top ranks. Common Pitfalls: Students often forget to add 1 in the relation bottom rank = N - top rank + 1 and instead use only N - top rank, which gives an incorrect result. Another error is to confuse whose rank is being converted and to accidentally swap the positions of Manoj and Satya. Writing the formula clearly each time and substituting step by step helps avoid these mistakes in exam conditions. Final Answer: The respective ranks of Manoj and Satya from the bottom of the class are 25th and 21st.

Question 8

The birthday of Sandhya falls on 15 August and the birthday of Minu falls on 25 June. If in a certain year Minu had her birthday on a Wednesday, what day of the week was Sandhya's birthday in that same year?

Accepted Answer

Introduction / Context: Calendar reasoning questions ask you to move forward or backward through dates and determine corresponding days of the week. Here, you are given two birthdays in the same year and the weekday for one of them. The task is to calculate the day of the week for the other birthday by counting the number of days between the two dates. Given Data / Assumptions: - The birthday of Minu is on 25 June. - The birthday of Sandhya is on 15 August of the same year. - In the year considered, 25 June falls on a Wednesday. - We treat the year as a normal year for month lengths, and both dates lie in the same year. Concept / Approach: The key approach is to calculate the exact number of days between 25 June and 15 August. Once this difference is known, we find the remainder when dividing by 7, because the day of the week repeats every seven days. By moving forward from Wednesday by this remainder, we obtain the weekday for 15 August. Careful counting of days across June, July, and August is essential. Step-by-Step Solution: Step 1: Determine days remaining in June after 25 June. June has 30 days, so there are 30 - 25 = 5 days left in June. Step 2: Count all days in July, which has 31 days. Step 3: Count days in August up to 15 August, which is 15 days. Step 4: Total number of days from 25 June to 15 August is 5 + 31 + 15 = 51 days. Step 5: To find the net shift in weekday, divide 51 by 7. We get 51 = 7 * 7 + 2, so the effective shift is 2 days forward. Step 6: Start from Wednesday, then move two days forward: Wednesday to Thursday is one day, Thursday to Friday is the second day. Step 7: Therefore, 15 August in that year falls on a Friday. Verification / Alternative check: As an alternative check, you can mark a short table of weekday shifts. Because 49 days correspond to exactly seven weeks, they do not change the weekday. The remaining 2 days shift the weekday from Wednesday to Friday. This method confirms that any error would have to come from miscounting the number of days between the two birthdays. Since the month lengths used are standard and the arithmetic 5 + 31 + 15 = 51 is correct, the resultant weekday Friday is reliable. Why Other Options Are Wrong: Monday: This would correspond to a shift of four days backward from Friday or a different remainder, which does not match the 51 day difference. Tuesday: This would require a shift of five days backward or two days backward from Thursday, which is not supported by the calculation. Saturday: This would correspond to a three day forward shift from Wednesday, which is larger than the remainder of two days. Common Pitfalls: A frequent mistake is to include or exclude the starting date incorrectly when counting days, which can change the total by one. Learners also sometimes misremember the number of days in July or June, which leads to a wrong remainder when dividing by seven. Another pitfall is to move in the wrong direction, going backward instead of forward. Writing down the months with their lengths and doing the arithmetic step by step helps avoid these issues. Final Answer: The day of the week on Sandhya's birthday in that year was Friday.

Question 9

The numbers 382, 473, 568, 728, 847 and 629 are written so that the digits of each number are arranged in ascending order. Which pair of these numbers then has the smallest difference between their new forms?

Accepted Answer

Introduction / Context: This question tests your ability to work with place values and digit rearrangement in three digit numbers. When digits are rearranged in a specific way, the new numbers can be compared to find differences. You are asked to follow a clear rule for re writing each number and then determine which pair exhibits the smallest difference between their transformed values. Given Data / Assumptions: - Original numbers: 382, 473, 568, 728, 847, and 629. - Transformation rule: write each number so that its digits are in ascending order. - Examples given: 382 becomes 238, 473 becomes 347. - We must compare the transformed values of the pairs listed in the options. Concept / Approach: The approach is to apply the same digit sorting rule to each number, then calculate the absolute difference between the transformed numbers in each given pair. The digits of a three digit number can be viewed as hundreds, tens, and units. Sorting them in ascending order changes the place value contribution of each digit, which changes the value of the number. Once all new forms are known, a simple subtraction identifies the pair with the smallest difference. Step-by-Step Solution: Step 1: Transform 382. Digits are 3, 8, and 2. In ascending order they are 2, 3, 8, giving 238. Step 2: Transform 473. Digits are 4, 7, and 3. In ascending order they are 3, 4, 7, giving 347. Step 3: Transform 568. Digits 5, 6, and 8 are already in ascending order, so the new form is 568. Step 4: Transform 728. Digits 7, 2, and 8 in ascending order are 2, 7, 8, giving 278. Step 5: Transform 847. Digits 8, 4, and 7 in ascending order are 4, 7, 8, giving 478. Step 6: Transform 629. Digits 6, 2, and 9 in ascending order are 2, 6, 9, giving 269. Step 7: Compute differences for the pairs in the options: - For 473 and 382: 347 minus 238 equals 109. - For 629 and 728: 278 minus 269 equals 9. - For 629 and 847: 478 minus 269 equals 209. - For 728 and 847: 478 minus 278 equals 200. Step 8: The smallest difference among these is 9, corresponding to the pair 629 and 728. Verification / Alternative check: As a quick mental check, focus only on the transformed numbers, not the originals. The transformed list is 238, 347, 568, 278, 478, and 269. If you sort these, the closest pair numerically appears around 269 and 278. These are the transformed forms of 629 and 728. Confirming this pair yields a difference of 9, and no other pair produces a smaller difference. This confirms that the chosen option is consistent and unique. Why Other Options Are Wrong: 473 and 382: Their new values differ by 109, which is much larger than 9. 629 and 847: The difference between 269 and 478 is 209, one of the largest differences. 728 and 847: Their transformed values 278 and 478 differ by 200, again far from the minimum. Common Pitfalls: Learners sometimes compare the original numbers instead of the transformed versions, which completely changes the result. Another common mistake is incorrect digit sorting, such as treating 473 as 437 rather than 347. Finally, some candidates forget to take absolute differences and may consider negative values or reversed subtraction, but since we care about magnitude, the order of subtraction does not matter as long as the absolute difference is used. Final Answer: The pair of numbers whose transformed forms have the smallest difference is 629 and 728.

Question 10

In a row of thirty boys, R is fourth from the right end and W is tenth from the left end. How many boys are there between R and W in the row?

Accepted Answer

Introduction / Context: Row arrangement and ranking questions require you to translate positions from right and left ends into absolute positions and then measure distances between them. In this question, you know the total number of boys in the row and the positions of R and W from opposite ends. The task is to find how many boys are seated strictly between them. Given Data / Assumptions: - Total number of boys in the row = 30. - R is fourth from the right end. - W is tenth from the left end. - All positions in the row are occupied by distinct boys. Concept / Approach: To compare positions from different ends, you first convert them to positions from a common side, usually from the left. If the total number of boys is N, then a position that is k-th from the right is at (N - k + 1) from the left. Once both positions are expressed from the same side, subtract the smaller from the larger and reduce by one to get the count of boys in between. This approach is standard for many exam questions of this type. Step-by-Step Solution: Step 1: Total number of boys N = 30. Step 2: R is fourth from the right. So his position from the left is N - 4 + 1 = 30 - 4 + 1 = 27. Step 3: W is tenth from the left, so W is at position 10 from the left. Step 4: We now have R at position 27 and W at position 10 when counted from the left. Step 5: To find how many boys sit between them, compute the difference between their positions and subtract 1. Step 6: Difference between 27 and 10 is 27 - 10 = 17. Step 7: The number of boys between R and W = 17 - 1 = 16. Verification / Alternative check: Imagine the row indexed from the left: positions 1 to 30. W sits at position 10. R sits at position 27. If you count from 11 up to 26, you get the boys between them. The number of such positions is 26 - 11 + 1, which equals 16. This matches the earlier calculation and verifies the result without relying solely on formulas. Why Other Options Are Wrong: 15: This would be correct if the positions differed by 16, which they do not. 17: This incorrectly takes the full difference between the positions without subtracting one for the endpoints. Cannot be determined: The total number of boys and both positions are given clearly, so the answer is uniquely determined. Common Pitfalls: A common error is forgetting to subtract one after taking the difference of positions. This leads to counting one of the end boys as being between themselves, which is not correct. Another mistake is miscalculating the left side position from the right side position by using N - k instead of N - k + 1. Writing out the positions and verifying them visually can prevent such slips in competitive exam settings. Final Answer: The number of boys sitting between R and W in the row is 16.

Question 11

For the numbers 567, 284, 696, 865 and 738, one is added to the last digit of each number and then the order of the digits is reversed. Which original number will become the fourth smallest number after this transformation?

Accepted Answer

Introduction / Context: This question blends digit manipulation with ordering of numbers. Each original three digit number undergoes a specific two step transformation: first the unit digit is increased by one, and then the entire three digit sequence is reversed. After performing this transformation on each number, you must determine which original number corresponds to the fourth smallest transformed value. Given Data / Assumptions: - Original numbers: 567, 284, 696, 865, 738. - Step 1: Add one to the last digit of each number. - Step 2: Reverse the order of the three digits of the resulting number. - None of the given numbers ends in 9, so we do not need to handle carrying over to a new digit. Concept / Approach: The approach is to apply the given two step operation carefully to each number. Then, list all transformed numbers and sort them in ascending order. Once the sorted order is clear, you can identify which transformed number is the fourth smallest and map it back to its original number. The critical part is consistent application of the same transformation rule to every number without arithmetic slips. Step-by-Step Solution: Step 1: Transform 567. Adding one to the last digit 7 gives 568. Reversing 568 yields 865. Step 2: Transform 284. Last digit 4 plus one gives 5, so we get 285. Reversing 285 yields 582. Step 3: Transform 696. Last digit 6 plus one gives 7, therefore the intermediate number is 697. Reversing 697 yields 796. Step 4: Transform 865. Last digit 5 plus one gives 6, so we get 866. Reversing 866 yields 668. Step 5: Transform 738. Last digit 8 plus one gives 9, so we get 739. Reversing 739 yields 937. Step 6: Collect the transformed numbers: 865 (from 567), 582 (from 284), 796 (from 696), 668 (from 865), 937 (from 738). Step 7: Arrange these values in ascending order: 582, 668, 796, 865, 937. Step 8: The fourth smallest among these is 865, which came from the original number 567. Verification / Alternative check: Given the small number of values, you can easily double check by rewriting a table that shows each original number next to its transformed version. Make sure that each reversal is correct: for example, reversing 568 should indeed give 865, not 856. After verifying each transformation, sort again mentally or on paper. The order 582, 668, 796, 865, 937 is straightforward and clearly places 865 fourth. This confirms that 567 is the correct original number. Why Other Options Are Wrong: 284: Its transformed value 582 is the smallest number, not the fourth smallest. 865: Its transformed value 668 is the second smallest, not the fourth smallest. 738: Its transformed value 937 is the largest, not the fourth smallest. Common Pitfalls: Some students reverse the digits first and then add one to the last digit, which reverses the intended order of operations and produces incorrect results. Others may misorder the transformed numbers because they mentally compare only the first digit instead of the full three digit values. Writing down each intermediate step and carefully sorting helps avoid these errors, especially under time constraints. Final Answer: The original number that becomes the fourth smallest after the described transformation is 567.

Question 12

In a row, Kumar is at seventh place from the left end and Pawan is at ninth place from the right end. After they interchange their positions, Kumar becomes eleventh from the left. How many persons are seated in the row?

Accepted Answer

Introduction / Context: Here you are given a typical ranking puzzle where two people in a row interchange their positions. You know their positions from different ends before and after the interchange. The problem tests your understanding of how to convert between left and right ranks and how to use that information to determine the total number of persons in the row. Given Data / Assumptions: - Kumar is seventh from the left before interchange. - Pawan is ninth from the right before interchange. - After interchange, Kumar becomes eleventh from the left. - The total number of persons in the row is fixed and is to be found. Concept / Approach: The main idea is that after interchange, each person occupies the other person's original position. So, when Kumar becomes eleventh from the left after the swap, he is now occupying the position that originally belonged to Pawan. That means Pawan's original position from the left must be the eleventh place. Using the relationship between left and right ranks in a row of N persons, we can form an equation to solve for N. Step-by-Step Solution: Step 1: Let N be the total number of persons in the row. Step 2: Before interchange, Kumar is seventh from the left, so his original left rank is 7. Step 3: Before interchange, Pawan is ninth from the right. His original position from the left is N - 9 + 1 = N - 8. Step 4: After they swap, Kumar takes Pawan's original position. According to the question, Kumar is now eleventh from the left. Step 5: Therefore Pawan's original position from the left, N - 8, must equal 11. Step 6: Set up and solve the equation: N - 8 = 11. Step 7: From this, N = 11 + 8 = 19. Step 8: Hence there are 19 persons seated in the row. Verification / Alternative check: Check consistency by computing Pawan's new position after the swap. Originally Kumar was seventh from the left. In a row of 19 persons, the seventh position from the left corresponds to right rank = N - 7 + 1 = 19 - 7 + 1 = 13. So, after swap, Pawan becomes thirteenth from the right. This is consistent and does not contradict any given information. Also, Pawan's original left rank N - 8 indeed becomes 11 when N equals 19, verifying the equation. Why Other Options Are Wrong: 20: If N were 20, Pawan's original left position would be 20 - 8 = 12, not 11. 21: For N = 21, Pawan's left position would be 13, which again does not match eleven. 27: This is far from the correct row size and produces an inconsistent left position for Pawan. Common Pitfalls: Many learners confuse whose position is being described after interchange and mistakenly apply the formula to the wrong person. Others incorrectly set the relationship N - 9 instead of N - 9 + 1, forgetting to add one. Carefully labeling original and new positions for each person and writing a clear equation avoids these misunderstandings in ranking problems. Final Answer: The total number of persons seated in the row is 19.

Question 13

For each of the numbers 517, 325, 639, 841 and 792, only the second and third digits within the number are interchanged. What is the first digit of the second highest number obtained after this change?

Accepted Answer

Introduction / Context: Digit interchange questions check your comfort with place value and simple manipulation of three digit numbers. Here you are asked to swap only the second and third digits of each number, then compare the resulting numbers. The final objective is to identify the first digit of the second highest transformed number, not the number itself. Given Data / Assumptions: - Original numbers: 517, 325, 639, 841, 792. - Operation: interchange only the second digit (tens place) and third digit (units place) of each number. - After transformation, we must find the second highest number and then identify its first digit. Concept / Approach: A three digit number has the form 100 * a + 10 * b + c, where a is the hundreds digit, b the tens digit, and c the units digit. Swapping b and c changes the number to 100 * a + 10 * c + b. We apply this to each given number, list all the new values, sort them in descending order, and then inspect the first digit of the second largest value. This systematic approach minimizes mistakes. Step-by-Step Solution: Step 1: Transform 517. Digits are a = 5, b = 1, c = 7. Swapping b and c gives 571. Step 2: Transform 325. Digits 3, 2, and 5 become 3, 5, and 2 after swapping, giving 352. Step 3: Transform 639. Digits 6, 3, and 9 become 6, 9, and 3, giving 693. Step 4: Transform 841. Digits 8, 4, and 1 become 8, 1, and 4, giving 814. Step 5: Transform 792. Digits 7, 9, and 2 become 7, 2, and 9, giving 729. Step 6: List all transformed numbers: 571, 352, 693, 814, 729. Step 7: Arrange these in descending order: 814, 729, 693, 571, 352. Step 8: The second highest transformed number is 729. Its first digit is 7. Verification / Alternative check: To verify, you can check that no transformed value exceeds 814 or lies between 814 and 729. Since 814 clearly has the greatest hundreds digit 8, it is the highest. Among the remaining numbers, 729 has hundreds digit 7, which is greater than 6 in 693 and 5 in 571, and greater than 3 in 352. Therefore 729 is indisputably the second highest. The first digit read from 729 is 7, which confirms the result. Why Other Options Are Wrong: 8: This is the first digit of the highest number 814, not the second highest. 2: No transformed number that could be second highest begins with 2, so this is not possible. 9: No transformed number begins with 9, so this option cannot be correct. Common Pitfalls: One error is to swap the first and second digits or to reverse the entire number instead of swapping only the last two digits. Another frequent mistake is to misorder the transformed numbers, especially if one relies only on quick visual comparison. Writing all transformed numbers clearly and comparing first digits, then second digits if required, ensures correct ranking and a reliable answer. Final Answer: The first digit of the second highest number after interchanging the second and third digits is 7.

Question 14

In the sequence of numbers 1 2 3 4 5 6 7 8 9 2 4 6 8 9 7 5 3 1 9 8 7 6 5 4 3 2 1, which number is third to the left of the number that lies exactly in the middle of the sequence?

Accepted Answer

Introduction / Context: In digit sequence reasoning, you are often asked to identify elements relative to a particular reference, such as the middle term or an element from one end. This question combines both ideas: you must first locate the exact middle term of a long sequence and then move a specified number of positions to the left to find the required number. Given Data / Assumptions: - The sequence is: 1 2 3 4 5 6 7 8 9 2 4 6 8 9 7 5 3 1 9 8 7 6 5 4 3 2 1. - We must identify the number that is third to the left of the middle element. - Positions are counted from the left, and the sequence length is odd, so there is a single exact middle term. Concept / Approach: First, determine the total number of elements in the sequence. For an odd length N, the middle position is (N + 1) divided by 2. Once the index of the middle term is found, move three positions to the left of that index to locate the desired element. Careful counting of positions is more important than any numeric pattern in the digits themselves for this question. Step-by-Step Solution: Step 1: Count the number of terms in the sequence. There are 27 terms. Step 2: The middle position for 27 terms is (27 + 1) / 2 = 14. So the 14th term is the middle element. Step 3: Write down the positions with their terms from the left to confirm: Positions 1 to 9: 1, 2, 3, 4, 5, 6, 7, 8, 9. Positions 10 to 18: 2, 4, 6, 8, 9, 7, 5, 3, 1. Positions 19 to 27: 9, 8, 7, 6, 5, 4, 3, 2, 1. Step 4: Identify the 14th term. From the listing, position 14 corresponds to the number 9. Step 5: Now we need the number that is third to the left of this 14th term. Step 6: One position left is 13th term (8), two positions left is 12th term (6), and three positions left is 11th term. Step 7: The 11th term in the sequence is 4. Step 8: Therefore, the number that is third to the left of the middle term is 4. Verification / Alternative check: You can verify by writing the sequence with index numbers above or below each digit. When you do this carefully, you will again find that the 14th position holds 9 and that positions 11, 12, and 13 hold 4, 6, and 8 respectively. Moving three steps left from position 14 lands you at position 11, which clearly contains 4. This cross check ensures that no counting or indexing error has occurred. Why Other Options Are Wrong: 3: The number 3 appears in the sequence but not in the required relative position. 5: This digit is present in several places, yet none of those positions match being third to the left of the middle term. 7: This digit also occurs, but confirming positions shows it is not three places left of the middle element. Common Pitfalls: Students sometimes miscount the total terms, leading to an incorrect middle position. Others confuse "third to the left" with "third from the left," which is a completely different reference. A careful habit is to mark the middle term first, then physically count left or right as required, rather than trying to jump directly to a mental answer. Final Answer: The number that is third to the left of the middle term of the sequence is 4.

Question 15

Among the integers from 9 to 54 inclusive, how many numbers are exactly divisible by 9 but not divisible by 3?

Accepted Answer

Introduction / Context: Divisibility questions test your understanding of basic number properties and relationships between factors. This problem asks you to examine numbers in a specified range and apply divisibility conditions involving the numbers 9 and 3. Recognising the relationship between divisibility by 9 and divisibility by 3 is the key insight here. Given Data / Assumptions: - We are considering all integers from 9 to 54 inclusive. - We want numbers that are divisible by 9. - Among those, we must retain only numbers that are not divisible by 3. - Standard divisibility rules for 3 and 9 apply. Concept / Approach: The crucial concept is that if a number is divisible by 9, it is always divisible by 3, because 9 itself is 3 multiplied by 3. In other words, divisibility by 9 implies divisibility by 3. Therefore, a number cannot simultaneously be divisible by 9 and not divisible by 3. Understanding this theoretical relationship means we do not actually need to list or test individual numbers in the range, because the condition is logically impossible. Step-by-Step Solution: Step 1: Recall that if a number is divisible by 9, it can be written as 9 * k for some integer k. Step 2: Since 9 = 3 * 3, 9 * k = 3 * (3 * k). Step 3: This shows that any number of the form 9 * k is also a multiple of 3, and hence divisible by 3. Step 4: The question asks for numbers that are divisible by 9 but not by 3. Step 5: From the previous steps, this is impossible because divisibility by 9 always includes divisibility by 3. Step 6: Therefore, within any range, including 9 to 54, no integer can satisfy the condition "divisible by 9 but not divisible by 3." Step 7: Hence, the required count of such numbers is zero. Verification / Alternative check: As a quick check, you can list the multiples of 9 between 9 and 54 inclusive: 9, 18, 27, 36, 45, 54. Each of these has a digit sum that is a multiple of 9 and therefore a multiple of 3, confirming that they are all divisible by 3 as well. Since every candidate multiple of 9 fails the "not divisible by 3" condition, the count remains zero. This practical verification matches the theoretical reasoning and confirms the answer. Why Other Options Are Wrong: 5: There are six multiples of 9 in the given range, not five, and all of them are divisible by 3. 6: This is the number of multiples of 9 but does not account for the divisibility by 3 condition. 9: This overestimates the number of multiples of 9 and still ignores the impossibility implied by the divisibility rules. Common Pitfalls: Some students misinterpret the condition and only count multiples of 9, forgetting to apply the additional restriction about 3. Others may mistakenly think that divisibility by 9 and divisibility by 3 are independent, when in fact divisibility by 9 is a stronger condition that automatically includes divisibility by 3. Remembering the hierarchy of factors helps to avoid this conceptual mistake. Final Answer: There are 0 numbers between 9 and 54 that are divisible by 9 but not divisible by 3.

Question 16

A bus for Shiridi leaves every 52 minutes from a bus stand. An enquiry officer tells a passenger that the last bus left 16 minutes ago and that the next bus will leave at 8:15 p.m. At what exact time did the enquiry officer give this information to …

Accepted Answer

Introduction / Context: Time and timetable problems often involve buses or trains leaving at fixed intervals. The key skill is to relate the fixed time gap between consecutive departures to the times given in the statement. In this question, you know the interval between buses, the time of the next bus, and how long ago the last bus left when the enquiry officer spoke. You must use these pieces to reconstruct the exact time of the enquiry. Given Data / Assumptions: - A bus for Shiridi leaves every 52 minutes from a bus stand. - The enquiry officer says the last bus left 16 minutes ago. - He also says the next bus will leave at 8:15 p.m. - Bus departures follow a strict 52 minute schedule without gaps or extra buses. Concept / Approach: When buses leave at a fixed interval, consecutive departure times form an arithmetic sequence with common difference equal to the interval. If the next bus is at 8:15 p.m., the previous one must have left exactly 52 minutes earlier. That previous departure is the same “last bus” that, according to the officer, left 16 minutes before he spoke. Once we know the last departure time, we simply add 16 minutes to get the time when the officer gave the information. Step-by-Step Solution: Step 1: Note that buses leave every 52 minutes. Step 2: The next bus is scheduled for 8:15 p.m. Step 3: The last bus before 8:15 p.m. must have left 52 minutes earlier. Step 4: Subtract 52 minutes from 8:15 p.m.: 8:15 p.m. minus 52 minutes equals 7:23 p.m. Step 5: Therefore, the last bus departed at 7:23 p.m. Step 6: The enquiry officer says this last bus left 16 minutes ago. Step 7: Add 16 minutes to 7:23 p.m. to find the enquiry time: 7:23 p.m. + 16 minutes = 7:39 p.m. Step 8: Thus, the officer must have given the information at 7:39 p.m. Verification / Alternative check: We can verify the logic by working forward from the enquiry time. If the officer spoke at 7:39 p.m. and the last bus left 16 minutes earlier, that departure occurred at 7:23 p.m. Adding the fixed interval of 52 minutes to 7:23 p.m. gives 8:15 p.m., which correctly matches the announced time of the next bus. Since all times are consistent, 7:39 p.m. is validated as the enquiry time. Why Other Options Are Wrong: 07:23 p.m: This is the time of the last bus departure, not the time of the enquiry. 06:25 p.m: From this time, adding 16 minutes would not land on a scheduled bus time that fits a 52 minute pattern ending at 8:15 p.m. 06:23 p.m: Similarly inconsistent with the 52 minute schedule and the given next departure time. Common Pitfalls: Students may mistakenly subtract 16 minutes from 8:15 p.m., mixing up “how long ago the bus left” with the interval between buses. Another common error is to forget that the 52 minute gap applies between consecutive buses, not between the enquiry and the next bus. Carefully separating “schedule interval” from “time since last event” prevents these confusions. Final Answer: The enquiry officer gave the information at 7:39 p.m.

Question 17

If the day before yesterday was Saturday, then what day of the week will it be on the day after tomorrow?

Accepted Answer

Introduction / Context: Day sequence questions check your grasp of relative time expressions like “day before yesterday” and “day after tomorrow.” Instead of needing a calendar, you simply need to track the order of weekdays correctly. Here you are told what day it was two days ago and are asked to find what day it will be two days from now, relative to today. Given Data / Assumptions: - The day before yesterday was Saturday. - Days of the week follow the standard repeating cycle: Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday. - No special holidays or jumps are involved; days progress normally one after another. Concept / Approach: The expressions can be decoded step by step. If “day before yesterday” is known, you can move forward one day to get “yesterday” and then another day to get “today.” From today, “tomorrow” is one step ahead and “day after tomorrow” is two steps ahead. This is simply a matter of moving along the weekday sequence in the correct direction without skipping or double counting any day. Step-by-Step Solution: Step 1: Let us start with the information that the day before yesterday was Saturday. Step 2: The day after the day before yesterday is yesterday. So, if the day before yesterday is Saturday, then yesterday is Sunday. Step 3: One day after Sunday is Monday. Therefore, today is Monday. Step 4: Tomorrow is the day after today, so tomorrow will be Tuesday. Step 5: The day after tomorrow is one more day ahead of Tuesday, which is Wednesday. Step 6: Hence, the day after tomorrow will fall on Wednesday. Verification / Alternative check: You can also visualize a short timeline of days. Mark “day before yesterday” as Saturday, “yesterday” as Sunday, “today” as Monday, “tomorrow” as Tuesday, and “day after tomorrow” as Wednesday in a simple list. Checking forward and backward confirms that Saturday is two days before Monday and Wednesday is two days after Monday, making the reasoning consistent and error free. Why Other Options Are Wrong: Friday: Friday is one day before Saturday, not two days after Monday. Thursday: This would be three days after Monday, which does not match “day after tomorrow.” Tuesday: Tuesday is only one day after Monday and represents “tomorrow,” not “day after tomorrow.” Common Pitfalls: Learners sometimes misinterpret “day before yesterday” as “yesterday” and shift all days by one. Another common mistake is to count the current day twice when moving forwards or backwards. Drawing a small sequence or speaking the days in order out loud often helps avoid such off by one errors in exam situations. Final Answer: If the day before yesterday was Saturday, the day after tomorrow will be Wednesday.

Question 18

In the series 2 8 3 8 2 4 8 2 4 8 6 8 2 8 2 4 8 3 8 2 8 6, how many times does the number 8 appear such that it is exactly divisible by both its immediate preceding number and its immediate succeeding number?

Accepted Answer

Introduction / Context: This question combines sequence scanning with number divisibility. You are asked to examine every occurrence of a particular number (8) in a series and test a specific condition involving the numbers immediately before and after it. Such problems are common in reasoning tests, where careful reading and systematic checking are more important than complex calculations. Given Data / Assumptions: - The series is: 2 8 3 8 2 4 8 2 4 8 6 8 2 8 2 4 8 3 8 2 8 6. - We focus only on positions where the middle number is 8. - For each such 8, the number before it and the number after it must both divide 8 exactly (8 % neighbour = 0). - Edge terms that do not have both a predecessor and a successor cannot be counted. Concept / Approach: The method is straightforward: scan through the series, locate each 8 that has both a previous and a next term, and then check divisibility. A number a is “exactly divisible” by another number b if the remainder of a / b is zero. Therefore, we test 8 % previous == 0 and 8 % next == 0. Whenever both conditions hold, we count that occurrence of 8. Finally, we add up how many such valid occurrences exist in the entire series. Step-by-Step Solution: Step 1: Write down the series with indices for clarity: Index: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Value: 2 8 3 8 2 4 8 2 4 8 6 8 2 8 2 4 8 3 8 2 8 6 Step 2: Identify positions where the value is 8 and has both neighbours: indices 2, 4, 7, 10, 12, 14, 17, 19, 21. Step 3: At index 2: neighbours are 2 and 3. 8 is divisible by 2 (8 % 2 = 0) but not by 3 (8 % 3 = 2), so do not count. Step 4: At index 4: neighbours are 3 and 2. 8 is not divisible by 3, so do not count. Step 5: At index 7: neighbours are 4 and 2. 8 % 4 = 0 and 8 % 2 = 0, so this 8 is valid and we count one. Step 6: At index 10: neighbours are 4 and 6. 8 is divisible by 4 but not by 6, so not counted. Step 7: At index 12: neighbours are 6 and 2. 8 is not divisible by 6, so not counted. Step 8: At index 14: neighbours are 2 and 2. 8 % 2 = 0 and 8 % 2 = 0, so this 8 is valid; count a second occurrence. Step 9: At indices 17, 19, and 21, at least one neighbour does not divide 8 exactly, so these are discarded. Step 10: The total count of valid 8s is 2. Verification / Alternative check: You can quickly re scan only the local triples where 8 is in the middle: (2, 8, 3), (3, 8, 2), (2, 4, 8), (4, 8, 6), (6, 8, 2), (2, 8, 2), (4, 8, 3), (3, 8, 2), (2, 8, 6). Among these, only (4, 8, 2) and (2, 8, 2) have both neighbours as exact divisors of 8. This confirms the answer without re-doing every index step. Why Other Options Are Wrong: five: Greatly overcounts because most occurrences of 8 fail the divisibility test with at least one neighbour. four: Also overestimates; only two triples satisfy the condition completely. one: Underestimates by acknowledging only one valid triple instead of two. Common Pitfalls: A common mistake is to count every 8 whose neighbours are smaller numbers, assuming they must be divisors, without actually checking the remainder. Another error is to ignore the requirement that both sides must divide 8, not just one side. Keeping the definition of exact divisibility in mind and checking each triple carefully prevents these errors. Final Answer: The number of times 8 appears with both neighbours exactly dividing it is two.

Question 19

In a row of boys, Rahul is 19th from the top and Ajay is 21st from the bottom. They interchange their positions and after swapping Ajay becomes 28th from the bottom. How many boys are there in the row in total?

Accepted Answer

Introduction / Context: This is a classic ranking and ordering puzzle involving two students who exchange their positions in a single row. You are given their positions from different ends before and after the interchange. The aim is to use these positions to determine the total number of boys in the row. Such problems test your ability to convert between ranks from the top and from the bottom and to set up simple equations. Given Data / Assumptions: - Rahul is 19th from the top initially. - Ajay is 21st from the bottom initially. - After they interchange positions, Ajay becomes 28th from the bottom. - There is a fixed total number of boys N in the row, with no vacant places. Concept / Approach: If the total number of boys is N, then a boy who is k-th from the bottom is at position N - k + 1 from the top. When two boys swap positions, each takes the other's original place. Here, after the swap, Ajay sits at Rahul's original position from the top. The information about Ajay's new position from the bottom gives a direct equation involving N that we can solve. This relation is the core of the method. Step-by-Step Solution: Step 1: Let N be the total number of boys in the row. Step 2: Initially, Rahul is 19th from the top; his top rank is 19. Step 3: Initially, Ajay is 21st from the bottom. His top rank is N - 21 + 1 = N - 20. Step 4: After they interchange positions, each boy occupies the other's original place. Step 5: So, after the swap, Ajay is now at Rahul's original position from the top, which is 19th from the top. Step 6: We are told that after the swap Ajay is 28th from the bottom. Step 7: The relation between Ajay's new bottom rank and the total N is: bottom rank = N - top rank + 1. Step 8: Substitute: 28 = N - 19 + 1, so 28 = N - 18. Step 9: Solve for N: N = 28 + 18 = 46. Step 10: Therefore, there are 46 boys in the row. Verification / Alternative check: We can confirm consistency by computing Ajay's original top position using N = 46 and his original bottom rank of 21. Top rank = 46 - 21 + 1 = 26. Thus initially Rahul is at position 19 and Ajay at position 26. After swapping, Ajay becomes 19th from the top and, with N = 46, his bottom rank is 46 - 19 + 1 = 28, which matches the given information. This cross-check supports the correctness of N = 46. Why Other Options Are Wrong: 47: For N = 47, Ajay's new bottom rank would be 47 - 19 + 1 = 29, not 28. 45: For N = 45, Ajay's new bottom rank would be 27, which again does not match the question. 48: For N = 48, Ajay's new bottom rank becomes 30, which contradicts the given data. Common Pitfalls: A frequent error is to forget the +1 in the conversion formula, writing N - k instead of N - k + 1. Another mistake is misidentifying which boy's position changes are being described after the swap, leading to the wrong equation. Clearly labelling positions “before” and “after” for each person prevents confusion and helps systematically reach the correct total. Final Answer: The total number of boys in the row is 46.

Question 20

The priest tells a devotee, 'The temple bell is rung at regular intervals of 45 minutes. The last bell was rung five minutes ago, and the next bell is due to be rung at 7:45 a.m.' At what time did the priest give this information to the devotee?

Accepted Answer

Introduction / Context: This question is another timetable and interval problem, framed around the ringing of a temple bell. You are given the regular interval between rings, the time of the next ring, and how long ago the last bell rang relative to when the priest spoke. The aim is to determine the exact time the priest made his statement by working backwards and forwards with these intervals. Given Data / Assumptions: - The bell is rung at regular intervals of 45 minutes. - The priest says the last bell was rung five minutes ago. - He also says the next bell is due at 7:45 a.m. - Bell timings strictly follow the 45 minute schedule. Concept / Approach: The ringing times of the bell form a sequence with a difference of 45 minutes. If we know one scheduled time (7:45 a.m. for the next bell), we can find the immediately previous scheduled time by subtracting 45 minutes. This previous scheduled time must be the “last bell” mentioned by the priest. Since that last bell rang five minutes before he spoke, we find the time of the priest's statement by adding 5 minutes to the time of the last ring. Step-by-Step Solution: Step 1: Note that the bell rings every 45 minutes. Step 2: The next scheduled bell is at 7:45 a.m. Step 3: The previous bell before 7:45 a.m. must have rung 45 minutes earlier. Step 4: Subtract 45 minutes from 7:45 a.m.: 7:45 a.m. minus 45 minutes equals 7:00 a.m. Step 5: Thus, the last bell was rung at 7:00 a.m. Step 6: According to the priest, the last bell was rung five minutes ago at the time he was speaking. Step 7: Therefore, the priest spoke five minutes after 7:00 a.m. Step 8: Add 5 minutes to 7:00 a.m. to get 7:05 a.m. Step 9: Hence, the priest gave this information at 7:05 a.m. Verification / Alternative check: We can verify by reconstructing the sequence of bell times around the statement. If a bell rang at 7:00 a.m., and the priest spoke at 7:05 a.m., then indeed the last bell was five minutes ago. Adding the 45 minute interval to 7:00 a.m. gives 7:45 a.m. as the next ringing time, which matches the priest's information. Since both conditions are satisfied simultaneously, 7:05 a.m. is consistent and correct. Why Other Options Are Wrong: 7.40 am: If the priest spoke at 7:40 a.m., five minutes earlier would be 7:35 a.m., which is not 45 minutes before 7:45 a.m. 7.00 am: This would mean the last bell and the time of speaking are the same, contradicting the statement that the bell was rung five minutes ago. 6.55 am: Five minutes earlier would be 6:50 a.m., and 45 minutes after 6:50 a.m. is 7:35 a.m., not 7:45 a.m. Common Pitfalls: Some students subtract the 5 minute gap from 7:45 a.m. instead of applying it to the previous bell time, mixing up the roles of the interval and the “time since last bell.” Others may subtract 45 minutes from the wrong reference point. Keeping track of which times are scheduled (7:00 a.m. and 7:45 a.m.) and which relate to the priest's speech (five minutes after the last bell) avoids these mistakes. Final Answer: The priest gave the information to the devotee at 7:05 a.m.

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