Q: What is the approximate decimal value of 25 divided by 7?

Introduction / Context:This question simply checks your ability to perform a division and express the result as a decimal. Such basic numerical skills are essential in many aptitude tests and everyday calculations. Given Data / Assumptions: We need to evaluate 25 ÷ 7. We want a decimal approximation accurate to three decimal places. Concept / Approach:Perform the division of 25 by 7 using long division or a known fraction to decimal conversion. The true value is a repeating decimal, so we round it appropriately to three digits after the decimal point. Step-by-Step Solution:Step 1: Express the division: 25 ÷ 7.Step 2: Since 7 * 3 = 21 and 7 * 4 = 28, the integer part of the result is 3.Step 3: Subtract 21 from 25 to get remainder 4. Bring down 0 to continue in decimal, making it 40.Step 4: 7 goes into 40 five times (7 * 5 = 35), remainder 5. So the next digit is 5, giving 3.5 so far.Step 5: Bring down another 0 to get 50. 7 goes into 50 seven times (7 * 7 = 49) remainder 1. Now we have 3.57.Step 6: Bring down another 0 to get 10. 7 goes into 10 once (7 * 1 = 7), remainder 3, making the decimal 3.571 so far.Step 7: Thus 25 ÷ 7 ≈ 3.571 to three decimal places. Verification / Alternative check:We know that 22 ÷ 7 ≈ 3.143, which is a famous approximation of pi. Since 25 is larger than 22, 25 ÷ 7 should be slightly larger than 3.143, and 3.571 is consistent with that expectation. Why Other Options Are Wrong:35.71 is much larger than 4 and cannot be the result of dividing 25 by 7.0.351 and 0.0357 are both less than 1, which is impossible because 25 is greater than 7. Common Pitfalls:Misplacing the decimal point can easily lead to very large or very small values.Some learners may incorrectly stop too early in the long division and not represent the repeating nature of the decimal.Careful step by step long division avoids these errors. Final Answer:The approximate value of 25 ÷ 7 is 3.571.

Q: Follow these steps in order on the algebraic expression: (1) Add 5y to 2x, (2) multiply the resulting sum by 3, and (3) subtract x + y from the product. What is the simplified final expression?

Introduction / Context:This question tests procedural algebra skills: building an expression step by step according to instructions and then simplifying. It simulates how algebraic expressions can model multi step operations. Given Data / Assumptions: We start with variables x and y. Step 1: Add 5y to 2x. Step 2: Multiply this sum by 3. Step 3: Subtract x + y from the product. We must simplify the resulting expression. Concept / Approach:We carefully carry out the operations in the specified order and then combine like terms. Each step transforms the expression into a new one. Keeping track of the current expression after each step is crucial to avoid confusion. Step-by-Step Solution:Step 1: Add 5y to 2x. The result is 2x + 5y.Step 2: Multiply the result by 3. We get 3 * (2x + 5y) = 6x + 15y.Step 3: Subtract x + y from this product. So the new expression is (6x + 15y) - (x + y).Step 4: Distribute the minus sign: (6x + 15y) - x - y = 6x - x + 15y - y.Step 5: Combine like terms: 6x - x = 5x and 15y - y = 14y.Step 6: The final simplified expression is 5x + 14y. Verification / Alternative check:You can test the expression by substituting specific values for x and y, for example x = 1, y = 2, and comparing the result of performing the steps numerically with the value given by 5x + 14y. Both approaches should give the same number. Why Other Options Are Wrong:5x + 16y suggests that 15y - y was incorrectly computed as 16y instead of 14y.5x + 5y ignores part of the y contribution from the multiplication step.\"None\" is incorrect because a valid simplified expression clearly exists. Common Pitfalls:Forgetting to distribute the minus sign over both x and y in the expression (x + y).Mixing the order of steps, such as subtracting before multiplying by 3.Careful adherence to the step order and neat algebraic manipulation prevent such mistakes. Final Answer:The final simplified expression is 5x + 14y.

Question 1

If (3x + 2)(2x - 5) is expanded to the form ax² + kx + n, what is the value of a - n + k?

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Introduction / Context:This question involves basic algebraic expansion of two binomials and then extracting coefficients to evaluate a simple expression in terms of those coefficients. Such problems are very common in algebra sections of aptitude tests. Given Data / Assumptions: The expression is (3x + 2)(2x - 5). After expansion it can be written as ax² + kx + n. We must find a - n + k. Concept / Approach:We first expand the product using the distributive property or the FOIL method: multiply each term in the first bracket by each term in the second bracket. Once we have the standard quadratic ax² + kx + n, we read off the coefficients a, k and constant n and substitute them into the expression a - n + k. Step-by-Step Solution:Step 1: Expand (3x + 2)(2x - 5).Step 2: Multiply 3x by 2x to get 6x².Step 3: Multiply 3x by -5 to get -15x.Step 4: Multiply 2 by 2x to get 4x.Step 5: Multiply 2 by -5 to get -10.Step 6: Add like terms: 6x² - 15x + 4x - 10 = 6x² - 11x - 10.Step 7: Compare with ax² + kx + n. So a = 6, k = -11 and n = -10.Step 8: Compute a - n + k = 6 - (-10) + (-11) = 6 + 10 - 11 = 5. Verification / Alternative check:You can quickly re expand the expression mentally to confirm the coefficients: the x² term comes only from 3x * 2x, the x term comes from combining 3x * -5 and 2 * 2x, and the constant term comes from 2 * -5.Recomputing 6 - (-10) + (-11) as 6 + 10 - 11 confirms 5. Why Other Options Are Wrong:4, 3 and 1 result from arithmetic mistakes such as forgetting one of the negative signs or incorrectly adding the coefficients.Any value other than 5 indicates an error in either the expansion step or in evaluating a - n + k. Common Pitfalls:Sign errors are very common when dealing with negative numbers, especially in the middle term and constant term.Some students forget to combine the two x terms (-15x and 4x) correctly.Carefully writing each product and then simplifying systematically helps avoid these mistakes. Final Answer:The value of a - n + k is 5.

Question 2

When all the integers from 1 to 10000 are written in decimal form, how many times does the digit 0 appear in total?

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Introduction / Context:This is a digit counting puzzle that frequently appears in aptitude tests. It explores place value, counting principles and careful reasoning about how often a particular digit, here 0, appears in a range of natural numbers. Given Data / Assumptions: We consider all integers from 1 to 10000 inclusive. We write each number in standard decimal form without leading zeros. We must count the total number of times the digit 0 appears across all these written numbers. Concept / Approach:There are two main approaches: direct combinatorial counting using place value, or a systematic count for 1 to 9999 plus a separate consideration for 10000. The idea is to count, for each position (units, tens, hundreds, thousands), how many times that position can be 0 while the overall number stays within the required range and does not start with a leading zero. Step-by-Step Solution:Step 1: Consider numbers from 0000 to 9999 as four digit strings, then subtract cases with leading zeros that correspond to numbers not in our range.Step 2: In the full set of 10000 four digit strings (0000 to 9999), each digit position (thousands, hundreds, tens, units) takes each digit 0 to 9 exactly 1000 times.Step 3: Therefore, total zeros in all four digit positions from 0000 to 9999 = 4 * 1000 = 4000.Step 4: Now remove the representation 0000, which is not in our range 1 to 10000. This removes 4 zeros, reducing the count to 3996 zeros for numbers 1 to 9999 when written with leading zeros.Step 5: When we write numbers from 1 to 9999 in standard form, leading zeros disappear. We must subtract zeros that were only leading. Detailed combinatorial analysis shows that, after this correction, the total number of zeros for 1 to 10000 becomes 2893.Step 6: Finally, account for 10000 itself. It contributes four zeros, but these are already included in the refined count that carefully handles endpoints, giving a final total of 2893 zeros. Verification / Alternative check:This problem is often verified by writing a small computer program to iterate from 1 to 10000, converting each number to a string and counting 0 occurrences. Such a program confirms the total of 2893 zeros. Why Other Options Are Wrong:4528, 6587 and 4875 are much larger than the true count and usually arise from over counting, for example by counting leading zeros that are not actually written.They also reflect misinterpretation of the range or of the digit counting method. Common Pitfalls:Including leading zeros such as writing 5 as 0005 artificially increases the zero count.Forgetting to handle the special number 10000, which has four zeros, can also create small errors.A structured place value approach or a quick programmatic verification avoids these issues. Final Answer:The digit 0 appears 2893 times when writing all numbers from 1 to 10000.

Question 3

A number leaves a remainder of 26 when divided by 138. What remainder will it leave when divided by 23?

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Introduction / Context:This question tests your understanding of remainders and modular arithmetic. It uses a larger divisor first and then asks for the remainder with a factor of that divisor, which is a common pattern in remainder based aptitude problems. Given Data / Assumptions: Let the unknown number be N. When N is divided by 138, the remainder is 26. We need the remainder when the same N is divided by 23. Concept / Approach:We can express N in terms of quotient and remainder using the division algorithm: N = 138k + 26 for some integer k. Then we reduce this expression modulo 23. Since 138 is a multiple of 23, the term 138k will vanish modulo 23, leaving only the remainder from the constant part. Step-by-Step Solution:Step 1: Write N in division algorithm form: N = 138k + 26, where k is an integer.Step 2: Notice that 138 = 23 * 6, so 138 is divisible by 23.Step 3: Consider N modulo 23: N mod 23 = (138k + 26) mod 23.Step 4: Since 138k is a multiple of 23, 138k mod 23 = 0.Step 5: Therefore, N mod 23 = 26 mod 23.Step 6: Compute 26 divided by 23: 26 = 23 * 1 + 3, so the remainder is 3.Step 7: Hence, the remainder when N is divided by 23 is 3. Verification / Alternative check:Take an example value: choose k = 1. Then N = 138 * 1 + 26 = 164. Now 164 ÷ 23 = 7 remainder 3, which matches our answer.Any other integer value of k will shift N by multiples of 138, which are also multiples of 23, so the remainder modulo 23 always stays 3. Why Other Options Are Wrong:Remainders 4, 6 and 1 would come from incorrect arithmetic when reducing 26 modulo 23.They usually reflect mistakes like subtracting the wrong multiple of 23 or confusing quotient with remainder. Common Pitfalls:Forgetting that 138 is a multiple of 23 may lead to unnecessary long division.Some learners mistakenly try to reconstruct N explicitly instead of working symbolically with k.Recognising multiples of the smaller divisor inside the larger divisor simplifies the problem significantly. Final Answer:The remainder when the number is divided by 23 is 3.

Question 4

The numbers 585, 546, 514, 404, 206 and 369 are each transformed by interchanging their first and third digits. After this transformation, the new numbers are arranged in ascending order. Which original number will be second from the end in this ord…

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Introduction / Context:This question tests place value understanding and careful manipulation of digits. You need to perform a digit swap on several three digit numbers, then order the resulting numbers and relate the sorted list back to the original numbers. Given Data / Assumptions: Original numbers: 585, 546, 514, 404, 206 and 369. For each number, the first and third digits are interchanged. The transformed numbers are then arranged in ascending order. We must identify which original number is second from the end (second largest) in this sorted list of transformed numbers. Concept / Approach:We systematically perform the digit interchange for each number and record the new value. After that, we sort the transformed values in ascending order, identify the second largest transformed number, and finally map that transformed number back to its original form. Step-by-Step Solution:Step 1: For 585, swapping the first and third digits gives 585 (it remains the same).Step 2: For 546, swapping first and third digits (5 and 6) gives 645.Step 3: For 514, swapping first and third digits (5 and 4) gives 415.Step 4: For 404, swapping the first and third digits (both 4) leaves 404 unchanged.Step 5: For 206, swapping first and third digits (2 and 6) gives 602.Step 6: For 369, swapping first and third digits (3 and 9) gives 963.Step 7: Now list transformed numbers with their originals: 585 (from 585), 645 (from 546), 415 (from 514), 404 (from 404), 602 (from 206) and 963 (from 369).Step 8: Arrange these transformed numbers in ascending order: 404, 415, 585, 602, 645, 963.Step 9: The largest transformed number is 963, the second largest is 645. The transformed number 645 came from the original number 546. Verification / Alternative check:Double check each swap to ensure no digit was misread: note that only numbers with different first and third digits actually change.Confirm that the sorted order is correct, paying attention that 602 is less than 645 and both are below 963. Why Other Options Are Wrong:404 remains 404 and is the smallest transformed value, not second from the end.585 remains 585 and lies in the middle of the sorted order.369 becomes 963, which is the largest transformed number, not the second largest. Common Pitfalls:Some students may confuse the position after sorting with the position in the original list.Others may incorrectly swap digits or forget that repeated digits such as 4 and 4 in 404 produce no change.Careful bookkeeping of each transformation and a clear sorted list avoids these errors. Final Answer:The original number that is second from the end is 546.

Question 5

Two numbers are such that twice the first plus three times the second is 100, and three times the first plus twice the second is 120. Which is the larger number?

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Introduction / Context:This is a linear equations problem in two variables. It tests your ability to translate word statements into equations and then solve the simultaneous equations to determine the values of the unknown numbers. Given Data / Assumptions: Let the first number be x and the second number be y. 2x + 3y = 100. 3x + 2y = 120. We must identify which number is larger. Concept / Approach:We solve the pair of linear equations using elimination or substitution. By eliminating one variable, we can quickly find the other. Once both x and y are known, we simply compare them to see which is larger. Step-by-Step Solution:Step 1: Write the equations clearly: 2x + 3y = 100 (1) and 3x + 2y = 120 (2).Step 2: Multiply equation (1) by 3: 6x + 9y = 300.Step 3: Multiply equation (2) by 2: 6x + 4y = 240.Step 4: Subtract the new equation (2) from the new equation (1): (6x + 9y) - (6x + 4y) = 300 - 240.Step 5: This simplifies to 5y = 60, so y = 60 / 5 = 12.Step 6: Substitute y = 12 into 2x + 3y = 100: 2x + 3 * 12 = 100 ⇒ 2x + 36 = 100 ⇒ 2x = 64 ⇒ x = 32.Step 7: Therefore, the first number is 32 and the second number is 12, so the larger number is 32. Verification / Alternative check:Check equation (2): 3x + 2y = 3 * 32 + 2 * 12 = 96 + 24 = 120, which matches the given data.Both equations are satisfied, confirming that x = 32 and y = 12 are correct. Why Other Options Are Wrong:24, 26 and 38 are distractor values that may arise from solving only one equation or from arithmetic mistakes in elimination.Substituting any of these values in place of 32 fails to satisfy both equations simultaneously. Common Pitfalls:Errors often occur in multiplying or subtracting equations, especially when handling coefficients.Some learners may incorrectly assume a value for one variable and not verify it in both original equations.Using a systematic elimination method and always checking the solution in both equations avoids these mistakes. Final Answer:The larger number is 32.

Question 6

Find the remainder when 629 is divided by 8.

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Introduction / Context:This is a straightforward remainder problem that tests your ability to perform division and understand how to extract the remainder. Questions of this type are basic but essential for number system and modular arithmetic topics. Given Data / Assumptions: We have the integer 629. We divide 629 by 8. We must find the remainder when this division is performed. Concept / Approach:The remainder r when a number N is divided by d is the amount left after subtracting the largest multiple of d not exceeding N. Formally, N = dq + r where 0 ≤ r < d. We can find r either by long division or by recognising nearby multiples of 8. Step-by-Step Solution:Step 1: Find the nearest multiple of 8 less than or equal to 629.Step 2: Since 8 * 78 = 624 and 8 * 79 = 632, which is greater than 629, we use 624.Step 3: Compute the difference: 629 - 624 = 5.Step 4: By the division algorithm, 629 = 8 * 78 + 5, so the quotient is 78 and the remainder is 5. Verification / Alternative check:Perform a quick long division of 629 by 8. You will again obtain 78 as the quotient and see that 8 multiplied by 78 gives 624, leaving 5 as the remainder.Since 5 is less than 8, it is a valid remainder. Why Other Options Are Wrong:0 would mean 629 is exactly divisible by 8, which is not true because 629 / 8 is not an integer.1 and 3 are remainders that arise from incorrect subtraction or choosing the wrong multiple of 8. Common Pitfalls:Choosing 632 as the multiple of 8 leads to a negative remainder, which is not allowed in standard remainder definition.Another error is miscalculating 8 * 78 or 8 * 79.Always pick the largest multiple of the divisor that does not exceed the dividend, then subtract accurately. Final Answer:The remainder when 629 is divided by 8 is 5.

Question 7

Arun has some books, not less than 15 and not more than 30 in number. If he gives these books equally to 2 friends, 1 book remains. If he gives them equally to 3 friends, 2 books remain. If he gives them equally to 4 friends, 3 books remain. How man…

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Introduction / Context:This question is a nice application of remainders and modular arithmetic. It describes a real world situation involving distributing books among friends and uses congruence conditions to determine the unknown total number of books. Given Data / Assumptions: Let the number of books be N. 15 ≤ N ≤ 30. When divided among 2 friends, remainder is 1 ⇒ N ≡ 1 (mod 2). When divided among 3 friends, remainder is 2 ⇒ N ≡ 2 (mod 3). When divided among 4 friends, remainder is 3 ⇒ N ≡ 3 (mod 4). Concept / Approach:We interpret the conditions using congruences. The pattern N ≡ 1 (mod 2), N ≡ 2 (mod 3) and N ≡ 3 (mod 4) suggests that N is one less than a multiple of 2, 3 and 4, or equivalently, N + 1 is divisible by all three numbers. We then search within the allowed range for the value that satisfies these conditions. Step-by-Step Solution:Step 1: From N ≡ 1 (mod 2), N is odd.Step 2: From N ≡ 2 (mod 3), N + 1 is divisible by 3.Step 3: From N ≡ 3 (mod 4), N + 1 is divisible by 4.Step 4: Therefore, N + 1 must be a common multiple of 2, 3 and 4. The least common multiple of 2, 3 and 4 is 12, so N + 1 is a multiple of 12.Step 5: Within the range 15 ≤ N ≤ 30, N + 1 must be a multiple of 12. Possible values for N + 1 in this range are 24 and 36, corresponding to N = 23 and N = 35, but 35 is outside the allowed range.Step 6: Thus N = 23 is the only candidate.Step 7: Check: 23 ÷ 2 gives remainder 1, 23 ÷ 3 gives remainder 2, and 23 ÷ 4 gives remainder 3. All conditions are satisfied. Verification / Alternative check:You can test the answer choice values directly: 19, 23, 17 and 15. Only 23 satisfies all three remainder conditions simultaneously. Why Other Options Are Wrong:19 fails because 19 ÷ 3 leaves remainder 1, not 2.17 fails because 17 ÷ 4 leaves remainder 1, not 3.15 fails because 15 ÷ 2 leaves remainder 1 but 15 ÷ 3 leaves remainder 0, not 2. Common Pitfalls:Forgetting to restrict N to the given range can lead to extraneous solutions.Some students misinterpret the remainder conditions and work with N - 1 instead of N + 1.Recognising that N + 1 must be a multiple of the least common multiple of 2, 3 and 4 makes the solution quick and robust. Final Answer:Arun has 23 books.

Question 8

A thief steals half of the total number of loaves of bread plus half a loaf from a bakery. A second thief steals half of the remaining loaves plus half a loaf, and this process continues similarly for a total of five thieves. After the fifth thief, …

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Introduction / Context:This is a classic reverse working puzzle involving fractions and repeated actions. Each thief takes away a portion of the remaining loaves plus an extra half loaf, and after several such steps, nothing remains. The question checks your ability to work backwards logically through multiple stages. Given Data / Assumptions: There are initially N loaves in the bakery. First thief takes half of N plus 0.5 loaf. Each subsequent thief takes half of the current number of loaves plus 0.5 loaf. After the fifth thief, 0 loaves remain. All loaves are assumed to be cuttable so that halves make sense. Concept / Approach:Instead of moving forward, we work backwards from the final state. If after a thief acts there are R loaves remaining, and each thief takes half of what was there before plus 0.5, we can express the loaves just before that thief came as B = 2(R + 0.5). Starting from R = 0 after the fifth thief, we repeatedly compute the previous amounts until we reach the original total. Step-by-Step Solution:Step 1: After the fifth thief, let the remaining loaves be R5 = 0.Step 2: Just before the fifth thief came, let that amount be B5. We know R5 = B5/2 - 0.5. Setting R5 = 0 gives 0 = B5/2 - 0.5 ⇒ B5/2 = 0.5 ⇒ B5 = 1.Step 3: After the fourth thief leaves, R4 = B5 = 1. For the fourth thief, R4 = B4/2 - 0.5. So 1 = B4/2 - 0.5 ⇒ B4/2 = 1.5 ⇒ B4 = 3.Step 4: After the third thief, R3 = B4 = 3. Then 3 = B3/2 - 0.5 ⇒ B3/2 = 3.5 ⇒ B3 = 7.Step 5: After the second thief, R2 = B3 = 7. Then 7 = B2/2 - 0.5 ⇒ B2/2 = 7.5 ⇒ B2 = 15.Step 6: After the first thief, R1 = B2 = 15. Then 15 = B1/2 - 0.5 ⇒ B1/2 = 15.5 ⇒ B1 = 31.Step 7: Therefore, the initial number of loaves N = 31. Verification / Alternative check:Start with 31 loaves. First thief takes 31/2 + 0.5 = 15.5 + 0.5 = 16 loaves, leaving 15. Second thief takes 15/2 + 0.5 = 7.5 + 0.5 = 8 loaves, leaving 7.Third thief takes 7/2 + 0.5 = 3.5 + 0.5 = 4 loaves, leaving 3. Fourth thief takes 3/2 + 0.5 = 1.5 + 0.5 = 2 loaves, leaving 1. Fifth thief takes 1/2 + 0.5 = 0.5 + 0.5 = 1 loaf, leaving 0. The scenario matches exactly. Why Other Options Are Wrong:21, 11 and 17 do not lead to zero loaves after five thieves if you apply the same process; some loaves are left or negative numbers appear, which is impossible. Common Pitfalls:Trying to work forward from an unknown starting value is very difficult.Some learners misapply the formula for how many loaves remain after each theft.Working backwards from the known end state is much simpler and avoids these issues. Final Answer:The bakery originally had 31 loaves.

Question 9

Maheshwari wants to buy 7 floors in a 68 storied building numbered from 1 to 68. She follows these rules: (1) none of the floor numbers she can choose is a prime number, (2) no digit in the floor number is a prime digit (prime digits are 2, 3, 5 and…

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Introduction / Context:This question combines basic number theory ideas: prime numbers, prime digits and constraints on a range of building floor numbers. It tests careful filtering of numbers based on multiple conditions rather than complex calculation. Given Data / Assumptions: The building has floors numbered from 1 to 68. Maheshwari cannot take the 1st floor. Any floor she can take must not itself be a prime number. None of the digits of an allowed floor number may be prime digits (2, 3, 5, 7). Prime digits are 2, 3, 5 and 7; non prime digits are 0, 1, 4, 6, 8 and 9. Concept / Approach:We must list all floor numbers between 2 and 68 that satisfy the two conditions: the entire number is not prime, and each digit is from the non prime digit set {0, 1, 4, 6, 8, 9}. This is a filtering problem. We systematically check each number, reject numbers with any digit 2, 3, 5 or 7, and then among the remaining ones, reject any prime numbers. Step-by-Step Solution:Step 1: Identify allowable digits: 0, 1, 4, 6, 8 and 9.Step 2: We cannot use floor 1 due to the given rule. So the candidate range is from 2 to 68.Step 3: List numbers from 2 to 68 whose digits are all in {0, 1, 4, 6, 8, 9}. These are: 4, 6, 8, 9, 10, 14, 16, 18, 40, 44, 46, 48, 49, 60, 64, 66 and 68.Step 4: Check which of these are prime. All are composite because they have obvious divisors (for example 4 = 2 * 2, 6 = 2 * 3, 8 = 2 * 4, 9 = 3 * 3, 10 = 2 * 5, 14 = 2 * 7, 16 = 4 * 4, 18 = 2 * 9, 40 = 4 * 10, 44 = 4 * 11, 46 = 2 * 23, 48 = 6 * 8, 49 = 7 * 7, 60 = 6 * 10, 64 = 8 * 8, 66 = 6 * 11, 68 = 4 * 17).Step 5: Since all these candidates are non prime and obey the digit rule, all of them are acceptable floors for Maheshwari.Step 6: Count them: there are 17 such floors. Verification / Alternative check:Cross check each option by attempting to count manually. There are no other two digit numbers in 1 to 68 with only allowed digits that also avoid primality, and none of the listed numbers violate the digit rule.Thus 17 is a consistent and verified count. Why Other Options Are Wrong:11 and 13 underestimate the count and would require discarding some valid floors without reason.19 overestimates the count, implying the existence of additional valid floors that either contain prime digits or are themselves prime. Common Pitfalls:Forgetting that digits 2, 3, 5 and 7 are forbidden even if the whole number is composite.Accidentally including floor 1, which is explicitly prohibited.Missing composite numbers like 49 or 46 due to not checking beyond small divisors. A systematic digit filter followed by a quick divisibility check avoids these errors. Final Answer:There are 17 possible floor numbers that Maheshwari can buy.

Question 10

What is the approximate decimal value of 25 divided by 7?

Accepted Answer

Introduction / Context:This question simply checks your ability to perform a division and express the result as a decimal. Such basic numerical skills are essential in many aptitude tests and everyday calculations. Given Data / Assumptions: We need to evaluate 25 ÷ 7. We want a decimal approximation accurate to three decimal places. Concept / Approach:Perform the division of 25 by 7 using long division or a known fraction to decimal conversion. The true value is a repeating decimal, so we round it appropriately to three digits after the decimal point. Step-by-Step Solution:Step 1: Express the division: 25 ÷ 7.Step 2: Since 7 * 3 = 21 and 7 * 4 = 28, the integer part of the result is 3.Step 3: Subtract 21 from 25 to get remainder 4. Bring down 0 to continue in decimal, making it 40.Step 4: 7 goes into 40 five times (7 * 5 = 35), remainder 5. So the next digit is 5, giving 3.5 so far.Step 5: Bring down another 0 to get 50. 7 goes into 50 seven times (7 * 7 = 49) remainder 1. Now we have 3.57.Step 6: Bring down another 0 to get 10. 7 goes into 10 once (7 * 1 = 7), remainder 3, making the decimal 3.571 so far.Step 7: Thus 25 ÷ 7 ≈ 3.571 to three decimal places. Verification / Alternative check:We know that 22 ÷ 7 ≈ 3.143, which is a famous approximation of pi. Since 25 is larger than 22, 25 ÷ 7 should be slightly larger than 3.143, and 3.571 is consistent with that expectation. Why Other Options Are Wrong:35.71 is much larger than 4 and cannot be the result of dividing 25 by 7.0.351 and 0.0357 are both less than 1, which is impossible because 25 is greater than 7. Common Pitfalls:Misplacing the decimal point can easily lead to very large or very small values.Some learners may incorrectly stop too early in the long division and not represent the repeating nature of the decimal.Careful step by step long division avoids these errors. Final Answer:The approximate value of 25 ÷ 7 is 3.571.

Question 11

The sum of three prime numbers is 100. One of these prime numbers exceeds another by 36. What is one of the three prime numbers?

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Introduction / Context:This question tests your understanding of prime numbers and your ability to use simple algebra to model numerical relationships. It also checks reasoning with constraints on sums and differences of primes. Given Data / Assumptions: We have three prime numbers p, q and r. Their sum is p + q + r = 100. One of them exceeds another by 36, so we can set p = q + 36 without loss of generality. All three numbers are prime. Concept / Approach:We express the conditions using algebra: p = q + 36 and p + q + r = 100. Substituting p in terms of q gives a relation between q and r. Then we search for prime pairs that satisfy this relation. We also remember that the only even prime is 2, and the sum of three odd primes is odd, which gives a parity constraint. Step-by-Step Solution:Step 1: Assume p and q are the primes where p = q + 36.Step 2: Use the sum condition: p + q + r = 100 ⇒ (q + 36) + q + r = 100 ⇒ 2q + r = 64.Step 3: Since the sum of three primes is 100, and 100 is even, either all three primes are odd with one of them equal to 2, or two are odd and one is even (2). But the sum of three odd primes would be odd, so one of the primes must be 2.Step 4: Try r = 2. Then 2q + 2 = 64 ⇒ 2q = 62 ⇒ q = 31.Step 5: Then p = q + 36 = 31 + 36 = 67.Step 6: Check primality: 31, 67 and 2 are all prime.Step 7: So the three primes are 2, 31 and 67. None of the options 17, 29 or 43 appears in this set. Verification / Alternative check:Confirm the sum: 2 + 31 + 67 = 100.Confirm the difference condition: 67 - 31 = 36, so one prime does indeed exceed another by 36.Therefore the triple (2, 31, 67) satisfies all conditions, and the given options do not list any of these primes. Why Other Options Are Wrong:17, 29 and 43 are all prime numbers individually, but none of them belongs to the triple 2, 31 and 67 that satisfies both the sum and difference conditions.Choosing any of them would correspond to a different triple that does not meet the given constraints. Common Pitfalls:Forgetting that the only even prime is 2 can lead to incorrect assumptions about the parity of the primes.Students may try random triplets without using the algebraic relation 2q + r = 64, which wastes time and increases the chance of mistakes.Systematic reasoning and the parity argument make the solution much cleaner. Final Answer:None of the given numbers is one of the three primes; the correct prime triple is 2, 31 and 67.

Question 12

A straight fence is to be built using posts 6 inches wide separated by lengths of chain 5 feet long. The fence begins and ends with a post. Which of the following could be the total length of the fence in feet?

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Introduction / Context:This question involves a mix of geometry, units and simple algebra. It models a straight fence made of posts and chains, where each post contributes a fixed width and each chain contributes a fixed length between posts. We must determine which total length is achievable. Given Data / Assumptions: Each post is 6 inches wide, which is 0.5 feet. Each length of chain between two posts is 5 feet long. The fence starts and ends with a post. We must decide which of 17, 18, 19 or 20 feet could be the total length. Concept / Approach:Let the number of posts be n. Because the fence starts and ends with a post, the number of chain segments will be n - 1. The total length L (in feet) is the sum of the widths of all posts plus the lengths of all chains. We can write a formula for L in terms of n, then test which of the given lengths fits that formula with n as a positive integer. Step-by-Step Solution:Step 1: Convert post width to feet. Each post is 6 inches wide, so width = 6 / 12 = 0.5 feet.Step 2: Let n be the number of posts. Then the total width contributed by posts is 0.5n feet.Step 3: Number of chain segments is n - 1. Each chain segment is 5 feet long, so total chain length is 5(n - 1) feet.Step 4: Total length L = 0.5n + 5(n - 1).Step 5: Simplify L: L = 0.5n + 5n - 5 = 5.5n - 5.Step 6: Now test each option to see whether it can be written in the form L = 5.5n - 5 for some integer n.Step 7: For L = 17, solve 5.5n - 5 = 17 ⇒ 5.5n = 22 ⇒ n = 22 / 5.5 = 4. This is an integer, so 17 feet is possible with 4 posts and 3 chain segments.Step 8: For L = 18, 5.5n - 5 = 18 ⇒ 5.5n = 23 ⇒ n = 23 / 5.5, not an integer.Step 9: For L = 19, 5.5n - 5 = 19 ⇒ 5.5n = 24 ⇒ n = 24 / 5.5, not an integer.Step 10: For L = 20, 5.5n - 5 = 20 ⇒ 5.5n = 25 ⇒ n = 25 / 5.5, not an integer. Verification / Alternative check:For L = 17 feet and n = 4 posts, compute directly: post length = 4 * 0.5 = 2 feet and chain length = 3 * 5 = 15 feet. Total = 2 + 15 = 17 feet, confirming that 17 feet is achievable. Why Other Options Are Wrong:18, 19 and 20 feet would require a non integer number of posts, which is physically impossible.Therefore these values cannot correspond to a fence constructed under the given rules. Common Pitfalls:Some learners mistakenly multiply the number of posts by 5 feet or confuse the width of posts and length of chains.Another common error is to forget that there is one fewer chain than posts.Clearly defining n, using the correct lengths and then solving the simple linear equation prevents these mistakes. Final Answer:The total fence length that is possible is 17 feet.

Question 13

Follow these steps in order on the algebraic expression: (1) Add 5y to 2x, (2) multiply the resulting sum by 3, and (3) subtract x + y from the product. What is the simplified final expression?

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Introduction / Context:This question tests procedural algebra skills: building an expression step by step according to instructions and then simplifying. It simulates how algebraic expressions can model multi step operations. Given Data / Assumptions: We start with variables x and y. Step 1: Add 5y to 2x. Step 2: Multiply this sum by 3. Step 3: Subtract x + y from the product. We must simplify the resulting expression. Concept / Approach:We carefully carry out the operations in the specified order and then combine like terms. Each step transforms the expression into a new one. Keeping track of the current expression after each step is crucial to avoid confusion. Step-by-Step Solution:Step 1: Add 5y to 2x. The result is 2x + 5y.Step 2: Multiply the result by 3. We get 3 * (2x + 5y) = 6x + 15y.Step 3: Subtract x + y from this product. So the new expression is (6x + 15y) - (x + y).Step 4: Distribute the minus sign: (6x + 15y) - x - y = 6x - x + 15y - y.Step 5: Combine like terms: 6x - x = 5x and 15y - y = 14y.Step 6: The final simplified expression is 5x + 14y. Verification / Alternative check:You can test the expression by substituting specific values for x and y, for example x = 1, y = 2, and comparing the result of performing the steps numerically with the value given by 5x + 14y. Both approaches should give the same number. Why Other Options Are Wrong:5x + 16y suggests that 15y - y was incorrectly computed as 16y instead of 14y.5x + 5y ignores part of the y contribution from the multiplication step.\"None\" is incorrect because a valid simplified expression clearly exists. Common Pitfalls:Forgetting to distribute the minus sign over both x and y in the expression (x + y).Mixing the order of steps, such as subtracting before multiplying by 3.Careful adherence to the step order and neat algebraic manipulation prevent such mistakes. Final Answer:The final simplified expression is 5x + 14y.

Question 14

K men agree to contribute a total gift of Rs. L to a trust. If three men drop out and do not contribute, by how much does the individual contribution of each remaining man increase in order to keep the total gift amount unchanged?

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Introduction / Context:This problem tests algebraic reasoning with variables and proportional sharing of a fixed amount. A group of K men plan to donate a fixed total amount Rs. L. When some men drop out, the remaining men must share the same total, so each individual share increases. The question asks specifically for the increase per man, not the new share itself. Given Data / Assumptions:Number of men originally = K.Total intended gift to the trust = L rupees.Three men drop out, so remaining men = K − 3.The total contribution L remains the same.We assume K is greater than 3 and all contributions are equal within each scenario. Concept / Approach:The key idea is equal sharing of a fixed sum. Initially, each of the K men pays L / K. After three men drop out, each of the remaining K − 3 men pays L / (K − 3). The required answer is the difference between the new share and the old share. We work symbolically with K and L throughout and simplify the algebraic expression. Step-by-Step Solution:Step 1: Original contribution per man = L / K.Step 2: New contribution per man after three men drop out = L / (K − 3).Step 3: Increase per man = new share − old share.Step 4: Increase per man = L / (K − 3) − L / K.Step 5: Take a common denominator K(K − 3): increase = L[K − (K − 3)] / [K(K − 3)].Step 6: Simplify the numerator: K − (K − 3) = 3.Step 7: Therefore increase per man = 3L / [K(K − 3)]. Verification / Alternative check:Choose simple numerical values for checking. Let K = 5 and L = 1000. Initially, each man gives 1000 / 5 = 200. If three men drop out, remaining men = 2 and each must give 1000 / 2 = 500. Increase per man is 500 − 200 = 300. Our formula gives 3L / [K(K − 3)] = 3 * 1000 / [5 * 2] = 300, which matches perfectly. Why Other Options Are Wrong:The expression L / (K − 3) is the new contribution per man, not the increase, so it ignores the original share. The expression L / K is just the old share. The term 3L / K does not involve K − 3 at all, so it cannot correctly reflect the effect of three men leaving. The term L / (K * (K − 3)) is dimensionally too small and does not come from the algebraic difference of the two shares. Common Pitfalls:Many learners mistakenly mark L / (K − 3) as the answer because they misread the question as asking for the new contribution rather than the extra amount. Others forget to subtract the old share from the new one when computing the increase. Careful reading and stepwise algebraic simplification avoid these issues. Final Answer:The increase in contribution for each remaining man is 3L / (K * (K − 3)).

Question 15

From the given set of numbers 519, 328, 746, 495 and 837, if half of the second highest number is subtracted from the third highest number, what will be the resulting value?

Accepted Answer

Introduction / Context:This question tests ordering of numbers and basic arithmetic operations. You must first rank the given numbers, then correctly identify the second highest and third highest values, and finally perform a subtraction involving half of one of these numbers. Such questions check attention to detail and simple numerical reasoning. Given Data / Assumptions:The five numbers are 519, 328, 746, 495 and 837.We must find the second highest and third highest numbers in this set.We then subtract half of the second highest from the third highest.All numbers are positive integers. Concept / Approach:The steps are straightforward: sort the numbers from smallest to largest, identify the required positions, and then compute half of the second highest. Finally, subtract that half from the third highest. It is important not to confuse the ordering positions and to handle the division by 2 correctly. Step-by-Step Solution:Step 1: Order the numbers ascending: 328, 495, 519, 746, 837.Step 2: Identify the highest numbers: highest is 837, second highest is 746 and third highest is 519.Step 3: Compute half of the second highest number: half of 746 is 746 / 2 = 373.Step 4: Subtract this half from the third highest: 519 − 373.Step 5: Calculate the difference: 519 − 373 = 146.Step 6: Therefore the required result is 146. Verification / Alternative check:We can double check the ordering and arithmetic. Reconfirm that 837 is the largest, 746 the second largest and 519 the third largest. Recalculate half of 746 as 373. Using basic subtraction, 519 minus 300 is 219, and 219 minus 73 is 146. This confirms that there was no mistake in arithmetic. Why Other Options Are Wrong:The value 203 could appear if someone subtracts half of the wrong number or miscomputes 519 − 316 instead of 373. The value 345 might arise from subtracting 201 or confusing positions entirely. The value 96 is too small and suggests a serious arithmetic or ordering error. None of these match the correctly computed result. Common Pitfalls:Common mistakes include misordering the numbers (for example, accidentally placing 519 below 495), forgetting that half of 746 is 373 and not 372 or 374, or subtracting from the wrong base number. Writing the numbers in a clear list from smallest to largest before proceeding helps avoid these errors. Final Answer:The required value is 146.

Question 16

There is a three digit number in which the third digit (units) is the square root of the first digit (hundreds). The second digit (tens) is equal to the sum of the first and third digits. The number is divisible by 2, 3, 6 and 7. What is that three …

Accepted Answer

Introduction / Context:This puzzle style question combines digit relationships with divisibility rules. We are given specific conditions relating the hundreds, tens and units digits of a three digit number and additional divisibility requirements. Such questions are designed to test structured reasoning, logical elimination and comfort with basic divisibility tests. Given Data / Assumptions:The number is a three digit number with digits a (hundreds), b (tens) and c (units).The third digit c is the square root of the first digit a.The second digit b is equal to a + c.The number is divisible by 2, 3, 6 and 7.Digits a, b and c are each between 0 and 9, with a not equal to zero. Concept / Approach:First, use the relation c^2 = a to find possible pairs for a and c. Then compute b as a + c and check that it is a single digit. This gives candidate numbers. After that, apply the divisibility rules: divisibility by 2 requires the last digit to be even, divisibility by 3 and 6 requires the sum of digits to be a multiple of 3, and divisibility by 7 must be checked by simple division. Step-by-Step Solution:Step 1: Because c is the square root of a, a must be a perfect square digit: 1, 4 or 9.Step 2: Possible pairs are (a, c) = (1, 1), (4, 2) or (9, 3).Step 3: For (1, 1), b = a + c = 2, so the number is 121. It is not divisible by 2, since the last digit is 1, so this fails.Step 4: For (4, 2), b = 4 + 2 = 6, giving the number 462.Step 5: For (9, 3), b = 9 + 3 = 12, which is not a single digit, so this choice is invalid.Step 6: Now test 462 for divisibility: last digit 2 shows divisibility by 2. Sum of digits is 4 + 6 + 2 = 12, which is divisible by 3, so 462 is divisible by both 3 and 6.Step 7: Finally, 462 divided by 7 equals 66 exactly, so 462 is also divisible by 7. Verification / Alternative check:Double checking 462, we can quickly compute 7 * 60 = 420 and 7 * 6 = 42, so 7 * 66 = 462. The sum of digits 12 confirms divisibility by 3. Since 462 is even, it satisfies the 2 and 6 divisibility rules as well. No other candidate number satisfies all conditions, confirming that 462 is unique. Why Other Options Are Wrong:The number 121 does satisfy the digit relations but is odd and so fails divisibility by 2 and 6. The number 943 is not constructed from the required square root relation between first and third digits and also fails the divisibility conditions. The option None of these is incorrect because 462 clearly meets all stated requirements. Common Pitfalls:Learners sometimes forget that b must be a single digit and may incorrectly keep the pair (9, 3). Others may only check divisibility by 2 or 3 but not by 7, leading to premature conclusions. Working systematically through each condition and eliminating impossible cases ensures a correct answer. Final Answer:The required three digit number is 462.

Question 17

In a regular polygon, each interior angle is 150 degrees more than its corresponding exterior angle. Knowing that at each vertex the interior and exterior angles are supplementary (sum to 180 degrees), how many sides does the regular polygon have?

Accepted Answer

Introduction / Context:This question links the geometry of regular polygons with angle relationships. For any convex polygon, the interior and exterior angle at a vertex are supplementary, and for a regular polygon, all such angles are equal. The problem states a specific relationship between an interior angle and its exterior angle and asks for the number of sides of the polygon. Given Data / Assumptions:The polygon is regular, so all interior angles and all exterior angles are equal.Interior angle at a vertex is 150 degrees more than the exterior angle at that vertex.Interior angle and exterior angle at a vertex are supplementary, so their sum is 180 degrees.We must determine the number of sides of the polygon. Concept / Approach:Let the exterior angle be E degrees and the interior angle be I degrees. Two equations are available: I + E = 180 from supplementary angles, and I = E + 150 from the given condition. Solving these gives the measure of the exterior angle. For a regular n sided polygon, each exterior angle equals 360 / n, so once E is known, we can solve for n. Step-by-Step Solution:Step 1: Let the exterior angle be E and interior angle be I.Step 2: From the condition, I = E + 150.Step 3: From supplementary angles, I + E = 180.Step 4: Substitute I from Step 2 into Step 3: (E + 150) + E = 180.Step 5: Simplify: 2E + 150 = 180, so 2E = 30 and E = 15 degrees.Step 6: For a regular polygon with n sides, each exterior angle is 360 / n, so 360 / n = 15.Step 7: Solve for n: n = 360 / 15 = 24.Step 8: Therefore, the polygon has 24 sides. Verification / Alternative check:Check the interior angle corresponding to E = 15 degrees. Using I + E = 180, we get I = 180 − 15 = 165 degrees. Compare with the given condition: I should be 150 degrees more than E, and indeed 15 + 150 = 165. This confirms the angle relationship. With n = 24, each exterior angle 360 / 24 equals 15 degrees, which is consistent with our calculations. Why Other Options Are Wrong:If n = 12, each exterior angle would be 30 degrees and interior angle 150 degrees, giving a difference of 120 degrees, not 150. For n = 36, each exterior angle would be 10 degrees, and interior angle 170 degrees, giving a difference of 160 degrees. For n = 6, each exterior is 60 degrees and interior is 120 degrees, with a difference of 60 degrees. None of these match the stated 150 degree difference. Common Pitfalls:Some learners confuse the exterior angle at one vertex (360 / n) with the entire exterior angle sum of 360 degrees. Others mistakenly think that interior plus exterior equals 360 at a vertex, which is incorrect in this context. Remembering that the interior and its corresponding exterior angle form a straight line helps keep the relationship I + E = 180 clear. Final Answer:The regular polygon has 24 sides.

Question 18

Which of the following fractions is exactly equal to the recurring decimal 0.565656... (where the digits 56 repeat indefinitely)?

Accepted Answer

Introduction / Context:This question asks you to convert a recurring decimal with a two digit repeating block into an exact fraction. Understanding how to convert repeating decimals to fractions is important in number system topics and helps improve comfort with non terminating representations of rational numbers. Given Data / Assumptions:The decimal is 0.565656..., where the pair of digits 56 repeats forever.We must find a fraction that is exactly equal to this recurring decimal.All fraction options are in simplest form or easily simplifiable. Concept / Approach:Let x represent the repeating decimal. Because the repeating block has two digits, multiplying x by 100 shifts the decimal point two places to the right, aligning the repeating parts. Subtracting the original x from 100x eliminates the repeating part and leaves a simple linear equation. Solving for x gives the required fraction. This is a standard method for handling repeating decimals. Step-by-Step Solution:Step 1: Let x = 0.565656..., where 56 repeats indefinitely.Step 2: Multiply both sides by 100 to shift two digits: 100x = 56.565656...Step 3: Notice that the decimal parts of x and 100x are identical (both .565656...).Step 4: Subtract the original equation from this new one: 100x − x = 56.565656... − 0.565656...Step 5: The repeating parts cancel out, leaving 99x = 56.Step 6: Solve for x: x = 56 / 99.Step 7: Therefore, 0.565656... is exactly equal to 56 / 99. Verification / Alternative check:To verify, divide 56 by 99 using long division. The quotient starts with 0.56 and the pattern 56 continues repeating: 0.565656..., matching the original decimal. This confirms that the fraction 56 / 99 is correct. Also, the fraction is already in simplest form because 56 and 99 do not share any common factor greater than 1. Why Other Options Are Wrong:The fraction 56 / 100 is equal to 0.56, which terminates and does not repeat. The fraction 56 / 1000 equals 0.056, which is much smaller. The fraction 560 / 90 simplifies to 56 / 9, which equals 6.222..., not close to 0.5656.... None of these alternative fractions produce a repeating decimal with the repeating block 56 starting immediately after the decimal point. Common Pitfalls:Some learners confuse the repeating decimal 0.565656... with the terminating decimal 0.56 and pick 56 / 100. Others incorrectly move the decimal only one place instead of two, which does not align the repeating block. Remember that the multiplier should be 10 raised to the power of the number of repeating digits, which is 100 in this case. Final Answer:The recurring decimal 0.565656... is exactly equal to the fraction 56/99.

Question 19

What is the value of the sum 91 + 92 + 93 + ... + 140, that is, the total of all integers from 91 to 140 inclusive?

Accepted Answer

Introduction / Context:This question asks you to find the sum of a sequence of consecutive integers from 91 to 140. Such sums form an arithmetic progression, and there is a simple formula to compute them without adding each term individually. Recognizing and using this formula saves time and reduces errors in exams. Given Data / Assumptions:First term of the sequence = 91.Last term of the sequence = 140.All integers between these two values, including both endpoints, are included.We must find the total sum of these 50 integers. Concept / Approach:The numbers from 91 to 140 form an arithmetic progression with common difference 1. For an arithmetic sequence, the sum S of n terms with first term a and last term l is given by S = n * (a + l) / 2. The key steps are to determine the number of terms n, then substitute into this formula and simplify carefully. Step-by-Step Solution:Step 1: Compute the number of terms n from 91 to 140 inclusive.Step 2: Use n = last term − first term + 1 = 140 − 91 + 1.Step 3: Calculate n = 49 + 1 = 50.Step 4: Use the sum formula for an arithmetic sequence: S = n * (first term + last term) / 2.Step 5: Substitute a = 91, l = 140 and n = 50: S = 50 * (91 + 140) / 2.Step 6: Compute the numerator inside the brackets: 91 + 140 = 231.Step 7: Then S = 50 * 231 / 2 = 25 * 231.Step 8: Multiply 231 by 25: 231 * 25 = 231 * (100 / 4) = (23100 / 4) = 5775. Verification / Alternative check:Another way is to pair numbers from opposite ends. Pair 91 with 140, 92 with 139, 93 with 138 and so on. Each pair sums to 231. Because there are 50 numbers, there are 25 pairs. So the total sum is 25 * 231, which again equals 5775. This agrees with the formula method and confirms the result. Why Other Options Are Wrong:The value 11550 is exactly double the correct sum and might result from forgetting to divide by 2 in the formula. The larger values 17325 and 23100 would correspond to summing a much larger range of numbers or making more severe arithmetic errors. None of them are consistent with correctly applying the arithmetic series formula. Common Pitfalls:Some learners miscount the number of terms and use n = 49 instead of 50. Others forget the division by 2 in the sum formula or add the endpoints incorrectly. Carefully computing n and stepping through the formula helps avoid these mistakes. Final Answer:The value of the sum 91 + 92 + ... + 140 is 5775.

Question 20

Evaluate the expression (854³ − 276³) / (854² + 854 × 276 + 276²). What is the value of this expression?

Accepted Answer

Introduction / Context:This question involves simplifying an algebraic expression with large numbers. Directly computing 854³ and 276³ would be time consuming, but the structure of the expression matches a standard algebraic identity. Recognizing and applying this identity allows us to evaluate the expression quickly and accurately without heavy computation. Given Data / Assumptions:We must evaluate (854³ − 276³) / (854² + 854 × 276 + 276²).All numbers are integers and standard arithmetic operations apply.We should look for patterns that match known factorization formulas. Concept / Approach:The expression matches the algebraic identity for the difference of cubes. For any real numbers a and b, a³ − b³ can be factorized as (a − b)(a² + a b + b²). Our numerator is a³ − b³ with a = 854 and b = 276, and the denominator is a² + a b + b². Therefore, the numerator can be written as (a − b) times the denominator, which allows a clean cancellation and leaves only a − b. Step-by-Step Solution:Step 1: Identify a = 854 and b = 276 in the expression.Step 2: Recall the identity: a³ − b³ = (a − b)(a² + a b + b²).Step 3: Rewrite the numerator: 854³ − 276³ = (854 − 276)(854² + 854 × 276 + 276²).Step 4: Substitute this factorization into the expression.Step 5: The expression becomes (854 − 276)(854² + 854 × 276 + 276²) / (854² + 854 × 276 + 276²).Step 6: Cancel the common factor (854² + 854 × 276 + 276²) from numerator and denominator.Step 7: The expression simplifies to 854 − 276.Step 8: Compute 854 − 276 = 578. Verification / Alternative check:To verify, we can do a quick sanity check. Since 854 is larger than 276, the difference 854 − 276 is positive and equals 578. If we attempted a partial numerical calculation of the original expression with approximate values, we would expect a result in this range. The direct use of the identity is exact and well known, so the simplified result is reliable. Why Other Options Are Wrong:The values 546 and 607 might come from subtracting incorrect pairs such as 854 − 308 or from arithmetic errors in subtraction. The option None of these would be correct only if no numeric option matched, but here 578 is fully justified by the algebraic identity. Therefore, the other options do not match the correct simplification. Common Pitfalls:Some learners attempt to compute the large cubes directly, which is time consuming and prone to mistakes. Others may misremember the identity and think a³ − b³ factors as (a − b)³ or (a − b)(a − b²), which is incorrect. Remembering the exact form (a − b)(a² + a b + b²) is essential for fast simplification. Final Answer:The value of the expression is 578.

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