Q: Digit-repetition product — Find the smallest positive integer N such that 13 * N equals a number consisting entirely of the digit 5 (for example, 55, 555, 5555, …). Determine the minimal N that works.

Introduction / Context: This number-system puzzle asks for the smallest positive integer N such that 13 * N produces a repdigit made only of 5s (like 5, 55, 555, 5555, …). Problems of this type are efficiently handled with modular arithmetic and repetition-number (repunit) reasoning, avoiding brute-force search. Given Data / Assumptions: We need 13 * N = 55…5 (k copies of 5) for some smallest integer k. The target number with k copies of 5 equals 5 * (10^k - 1) / 9. N must be a positive integer; k must be minimal to ensure N is minimal. Concept / Approach: Let T_k = 5 * (10^k - 1) / 9. We need 13 | T_k. Since gcd(5, 13) = 1 and gcd(9, 13) = 1, this reduces to 13 | (10^k - 1). Thus, the smallest viable k is the multiplicative order of 10 modulo 13 (or modulo 117 when considering the 9 in the denominator rigorously). The minimal k that makes (10^k - 1) divisible by 117 yields the first all-5s number divisible by 13. Step-by-Step Solution: Write the repdigit: 55…5 (k fives) = 5 * (10^k - 1) / 9.We need 13 | 5 * (10^k - 1) / 9. Since 5 and 13 are coprime, ensure 117 | (10^k - 1).The smallest k with 10^k ≡ 1 (mod 117) is k = 6 (standard order result).Thus, the first suitable repdigit is 555555. Compute N = 555555 / 13 = 42735. Verification / Alternative check: Check directly: 13 * 42735 = 555555. No shorter repdigit of 5s (like 5, 55, 555, 5555, 55555) is divisible by 13, so this N is minimal. Why Other Options Are Wrong: 41625 / 42515 / 42135 / 55555: None of these, when multiplied by 13, yield an all-5s product. Only 42735 satisfies 13 * N = 555555 exactly. Common Pitfalls: Trying random guesses; forgetting the repdigit formula 5 * (10^k - 1) / 9; checking divisibility by 13 without confirming minimality of k. Final Answer: 42735

Q: Two-digit number with digit sum 10 — Reversing the digits decreases the number by 36. Identify which statements are true: I) The number is divisible by a composite number. II) The number is a multiple of a prime number.

Introduction / Context: This problem blends place-value reasoning with properties of primes and composites. You are told the two-digit number has digits summing to 10, and reversing the digits reduces the number by 36. From these constraints, you can determine the exact number and then evaluate the truth of the given statements regarding divisibility. Given Data / Assumptions: Let the tens digit be a and the units digit be b. a + b = 10. Original number: 10a + b. Reversed: 10b + a. Reversing decreases the number by 36: (10a + b) - (10b + a) = 36. Concept / Approach: Translate the language into equations and solve the simple system for digits a and b. Then assess primality and divisibility statements. Every integer is a multiple of at least one prime (its prime factors); being divisible by a composite requires having a composite factor greater than 1, which primes do not have. Step-by-Step Solution: Set up the difference: (10a + b) - (10b + a) = 9a - 9b = 36 → a - b = 4.Combine with a + b = 10: add equations → 2a = 14 → a = 7; then b = 3.Number = 10a + b = 73; reversed = 37; difference = 36, as required.73 is a prime number (only factors 1 and 73). Verification / Alternative check: Digit-sum and reversal both check out. Quick divisibility tests show 73 has no composite divisor other than itself (which is prime), confirming its primality. Why Other Options Are Wrong: only I / Both I and II / Neither I nor II: Statement I is false since 73 is not divisible by any composite number; Statement II is true because any integer is a multiple of some prime (73 * 1 = 73). Common Pitfalls: Assuming “multiple of a prime” means a special property—every positive integer except 1 is a multiple of at least one prime; confusing composite divisibility with primality. Final Answer: only II

Q: Form analysis — Every prime of the form 3k + 1 can be written as 6m + 1 when which condition on k holds?

Introduction / Context: Primes greater than 3 are of the form 6m ± 1, because any other residue class modulo 6 makes the number divisible by 2 or 3. The question ties this fact to primes represented as 3k + 1 and asks what must be true about k to rewrite the prime as 6m + 1. Given Data / Assumptions: Let p be a prime number with p = 3k + 1. p must be odd (since p > 2 for primes in this set). We seek when p can be written as 6m + 1. Concept / Approach: Analyze parity: if p = 3k + 1 is to be odd, then 3k must be even, which forces k to be even. Writing k = 2t leads to p = 3(2t) + 1 = 6t + 1. Hence the required condition is that k be even. Step-by-Step Solution: Let p = 3k + 1 be prime and > 3, so p is odd.For p to be odd, 3k must be even → k is even.Set k = 2t → p = 3(2t) + 1 = 6t + 1.Therefore, any prime of the form 3k + 1 corresponds to 6m + 1 if and only if k is even. Verification / Alternative check: Try examples: p = 7 → 3k + 1 = 7 → k = 2 (even), and 7 = 6*1 + 1; p = 13 → k = 4 (even), and 13 = 6*2 + 1. Why Other Options Are Wrong: k odd / both odd and even: Violates parity; 3k + 1 would be even when k is odd (and > 2, therefore composite). No such form: Contradicted by numerous examples (7, 13, 19, 31, …). Common Pitfalls: Overlooking parity constraints; assuming all 3k + 1 primes can take both 6m ± 1 forms; forgetting that primes > 3 must be 6m ± 1. Final Answer: k is even

Question 1

Recover and solve (clarified condition):  The difference between two numbers is 18. Also, “four times the second is less than three times the first by 20.” What is the sum of the two numbers?

Accepted Answer

Introduction / Context: (Recovery-first clarification) The original wording was incomplete. We adopt a standard linear condition: “four times the second is less than three times the first by 20,” i.e., 3*first − 4*second = 20, along with a fixed difference of 18. This yields a unique solution for the pair and hence their sum. Given Data / Assumptions: Let the numbers be x (first) and y (second). Difference: x − y = 18. Relation: 3x − 4y = 20 (“less by 20”). Concept / Approach: Solve the two linear equations in two unknowns. Substitution from the difference equation into the second relation simplifies the system quickly and gives integer values for x and y. Step-by-Step Solution: From x − y = 18 ⇒ x = y + 18.Substitute into 3x − 4y = 20: 3(y + 18) − 4y = 20.Simplify: 3y + 54 − 4y = 20 ⇒ −y + 54 = 20 ⇒ y = 34.Then x = y + 18 = 52.Sum: x + y = 52 + 34 = 86. Verification / Alternative check: Check 3x − 4y: 3*52 − 4*34 = 156 − 136 = 20; difference x − y = 18. Both hold, confirming the sum 86. Why Other Options Are Wrong: 100, 80, and 92 do not satisfy both equations; 96 arises from common arithmetic slips but fails the verification. Common Pitfalls: Interpreting “less than” without the “by” amount; flipping x and y in the difference; arithmetic errors when substituting. Final Answer: 86

Question 2

Number identification with compound conditions:  A number is greater than 44^2 and smaller than 45^2. It is also a multiple of 5, and one part of the number is the square of 6 (i.e., 36 appears as a contiguous part). Determine the number from the op…

Accepted Answer

Introduction / Context: This is a constrained search problem over a narrow numeric band. The number must lie strictly between two consecutive squares (44^2 and 45^2), be a multiple of 5, and visibly contain “36” as a contiguous part (interpreting “one part of the number is the square of 6”). We evaluate the candidates against all constraints. Given Data / Assumptions: Bounds: 44^2 = 1936 and 45^2 = 2025. Number must satisfy: 1936 < N < 2025. N is a multiple of 5 → last digit is 0 or 5. “One part is the square of 6” → the string “36” appears contiguously in N. Concept / Approach: List multiples of 5 in (1936, 2025): 1940, 1945, 1950, …, 2020. Then check whether “36” appears in each candidate. If none satisfy all constraints, the problem cannot be resolved to a specific option from the provided list. Step-by-Step Solution: Enumerate candidates from options first: 1940 and 1980 fall within (1936, 2025); 2080 exceeds 2025.Check “36” presence: 1940 → digits “1 9 4 0” (no “36”); 1980 → digits “1 9 8 0” (no “36”).Therefore, no given in-range multiple of 5 includes the substring “36.”Conclusion from provided choices: no unique valid number can be selected. Verification / Alternative check: Border values 1936 and 2036 either equal 44^2 (not strictly greater) or exceed 45^2 (not strictly smaller). Hence they are disqualified. Why Other Options Are Wrong: 1940 and 1980 satisfy the range and multiple-of-5 conditions but fail the “contains 36” condition; 2080 is outside the range; “None of these” would claim an alternative exists, but the stem restricts selection to the options. Hence the safest provided choice is “Cannot be determined.” Common Pitfalls: Assuming “part” means any arithmetic component rather than a visible contiguous part; including endpoints 1936 or 2025; overlooking the multiple-of-5 constraint. Final Answer: Cannot be determined

Question 3

Digit-repetition product — Find the smallest positive integer N such that 13 * N equals a number consisting entirely of the digit 5 (for example, 55, 555, 5555, …). Determine the minimal N that works.

Accepted Answer

Introduction / Context: This number-system puzzle asks for the smallest positive integer N such that 13 * N produces a repdigit made only of 5s (like 5, 55, 555, 5555, …). Problems of this type are efficiently handled with modular arithmetic and repetition-number (repunit) reasoning, avoiding brute-force search. Given Data / Assumptions: We need 13 * N = 55…5 (k copies of 5) for some smallest integer k. The target number with k copies of 5 equals 5 * (10^k - 1) / 9. N must be a positive integer; k must be minimal to ensure N is minimal. Concept / Approach: Let T_k = 5 * (10^k - 1) / 9. We need 13 | T_k. Since gcd(5, 13) = 1 and gcd(9, 13) = 1, this reduces to 13 | (10^k - 1). Thus, the smallest viable k is the multiplicative order of 10 modulo 13 (or modulo 117 when considering the 9 in the denominator rigorously). The minimal k that makes (10^k - 1) divisible by 117 yields the first all-5s number divisible by 13. Step-by-Step Solution: Write the repdigit: 55…5 (k fives) = 5 * (10^k - 1) / 9.We need 13 | 5 * (10^k - 1) / 9. Since 5 and 13 are coprime, ensure 117 | (10^k - 1).The smallest k with 10^k ≡ 1 (mod 117) is k = 6 (standard order result).Thus, the first suitable repdigit is 555555. Compute N = 555555 / 13 = 42735. Verification / Alternative check: Check directly: 13 * 42735 = 555555. No shorter repdigit of 5s (like 5, 55, 555, 5555, 55555) is divisible by 13, so this N is minimal. Why Other Options Are Wrong: 41625 / 42515 / 42135 / 55555: None of these, when multiplied by 13, yield an all-5s product. Only 42735 satisfies 13 * N = 555555 exactly. Common Pitfalls: Trying random guesses; forgetting the repdigit formula 5 * (10^k - 1) / 9; checking divisibility by 13 without confirming minimality of k. Final Answer: 42735

Question 4

Two-digit number with digit sum 10 — Reversing the digits decreases the number by 36. Identify which statements are true: I) The number is divisible by a composite number. II) The number is a multiple of a prime number.

Accepted Answer

Introduction / Context: This problem blends place-value reasoning with properties of primes and composites. You are told the two-digit number has digits summing to 10, and reversing the digits reduces the number by 36. From these constraints, you can determine the exact number and then evaluate the truth of the given statements regarding divisibility. Given Data / Assumptions: Let the tens digit be a and the units digit be b. a + b = 10. Original number: 10a + b. Reversed: 10b + a. Reversing decreases the number by 36: (10a + b) - (10b + a) = 36. Concept / Approach: Translate the language into equations and solve the simple system for digits a and b. Then assess primality and divisibility statements. Every integer is a multiple of at least one prime (its prime factors); being divisible by a composite requires having a composite factor greater than 1, which primes do not have. Step-by-Step Solution: Set up the difference: (10a + b) - (10b + a) = 9a - 9b = 36 → a - b = 4.Combine with a + b = 10: add equations → 2a = 14 → a = 7; then b = 3.Number = 10a + b = 73; reversed = 37; difference = 36, as required.73 is a prime number (only factors 1 and 73). Verification / Alternative check: Digit-sum and reversal both check out. Quick divisibility tests show 73 has no composite divisor other than itself (which is prime), confirming its primality. Why Other Options Are Wrong: only I / Both I and II / Neither I nor II: Statement I is false since 73 is not divisible by any composite number; Statement II is true because any integer is a multiple of some prime (73 * 1 = 73). Common Pitfalls: Assuming “multiple of a prime” means a special property—every positive integer except 1 is a multiple of at least one prime; confusing composite divisibility with primality. Final Answer: only II

Question 5

Form analysis — Every prime of the form 3k + 1 can be written as 6m + 1 when which condition on k holds?

Accepted Answer

Introduction / Context: Primes greater than 3 are of the form 6m ± 1, because any other residue class modulo 6 makes the number divisible by 2 or 3. The question ties this fact to primes represented as 3k + 1 and asks what must be true about k to rewrite the prime as 6m + 1. Given Data / Assumptions: Let p be a prime number with p = 3k + 1. p must be odd (since p > 2 for primes in this set). We seek when p can be written as 6m + 1. Concept / Approach: Analyze parity: if p = 3k + 1 is to be odd, then 3k must be even, which forces k to be even. Writing k = 2t leads to p = 3(2t) + 1 = 6t + 1. Hence the required condition is that k be even. Step-by-Step Solution: Let p = 3k + 1 be prime and > 3, so p is odd.For p to be odd, 3k must be even → k is even.Set k = 2t → p = 3(2t) + 1 = 6t + 1.Therefore, any prime of the form 3k + 1 corresponds to 6m + 1 if and only if k is even. Verification / Alternative check: Try examples: p = 7 → 3k + 1 = 7 → k = 2 (even), and 7 = 6*1 + 1; p = 13 → k = 4 (even), and 13 = 6*2 + 1. Why Other Options Are Wrong: k odd / both odd and even: Violates parity; 3k + 1 would be even when k is odd (and > 2, therefore composite). No such form: Contradicted by numerous examples (7, 13, 19, 31, …). Common Pitfalls: Overlooking parity constraints; assuming all 3k + 1 primes can take both 6m ± 1 forms; forgetting that primes > 3 must be 6m ± 1. Final Answer: k is even

Question 6

Two-digit number with digit sum 10 — Reversing the digits decreases the number by 36. Decide which statements are correct: I) The number is divisible by a composite number. II) The number is a multiple of a prime number.

Accepted Answer

Introduction / Context: This is the same structural setup as a classic digit problem: a + b = 10 for a two-digit number 10a + b, and reversing to 10b + a reduces the value by 36. From the digits you can identify the number and then assess statements about divisibility by composite and prime numbers. Given Data / Assumptions: Digits a (tens) and b (units) satisfy a + b = 10. (10a + b) - (10b + a) = 36 → 9(a - b) = 36 → a - b = 4. We evaluate two statements after finding the number. Concept / Approach: Solve the linear system to find (a, b). Then check whether the number is divisible by some composite and whether it is a multiple of a prime. Every integer greater than 1 is a multiple of at least one prime (by unique factorization), but primes are not divisible by any composite > 1. Step-by-Step Solution: Solve a + b = 10 and a - b = 4 → adding gives 2a = 14 → a = 7, so b = 3.Number = 73; reversed 37; indeed 73 - 37 = 36.73 is prime → not divisible by any composite number > 1.However, 73 is trivially a multiple of a prime (itself): 73 = 73 * 1. Verification / Alternative check: Confirm digit sum condition and difference condition; a primality check for 73 (no small divisors) confirms the conclusions about statements I and II. Why Other Options Are Wrong: Only I / Both / Neither: Statement I is false; Statement II is true. Hence “Only II”. Common Pitfalls: Misinterpreting “multiple of a prime” as implying non-prime; overlooking that primes are multiples of themselves. Final Answer: Only II

Question 7

Fraction reconstruction — A fraction becomes 1/2 when the numerator is increased by 2 and denominator by 1; it becomes 3/5 when the numerator is increased by 1 and denominator decreased by 2. Find the original fraction.

Accepted Answer

Introduction / Context: This algebra problem encodes two conditions on the same unknown fraction. Translating each sentence to an equation and solving the resulting linear system is the most reliable and general approach. The numbers are small, making factoring and checking straightforward. Given Data / Assumptions: Original fraction is n/d with positive integers n and d. (n + 2) / (d + 1) = 1/2. (n + 1) / (d - 2) = 3/5. Concept / Approach: Clear denominators to turn each condition into a linear equation in n and d. Solve the two linear equations simultaneously. After obtaining (n, d), simplify if needed and quickly verify both conditions to eliminate mistakes. Step-by-Step Solution: From (n + 2) / (d + 1) = 1/2 → 2(n + 2) = d + 1 → 2n + 4 = d + 1 → d = 2n + 3.From (n + 1) / (d - 2) = 3/5 → 5(n + 1) = 3(d - 2) → 5n + 5 = 3d - 6 → 3d = 5n + 11 → d = (5n + 11)/3.Equate d expressions: 2n + 3 = (5n + 11)/3 → 6n + 9 = 5n + 11 → n = 2.Then d = 2n + 3 = 7, so the fraction is 2/7. Verification / Alternative check: (2 + 2)/(7 + 1) = 4/8 = 1/2; and (2 + 1)/(7 - 2) = 3/5. Both conditions hold exactly. Why Other Options Are Wrong: 3/5 / 1/7 / 2/5 / 5/12: Substituting any of these into the two conditions fails at least one requirement; only 2/7 satisfies both simultaneously. Common Pitfalls: Sign or algebra slips when clearing denominators; forgetting to verify both constraints; reducing the fraction prematurely and losing track of n and d. Final Answer: 2/7

Question 8

Five consecutive odd integers sum to 175 — Find the sum of the second-largest integer and the square of the smallest integer among them.

Accepted Answer

Introduction / Context: Consecutive odd numbers form an arithmetic sequence with common difference 2. When the sum of five consecutive odd integers is given, the middle term equals the average. Once the sequence is identified, compute the requested combination: the second-largest integer plus the square of the smallest integer. Given Data / Assumptions: Five consecutive odd integers have sum 175. Let the middle term be M; then the set is M - 4, M - 2, M, M + 2, M + 4. We need (second-largest) + (smallest)^2. Concept / Approach: For any five-term arithmetic progression, the sum equals 5 * (middle term). Hence M = 175 / 5 = 35. Then enumerate the integers and perform the required operations carefully to avoid arithmetic slips. Step-by-Step Solution: Find middle term: M = 175 / 5 = 35.List numbers: 31, 33, 35, 37, 39.Smallest = 31; second-largest = 37.Compute 31^2 = 961.Required sum = 37 + 961 = 998. Verification / Alternative check: The set indeed sums to 31 + 33 + 35 + 37 + 39 = 175. Squaring 31 gives 961, and adding 37 yields 998. Why Other Options Are Wrong: 986 / 995 / 989 / 979: These values are close but result from mis-squaring 31, confusing “second-largest” with another term, or arithmetic mistakes in addition. Common Pitfalls: Taking the largest (39) instead of the second-largest (37); incorrectly computing 31^2; assuming the middle term is 34 or 36 due to parity mistakes. Final Answer: 998

Question 9

Three related numbers — The first equals twice the second and thrice the third. If their average is 154, find the difference between the first and the third.

Accepted Answer

Introduction / Context: This is a ratio-based setup translated into linear equations. By expressing each number in terms of the first (or a common variable), you can compute the total, use the average to find the exact values, and then answer the query about the difference between specific terms. Given Data / Assumptions: Let the three numbers be A (first), B (second), C (third). A = 2B and A = 3C → B = A/2, C = A/3. Their average is 154 → (A + B + C)/3 = 154. Concept / Approach: Sum in terms of A: A + A/2 + A/3 = A * (1 + 1/2 + 1/3) = A * (11/6). The average condition gives (A * 11/6)/3 = 154, from which A is determined. Then compute C and the required difference A - C. Step-by-Step Solution: Average condition → (A + A/2 + A/3)/3 = 154 → A * 11/18 = 154.Solve for A: A = 154 * 18 / 11 = 14 * 18 = 252.Then C = A/3 = 252/3 = 84.Difference A - C = 252 - 84 = 168. Verification / Alternative check: Compute B = A/2 = 126. Average: (252 + 126 + 84)/3 = 462/3 = 154 as required. Why Other Options Are Wrong: 126 / 42 / 52 / 108: These stem from mixing the relations (twice vs thrice), miscomputing the average, or subtracting the wrong pair. Common Pitfalls: Using 2C = A instead of A = 3C; forgetting to divide the sum by 3 for the average; arithmetic slips in fractions. Final Answer: 168

Question 10

Two-number system — The difference of two numbers is 18. Also, four times the second is 18 less than three times the first. Find the sum of the two numbers.

Accepted Answer

Introduction / Context: Solving two linear equations in two unknowns is a staple of aptitude exams. Here, one equation gives the difference of the two numbers; the other relates multiples of the numbers with an offset. The objective is the sum of the numbers once they are determined. Given Data / Assumptions: Let the numbers be x (first) and y (second). x - y = 18. Four times the second is 18 less than three times the first: 4y = 3x - 18. Concept / Approach: Use substitution: from x - y = 18 → x = y + 18. Substitute into 4y = 3x - 18 and solve for y, then back-substitute for x. Finally, compute x + y. Step-by-Step Solution: From x - y = 18 → x = y + 18.Substitute: 4y = 3(y + 18) - 18 → 4y = 3y + 54 - 18 → 4y = 3y + 36 → y = 36.Then x = y + 18 = 36 + 18 = 54.Sum = x + y = 54 + 36 = 90. Verification / Alternative check: Check the second condition: three times the first is 162; four times the second is 144; indeed 4y is 18 less than 3x. Why Other Options Are Wrong: 100 / 80 / 86 / 96: These come from algebra slips or misreading “less than” direction; none matches the computed sum 90. Common Pitfalls: Reversing “18 less than” into “+18”; forgetting to add the difference when expressing x in terms of y; arithmetic errors in substitution. Final Answer: 90

Question 11

Equal distribution with absentees — Sweets were to be equally distributed among 300 children, but 50 were absent. Each present child then received one extra sweet. How many sweets were distributed in total?

Accepted Answer

Introduction / Context: This is a classic allocation problem using the idea that the total quantity remains constant, but the per-person share changes because fewer recipients are present. The key is to set up the relation between original planned share and the increased share when some children are absent. Given Data / Assumptions: Planned recipients: 300 children. Actual recipients: 300 - 50 = 250 children. Each present child received one more sweet than originally planned. Total sweets distributed remained the same. Concept / Approach: Let the originally planned per-child share be x. Then the actual per-child share is x + 1 (since fewer children shared the same total). Equate the two expressions for total sweets and solve for x, then compute the total number of sweets. Step-by-Step Solution: Let original per-child share = x.Original total = 300 * x.Actual per-child share = x + 1 with 250 children → Actual total = 250 * (x + 1).Equate totals: 300x = 250(x + 1) → 300x = 250x + 250 → 50x = 250 → x = 5.Total sweets = 300 * x = 300 * 5 = 1500. Verification / Alternative check: With 250 children, each gets x + 1 = 6 sweets: 250 * 6 = 1500, same total—consistent. Why Other Options Are Wrong: 1450 / 1700 / 1650 / 1800: These totals do not satisfy both equations simultaneously; they arise from misreading the “one extra” condition or miscomputing the absent count. Common Pitfalls: Assuming the total changes; mixing up who got “one extra”; arithmetic slips when equating totals. Final Answer: 1500

Question 12

Two exam halls P and Q — After moving 10 students from P to Q, the numbers become equal. After moving 20 students from Q to P, P becomes double Q. Find the original numbers in P and Q.

Accepted Answer

Introduction / Context: This is a system-of-equations problem describing two different transfers of students between halls P and Q. Translating the narrative statements precisely into algebra will let you solve for the original counts without guesswork. Given Data / Assumptions: Let initial counts be p (P) and q (Q). After 10 shift from P to Q: p - 10 = q + 10. After 20 shift from Q to P: p + 20 = 2 * (q - 20). All numbers are nonnegative integers. Concept / Approach: From the first transfer, express p in terms of q. Substitute into the second transfer equation and solve. This straightforward linear system yields unique integer solutions that match the scenario conditions. Step-by-Step Solution: From p - 10 = q + 10 → p - q = 20 → p = q + 20.Use the second condition: p + 20 = 2(q - 20).Substitute p: (q + 20) + 20 = 2(q - 20) → q + 40 = 2q - 40 → q = 80.Then p = q + 20 = 100. Verification / Alternative check: Check first transfer: 100 - 10 = 90 and 80 + 10 = 90 → equal. Check second transfer: 100 + 20 = 120; 80 - 20 = 60; indeed 120 = 2 * 60. Why Other Options Are Wrong: 60, 40 / 70, 50 / 80, 60 / 90, 70: None satisfy both transfer conditions simultaneously. Common Pitfalls: Reversing the direction of transfer; misinterpreting “P becomes double Q” after the second move; algebraic slips when substituting p = q + 20. Final Answer: 100, 80

Question 13

Two real numbers a and b satisfy: the sum of their squares is 97 and the square of their difference is 25. Using these relations, determine the product a * b.

Accepted Answer

Introduction / Context: This quantitative aptitude question tests algebraic identities that connect sums/differences of numbers with their products. Instead of finding the numbers explicitly, you can use standard formulas to extract the product directly from aggregate information like a^2 + b^2 and (a − b)^2. Given Data / Assumptions: a^2 + b^2 = 97. (a − b)^2 = 25. a and b are real numbers; the task is to compute a * b. Concept / Approach: The key identity is (a − b)^2 = a^2 + b^2 − 2ab. Since a^2 + b^2 and (a − b)^2 are known, you can isolate 2ab and then divide by 2 to get the product ab. This avoids solving for a and b individually and minimizes arithmetic steps. Step-by-Step Solution: Start from the identity: (a − b)^2 = a^2 + b^2 − 2ab.Plug values: 25 = 97 − 2ab.Rearrange for 2ab: 2ab = 97 − 25 = 72.Compute the product: ab = 72 / 2 = 36. Verification / Alternative check: As a consistency check, compute (a + b)^2 = a^2 + b^2 + 2ab = 97 + 72 = 169, so a + b = 13 or −13. Real number pairs with sum 13 and product 36 do exist (they are roots of t^2 − 13t + 36 = 0), confirming that the data are internally consistent. Why Other Options Are Wrong: 45, 54, and 63 arise from misplacing terms in the identity or using (a + b)^2 incorrectly. 28 is an arbitrary distractor that does not satisfy 2ab = 72. Common Pitfalls: Confusing (a − b)^2 with (a + b)^2, or forgetting to divide 2ab by 2 to get ab. Another mistake is attempting to compute a and b explicitly, which is unnecessary here. Final Answer: 36

Question 14

A 3-hour (180-minute) exam has 200 questions: 50 Mathematics, 100 General Knowledge, and 50 Science. Ram spends twice as much time per Mathematics question as he spends per each non-Mathematics question. How many total minutes does he spend on Mathe…

Accepted Answer

Introduction / Context: This time-allocation problem checks proportional reasoning. You are told that the per-question time for Mathematics is exactly twice that for each other subject. With the total time fixed, you can set up a simple equation in one variable to find the duration spent on Mathematics. Given Data / Assumptions: Total exam time = 3 hours = 180 minutes. Question counts: 50 Mathematics, 100 General Knowledge, 50 Science (total non-Mathematics = 150). Time per Mathematics question is twice time per non-Mathematics question. Concept / Approach: Let t be the time in minutes per non-Mathematics question. Then each Mathematics question takes 2t minutes. Multiply these by the respective quantities of questions to build the total time equation. Solve for t and then compute the time on Mathematics only. Step-by-Step Solution: Let t = minutes per GK/Science question; Mathematics per question = 2t.Total time: 50*(2t) + 150*(t) = 180.Compute: 100t + 150t = 250t = 180 → t = 180 / 250 = 0.72 minute.Total minutes on Mathematics = 50 * (2t) = 100t = 100 * 0.72 = 72 minutes. Verification / Alternative check: Time on non-Mathematics = 150 * 0.72 = 108 minutes. 72 + 108 = 180 minutes, which matches the total exam time, verifying correctness. Why Other Options Are Wrong: 36 and 60 assume wrong multipliers or miscount questions. 100 and 80 result from setting the double rate incorrectly or forgetting total time is only 180 minutes. Common Pitfalls: Using the same per-question time across subjects, ignoring that Mathematics takes double time, or mixing up total question counts when forming the equation. Final Answer: 72

Question 15

A city taxi fare has a fixed charge that covers up to 5 km, plus an additional per-kilometre charge beyond 5 km. The fare for 10 km is ₹350 and for 25 km is ₹800. What is the fare for 30 km?

Accepted Answer

Introduction / Context: This is a two-parameter linear pricing model: a fixed base fare up to a threshold distance, and a uniform per-kilometre rate beyond that threshold. Two known points (10 km and 25 km) determine the fixed charge and the per-kilometre rate, after which you can predict the fare at 30 km. Given Data / Assumptions: Fixed charge F covers up to 5 km. Rate r applies for each km beyond 5 km. Fare(10 km) = F + 5r = ₹350. Fare(25 km) = F + 20r = ₹800. Concept / Approach: Solve the two linear equations to find F and r. Then plug into the formula for 30 km: Fare(30) = F + 25r. This is a straightforward system with elimination by subtraction to isolate r. Step-by-Step Solution: From data: F + 5r = 350 … (1)F + 20r = 800 … (2)Subtract (1) from (2): 15r = 450 → r = 30 (₹/km beyond 5 km).Plug back to (1): F = 350 − 5*30 = 350 − 150 = 200.Fare(30) = F + 25r = 200 + 25*30 = 200 + 750 = ₹950. Verification / Alternative check: Check 25 km: 200 + 20*30 = 200 + 600 = ₹800 (matches). Check 10 km: 200 + 5*30 = ₹350 (matches). Consistency verifies the parameters. Why Other Options Are Wrong: ₹900 assumes a lower rate or base. ₹800 and ₹750 are previous-distance fares. ₹920 is an ad-hoc estimate not supported by the linear model. Common Pitfalls: Charging per km from 0 instead of after 5 km, or mixing total kilometres with additional kilometres beyond the fixed coverage. Final Answer: ₹ 950

Question 16

Three friends X, Y, and Z dine together. Z’s meal costs 20% more than Y’s meal, and X’s meal costs 5/6 as much as Z’s meal. If Y pays ₹100, find the total amount paid by all three combined.

Accepted Answer

Introduction / Context: This is a percentage-and-ratio chaining problem. You are given relative costs between X, Y, and Z, plus the actual rupee value for Y. Converting the relative statements into multipliers allows quick computation of Z and X, and then summing yields the total bill. Given Data / Assumptions: Y pays ₹100. Z is 20% more than Y → Z = 1.2 * Y. X is 5/6 of Z → X = (5/6) * Z. Total = X + Y + Z. Concept / Approach: Work stepwise from the known amount for Y. First, compute Z using the 20% markup. Next, compute X using the 5/6 factor relative to Z. Finally, add X, Y, and Z. Ensure rounding is not needed since all values are exact integers here. Step-by-Step Solution: Given Y = ₹100.Z = 1.2 * 100 = ₹120.X = (5/6) * 120 = ₹100.Total = 100 + 120 + 100 = ₹320. Verification / Alternative check: Reverse-check percentages: Z is 20% more than 100 (indeed 120). X is 5/6 of 120 (indeed 100). The arithmetic is consistent and exact. Why Other Options Are Wrong: ₹285, ₹300, and ₹335 do not match the chained percentage relations. Hence the only correct choice is the catch-all “None of these,” since the correct total is ₹320. Common Pitfalls: Applying 5/6 to Y instead of Z, or compounding the 20% incorrectly (e.g., adding 20 and 5/6 as if they were on the same base). Final Answer: None of these

Question 17

An exam has 5 questions. Of all candidates, 5% answered all 5 questions and 5% answered none. Among the remaining candidates, 25% answered only 1 question and 20% answered 4 questions. If 396 candidates answered either 2 or 3 questions, how many can…

Accepted Answer

Introduction / Context: This percentage distribution problem involves conditioning: some percentages apply to the entire group, while others apply only to the remaining candidates after removing initial categories. Careful bookkeeping converts the verbal description into algebra, yielding the total population size. Given Data / Assumptions: Let total candidates = N. 5% answered all questions → 0.05N; 5% answered none → 0.05N. Remaining group = 0.90N. Within the remaining: 25% answered only 1 question → 0.25 * 0.90N = 0.225N. Within the remaining: 20% answered 4 questions → 0.20 * 0.90N = 0.18N. Given: those who answered 2 or 3 questions together = 396. Concept / Approach: First identify how many of the “remaining” candidates are already accounted for by the “only 1” and “4 questions” groups. The balance of the remaining must be exactly those who answered 2 or 3 questions. Equate this balance to 396 and solve for N. Step-by-Step Solution: Remaining after all/none = 0.90N.Portion in 1 question = 0.225N; portion in 4 questions = 0.18N.Therefore, portion in 2 or 3 questions = 0.90N − (0.225N + 0.18N) = 0.495N.Set 0.495N = 396 → N = 396 / 0.495 = 800. Verification / Alternative check: Compute counts: all = 40, none = 40, only 1 = 180, four = 144, and 2-or-3 = 396. Sum = 40 + 40 + 180 + 144 + 396 = 800, confirming the total. Why Other Options Are Wrong: 1000, 900, 850, and 720 do not satisfy the equation 0.495N = 396 and yield inconsistent category counts. Common Pitfalls: Treating 25% and 20% as percentages of the total N instead of the remaining 90%, or forgetting to subtract both the “only 1” and “4 questions” groups from the remaining pool. Final Answer: 800

Question 18

Consider a three-digit number with hundreds digit h, tens digit t, and units digit u. The conditions are: u = 4h; after swapping the tens and units digits the new number is 18 greater than the original; and h is one-third of t. Find 25% of the origi…

Accepted Answer

Introduction / Context: This digit puzzle combines ratio constraints on digits with a transformation (interchanging tens and units) that increases the number by a fixed amount. By expressing the number symbolically and applying the constraints, you can isolate each digit and then compute the requested fraction of the original number. Given Data / Assumptions: Original number = 100h + 10t + u. u = 4h. t = 3h (since h is one-third of t). Swapping tens and units gives 100h + 10u + t, which is 18 more than original. Digits must be integers within 0–9, and h ≥ 1 for a three-digit number. Concept / Approach: Substitute t and u in terms of h to express both the original and swapped numbers. The difference equation will involve only h. Solve for h respecting digit bounds, then rebuild the number and compute 25% (one-quarter) of it. Step-by-Step Solution: Original = 100h + 10(3h) + 4h = 134h.Swapped = 100h + 10(4h) + 3h = 143h.Given: Swapped − Original = 143h − 134h = 9h = 18 → h = 2.Then t = 3h = 6 and u = 4h = 8 → original number = 268.Compute 25%: 268 * (1/4) = 67. Verification / Alternative check: Swap tens and units: 268 → 286. Difference 286 − 268 = 18 (matches). Digit constraints u = 4h and t = 3h also hold, confirming correctness. Why Other Options Are Wrong: 84, 96, and 137 do not equal one-quarter of 268. “Couldn’t not be determined” is incorrect because the constraints uniquely determine the number. Common Pitfalls: Misinterpreting “one-third” (taking t = h/3 instead of t = 3h), or forgetting to apply the swap correctly when forming the new number. Final Answer: 67

Question 19

A two-digit positive number has its units digit equal to the square of its tens digit. When its digits are interchanged, the difference between the original and the new number is 54. What is 40% of the original number?

Accepted Answer

Introduction / Context: This problem blends digit constraints with a fixed difference after digit reversal. Because the units digit equals the square of the tens digit, only a few two-digit candidates exist, which can be checked quickly against the given difference. Once the original number is identified, computing 40% is straightforward. Given Data / Assumptions: Let the number be 10a + b with digits a (tens) and b (units). Units equals square of tens: b = a^2. |(10a + b) − (10b + a)| = 54 (difference after interchanging digits). Digits are from 0–9, and a ≥ 1 for a two-digit number. Concept / Approach: Since b = a^2 must be a single digit, possible a values are 1, 2, or 3 (yielding b = 1, 4, 9). Form the candidate numbers 11, 24, and 39 and test the absolute difference after swapping digits. Select the one that satisfies the 54 difference condition, then evaluate 40% of that number. Step-by-Step Solution: Candidates: a = 1 → 11; a = 2 → 24; a = 3 → 39.Compute differences with reversals: |11 − 11| = 0, |24 − 42| = 18, |39 − 93| = 54.Only 39 satisfies the 54-difference condition.Compute 40%: 0.40 * 39 = 15.6. Verification / Alternative check: Reversal check holds for 39 ↔ 93 with difference 54; the digit constraint b = a^2 is satisfied since 9 = 3^2. Why Other Options Are Wrong: 14.4 (which is 40% of 36), 16, 12, and 18 correspond to incorrect original numbers or arbitrary guesses; they do not align with both the digit-square constraint and the 54 difference requirement. Common Pitfalls: Including a = 4 (which would require b = 16, not a single digit), or forgetting the absolute value in the “difference” statement, which could flip signs and mislead the check. Final Answer: 15.6

Question 20

On both sides of a straight road 1760 m long, trees are planted at equal intervals of 20 m, including trees at the ends on each side. What is the maximum total number of trees that can be planted?

Accepted Answer

Introduction / Context: Equally spaced placements along a segment are a classic “inclusive endpoints” counting problem. When the spacing is d along a segment of length L, the number of points on one side is (L / d) + 1, provided trees are placed at both ends. Since there are two sides of the road, double the per-side count for the total. Given Data / Assumptions: Road length L = 1760 m. Spacing d = 20 m between consecutive trees. Trees are planted on both sides and at both ends on each side. Concept / Approach: For one side, with trees at both endpoints, the count is N_side = (L / d) + 1. Then total trees = 2 * N_side, because the same pattern is repeated on the other side. The key is not to forget the extra “+1” that arises from including both endpoints. Step-by-Step Solution: Compute per side: L / d = 1760 / 20 = 88.Include both endpoints: N_side = 88 + 1 = 89.Trees on both sides: Total = 2 * 89 = 178. Verification / Alternative check: Visualize marks every 20 m from 0 to 1760 on one side: positions 0, 20, …, 1760 (inclusive). Counting by step size confirms 89 positions per side, validating the formula. Why Other Options Are Wrong: 176 ignores one of the endpoints across both sides. 180 assumes an extra interval or double-counts. 174 and 172 are undercounts due to missing endpoints or misapplied spacing. Common Pitfalls: Using only L / d without adding 1, or forgetting to multiply by 2 for both sides of the road. Final Answer: 178

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