Q: Solve for the missing radicand — In the equation 392 / sqrt(x) = 28, determine the value of x. Show each manipulation of the equation clearly.

Introduction / Context: This question checks your fluency with algebraic manipulation involving square roots and simple fractions. The structure 392 / sqrt(x) = 28 invites you to isolate the radical term and then square appropriately to solve for x without introducing extraneous solutions. Given Data / Assumptions: Equation: 392 / sqrt(x) = 28. sqrt(x) denotes the principal (non-negative) square root. x is positive so that sqrt(x) is defined in real numbers. Concept / Approach: Isolate sqrt(x) by multiplying both sides by sqrt(x) and dividing by 28, or equivalently by cross-multiplication. Once sqrt(x) is expressed as a number, square both sides to remove the radical and obtain x. Always verify by substituting back into the original equation to avoid sign or arithmetic slips. Step-by-Step Solution: Start with 392 / sqrt(x) = 28.Rearrange: sqrt(x) = 392 / 28.Compute 392 / 28 = 14.Square both sides: x = 14^2 = 196. Verification / Alternative check: Substitute x = 196: sqrt(196) = 14, and 392 / 14 = 28. The original equation is satisfied exactly, confirming the solution. Why Other Options Are Wrong: 144: Would imply sqrt(x) = 12, giving 392 / 12 = 32.666..., not 28. 24 / 48: These are not valid values of x in this context; they would make sqrt(x) non-integer and yield a non-28 quotient. 784: sqrt(784) = 28 ⇒ 392 / 28 = 14, not 28. Common Pitfalls: Squaring the entire original fraction prematurely; inverting 392/28 incorrectly; forgetting that sqrt(x) must be non-negative in the real-number context. Final Answer: 196

Q: Prime numbers — Compute the sum of the first four prime numbers (in increasing order). Show the primes you add to avoid mistakes.

Introduction / Context: This checks recognition of prime numbers and accurate addition. Primes are integers greater than 1 with exactly two distinct positive divisors: 1 and the number itself. Listing the earliest primes and adding them carefully produces the correct sum quickly. Given Data / Assumptions: We need the first four prime numbers in ascending order. Definition of prime excludes 1 and includes 2 as the only even prime. Perform a direct sum once identified. Concept / Approach: Identify the primes: 2, 3, 5, 7 are the smallest four primes. Then add them explicitly. Writing the list prevents the common error of including 1 or skipping 7 by mistake. Ensuring fundamental definitions avoids traps in multiple-choice settings. Step-by-Step Solution: List primes: 2, 3, 5, 7.Add: 2 + 3 = 5.Add next: 5 + 5 = 10.Add last: 10 + 7 = 17. Verification / Alternative check: Double-check primality: each of 2, 3, 5, 7 has no positive divisors other than 1 and itself. Re-sum quickly to confirm 17. Why Other Options Are Wrong: 10 / 11 / 16 / 19: Each corresponds to a mis-addition or an incorrect set of early primes (for example, accidentally including 1 or omitting 7). Common Pitfalls: Counting 1 as a prime; forgetting that 2 is prime; adding too fast and arriving at 16 instead of 17. Final Answer: 17

Q: Prime identification — Which is the first (smallest) prime number according to the standard definition of primes?

Introduction / Context: Understanding the definition of a prime number is foundational. A prime is an integer greater than 1 that has exactly two distinct positive divisors: 1 and the number itself. This simple definition immediately identifies the smallest prime and avoids common misconceptions found in early number theory questions. Given Data / Assumptions: A prime must be greater than 1. It must have exactly two distinct positive divisors. We are to choose the smallest number fulfilling these conditions. Concept / Approach: Check the candidates in increasing order: 0, 1, 2, 3, 5. Values 0 and 1 do not satisfy the prime definition. The number 2 is divisible only by 1 and 2, and is thus the smallest prime. The next prime is 3; however, the question asks for the first prime only, which is 2. Step-by-Step Solution: Evaluate 0: has infinitely many divisors; not prime.Evaluate 1: has only one positive divisor (1); not prime.Evaluate 2: divisors are 1 and 2; prime.Therefore the first prime is 2. Verification / Alternative check: List small primes: 2, 3, 5, 7, 11. The smallest is 2, confirming the choice. Why Other Options Are Wrong: 0 and 1: Do not meet prime criteria. 3 and 5: These are primes, but not the first; they are larger than 2. Common Pitfalls: Thinking 1 is prime; forgetting that 2 is the only even prime; confusing definitions that apply to composites and primes. Final Answer: 2

Q: Proportion solving — In the proportion 10 / 11 = 110 / x, find the value of x using cross-multiplication for exact equality of ratios.

Introduction / Context: Proportions equate two ratios. The standard method for finding a missing term is cross-multiplication, which preserves equality when all terms are non-zero. This is routinely tested in aptitude arithmetic for speed and accuracy. Given Data / Assumptions: Proportion: 10 / 11 = 110 / x. All quantities are positive and non-zero. We solve for x. Concept / Approach: Use cross-multiplication: (10) * (x) = (11) * (110). Then isolate x by dividing both sides by 10. This direct manipulation is both quick and reliable. Finally, verify by substituting x back to confirm equal ratios. Step-by-Step Solution: Apply cross-multiplication: 10 * x = 11 * 110.Compute right-hand side: 11 * 110 = 1210.Solve for x: x = 1210 / 10 = 121.Therefore, x = 121. Verification / Alternative check: Replace x with 121: 10 / 11 = 0.90909..., and 110 / 121 = 0.90909..., confirming the proportion holds exactly. Why Other Options Are Wrong: 111 / 1100 / 100 / 11: None satisfy the equality 10 / 11 = 110 / x when substituted; the ratios differ slightly or substantially. Common Pitfalls: Swapping numerator and denominator during cross-multiplication; dividing the wrong side; arithmetic slips when multiplying 11 by 110. Final Answer: 121

Q: Guaranteed divisibility — The product of three consecutive even natural numbers is always divisible by what largest natural number? Justify using factor patterns.

Introduction / Context: Problems about guaranteed divisibility rely on number patterns. For three consecutive even numbers, certain prime powers and factors must appear in their product regardless of where the sequence starts. The question asks for the largest natural number that always divides such a product. Given Data / Assumptions: Let the consecutive even numbers be 2n, 2n + 2, 2n + 4 for some integer n. We analyze the product P = (2n)(2n + 2)(2n + 4). We seek the largest fixed divisor valid for all integers n. Concept / Approach: Factor out common elements and inspect guaranteed multiples. Note that P = 2 * 2 * 2 * n * (n + 1) * (n + 2) = 8 * n * (n + 1) * (n + 2). Among n, n + 1, and n + 2, there is always exactly one multiple of 3. Also, at least one of these three consecutive integers is even, contributing at least one extra factor 2 beyond the 8 already factored out. Therefore, the product is always divisible by 8 * 2 * 3 = 48. Step-by-Step Solution: Write P = 8 * n * (n + 1) * (n + 2).Among three consecutive integers, one is a multiple of 3 → factor 3 present.Among these three, at least one is even → at least one extra factor 2 beyond the initial 8.Therefore guaranteed factor = 8 * 2 * 3 = 48. Verification / Alternative check: Test small cases: (2, 4, 6) product = 48; (4, 6, 8) product = 192, which is 48 * 4; (6, 8, 10) product = 480 = 48 * 10. Each case is divisible by 48, supporting the general proof. Why Other Options Are Wrong: 16 / 24: Too small; they always divide the product, but larger guaranteed factors exist. 96: Not always guaranteed (counterexample 2 * 4 * 6 = 48). 12: Also divides, but not the largest guaranteed divisor. Common Pitfalls: Overlooking the extra factor of 2 from n(n + 1)(n + 2); assuming a fixed multiple of 4 inside n(n + 1)(n + 2) when it is not guaranteed for all n. Final Answer: 48

Q: Reverse division with conditions — In a division, the divisor equals 12 times the quotient and 5 times the remainder. If the remainder is 48, find the dividend using the division identity.

Introduction / Context: Division problems with constraints on the divisor, quotient, and remainder are classic applications of the division identity: dividend = divisor * quotient + remainder. Here, you are given relationships between divisor, quotient, and remainder and must compute the dividend directly and cleanly. Given Data / Assumptions: Let divisor = d, quotient = q, remainder = r. Given: d = 12 * q and d = 5 * r. Given remainder r = 48. Standard condition: 0 ≤ r < d. Concept / Approach: Use the given relationships to resolve d and q. From d = 5 * r with r = 48, get d. Then from d = 12 * q, obtain q. Finally, use dividend = d * q + r to compute the desired number. This sequence avoids guesswork and stays faithful to the constraints. Step-by-Step Solution: From d = 5 * r and r = 48, compute d = 5 * 48 = 240.From d = 12 * q, compute q = d / 12 = 240 / 12 = 20.Apply the division identity: dividend = d * q + r = 240 * 20 + 48 = 4800 + 48 = 4848.Thus, the dividend is 4848. Verification / Alternative check: Divide 4848 by 240: 4848 = 240 * 20 + 48, so quotient 20 and remainder 48. The relationships d = 12 * q (240 = 12 * 20) and d = 5 * r (240 = 5 * 48) hold perfectly. Why Other Options Are Wrong: 240 / 576 / 4800 / 5280: These do not satisfy dividend = d * q + r under the given constraints or correspond to intermediate values (divisor/product) rather than the actual dividend. Common Pitfalls: Confusing which variable is 12 times which; forgetting to add the remainder after the product; violating the condition r < d. Final Answer: 4848

Q: Quadratic from a product — A number is multiplied by three-fourths of itself and the result is 10800. Find the number, showing how the quadratic simplifies cleanly.

Introduction / Context: Phrases like “multiplied by three-fourths of itself” translate into simple algebraic equations. Interpreting the statement accurately as a product of the number and a fraction of itself leads to a quadratic equation that is straightforward to solve by isolating the square term. Given Data / Assumptions: Let the unknown number be n. Equation: n * (3/4 * n) = 10800. We seek the positive real solution appropriate for a “number” in standard aptitude context. Concept / Approach: Combine factors: n * (3/4 * n) = (3/4) * n^2. Isolate n^2 by multiplying both sides by 4/3. Then take the principal square root to find n. Finally, verify by substitution to ensure no arithmetic errors occurred during scaling. Step-by-Step Solution: Write equation: (3/4) * n^2 = 10800.Multiply both sides by 4/3: n^2 = 10800 * (4/3) = 14400.Take square root: n = sqrt(14400) = 120 (positive branch).Therefore, the number is 120. Verification / Alternative check: Compute three-fourths of 120: (3/4) * 120 = 90. Then n * that = 120 * 90 = 10800, which matches the given result exactly. Why Other Options Are Wrong: 210 / 180 / 160 / 240: Substituting any of these into (3/4) * n^2 does not yield 10800. Only n = 120 satisfies the equation precisely. Common Pitfalls: Misreading “three-fourths of itself” as “n + 3/4” times n; forgetting to multiply both sides by 4/3; taking the wrong square root magnitude. Final Answer: 120

Question 1

Number System — Find the positive number x that satisfies 80 / x = x / 20. Choose the exact value.

Accepted Answer

Introduction / Context: This proportion is a classic “means equals extremes” setup. Cross-multiplication yields a simple quadratic in x^2 with a perfect-square result, so the solution is immediate once the relation is written correctly. Given Data / Assumptions: Equation: 80 / x = x / 20. x is taken as positive in typical number system contexts. Basic algebraic operations (cross-multiplication) apply. Concept / Approach: Cross-multiply to eliminate denominators and solve for x^2. Recognize 1600 as a perfect square to find x without decimals. Always consider the sign context; here, a positive length/value is expected. Step-by-Step Solution: 1) Cross-multiply: 80 * 20 = x * x.2) Compute left side: 1600 = x^2.3) Take square root: x = sqrt(1600) = 40 (positive solution).4) Therefore, x = 40. Verification / Alternative check: Plug back: 80/40 = 2 and 40/20 = 2. Both sides equal, confirming x = 40. Why Other Options Are Wrong: 400, 800, and 1600 are results of misplacing the square root; 20 yields 80/20 = 4 while 20/20 = 1, not equal. Common Pitfalls: Forgetting to take the square root after obtaining x^2; mixing 80 and 20 when cross-multiplying; introducing negative roots where the context implies positivity. Final Answer: 40

Question 2

Number System — Find the number x that satisfies (x/21) × (x/189) = 1. Choose the value that makes the product exactly one.

Accepted Answer

Introduction / Context: This problem is a straightforward algebraic equation in multiplicative form. Recognizing that the product equals 1 allows us to combine factors into x^2 and solve using square roots. Integer factorization helps confirm the perfect-square nature of the result. Given Data / Assumptions: (x/21) × (x/189) = 1. x is real; we seek the positive value consistent with the context. 21 and 189 are positive integers with simple factorization. Concept / Approach: Multiply the numerators and denominators: (x × x) / (21 × 189) = 1 → x^2 = 21 × 189. Factor the right-hand side to test if it is a perfect square and then take the square root to find x. Step-by-Step Solution: 1) Combine: x^2 / (21 × 189) = 1 → x^2 = 21 × 189.2) Compute 21 × 189 = 21 × (200 − 11) = 4200 − 231 = 3969.3) Note 3969 = 63^2 (since 60^2 = 3600 and 63^2 = 3969).4) Therefore, x = 63 (taking the positive root consistent with the setting). Verification / Alternative check: Substitute x = 63: (63/21) × (63/189) = 3 × 1/3 = 1. The identity holds exactly. Why Other Options Are Wrong: 21 gives product 1/3; 147 gives 7/3; 3969 or 84 produce values far from 1. Only 63 satisfies the equation. Common Pitfalls: Not combining the fractions first; taking x^2 = 210 or 189 by mistake; forgetting to verify by substitution. Final Answer: 63

Question 3

Divisibility adjustment — What is the least number that must be subtracted from 13601 so that the result is exactly divisible by 87? Compute the minimal subtraction using the remainder concept.

Accepted Answer

Introduction / Context: This problem tests fast divisibility reasoning. Instead of performing long division fully, you can use the idea of remainders: if a number leaves a remainder r when divided by a divisor d, then subtracting r creates a number that is exactly divisible by d. The goal is to find the smallest subtraction that achieves divisibility by 87 for the number 13601. Given Data / Assumptions: Dividend under consideration: 13601. Divisor: 87. We seek the least non-negative integer k such that (13601 - k) is divisible by 87. Concept / Approach: By the remainder theorem of division: if N = 87 * q + r with 0 ≤ r < 87, then N - r = 87 * q is exactly divisible by 87. Therefore the minimal subtraction is simply the remainder r when 13601 is divided by 87. No larger subtraction is needed because r is, by definition, the smallest non-negative offset that lands on a multiple of 87. Step-by-Step Solution: Divide 13601 by 87 to obtain the remainder r.Compute r = 13601 mod 87.The computation yields r = 29.Therefore, subtract k = 29: 13601 - 29 = 13572, which is divisible by 87. Verification / Alternative check: Check divisibility directly: 13572 / 87 = 156 exactly, so the subtraction 29 is correct and minimal. Any smaller positive subtraction would leave a non-zero remainder. Why Other Options Are Wrong: 49 / 23 / 31 / 58: These do not equal the true remainder 29. Subtracting any of these does not yield an exact multiple of 87 or is not the least such subtraction. Common Pitfalls: Subtracting the next multiple difference in the wrong direction; mixing up the idea of “least addition” versus “least subtraction”; performing unnecessary long division without focusing on the remainder. Final Answer: 29

Question 4

Solve for the missing radicand — In the equation 392 / sqrt(x) = 28, determine the value of x. Show each manipulation of the equation clearly.

Accepted Answer

Introduction / Context: This question checks your fluency with algebraic manipulation involving square roots and simple fractions. The structure 392 / sqrt(x) = 28 invites you to isolate the radical term and then square appropriately to solve for x without introducing extraneous solutions. Given Data / Assumptions: Equation: 392 / sqrt(x) = 28. sqrt(x) denotes the principal (non-negative) square root. x is positive so that sqrt(x) is defined in real numbers. Concept / Approach: Isolate sqrt(x) by multiplying both sides by sqrt(x) and dividing by 28, or equivalently by cross-multiplication. Once sqrt(x) is expressed as a number, square both sides to remove the radical and obtain x. Always verify by substituting back into the original equation to avoid sign or arithmetic slips. Step-by-Step Solution: Start with 392 / sqrt(x) = 28.Rearrange: sqrt(x) = 392 / 28.Compute 392 / 28 = 14.Square both sides: x = 14^2 = 196. Verification / Alternative check: Substitute x = 196: sqrt(196) = 14, and 392 / 14 = 28. The original equation is satisfied exactly, confirming the solution. Why Other Options Are Wrong: 144: Would imply sqrt(x) = 12, giving 392 / 12 = 32.666..., not 28. 24 / 48: These are not valid values of x in this context; they would make sqrt(x) non-integer and yield a non-28 quotient. 784: sqrt(784) = 28 ⇒ 392 / 28 = 14, not 28. Common Pitfalls: Squaring the entire original fraction prematurely; inverting 392/28 incorrectly; forgetting that sqrt(x) must be non-negative in the real-number context. Final Answer: 196

Question 5

Prime numbers — Compute the sum of the first four prime numbers (in increasing order). Show the primes you add to avoid mistakes.

Accepted Answer

Introduction / Context: This checks recognition of prime numbers and accurate addition. Primes are integers greater than 1 with exactly two distinct positive divisors: 1 and the number itself. Listing the earliest primes and adding them carefully produces the correct sum quickly. Given Data / Assumptions: We need the first four prime numbers in ascending order. Definition of prime excludes 1 and includes 2 as the only even prime. Perform a direct sum once identified. Concept / Approach: Identify the primes: 2, 3, 5, 7 are the smallest four primes. Then add them explicitly. Writing the list prevents the common error of including 1 or skipping 7 by mistake. Ensuring fundamental definitions avoids traps in multiple-choice settings. Step-by-Step Solution: List primes: 2, 3, 5, 7.Add: 2 + 3 = 5.Add next: 5 + 5 = 10.Add last: 10 + 7 = 17. Verification / Alternative check: Double-check primality: each of 2, 3, 5, 7 has no positive divisors other than 1 and itself. Re-sum quickly to confirm 17. Why Other Options Are Wrong: 10 / 11 / 16 / 19: Each corresponds to a mis-addition or an incorrect set of early primes (for example, accidentally including 1 or omitting 7). Common Pitfalls: Counting 1 as a prime; forgetting that 2 is prime; adding too fast and arriving at 16 instead of 17. Final Answer: 17

Question 6

Prime identification — Which is the first (smallest) prime number according to the standard definition of primes?

Accepted Answer

Introduction / Context: Understanding the definition of a prime number is foundational. A prime is an integer greater than 1 that has exactly two distinct positive divisors: 1 and the number itself. This simple definition immediately identifies the smallest prime and avoids common misconceptions found in early number theory questions. Given Data / Assumptions: A prime must be greater than 1. It must have exactly two distinct positive divisors. We are to choose the smallest number fulfilling these conditions. Concept / Approach: Check the candidates in increasing order: 0, 1, 2, 3, 5. Values 0 and 1 do not satisfy the prime definition. The number 2 is divisible only by 1 and 2, and is thus the smallest prime. The next prime is 3; however, the question asks for the first prime only, which is 2. Step-by-Step Solution: Evaluate 0: has infinitely many divisors; not prime.Evaluate 1: has only one positive divisor (1); not prime.Evaluate 2: divisors are 1 and 2; prime.Therefore the first prime is 2. Verification / Alternative check: List small primes: 2, 3, 5, 7, 11. The smallest is 2, confirming the choice. Why Other Options Are Wrong: 0 and 1: Do not meet prime criteria. 3 and 5: These are primes, but not the first; they are larger than 2. Common Pitfalls: Thinking 1 is prime; forgetting that 2 is the only even prime; confusing definitions that apply to composites and primes. Final Answer: 2

Question 7

Proportion solving — In the proportion 10 / 11 = 110 / x, find the value of x using cross-multiplication for exact equality of ratios.

Accepted Answer

Introduction / Context: Proportions equate two ratios. The standard method for finding a missing term is cross-multiplication, which preserves equality when all terms are non-zero. This is routinely tested in aptitude arithmetic for speed and accuracy. Given Data / Assumptions: Proportion: 10 / 11 = 110 / x. All quantities are positive and non-zero. We solve for x. Concept / Approach: Use cross-multiplication: (10) * (x) = (11) * (110). Then isolate x by dividing both sides by 10. This direct manipulation is both quick and reliable. Finally, verify by substituting x back to confirm equal ratios. Step-by-Step Solution: Apply cross-multiplication: 10 * x = 11 * 110.Compute right-hand side: 11 * 110 = 1210.Solve for x: x = 1210 / 10 = 121.Therefore, x = 121. Verification / Alternative check: Replace x with 121: 10 / 11 = 0.90909..., and 110 / 121 = 0.90909..., confirming the proportion holds exactly. Why Other Options Are Wrong: 111 / 1100 / 100 / 11: None satisfy the equality 10 / 11 = 110 / x when substituted; the ratios differ slightly or substantially. Common Pitfalls: Swapping numerator and denominator during cross-multiplication; dividing the wrong side; arithmetic slips when multiplying 11 by 110. Final Answer: 121

Question 8

Guaranteed divisibility — The product of three consecutive even natural numbers is always divisible by what largest natural number? Justify using factor patterns.

Accepted Answer

Introduction / Context: Problems about guaranteed divisibility rely on number patterns. For three consecutive even numbers, certain prime powers and factors must appear in their product regardless of where the sequence starts. The question asks for the largest natural number that always divides such a product. Given Data / Assumptions: Let the consecutive even numbers be 2n, 2n + 2, 2n + 4 for some integer n. We analyze the product P = (2n)(2n + 2)(2n + 4). We seek the largest fixed divisor valid for all integers n. Concept / Approach: Factor out common elements and inspect guaranteed multiples. Note that P = 2 * 2 * 2 * n * (n + 1) * (n + 2) = 8 * n * (n + 1) * (n + 2). Among n, n + 1, and n + 2, there is always exactly one multiple of 3. Also, at least one of these three consecutive integers is even, contributing at least one extra factor 2 beyond the 8 already factored out. Therefore, the product is always divisible by 8 * 2 * 3 = 48. Step-by-Step Solution: Write P = 8 * n * (n + 1) * (n + 2).Among three consecutive integers, one is a multiple of 3 → factor 3 present.Among these three, at least one is even → at least one extra factor 2 beyond the initial 8.Therefore guaranteed factor = 8 * 2 * 3 = 48. Verification / Alternative check: Test small cases: (2, 4, 6) product = 48; (4, 6, 8) product = 192, which is 48 * 4; (6, 8, 10) product = 480 = 48 * 10. Each case is divisible by 48, supporting the general proof. Why Other Options Are Wrong: 16 / 24: Too small; they always divide the product, but larger guaranteed factors exist. 96: Not always guaranteed (counterexample 2 * 4 * 6 = 48). 12: Also divides, but not the largest guaranteed divisor. Common Pitfalls: Overlooking the extra factor of 2 from n(n + 1)(n + 2); assuming a fixed multiple of 4 inside n(n + 1)(n + 2) when it is not guaranteed for all n. Final Answer: 48

Question 9

Reverse division with conditions — In a division, the divisor equals 12 times the quotient and 5 times the remainder. If the remainder is 48, find the dividend using the division identity.

Accepted Answer

Introduction / Context: Division problems with constraints on the divisor, quotient, and remainder are classic applications of the division identity: dividend = divisor * quotient + remainder. Here, you are given relationships between divisor, quotient, and remainder and must compute the dividend directly and cleanly. Given Data / Assumptions: Let divisor = d, quotient = q, remainder = r. Given: d = 12 * q and d = 5 * r. Given remainder r = 48. Standard condition: 0 ≤ r < d. Concept / Approach: Use the given relationships to resolve d and q. From d = 5 * r with r = 48, get d. Then from d = 12 * q, obtain q. Finally, use dividend = d * q + r to compute the desired number. This sequence avoids guesswork and stays faithful to the constraints. Step-by-Step Solution: From d = 5 * r and r = 48, compute d = 5 * 48 = 240.From d = 12 * q, compute q = d / 12 = 240 / 12 = 20.Apply the division identity: dividend = d * q + r = 240 * 20 + 48 = 4800 + 48 = 4848.Thus, the dividend is 4848. Verification / Alternative check: Divide 4848 by 240: 4848 = 240 * 20 + 48, so quotient 20 and remainder 48. The relationships d = 12 * q (240 = 12 * 20) and d = 5 * r (240 = 5 * 48) hold perfectly. Why Other Options Are Wrong: 240 / 576 / 4800 / 5280: These do not satisfy dividend = d * q + r under the given constraints or correspond to intermediate values (divisor/product) rather than the actual dividend. Common Pitfalls: Confusing which variable is 12 times which; forgetting to add the remainder after the product; violating the condition r < d. Final Answer: 4848

Question 10

Quadratic from a product — A number is multiplied by three-fourths of itself and the result is 10800. Find the number, showing how the quadratic simplifies cleanly.

Accepted Answer

Introduction / Context: Phrases like “multiplied by three-fourths of itself” translate into simple algebraic equations. Interpreting the statement accurately as a product of the number and a fraction of itself leads to a quadratic equation that is straightforward to solve by isolating the square term. Given Data / Assumptions: Let the unknown number be n. Equation: n * (3/4 * n) = 10800. We seek the positive real solution appropriate for a “number” in standard aptitude context. Concept / Approach: Combine factors: n * (3/4 * n) = (3/4) * n^2. Isolate n^2 by multiplying both sides by 4/3. Then take the principal square root to find n. Finally, verify by substitution to ensure no arithmetic errors occurred during scaling. Step-by-Step Solution: Write equation: (3/4) * n^2 = 10800.Multiply both sides by 4/3: n^2 = 10800 * (4/3) = 14400.Take square root: n = sqrt(14400) = 120 (positive branch).Therefore, the number is 120. Verification / Alternative check: Compute three-fourths of 120: (3/4) * 120 = 90. Then n * that = 120 * 90 = 10800, which matches the given result exactly. Why Other Options Are Wrong: 210 / 180 / 160 / 240: Substituting any of these into (3/4) * n^2 does not yield 10800. Only n = 120 satisfies the equation precisely. Common Pitfalls: Misreading “three-fourths of itself” as “n + 3/4” times n; forgetting to multiply both sides by 4/3; taking the wrong square root magnitude. Final Answer: 120

Question 11

Consecutive odd integers — The product of two consecutive odd numbers is 19043. Determine the smaller odd integer, using a neat square-approximation check.

Accepted Answer

Introduction / Context: Products of consecutive odd integers can be solved either by forming a quadratic or by using a square-approximation strategy. Since consecutive odd numbers are centered around their average, comparing the target product to a nearby perfect square quickly leads to the correct pair without heavy computation. Given Data / Assumptions: Let the two consecutive odd numbers be x and x + 2. Product: x(x + 2) = 19043. We seek the smaller integer x. Concept / Approach: Form the quadratic: x^2 + 2x - 19043 = 0. Alternatively, note that for consecutive odd numbers, the product is close to the square of the middle number (x + 1). Compute sqrt(19043) to locate the vicinity and test the nearest odd pair. This avoids full quadratic solving in most test settings. Step-by-Step Solution: Estimate sqrt(19043) ≈ 138 (since 138^2 = 19044).Consecutive odd numbers around this are 137 and 139.Compute 137 * 139 = 19043.Therefore, the smaller odd integer is 137. Verification / Alternative check: Quadratic method: x^2 + 2x - 19043 = 0 has integer roots at x = 137 and x = -139 (discard negative for this context). The relevant consecutive odd pair is (137, 139). Why Other Options Are Wrong: 131 / 133 / 129 / 135: Their paired products with the next odd number are far from 19043 and do not match the target. Common Pitfalls: Testing even numbers; forgetting that the numbers differ by 2; approximating the square root too loosely and not checking the nearest odd pair. Final Answer: 137

Question 12

Fractional difference — If three-fourths of a number is subtracted from the number, the result is 163. Find the original number clearly and verify.

Accepted Answer

Introduction / Context: Linear equations with fractional coefficients appear frequently in aptitude tests. This problem states that removing three-fourths of a number from itself leaves 163. Recognizing that the remaining fraction is one-fourth leads to an immediate and accurate solution. Given Data / Assumptions: Let the number be n. Statement: n - (3/4) * n = 163. All arithmetic is over real numbers; we expect a positive integer result. Concept / Approach: Combine like terms on the left. Since n - (3/4) * n = (1/4) * n, the equation becomes a one-step calculation after isolating n by multiplying both sides by 4. This eliminates fractions and yields the answer directly. Always confirm with a quick substitution at the end. Step-by-Step Solution: Start with n - (3/4) * n = 163.Compute left-hand side: (1/4) * n = 163.Multiply both sides by 4: n = 163 * 4 = 652.Therefore, the original number is 652. Verification / Alternative check: Compute three-fourths of 652: (3/4) * 652 = 489. Then n - (3/4) * n = 652 - 489 = 163, exactly matching the condition. Why Other Options Are Wrong: 625 / 562 / 632 / 544: Substituting any of these values into n - (3/4) * n does not yield 163; they produce different results. Common Pitfalls: Misreading “three-fourths” as “three-fourth” of something else; forgetting that n - (3/4) * n equals (1/4) * n; arithmetic slips when multiplying 163 by 4. Final Answer: 652

Question 13

A positive number x satisfies: when x is increased by 10, the result equals 200 times its reciprocal (that is, x + 10 = 200 * (1 / x)). What is the value of x?

Accepted Answer

Introduction / Context: This problem checks comfort with translating a verbal condition into an algebraic equation involving a reciprocal. Once the equation is set, solving a simple quadratic yields the positive value for the number. Such questions are common in aptitude tests and reinforce manipulation of fractions and quadratics. Given Data / Assumptions: The number is positive and denoted by x. Condition: x + 10 = 200 * (1 / x). We seek the positive solution only, because the prompt specifies a positive number. Concept / Approach: Clear the denominator to avoid fractions, then bring terms to one side and factor or use the quadratic formula. The reciprocal relation typically produces a quadratic in x. Choose the root that satisfies the positivity constraint. Step-by-Step Solution: Given x + 10 = 200 * (1 / x).Multiply both sides by x: x^2 + 10x = 200.Rearrange: x^2 + 10x − 200 = 0.Solve the quadratic: Discriminant D = 10^2 + 4 * 200 = 100 + 800 = 900, so sqrt(D) = 30.Roots: x = (−10 ± 30) / 2 → x = 10 or x = −20.Since x is positive, select x = 10. Verification / Alternative check: Check the condition: 10 + 10 = 20 and 200 * (1 / 10) = 20. Both sides match, confirming the result. Why Other Options Are Wrong: Values 100, 20, 200, and 5 do not satisfy x + 10 = 200 / x when substituted. Common Pitfalls: Forgetting to multiply through by x leads to errors with the reciprocal. Selecting the negative root violates the stated condition that the number is positive. Final Answer: 10

Question 14

Two numbers have a sum of 10 and a product of 20. What is the value of the sum of their reciprocals, that is, (1 / a) + (1 / b)?

Accepted Answer

Introduction / Context: The task is to compute the sum of reciprocals of two numbers when their sum and product are known. This is a standard identity application in algebra and aptitude tests, allowing a quick computation without explicitly finding the two numbers themselves. Given Data / Assumptions: a + b = 10. ab = 20. We need (1 / a) + (1 / b). Concept / Approach: Use the identity (1 / a) + (1 / b) = (a + b) / (ab). This follows from putting the two fractions over a common denominator ab. Then, insert the given sum and product directly into the identity to compute a concise result. Step-by-Step Solution: Start with (1 / a) + (1 / b) = (a + b) / (ab).Substitute given values: (a + b) / (ab) = 10 / 20.Compute the quotient: 10 / 20 = 1/2.Therefore, the sum of reciprocals is 1/2. Verification / Alternative check: If desired, solve t^2 − 10t + 20 = 0 for actual numbers; they will yield a + b = 10 and ab = 20, and their reciprocals will still sum to 1/2 by the same identity. Why Other Options Are Wrong: 1, 3/5, 11/6, and 3/2 do not equal (a + b) / (ab) with the given values. They arise from arithmetic slips or incorrect identities. Common Pitfalls: Attempting to find each number individually is unnecessary and can introduce errors. Forgetting the identity or inverting the ratio (ab)/(a + b) is another common mistake. Final Answer: 1/2

Question 15

Let n be a positive integer such that five times n equals three less than twice the square of n (that is, 5n = 2n^2 − 3). What is the value of n?

Accepted Answer

Introduction / Context: Here we turn a verbal relation into a quadratic equation in a positive integer. Recognizing and solving the quadratic cleanly, then applying the positivity constraint, identifies the valid integer solution. This is a direct test of forming and solving quadratic equations. Given Data / Assumptions: n is a positive integer. Equation: 5n = 2n^2 − 3. We need the value of n that satisfies the equation. Concept / Approach: Rearrange the equation into standard quadratic form an^2 + bn + c = 0, then factor or use the quadratic formula. Discard any non-integer or negative solution that does not meet the stated condition on n. Step-by-Step Solution: Move all terms to one side: 2n^2 − 5n − 3 = 0.Compute discriminant: D = (−5)^2 − 4 * 2 * (−3) = 25 + 24 = 49.Square root of D: sqrt(49) = 7.Solutions: n = (5 ± 7) / (2 * 2) = (5 ± 7) / 4.Hence n = 12 / 4 = 3 or n = −2 / 4 = −1/2.Since n is a positive integer, n = 3. Verification / Alternative check: Check: Left side 5n = 15; right side 2n^2 − 3 = 2 * 9 − 3 = 15. The equality holds. Why Other Options Are Wrong: 13, 23, and 33 do not satisfy the equation. 6 does not satisfy it either when substituted. Common Pitfalls: Sign errors when moving terms, miscomputing the discriminant, or accepting a negative or fractional root despite the integer requirement. Final Answer: 3

Question 16

A person distributes property as follows: one-fourth to the daughter, one-half to the sons, and one-fifth to charity. What fraction of the property has been given away in total?

Accepted Answer

Introduction / Context: This is a fraction addition problem with different denominators. It tests the ability to find a common denominator, sum the parts, and interpret the result as the total fraction distributed. Such distribution problems appear routinely in aptitude and arithmetic reasoning tests. Given Data / Assumptions: Fractions to sum: 1/4, 1/2, and 1/5. Each fraction represents a portion of the whole property. We assume the whole is 1. Concept / Approach: Find the least common denominator (LCD) of 4, 2, and 5, which is 20. Convert each fraction to twentieths and add the numerators. The resulting fraction is the total given away. The remaining part, if needed, would be 1 minus this sum. Step-by-Step Solution: Convert to twentieths: 1/4 = 5/20, 1/2 = 10/20, 1/5 = 4/20.Add the numerators: 5 + 10 + 4 = 19.Total distributed = 19/20 of the property.Therefore, the person has given away 19/20. Verification / Alternative check: Compute the remainder: 1 − 19/20 = 1/20. This means 1/20 is left, which is consistent and less than the whole. Why Other Options Are Wrong: 1/20 represents the remainder, not the given total. 9/10 and 17/20 undercount the distribution. 1/10 is much too small and reflects an arithmetic error. Common Pitfalls: Using 10 instead of 20 as the LCD, forgetting to convert 1/5 properly, or adding denominators directly which is incorrect. Final Answer: 19/20

Question 17

A chocolate bar has 12 equal pieces. Manju gives 1/4 of the bar to Anju, 1/3 of the bar to Sujata, and 1/6 of the bar to Fiza. How many pieces of chocolate are left with Manju?

Accepted Answer

Introduction / Context: This problem combines fraction operations with a concrete total number of items. By summing the fractional parts given away and applying the result to the total piece count, we can quickly compute the remaining pieces. Given Data / Assumptions: Total pieces = 12. Given away: 1/4 + 1/3 + 1/6 of the entire bar. All pieces are equal and divisible according to the stated fractions. Concept / Approach: First, sum the fractions to find the total fraction given. Then, subtract from 1 to find the remaining fraction. Finally, multiply the remaining fraction by 12 to get the number of pieces left. Using a common denominator streamlines the addition. Step-by-Step Solution: Find a common denominator for 1/4, 1/3, and 1/6. The least common denominator is 12.Convert: 1/4 = 3/12, 1/3 = 4/12, 1/6 = 2/12.Add: 3/12 + 4/12 + 2/12 = 9/12 = 3/4.Remaining fraction = 1 − 3/4 = 1/4.Remaining pieces = 1/4 of 12 = 3 pieces. Verification / Alternative check: Compute given pieces directly: 1/4 of 12 = 3, 1/3 of 12 = 4, 1/6 of 12 = 2. Total given = 3 + 4 + 2 = 9. Leftover = 12 − 9 = 3. Matches the fraction method. Why Other Options Are Wrong: 1, 2, 4, and 5 do not match the leftover count when the fractions are correctly applied to 12. Common Pitfalls: Adding denominators directly or using an incorrect common denominator; misreading the question as adding fractional pieces rather than fractions of the whole bar. Final Answer: 3

Question 18

A student was supposed to multiply a number by 3/2 but instead divided the number by 3/2 (effectively multiplying by 2/3). His incorrect result was 10 less than the correct result. What was the original number?

Accepted Answer

Introduction / Context: This classic error-analysis problem compares a correct operation with an incorrect reciprocal operation. By expressing both results in terms of the original number, the difference becomes a simple linear equation that reveals the original value. Given Data / Assumptions: Original number = N. Correct result = N * (3/2). Incorrect result = N / (3/2) = N * (2/3). Correct result minus incorrect result = 10. Concept / Approach: Write the difference explicitly and solve for N. Because both expressions are proportional to N, we simply subtract the coefficients and divide to isolate N. Careful fraction arithmetic is essential to avoid sign or factor mistakes. Step-by-Step Solution: Compute difference: N * (3/2) − N * (2/3) = 10.Factor N: N * (3/2 − 2/3) = 10.Find the bracket: 3/2 − 2/3 = (9/6 − 4/6) = 5/6.So (5/6) * N = 10, hence N = 10 * (6/5) = 12. Verification / Alternative check: Correct result: 12 * (3/2) = 18. Incorrect result: 12 * (2/3) = 8. Difference = 18 − 8 = 10, which matches the condition. Why Other Options Are Wrong: 10, 15, 20, and 24 do not produce a difference of 10 between the correct and erroneous operations. Common Pitfalls: Dividing by 3/2 twice, mixing up which result is larger, or computing 3/2 − 2/3 incorrectly by adding denominators. Final Answer: 12

Question 19

Find the number N such that (square root of N) divided by 19 equals 4; that is, sqrt(N) / 19 = 4. Determine the value of N.

Accepted Answer

Introduction / Context: This item assesses comfort with undoing a square root relation presented in a fractional form. By isolating the square root term and then squaring both sides, we recover the original number. It is a straightforward arithmetic manipulation once the equation is read correctly. Given Data / Assumptions: We are given sqrt(N) / 19 = 4. N is nonnegative since a real square root is implied. We need to find N. Concept / Approach: Multiply both sides by 19 to isolate sqrt(N), then square both sides to eliminate the square root. This yields N directly. Always check by substitution to avoid errors from misreading the original expression. Step-by-Step Solution: Given sqrt(N) / 19 = 4.Multiply both sides by 19: sqrt(N) = 4 * 19 = 76.Square both sides: N = 76^2.Compute 76^2 = (70 + 6)^2 = 70^2 + 2 * 70 * 6 + 6^2 = 4900 + 840 + 36 = 5776. Verification / Alternative check: Check: sqrt(5776) = 76 and 76 / 19 = 4. The original relation holds exactly. Why Other Options Are Wrong: 76 is the square root, not N. 304 and 1296 do not satisfy the equation. 361 is 19^2, which would make sqrt(N) = 19, yielding 1 when divided by 19, not 4. Common Pitfalls: Confusing N with sqrt(N), or squaring 19 incorrectly. Some may multiply 4 * 19 and stop without squaring, which is incomplete. Final Answer: 5776

Question 20

Evaluate the missing value: 4050 divided by sqrt(N) equals 450. That is, 4050 / sqrt(N) = 450. Find N.

Accepted Answer

Introduction / Context: This question involves solving a simple equation with a square root in the denominator. The strategy is to isolate the square root, compute its value, and then square to obtain N. It checks manipulation of radicals and proportional reasoning. Given Data / Assumptions: Equation: 4050 / sqrt(N) = 450. N is positive to make sqrt(N) real and nonzero. We seek N. Concept / Approach: Cross-multiply to isolate sqrt(N). Once sqrt(N) is expressed numerically, square to find N. Verify by substituting back to the original equation to ensure no arithmetic slip occurred. Step-by-Step Solution: Multiply both sides by sqrt(N): 4050 = 450 * sqrt(N).Divide both sides by 450: sqrt(N) = 4050 / 450 = 9.Square both sides: N = 9^2 = 81. Verification / Alternative check: Substitute: 4050 / sqrt(81) = 4050 / 9 = 450. This matches the given relation. Why Other Options Are Wrong: 49, 100, 9, and 36 produce sqrt(N) values that do not yield 450 when dividing 4050. Common Pitfalls: Computing 4050 / 450 incorrectly or forgetting to square 9 to get N. Mixing up whether the square root is in the numerator or denominator can also cause errors. Final Answer: 81

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