Question 1

Sum and difference of squares — The sum of two numbers is 29 and the difference of their squares is 145. What is the difference between the numbers?

Accepted Answer

Introduction / Context:This problem is tailor-made for the difference-of-squares identity. Instead of determining each number, you can infer the difference directly from the product of sum and difference equal to the difference of squares. Given Data / Assumptions: a + b = 29. a^2 - b^2 = 145. We need a - b. Concept / Approach:Recall that a^2 - b^2 = (a - b)(a + b). With the sum a + b known, divide the given difference of squares by the sum to get a - b directly. This avoids extra algebra and speeds up the solution. Step-by-Step Solution:Use identity: a^2 - b^2 = (a - b)(a + b).Substitute numbers: 145 = (a - b) * 29.Compute (a - b) = 145 / 29 = 5.Therefore, the difference between the numbers is 5. Verification / Alternative check:If desired, solve for a and b: a = (29 + 5)/2 = 17, b = (29 - 5)/2 = 12. Then 17^2 - 12^2 = 289 - 144 = 145, confirming the calculation. Why Other Options Are Wrong: 13/8/11/7: None equals 145 / 29; only 5 makes the identity hold. Common Pitfalls:Adding or subtracting the given numbers prematurely; forgetting to divide by the sum; arithmetic errors when dividing 145 by 29 (remember 29 * 5 = 145). Final Answer:5

Question 2

Using identities — The difference of two numbers is 5 and the difference of their squares is 135. What is the sum of the numbers?

Accepted Answer

Introduction / Context:Another direct application of the difference-of-squares identity. When you know a - b and a^2 - b^2, you can immediately compute a + b by rearranging the identity without solving for a and b individually. Given Data / Assumptions: a - b = 5. a^2 - b^2 = 135. We need a + b. Concept / Approach:Use a^2 - b^2 = (a - b)(a + b). Solve for a + b by dividing the known difference of squares by the known difference of the numbers. This is a fast and reliable technique in timed exams. Step-by-Step Solution:Identity: a^2 - b^2 = (a - b)(a + b).Substitute: 135 = 5 * (a + b).Compute a + b = 135 / 5 = 27.Hence, the sum is 27. Verification / Alternative check:Optionally solve: Let a + b = 27 and a - b = 5 → a = 16, b = 11. Then a^2 - b^2 = 256 - 121 = 135, confirming the result. Why Other Options Are Wrong: 25/30/32/35: None equals 135 / 5; these values do not satisfy the identity with the given numbers. Common Pitfalls:Multiplying instead of dividing by the known difference; mixing up (a + b) with (a - b); making arithmetic slips when dividing 135 by 5. Final Answer:27

Question 3

Find the fraction (in lowest terms): If 1 is added to the denominator of a fraction, the value becomes 1/2; and if 1 is added to the numerator, the value becomes exactly 1. What is the original fraction?

Accepted Answer

Introduction / Context:This problem asks you to determine a rational number (a fraction) from two simple transformations. When 1 is added to the denominator, the fraction equals 1/2. When 1 is added to the numerator, the fraction equals 1. Translating each verbal condition into an equation and solving the resulting system is a classic aptitude skill. Given Data / Assumptions: The fraction is x / y with x and y as positive integers and y ≠ 0. Condition A: x / (y + 1) = 1/2. Condition B: (x + 1) / y = 1. We seek the simplest form of the fraction. Concept / Approach:Convert each sentence into an algebraic equation, then solve simultaneously. From Condition B, (x + 1) / y = 1 implies y = x + 1. Substitute this into Condition A to find x. Once x is known, compute y and write the fraction x / y in lowest terms. Step-by-Step Solution: From (x + 1) / y = 1, obtain y = x + 1.Use x / (y + 1) = 1/2. Substitute y = x + 1 to get x / (x + 2) = 1/2.Cross-multiply: 2x = x + 2, which gives x = 2.Then y = x + 1 = 3, so the original fraction is 2/3. Verification / Alternative check:Add 1 to the denominator: 2 / (3 + 1) = 2/4 = 1/2 (matches). Add 1 to the numerator: (2 + 1) / 3 = 3/3 = 1 (matches). Everything is consistent. Why Other Options Are Wrong:4/7, 5/9, 10/11, and 3/5 do not satisfy both conditions; testing either condition fails for each distractor. Common Pitfalls:Mixing up which operation (adding 1) applies to the numerator versus the denominator; forgetting to substitute y = x + 1 correctly; or reducing too early and losing track of variables. Final Answer:2/3

Question 4

Solve for the number: Four-fifths (4/5) of a number exceeds two-thirds (2/3) of the same number by 8. What is the number?

Accepted Answer

Introduction / Context:Many aptitude questions compare two fractional parts of the same unknown and provide the difference between them. The key is to set up a linear equation using the difference of fractions and solve cleanly. Here, 4/5 of a number is greater than 2/3 of that number by 8 units. Given Data / Assumptions: Let the number be n (real and positive). (4/5)n − (2/3)n = 8. We are to find n. Concept / Approach:Compute the difference of the two fractions as a single fraction of n, then equate to 8. Solving for n requires only basic fraction arithmetic and cross-multiplication. Using least common multiples for denominators speeds up the calculation. Step-by-Step Solution: Compute the coefficient: 4/5 − 2/3 = (12/15 − 10/15) = 2/15.Set up the equation: (2/15) * n = 8.Solve for n: n = 8 * (15/2) = 60.Therefore, the required number is 60. Verification / Alternative check:Check directly: 4/5 of 60 is 48; 2/3 of 60 is 40; 48 − 40 = 8. The condition is satisfied exactly. Why Other Options Are Wrong:30, 90, and 48 do not satisfy the given difference when substituted. “None of these” is unnecessary since a valid solution exists. Common Pitfalls:Subtracting the fractions incorrectly (e.g., 4/5 − 2/3 = 2/5, which is wrong); forgetting to multiply by the reciprocal when isolating n. Final Answer:60

Question 5

If one-fourth of one-third of one-half of a number equals 15, what is the original number?

Accepted Answer

Introduction / Context:This problem is a straightforward application of multiplying fractions to represent successive parts of a number. It assesses your ability to translate nested fractional statements into a single fraction and then invert that operation to recover the original number. Given Data / Assumptions: Let the number be N. (1/4) * (1/3) * (1/2) * N = 15. All quantities are real; N is positive by context. Concept / Approach:Multiply the fractions to get a single factor applied to N. Then solve for N by dividing 15 by that factor (equivalently multiplying by its reciprocal). Order does not matter because multiplication is associative and commutative. Step-by-Step Solution: Combine fractions: (1/4) * (1/3) * (1/2) = 1 / (4 * 3 * 2) = 1/24.Set up: (1/24) * N = 15.Solve for N: N = 15 * 24 = 360.Hence, the number is 360. Verification / Alternative check:Compute backward: half of 360 is 180; one-third of 180 is 60; one-fourth of 60 is 15. The chain matches the given statement exactly. Why Other Options Are Wrong:72, 120, 180, and 240 each produce a final value different from 15 when the same fractional steps are applied. Common Pitfalls:Adding the fractions instead of multiplying, or taking the sequence in reverse without checking each stage. Final Answer:360

Question 6

A number satisfies: its one-fifth increased by 5 equals its one-fourth decreased by 5. What is the number?

Accepted Answer

Introduction / Context:Here you are given a linear relationship between two different fractions of the same unknown number, with constant adjustments. Problems like this are common in arithmetic reasoning and require careful setup of a simple equation and accurate fraction manipulation. Given Data / Assumptions: Let the number be N. N/5 + 5 = N/4 − 5. N is real and positive. Concept / Approach:Isolate N by moving all N terms to one side and constants to the other. The coefficient of N will involve the difference between 1/4 and 1/5. Solve by multiplying through by the common denominator to avoid fractional clutter. Step-by-Step Solution: Start: N/5 + 5 = N/4 − 5.Bring N terms to the right and constants to the left: 5 + 5 = N/4 − N/5.Compute right side: N * (1/4 − 1/5) = N * (5/20 − 4/20) = N/20.So 10 = N/20 → N = 10 * 20 = 200. Verification / Alternative check:Check: one-fifth of 200 is 40; 40 + 5 = 45. One-fourth of 200 is 50; 50 − 5 = 45. Both sides equal 45, confirming correctness. Why Other Options Are Wrong:160, 180, 220, and 240 do not satisfy both sides simultaneously when substituted into the equation. Common Pitfalls:Subtracting the wrong way around (1/5 − 1/4) or forgetting to move constants to the other side before combining fractional coefficients. Final Answer:200

Question 7

Two numbers have product 120 and the sum of their squares is 289. Find the sum of the two numbers.

Accepted Answer

Introduction / Context:Instead of solving for each number individually, this problem invites you to use algebraic identities that connect sums, products, and sums of squares. Recognizing and applying these identities leads to a direct solution for the required sum. Given Data / Assumptions: Let the numbers be a and b. ab = 120. a^2 + b^2 = 289. We need S = a + b. Concept / Approach:Use the identity (a + b)^2 = a^2 + b^2 + 2ab. This provides a direct route to the square of the sum without finding a or b individually. Take the positive square root for the sum since a + b is nonnegative in typical aptitude contexts (and options reflect a positive result). Step-by-Step Solution: Compute S^2 = (a + b)^2 = (a^2 + b^2) + 2ab.Substitute values: S^2 = 289 + 2 * 120 = 289 + 240 = 529.Take the square root: S = sqrt(529) = 23.Therefore, the sum of the two numbers is 23. Verification / Alternative check:If desired, solve the quadratic t^2 − 23t + 120 = 0 to find potential pairs (t represents either number). The roots multiply to 120 and sum to 23, confirming the result. Why Other Options Are Wrong:20 and 22 do not satisfy the identity with the given product and sum of squares. 169 is not a plausible sum here; it appears to confuse the sum of squares with the sum itself. Common Pitfalls:Forgetting the 2ab term in the identity or taking the wrong square root without checking feasibility. Final Answer:23

Question 8

The sum of squares of two numbers is 80 and the square of their difference is 36. What is the product of the two numbers?

Accepted Answer

Introduction / Context:This question rewards familiarity with the identity for the square of a difference and its relationship to the product of two numbers. With two aggregate pieces of information—sum of squares and square of difference—you can extract the product directly without finding the individual numbers first. Given Data / Assumptions: a^2 + b^2 = 80. (a − b)^2 = 36. We need ab. Concept / Approach:Use (a − b)^2 = a^2 + b^2 − 2ab. Since both a^2 + b^2 and (a − b)^2 are known, rearrange to solve for ab. This approach avoids solving any quadratic for a and b explicitly. Step-by-Step Solution: Start with (a − b)^2 = a^2 + b^2 − 2ab.Substitute numbers: 36 = 80 − 2ab.Rearrange: 2ab = 80 − 36 = 44.Therefore, ab = 44 / 2 = 22. Verification / Alternative check:If needed, note that (a + b)^2 = a^2 + b^2 + 2ab = 80 + 44 = 124, so a + b = sqrt(124). This is consistent with real values for a and b; no contradiction arises. Why Other Options Are Wrong:44 is 2ab, not ab. Values 58 and 116 do not align with the given identities. 18 is arbitrary here. Common Pitfalls:Stopping at 2ab = 44 and reporting 44 instead of dividing by 2; mixing up the identities for (a − b)^2 and (a + b)^2. Final Answer:22

Question 9

The sum of two numbers equals twice their difference. If one number is 10, what is the other number?

Accepted Answer

Introduction / Context:Verbal equations involving sums and differences require careful interpretation. Here, the sum of two numbers equals twice their difference. One of the numbers is given as 10; we must determine the other number consistent with this relationship. Given Data / Assumptions: Known number = 10; unknown number = t. Sum equals twice the (positive) difference, so 10 + t = 2 * |10 − t|. We consider cases depending on whether t ≥ 10 or t ≤ 10. Concept / Approach:The absolute value in the difference requires case analysis. Solve separately for t ≥ 10 (difference is t − 10) and for t ≤ 10 (difference is 10 − t), then check which solution matches the options and the original equation. Step-by-Step Solution: Case 1 (t ≥ 10): 10 + t = 2(t − 10) → 10 + t = 2t − 20 → 30 = t.Case 2 (t ≤ 10): 10 + t = 2(10 − t) → 10 + t = 20 − 2t → 3t = 10 → t = 10/3.Both algebraic solutions satisfy the equation structure, but the question expects a single result from the given options; 30 is listed and valid.Thus, the other number is 30. Verification / Alternative check:For t = 30: sum = 40; difference = 20; twice the difference = 40; equality holds. For t = 10/3, equality also holds, but that choice is not provided in the same exact form among the options (spacing aside), and typical test intent favors the integer solution. Why Other Options Are Wrong:31/ 3 (10 1/3) does not satisfy the intended integer solution pattern; 1/3 and 41/ 4 are extraneous; 20 fails the relation. Common Pitfalls:Dropping the absolute value and missing case analysis; choosing a plausible but unlisted fractional value; arithmetic slips when moving terms. Final Answer:30

Question 10

If four-fifths (4/5) of a certain number is 64, what is half of that number?

Accepted Answer

Introduction / Context:Scaling numbers using fractions is a frequent aptitude task. Given a fractional value of an unknown, you can recover the whole and then compute any other required fraction, such as one-half in this problem. Given Data / Assumptions: (4/5) * N = 64. We need (1/2) * N. N is positive. Concept / Approach:Solve for N directly by multiplying both sides by the reciprocal of 4/5, which is 5/4. Once N is known, take half to answer the question. Always keep operations symbolic to avoid arithmetic mistakes, then substitute numbers. Step-by-Step Solution: Find N: N = 64 * (5/4) = 16 * 5 = 80.Compute half: (1/2) * N = 80 / 2 = 40.Therefore, half of the number is 40. Verification / Alternative check:Check the given: (4/5) * 80 = 64; correct. The requested half is 40; consistent. Why Other Options Are Wrong:32 and 16 are smaller fractions (1/2 of 64 or 1/4 of 64) misapplied; 80 is the full number, not half; 64 merely restates the given 4/5 value. Common Pitfalls:Multiplying by 4/5 instead of its reciprocal; halving 64 directly (confusing the target with the given part). Final Answer:40

Question 11

Three-fourths of a number exceeds two-thirds of the same number by 5. What is the number?

Accepted Answer

Introduction / Context:This is another linear equation built from the difference of two fractional parts of a number. Solving requires careful fraction subtraction and then isolating the unknown. It checks comfort with least common multiples and proportional reasoning. Given Data / Assumptions: (3/4)N − (2/3)N = 5. N is real and positive. We must find N. Concept / Approach:Subtract the fractions with a common denominator, then equate the result to 5 and solve for N. Using the identity (a/b − c/d) = (ad − bc) / bd helps to avoid errors. Step-by-Step Solution: Compute the difference: 3/4 − 2/3 = (9/12 − 8/12) = 1/12.Set the equation: (1/12) * N = 5.Solve: N = 5 * 12 = 60.Therefore, the required number is 60. Verification / Alternative check:Check: (3/4)*60 = 45; (2/3)*60 = 40; 45 − 40 = 5. The condition holds perfectly. Why Other Options Are Wrong:48, 72, 84, and 36 yield differences other than 5 when substituted into the expression (3/4)N − (2/3)N. Common Pitfalls:Computing 3/4 − 2/3 incorrectly (e.g., 1/12 reversed in sign) or multiplying by the wrong reciprocal when isolating N. Final Answer:60

Question 12

What constant must be added to both numerator and denominator of 3/4 so that the resulting fraction equals 11/12?

Accepted Answer

Introduction / Context:Adjusting a fraction by adding the same constant to its numerator and denominator changes its value in a predictable way. The task is to find the constant that transforms 3/4 into 11/12. Cross-multiplication is the cleanest route to a solution. Given Data / Assumptions: We need k such that (3 + k) / (4 + k) = 11/12. k is a real number; in typical aptitude settings an integer solution is expected. Concept / Approach:Set up the proportion and cross-multiply to produce a linear equation in k. Solve for k and verify by substitution back into the fraction. This avoids any guesswork and works for any target ratio. Step-by-Step Solution: Start with (3 + k) / (4 + k) = 11/12.Cross-multiply: 12(3 + k) = 11(4 + k).Expand: 36 + 12k = 44 + 11k.Solve: k = 44 − 36 = 8. Verification / Alternative check:Substitute back: (3 + 8)/(4 + 8) = 11/12; numerically 11/12 equals 11/12. Verified. Why Other Options Are Wrong:5, 6, 7, and 9 yield (3 + k)/(4 + k) values different from 11/12 upon substitution. Common Pitfalls:Adding to only the numerator or only the denominator; arithmetic slips when cross-multiplying; or canceling terms prematurely. Final Answer:8

Question 13

Ratio and Average — Three Numbers Three numbers are in the ratio 4 : 5 : 6. Their arithmetic average is 25. Based on this information, determine the largest of the three numbers.

Accepted Answer

Introduction / Context:This problem uses the standard technique for ratio sets: represent the three numbers as multiples of a common factor and then use the average to determine that factor. It tests fluency with averages, ratios, and quick algebra—skills that are common in competitive aptitude exams. Given Data / Assumptions: The numbers are proportional to 4 : 5 : 6. Let the numbers be 4k, 5k, and 6k for some positive k. Their average (arithmetic mean) is 25. We are asked to find the largest number among the three. Concept / Approach:Average of three quantities equals (sum of the three)/3. When each number is written as a multiple of k, the sum becomes (4k + 5k + 6k) = 15k. Hence the average is (15k) / 3 = 5k. Equating 5k to the given average provides k directly, and then each term follows by simple multiplication. Step-by-Step Solution:Let the numbers be 4k, 5k, 6k.Average = (4k + 5k + 6k)/3 = 15k/3 = 5k.Set 5k = 25 → k = 5.Largest number = 6k = 6 * 5 = 30. Verification / Alternative check:Numbers are 20, 25, 30. Their average is (20 + 25 + 30)/3 = 75/3 = 25, confirming consistency with the given average. Why Other Options Are Wrong:42 and 36 correspond to incorrect k values; 32 and 28 also result from misreading the ratio or mixing average with sum. Only 30 aligns with k = 5. Common Pitfalls:Using 4 + 5 + 6 = 15 directly as the largest term; dividing 25 by 3 instead of equating it to 5k; forgetting that the largest corresponds to the largest ratio term (6k). Final Answer:30

Question 14

Consecutive Odd Integers — Sum Given The sum of three consecutive odd integers is 57. Identify the middle (second) integer among them.

Accepted Answer

Introduction / Context:Consecutive odd integers are evenly spaced by 2. When three such numbers are added, the middle one equals their average, which dramatically simplifies the calculation. This question examines your ability to translate a verbal description into a compact algebraic form and compute accurately. Given Data / Assumptions: There are three consecutive odd integers. Their sum is 57. We need the middle integer. Concept / Approach:Let the middle odd integer be n. Then the three consecutive odds are n − 2, n, and n + 2. Their sum becomes (n − 2) + n + (n + 2) = 3n. Setting 3n = 57 yields n directly. This approach avoids listing numbers and reduces the task to a single division. Step-by-Step Solution:Let the three numbers be n − 2, n, n + 2.Sum = 3n = 57.Solve for n: n = 57 / 3 = 19.Therefore, the middle integer is 19. Verification / Alternative check:List them explicitly: 17, 19, 21. Sum = 17 + 19 + 21 = 57. The middle is indeed 19. Why Other Options Are Wrong:21 and 23 are neighboring odd numbers but would yield sums 63 and 67 respectively; 17 is the smallest, not the middle; 15 lies outside the valid triple for sum 57. Common Pitfalls:Using step 1 instead of step 2 (confusing consecutive integers with consecutive odd integers); adding wrong or mixing positions (smallest vs middle). Final Answer:19

Question 15

Equation from Verbal Statement — “Dividing by 6 diminishes the number by 40” A number, when divided by 6, is diminished by 40 (i.e., the result is 40 less than the original number). Find the number.

Accepted Answer

Introduction / Context:Translating words into equations is a frequent aptitude skill. The phrase “a number when divided by 6 is diminished by 40” means the quotient equals the original number minus 40. Setting up this equation unlocks the value immediately with basic algebra. Given Data / Assumptions: Let the number be N. “Divided by 6” produces N/6. “Is diminished by 40” means equals N − 40. Concept / Approach:Form the equation N/6 = N − 40. Solve by clearing the denominator and isolating N. Always re-check by substituting back to ensure the verbal condition is satisfied exactly, avoiding misinterpretations of “diminished by.” Step-by-Step Solution:Set N/6 = N − 40.Multiply both sides by 6: N = 6N − 240.Rearrange: 5N = 240 → N = 240/5 = 48.Hence, the number is 48. Verification / Alternative check:Compute N/6 = 48/6 = 8 and N − 40 = 48 − 40 = 8. The two sides match precisely, confirming N = 48. Why Other Options Are Wrong:72, 84, 60, and 96 produce quotients and differences that do not match under the stated relationship. Common Pitfalls:Misreading the sentence as “N divided by 6 is 40 less than 6” (nonsense) or as “N divided by 6 equals 40,” which would give N = 240 but ignores “less than the original number.” Final Answer:48

Question 16

Split a Total with Reciprocal Condition Divide 50 into two positive parts such that the sum of their reciprocals equals 1/12. What are the two parts?

Accepted Answer

Introduction / Context:This is a classic two-part split constrained by a condition on reciprocals. The key is to convert the verbal condition into an algebraic equation in a single variable and solve a simple quadratic. Such questions appear frequently in number system sections of aptitude tests. Given Data / Assumptions: Total sum of the two parts is 50. Let the parts be x and 50 − x, with x > 0 and 50 − x > 0. 1/x + 1/(50 − x) = 1/12. Concept / Approach:Add the reciprocals using a common denominator and equate to 1/12. This yields a quadratic equation in x. Solve the quadratic to obtain two positive solutions, which are complementary parts summing to 50. Check both solutions for validity and verify the reciprocal condition numerically. Step-by-Step Solution:Start: 1/x + 1/(50 − x) = (50) / (x(50 − x)) = 1/12.Therefore, x(50 − x) = 600.Expand: 50x − x^2 = 600 → x^2 − 50x + 600 = 0.Solve: Discriminant D = 2500 − 2400 = 100; x = (50 ± 10)/2 → x = 20 or 30. Verification / Alternative check:Check 20 and 30: 1/20 + 1/30 = (3 + 2)/60 = 5/60 = 1/12. Condition satisfied exactly. Why Other Options Are Wrong:24, 26 give 1/24 + 1/26 ≠ 1/12; similarly 28, 22 and 36, 14 and 18, 32 do not satisfy the reciprocal sum constraint. Common Pitfalls:Algebra sign errors when forming the quadratic; forgetting both roots are valid and symmetric around 25; mixing reciprocal sum with arithmetic sum or difference. Final Answer:20, 30

Question 17

Simple Ratio — Find the Antecedent Two numbers are in the ratio 2 : 3. If the consequent (the second term) is 24, determine the antecedent (the first term).

Accepted Answer

Introduction / Context:Understanding the terms antecedent (first) and consequent (second) is fundamental in ratio questions. Once you recognize which term is fixed, you can scale the entire ratio by the same factor to find the unknown counterpart accurately and quickly. Given Data / Assumptions: Ratio of two numbers: 2 : 3. Consequent (second term) = 24. Find the antecedent (first term). Concept / Approach:Let the numbers be 2k and 3k. If 3k = 24, then k = 24/3 = 8. The antecedent is 2k = 16. This direct scaling avoids any unnecessary computations and ensures proportional consistency. Step-by-Step Solution:Represent numbers as 2k and 3k.Set 3k = 24 → k = 8.Antecedent = 2k = 2 * 8 = 16.Answer: 16. Verification / Alternative check:Check ratio: 16 : 24 reduces by dividing both by 8 to 2 : 3, confirming the correct scaling factor. Why Other Options Are Wrong:36 and 48 would correspond to wrong multipliers for the first term; 72 is 3 times 24, mixing positions; 12 corresponds to k = 6, which would make the consequent 18, not 24. Common Pitfalls:Confusing antecedent with consequent; scaling the wrong term; failing to simplify the resulting ratio to confirm correctness. Final Answer:16

Question 18

Forming and Solving a Quadratic from Words If a number is subtracted from the square of its one-half, the result is 48. What is the square root of that number?

Accepted Answer

Introduction / Context:This question translates a verbal algebraic relationship into an equation involving a square term. Careful parsing of “square of its one-half” and “subtracted from” ensures the correct order of operations. Solving the resulting quadratic yields the number; then take its square root as requested. Given Data / Assumptions: Let the number be N. Square of its one-half is (N/2)^2 = N^2/4. N^2/4 − N = 48. We need sqrt(N). Concept / Approach:Multiply both sides by 4 to clear the denominator, then solve the quadratic by the standard formula or factoring. Consider both roots, but choose the value that makes sense for the context (typically positive for a “number” unless otherwise stated). Finally, compute the square root of N as asked. Step-by-Step Solution:Start: N^2/4 − N = 48.Multiply by 4: N^2 − 4N = 192 → N^2 − 4N − 192 = 0.Discriminant: 16 + 768 = 784; sqrt = 28.N = (4 ± 28)/2 → N = 16 or N = −12.Take the sensible positive value: N = 16 → sqrt(N) = 4. Verification / Alternative check:Check N = 16: (16/2)^2 − 16 = 8^2 − 16 = 64 − 16 = 48 (works). The negative root is extraneous for a natural-number context. Why Other Options Are Wrong:5, 6, 8, 12 are not sqrt(16); they arise from arithmetic slips or misreading the phrase order. Common Pitfalls:Interpreting “square of its one-half” as (N^2)/2; subtracting in the wrong order; discarding the quadratic step too soon. Final Answer:4

Question 19

Three Consecutive Integers — Invariant Difference Given the sum of three consecutive integers (not necessarily provided explicitly), what is the constant difference between the first and the third integer in any such triple?

Accepted Answer

Introduction / Context:The question focuses on a structural property of consecutive integers. Regardless of the actual values or their sum, three consecutive integers always have fixed spacing. Recognizing this invariant lets you answer without computing any specific numbers, which is an important reasoning shortcut in aptitude tests. Given Data / Assumptions: The integers are consecutive; call them n − 1, n, n + 1. “Consecutive” means the step size is 1 between neighbors. We are asked for the difference between the first and the third. Concept / Approach:If the three numbers are (n − 1), n, and (n + 1), then the first is n − 1 and the third is n + 1. Their difference equals (n + 1) − (n − 1) = 2. This result does not depend on the specific n or the actual sum; it is purely structural and therefore universal to any such triple. Step-by-Step Solution:Represent the triple: first = n − 1, middle = n, third = n + 1.Compute difference: (n + 1) − (n − 1) = 2.Conclude: The difference is always 2.No additional numeric data are required. Verification / Alternative check:Try sample triples: (5, 6, 7) → 7 − 5 = 2; (−3, −2, −1) → (−1) − (−3) = 2. The result is consistent across all cases. Why Other Options Are Wrong:“One” or “Three” contradict the exact spacing; “Either (A) or (B)” is logically impossible for a fixed invariant; “Zero” would require the first and third to be equal, which never occurs for consecutive integers. Common Pitfalls:Confusing “consecutive integers” with “consecutive odd/even integers,” where spacing differs; overcomplicating by trying to use the sum when it is unnecessary. Final Answer:Two

Question 20

Linear Condition — Reduced to One-Third After Subtracting 48 A number becomes one-third of itself when 48 is subtracted from it. What is two-thirds of that original number?

Accepted Answer

Introduction / Context:Here you translate a verbal condition into a simple linear equation and then compute a requested fraction of the found number. This checks your comfort with algebraic manipulation and careful reading—especially verifying that the requested final quantity appears among the options. Given Data / Assumptions: Let the number be N. Subtracting 48 yields one-third of itself: N − 48 = N/3. We must find 2N/3 (two-thirds of N). Concept / Approach:Solve the linear equation to get N. Then compute the requested fraction 2N/3. Always compare the derived value to the choices; if it is not listed explicitly, the correct selection is “None of these.” Step-by-Step Solution:Start: N − 48 = N/3.Rearrange: N − N/3 = 48 → (2N/3) = 48.Therefore, 2N/3 = 48 immediately.Hence N = (48 * 3)/2 = 72, and two-thirds of N is 48. Verification / Alternative check:Check the original statement with N = 72: 72 − 48 = 24 and N/3 = 24, consistent. Thus 2N/3 = 48 is correct. Why Other Options Are Wrong:24, 72, and 36 represent N/3, N, or other misread values. Since 48 (the true two-thirds) is not among the listed options, “None of these” is the only correct choice. Common Pitfalls:Selecting 72 (the value of N) instead of two-thirds of N; stopping at N without computing the asked quantity; overlooking that the exact computed result is absent from the choices. Final Answer:None of these

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