Question 1

Partnership profit sharing: A, B, and C start a business by investing ₹ 28,000, ₹ 35,000, and ₹ 14,000, respectively. At the end of one full year, the total profit is ₹ 5,225. What is A's share of the profit, assuming profit is divided in the ratio …

Accepted Answer

Introduction / Context: Partnership problems in aptitude test how profits are divided among partners when capital contributions and time periods are known. Here, A, B, and C invest different amounts for a full year, so profit sharing is directly proportional to their invested capitals. We must compute A's share from the total profit of ₹ 5,225 using capital ratios. Given Data / Assumptions: A invests ₹ 28,000 for the entire year. B invests ₹ 35,000 for the entire year. C invests ₹ 14,000 for the entire year. Total profit at year end = ₹ 5,225. All capitals remain invested for the same duration (1 year). Profit is divided in the ratio of (capital * time). Concept / Approach: When time is equal, profit shares are in the ratio of capitals invested. Reduce the ratio to simplest terms, find total ratio units, then compute each partner's share = (individual ratio / total ratio) * total profit. Step-by-Step Solution: Capitals: 28,000 : 35,000 : 14,000. Divide by 7,000 to simplify → 4 : 5 : 2. Total ratio units = 4 + 5 + 2 = 11. Value per unit = 5,225 / 11 = 475. A's share = 4 * 475 = ₹ 1,900. Verification / Alternative check: Compute all shares: A = 4*475 = 1,900; B = 5*475 = 2,375; C = 2*475 = 950. Sum = 1,900 + 2,375 + 950 = 5,225, matching the total profit. Why Other Options Are Wrong: ₹ 1,740, ₹ 1,775, ₹ 1,850, and ₹ 1,650 do not equal 4/11 of ₹ 5,225. They would break the proportionality implied by the 4 : 5 : 2 capital ratio. Common Pitfalls: Dividing profit directly by the number of partners instead of using capital ratios. Forgetting to simplify the ratio or mis-adding ratio parts. Confusing total capital (sum) with the ratio approach. Final Answer: ₹ 1,900

Question 2

Partnership difference in shares: A, B, and C invest ₹ 45,000, ₹ 55,000, and ₹ 60,000, respectively, for one full year. The total profit at year end is ₹ 11,200. How much more does B receive than A from the profit distribution (capital-time proporti…

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Introduction / Context: In capital-time proportional partnerships with equal time periods, profit shares are distributed in the ratio of capitals. This question asks for the difference between B's and A's shares, which is a neat way to avoid calculating each partner's entire share if you handle ratios correctly. Given Data / Assumptions: A invests ₹ 45,000; B invests ₹ 55,000; C invests ₹ 60,000. Investment period for all = 1 year. Total profit = ₹ 11,200. Profit share ∝ capital * time (time equal, so ∝ capital). Concept / Approach: Convert capitals to a simple ratio and compute the value per unit of ratio. The difference in shares between B and A is (B's ratio − A's ratio) * value per unit. This is faster and reduces arithmetic errors. Step-by-Step Solution: Capitals: 45 : 55 : 60 (in thousands → ratio units 45, 55, 60). Total ratio units = 45 + 55 + 60 = 160. Value per unit = 11,200 / 160 = 70. Difference B − A = (55 − 45) * 70 = 10 * 70 = ₹ 700. Verification / Alternative check: Compute shares explicitly: A = 45*70 = 3,150; B = 55*70 = 3,850; C = 60*70 = 4,200. Total = 3,150 + 3,850 + 4,200 = 11,200. Difference = 3,850 − 3,150 = 700. Why Other Options Are Wrong: ₹ 750, ₹ 710, ₹ 780, ₹ 840 correspond to incorrect unit values or wrong ratio differences. Common Pitfalls: Dividing ₹ 11,200 equally among partners instead of using ratio 45 : 55 : 60. Arithmetic slips when computing value per unit or subtracting ratio parts. Final Answer: ₹ 700

Question 3

Ratio equivalence: If 2A = 3B = 4C (all positive), determine the simplified ratio A : B : C.

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Introduction / Context: Problems that state an expression like 2A = 3B = 4C use a common equal value to relate different variables. The goal is to express A, B, and C as multiples of that common value and then reduce to the simplest whole-number ratio A : B : C. Given Data / Assumptions: 2A = 3B = 4C (assume all quantities are positive for ratio interpretation). We want A : B : C in least integer terms. Concept / Approach: Let the common value be k. Then 2A = k, 3B = k, and 4C = k. Solve for A, B, C in terms of k, then scale to remove fractions and simplify. Step-by-Step Solution: Let 2A = 3B = 4C = k. Then A = k/2, B = k/3, C = k/4. A : B : C = (k/2) : (k/3) : (k/4). Cancel k and clear denominators by multiplying by LCM(2,3,4) = 12. Ratio becomes 12*(1/2) : 12*(1/3) : 12*(1/4) = 6 : 4 : 3. Verification / Alternative check: Pick k = 12 for convenience. Then A = 6, B = 4, C = 3. Check: 2A = 12, 3B = 12, 4C = 12; all equal. Thus A : B : C = 6 : 4 : 3 is consistent. Why Other Options Are Wrong: 2 : 3 : 4 or permutations do not satisfy 2A = 3B = 4C simultaneously when scaled. 12 : 8 : 6 is correct in proportion but not simplified; the question expects the least integer ratio. Common Pitfalls: Treating 2A, 3B, and 4C individually instead of introducing a common value k. Forgetting to clear denominators to get integer ratios. Final Answer: 6 : 4 : 3

Question 4

Fourth proportional: Find the fourth proportional to 4, 16, and 7 (i.e., determine x such that 4 : 16 = 7 : x).

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Introduction / Context: The fourth proportional to three numbers a, b, c is a number x that satisfies a : b = c : x. Such problems are direct applications of the property of equal ratios and cross multiplication. Given Data / Assumptions: a = 4, b = 16, c = 7. Find x such that 4 : 16 = 7 : x. Concept / Approach: Use a : b = c : x → a/b = c/x. Cross multiply to solve for x: a*x = b*c → x = (b*c)/a. Step-by-Step Solution: Set up proportion: 4 : 16 = 7 : x. Cross multiplication: 4x = 16 * 7. Compute 16 * 7 = 112. Thus x = 112 / 4 = 28. Verification / Alternative check: Check ratios as decimals: 4/16 = 0.25. With x = 28, 7/28 = 0.25. Ratios match, confirming x = 28. Why Other Options Are Wrong: 29, 22, 25, 32 do not satisfy 7 : x = 4 : 16. Only 28 yields identical ratios on both sides. Common Pitfalls: Inverting the ratio (using 16 : 4) and getting 7 : x wrong. Avoid arithmetic mistakes when multiplying 16 and 7. Final Answer: 28

Question 5

Duplicate ratio: What is the duplicate ratio (i.e., the squared ratio) of 2 : 7?

Accepted Answer

Introduction / Context: Classical ratio terminology: the duplicate ratio of a : b is the ratio of their squares, a^2 : b^2. Similarly, sub-duplicate refers to square roots, and triplicate refers to cubes. This problem asks for the duplicate ratio of 2 : 7. Given Data / Assumptions: Base ratio = 2 : 7. Duplicate ratio means square each term. Concept / Approach: If the ratio is a : b, then the duplicate ratio is a^2 : b^2. No simplification is necessary if the squared terms are already co-prime or in simplest form. Step-by-Step Solution: Start with 2 : 7. Square both terms: 2^2 : 7^2 = 4 : 49. The pair (4, 49) shares no common factor other than 1, so it is already in simplest terms. Verification / Alternative check: Numerically, 2/7 ≈ 0.2857. The duplicate ratio is (2/7)^2 ≈ 0.08163. As a ratio of integers with the same value, 4/49 ≈ 0.08163, confirming correctness. Why Other Options Are Wrong: 49 : 4 is the inverse duplicate ratio. 4 : 14 mixes squaring with doubling. 8 : 343 is the triplicate ratio (2^3 : 7^3). 14 : 49 mixes multiplication with squaring. Common Pitfalls: Confusing duplicate with triplicate ratio, or accidentally inverting the order. Remember: duplicate = square each term, keep the original order. Final Answer: 4 : 49

Question 6

Sub-duplicate ratio: Find the sub-duplicate ratio (i.e., the ratio of square roots) of 81 : 64.

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Introduction / Context: Sub-duplicate ratio refers to the ratio of the square roots of the given numbers. So, for p : q, the sub-duplicate ratio is √p : √q. This is standard terminology in ratio and proportion problems. Given Data / Assumptions: Given ratio = 81 : 64. We assume both are non-negative, perfect squares. Concept / Approach: Compute the square roots and express their ratio in the simplest whole-number form. For 81 and 64, the roots are integers, making the computation straightforward. Step-by-Step Solution: √81 = 9 and √64 = 8. Therefore, sub-duplicate ratio = 9 : 8. This is already simplified. Verification / Alternative check: Check by squaring the result: (9 : 8) duplicated (squared) gives 81 : 64, confirming that (9 : 8) is indeed the sub-duplicate ratio. Why Other Options Are Wrong: 8 : 9 is the inverse order. 4 : 9 and 7 : 8 are unrelated to square roots of 81 and 64. 3 : 4 would be correct for 9 : 16, not for 81 : 64. Common Pitfalls: Inverting the ratio or using cube roots instead of square roots. Carefully match each term with its square root. Final Answer: 9 : 8

Question 7

Triplicate ratio: Determine the triplicate ratio (i.e., the ratio of cubes) of 7 : 5.

Accepted Answer

Introduction / Context: The triplicate ratio of a : b is a^3 : b^3. This extends the idea of duplicate (squared) ratios to cubes and appears often in classical proportion problems and speed/volume scaling analogies. Given Data / Assumptions: Base ratio = 7 : 5. Triplicate ratio means cube each term. Concept / Approach: Compute 7^3 and 5^3 and write the ratio in simplest integer form, preserving the original order. Step-by-Step Solution: 7^3 = 343 and 5^3 = 125. Triplicate ratio = 343 : 125. No further simplification is possible since 343 and 125 share no common factor. Verification / Alternative check: If the duplicate (squared) ratio would be 49 : 25, the triplicate is indeed 343 : 125. Both follow the nth-power definition of these classical terms. Why Other Options Are Wrong: 125 : 343 is the inverse order. 344 : 125 and 343 : 126 are numerically off and do not represent cubes of 7 and 5. 49 : 25 is the duplicate (squared) ratio, not the triplicate ratio. Common Pitfalls: Inverting the order or mixing duplicate and triplicate definitions. Always cube each term and keep the original order. Final Answer: 343 : 125

Question 8

Inverse ratio: What is the inverse (reciprocal) ratio of 17 : 19?

Accepted Answer

Introduction / Context: The inverse (or reciprocal) ratio of a : b is b : a. These appear in proportion manipulations, especially when converting between direct and inverse relationships. Given Data / Assumptions: Ratio given = 17 : 19. Concept / Approach: Inverse ratio simply swaps the order: a : b → b : a. No further computation is needed unless simplification is required. Step-by-Step Solution: Given 17 : 19. Inverse ratio = 19 : 17. Verification / Alternative check: 17/19 ≈ 0.8947; the reciprocal is 19/17 ≈ 1.1176, which is indeed the inverse ratio expressed as a fraction. Why Other Options Are Wrong: 18 : 17 and 17 : 18 change the numbers; 19 : 5 is unrelated; 1 : 1 would be the inverse only if a = b. Common Pitfalls: Confusing inverse ratio with subtracting or adding terms. Inversion is a swap, not an arithmetic operation on terms. Final Answer: 19 : 17

Question 9

Componendo property in ratio: If a : b = b : c (i.e., a/b = b/c), then which of the following equals a^4 : b^4?

Accepted Answer

Introduction / Context: From a : b = b : c, it follows that a/b = b/c, giving a key identity: b^2 = a c. The question asks you to express a^4 : b^4 in another equivalent form using this relationship. Such manipulations are common in algebraic ratio problems. Given Data / Assumptions: a : b = b : c ⇒ a/b = b/c. All variables are positive for ratio interpretation. Concept / Approach: From a/b = b/c, cross multiplication gives a c = b^2. Then manipulate a^4 : b^4 by expressing b^4 in terms of a and c: b^4 = (b^2)^2 = (a c)^2 = a^2 c^2. Step-by-Step Solution: Given a c = b^2. Compute b^4 = (b^2)^2 = (a c)^2 = a^2 c^2. Therefore, a^4 : b^4 = a^4 : (a^2 c^2) = a^2 : c^2. Verification / Alternative check: Try sample values satisfying a : b = b : c. Let a = 1, b = 2, c = 4 (since 1/2 = 2/4). Then a^4 : b^4 = 1 : 16. Also a^2 : c^2 = 1 : 16, confirming the identity. Why Other Options Are Wrong: ac : b^2 equals 1 : 1 under the same condition, not a^4 : b^4. c^2 : a^2 is the inverse. b^2 : ac again gives 1 : 1. ab : c^2 has no direct equivalence here. Common Pitfalls: Forgetting that b^2 = a c and thus b^4 = a^2 c^2; mixing up which terms square or invert leads to incorrect options. Final Answer: a^2 : c^2

Question 10

Chain ratios: If a : b = 3 : 5 and b : c = 4 : 7, determine a : c.

Accepted Answer

Introduction / Context: Combining two ratios with a common middle term (here b) requires equalizing that middle term. This is the standard method to derive a chain ratio such as a : c from a : b and b : c. Given Data / Assumptions: a : b = 3 : 5 and b : c = 4 : 7. Concept / Approach: Adjust the ratios so that the b term is the same in both. Then multiply across to get a consistent a : b : c, and finally extract a : c. Step-by-Step Solution: From a : b = 3 : 5, take b = 5k → a = 3k. From b : c = 4 : 7, take b = 4m → c = 7m. Equalize b: 5k = 4m. Choose k = 4 and m = 5 → b = 20. Then a = 3k = 12 and c = 7m = 35. Therefore, a : c = 12 : 35. Verification / Alternative check: Build the full triple ratio: a : b : c = 12 : 20 : 35. Check a : b = 12 : 20 = 3 : 5 and b : c = 20 : 35 = 4 : 7, confirming consistency. Why Other Options Are Wrong: 11 : 35 and 35 : 11 are not derived by equalizing b. 35 : 12 flips order and values. 3 : 7 would be correct only if a : b and b : c were compatible with that direct relation, which they are not without scaling. Common Pitfalls: Failing to equalize the common term or incorrectly cross-multiplying. Always align the shared term before concluding the end ratios. Final Answer: 12 : 35

Question 11

Linked equations to combined ratio: If 4a = 5b and 7b = 9c, find the simplified ratio a : b : c.

Accepted Answer

Introduction / Context: Converting linear relations (like 4a = 5b and 7b = 9c) into a three-term ratio a : b : c requires aligning the common variable b and then computing consistent values for a and c. This is a standard algebraic technique for ratio building. Given Data / Assumptions: 4a = 5b. 7b = 9c. All terms are positive, allowing ratio construction. Concept / Approach: From 4a = 5b → a : b = 5 : 4. From 7b = 9c → b : c = 9 : 7. Combine by making b equal in both pairs and then read off a, b, c. Step-by-Step Solution: From 4a = 5b → a : b = 5 : 4. From 7b = 9c → b : c = 9 : 7. Equalize b: choose b = LCM(4, 9) = 36. If b = 36, then from a : b = 5 : 4 → a = (5/4) * 36 = 45. From b : c = 9 : 7 → c = (7/9) * 36 = 28. Hence, a : b : c = 45 : 36 : 28. Verification / Alternative check: Check the original equations: 4a = 4*45 = 180; 5b = 5*36 = 180 (ok). Also 7b = 7*36 = 252; 9c = 9*28 = 252 (ok). Why Other Options Are Wrong: Other permutations do not satisfy both equations simultaneously. 44 : 33 : 28 is not consistent with 4a = 5b. Common Pitfalls: Not equalizing b properly or mixing the roles of b and c when translating ratios. Always verify by substituting back into the original equations. Final Answer: 45 : 36 : 28

Question 12

Equal fractional forms: If a/7 = b/9 = c/11, determine the simplified ratio a : b : c.

Accepted Answer

Introduction / Context: When several fractions are equal, each numerator is proportional to its denominator. This is a standard method to derive direct ratios from equal fractional relationships. Given Data / Assumptions: a/7 = b/9 = c/11 = k (say), with positive quantities for ratio interpretation. Concept / Approach: Express a, b, c in terms of k and their denominators. Then form the ratio and simplify if needed. Step-by-Step Solution: Let a/7 = b/9 = c/11 = k. Then a = 7k, b = 9k, c = 11k. Therefore, a : b : c = 7 : 9 : 11. Verification / Alternative check: Substitute sample k = 1 to see a = 7, b = 9, c = 11, which obviously satisfy the original equal fractions. Why Other Options Are Wrong: Any permutation that does not keep denominators aligned with numerators will be incorrect. 14 : 18 : 22 is a correct multiple but not the simplest form the question expects. Common Pitfalls: Inverting denominators or mixing the order. Keep a aligned with 7, b with 9, and c with 11. Final Answer: 7 : 9 : 11

Question 13

Percent-to-ratio conversion: If 30% of A equals 20% of B, determine the ratio A : B in simplest form.

Accepted Answer

Introduction / Context: Converting percentage statements into ratios is common. Here, 30% of A equals 20% of B. We must express A : B as a simplest whole-number ratio by translating the percentages into decimal multipliers and then simplifying. Given Data / Assumptions: 30% of A = 20% of B; assume positive values for A and B. Concept / Approach: Translate percentages into multipliers: 30% = 0.30 and 20% = 0.20. Equate 0.30*A = 0.20*B and isolate A/B to get the ratio. Step-by-Step Solution: 0.30 A = 0.20 B. Divide both sides by 0.10 to remove decimals → 3 A = 2 B. Rearrange: A / B = 2 / 3. Therefore, A : B = 2 : 3. Verification / Alternative check: Let A = 2 and B = 3. Then 30% of A = 0.6, and 20% of B = 0.6; equal, confirming the ratio. Why Other Options Are Wrong: 1 : 3, 3 : 2, 3 : 1, 5 : 3 do not satisfy 0.30*A = 0.20*B when tested. Common Pitfalls: Forgetting to convert percentages correctly or flipping the ratio when isolating A/B. Keep track of which variable is in the numerator. Final Answer: 2 : 3

Question 14

Composite ratios with division: Given P : Q : R = 2 : 3 : 4, compute P/Q : Q/R : R/P in simplest integer terms.

Accepted Answer

Introduction / Context: Transforming a three-term ratio into derived ratios like P/Q, Q/R, and R/P requires dividing corresponding terms, then simplifying to a clean integer ratio. This builds fluency with fractional manipulation in proportional reasoning. Given Data / Assumptions: P : Q : R = 2 : 3 : 4 with all positive terms. Concept / Approach: Compute each fraction: P/Q = 2/3, Q/R = 3/4, and R/P = 4/2 = 2. Then clear denominators by multiplying by a common multiple to get an integer ratio. Step-by-Step Solution: P/Q = 2/3; Q/R = 3/4; R/P = 2. Choose 12 as a convenient common multiple. (2/3)*12 = 8; (3/4)*12 = 9; (2)*12 = 24. Therefore, P/Q : Q/R : R/P = 8 : 9 : 24. Verification / Alternative check: Converting back to fractions: 8/12 = 2/3, 9/12 = 3/4, 24/12 = 2, matching the original computed values. Why Other Options Are Wrong: Options that permute or mis-scale these values do not correspond to the defined fractions; only 8 : 9 : 24 preserves all three relationships simultaneously. Common Pitfalls: Mixing up the order (e.g., writing Q/P instead of P/Q) or failing to multiply by a common factor to get integers. Final Answer: 8 : 9 : 24

Question 15

Linked variable definitions to ratio: If A = B/4 and B = C/2, find the simplest whole-number ratio A : B : C.

Accepted Answer

Introduction / Context: Expressing variables in terms of a common parameter often simplifies ratio problems. Here, A and B are defined in terms of each other, and B is defined in terms of C, so we align all three to construct the ratio A : B : C. Given Data / Assumptions: A = B/4 and B = C/2. Concept / Approach: Use the second relation to express B and A in terms of C, then read off the ratio and simplify to the least integers. Step-by-Step Solution: From B = C/2, we get C = 2B. From A = B/4, we get A expressed via B. Hence A : B : C = (B/4) : B : (2B) = 1/4 : 1 : 2. Multiply by 4 to remove fractions → 1 : 4 : 8. Verification / Alternative check: Put B = 4 (convenient). Then A = 1 and C = 8. Relations hold: A = B/4 → 1 = 4/4; B = C/2 → 4 = 8/2. Why Other Options Are Wrong: 4 : 2 : 1 or 1 : 2 : 4 invert and mis-scale the relationships. 8 : 4 : 1 reverses the order. 2 : 4 : 8 scales inconsistently with A = B/4. Common Pitfalls: Inverting the direction of relationships (e.g., assuming B = 4A instead of A = B/4) and mixing up order when writing ratios. Final Answer: 1 : 4 : 8

Question 16

Decimal-to-ratio conversion: If 0.8 × A = 0.09 × B, determine the simplest ratio A : B.

Accepted Answer

Introduction / Context: Converting linear equations with decimals to clean ratios is a staple aptitude skill. Here, we solve for A/B from the equation 0.8A = 0.09B and then write A : B in lowest integer terms. Given Data / Assumptions: 0.8A = 0.09B; A and B are positive. Concept / Approach: Isolate A/B: A/B = 0.09 / 0.8. Then express as a fraction and reduce to the simplest whole-number ratio. Step-by-Step Solution: A/B = 0.09 / 0.8. Convert to fractions: 0.09 = 9/100, 0.8 = 8/10 = 4/5. A/B = (9/100) / (4/5) = (9/100) * (5/4) = 45/400 = 9/80. Hence, A : B = 9 : 80. Verification / Alternative check: Take A = 9 and B = 80. Then 0.8A = 7.2 and 0.09B = 7.2, satisfying the equation. Why Other Options Are Wrong: 3 : 80, 7 : 80, 80 : 9, 9 : 8 are not equal to 9 : 80 and will not satisfy 0.8A = 0.09B. Common Pitfalls: Mishandling decimals or forgetting to invert the divisor when dividing fractions. Carefully convert decimals to fractions before simplifying. Final Answer: 9 : 80

Question 17

Reciprocal ratios: If 1/x : 1/y : 1/z = 2 : 3 : 5, determine the ratio x : y : z in simplest integer form.

Accepted Answer

Introduction / Context: Ratios involving reciprocals invert the usual proportionality. If 1/x : 1/y : 1/z is known, then x : y : z is proportional to the reciprocals of those terms. Converting correctly is vital for many algebraic and rate problems. Given Data / Assumptions: 1/x : 1/y : 1/z = 2 : 3 : 5 with positive variables. Concept / Approach: Let 1/x = 2k, 1/y = 3k, 1/z = 5k. Then x = 1/(2k), y = 1/(3k), z = 1/(5k). Therefore, x : y : z = (1/2) : (1/3) : (1/5). Clear denominators to get integers. Step-by-Step Solution: x : y : z = (1/2) : (1/3) : (1/5). Multiply all by LCM(2,3,5) = 30. x : y : z = 30*(1/2) : 30*(1/3) : 30*(1/5) = 15 : 10 : 6. Verification / Alternative check: Take x = 15, y = 10, z = 6. Then 1/x : 1/y : 1/z = (1/15) : (1/10) : (1/6) = 2 : 3 : 5 after multiplying by 30, confirming consistency. Why Other Options Are Wrong: Options that permute the order (e.g., 6 : 15 : 10) or mix terms do not correspond to the correct inversion step from reciprocals. Common Pitfalls: Forgetting that the ratio of reciprocals inverts the direction of proportionality. Always invert and then clear denominators carefully. Final Answer: 15 : 10 : 6

Question 18

Adjusted ratio with translated terms: If a/3 = b/8, find the ratio (a + 3) : (b + 8) in simplest form.

Accepted Answer

Introduction / Context: When two variables are proportional via a/3 = b/8, we can parametrize them and then evaluate expressions like (a + constant) : (b + constant). This technique shows up in many ratio translation problems. Given Data / Assumptions: a/3 = b/8 with positive values for a and b. Concept / Approach: From a/3 = b/8 = k, write a = 3k and b = 8k. Then substitute into (a + 3) : (b + 8) and simplify. Step-by-Step Solution: Let a = 3k and b = 8k. Compute a + 3 = 3k + 3 = 3(k + 1). Compute b + 8 = 8k + 8 = 8(k + 1). Therefore, (a + 3) : (b + 8) = 3(k + 1) : 8(k + 1) = 3 : 8. Verification / Alternative check: Choose k = 1 → a = 3, b = 8. Then (a + 3) : (b + 8) = 6 : 16 = 3 : 8, confirming the result. Why Other Options Are Wrong: 8 : 3 inverts the order; 5 : 8 or 11 : 16 do not arise from consistent parametrization; 3 : 5 truncates scaling incorrectly. Common Pitfalls: Forgetting to add constants to the correct variable or cancelling incorrectly; always factor out the common (k + 1) before cancelling. Final Answer: 3 : 8

Question 19

Linked fractional equalities to overall ratio: If (1/2) of A = (2/5) of B = (1/3) of C, find the ratio A : B : C in simplest whole-number terms.

Accepted Answer

Introduction / Context: When multiple proportional equalities are chained through constants on each variable, parametrization is the cleanest method. Here, three fractions of A, B, and C are equal, and we must convert that into a simple ratio A : B : C. Given Data / Assumptions: (1/2)A = (2/5)B = (1/3)C = k (say), with all variables positive. Concept / Approach: Express each variable in terms of k, then write the ratio and simplify to the least integers. Specifically: A = 2k, B = (5/2)k, C = 3k. Step-by-Step Solution: From (1/2)A = k ⇒ A = 2k. From (2/5)B = k ⇒ B = (5/2)k. From (1/3)C = k ⇒ C = 3k. Therefore, A : B : C = 2 : (5/2) : 3. Multiply by 2 to clear the fraction → 4 : 5 : 6. Verification / Alternative check: Let k = 2 for convenience. Then A = 4, B = 5, C = 6. Check: (1/2)A = 2; (2/5)B = 2; (1/3)C = 2; all equal, confirming the ratio. Why Other Options Are Wrong: Permutations like 6 : 4 : 5 or 5 : 4 : 6 scramble the order. The ratio must align with A, B, C derived from their respective equations. Common Pitfalls: Mixing up which constant multiplies which variable and failing to eliminate fractional components correctly. Final Answer: 4 : 5 : 6

Question 20

Implication of product constraint: If x·y = 36, which of the following proportional statements is always correct?

Accepted Answer

Introduction / Context: Proportional equations can encode multiplicative relationships compactly. Here we are told that x·y = 36, and we must identify which given proportion necessarily follows from that fact for any positive x and y satisfying the product constraint. Given Data / Assumptions: x·y = 36. We interpret the proportion a : b = c : d as a/b = c/d with positive quantities. Concept / Approach: Translate each option into an equality of fractions and cross multiply to see what condition it encodes. The correct choice will be the one that is equivalent to x·y = 36 for all positive solutions, without adding extra restrictions. Step-by-Step Solution: Option A: x : 9 = 4 : y ⇒ x/9 = 4/y ⇒ x*y = 36 (after cross multiplication x*y = 9*4 ). This is identically the given constraint. Option B: 9 : x = 4 : y ⇒ 9/x = 4/y ⇒ 9y = 4x ⇒ With y = 36/x, we get 9*(36/x) = 4x ⇒ 324/x = 4x ⇒ x^2 = 81 ⇒ x = 9 (or −9), which adds an extra restriction and is not always true. Option C: x : 17 = y : 7 ⇒ x/17 = y/7 ⇒ 7x = 17y; using y = 36/x typically fails except for one special pair. Option D: x : 6 = y : 6 ⇒ x/6 = y/6 ⇒ x = y, which forces x = y = 6 under xy = 36, not general. Option E: x : 4 = 9 : y ⇒ x/4 = 9/y ⇒ x*y = 36 (same as A). Though algebraically equivalent, the question expects a single best match; Option A is the standard presentation given the set. Verification / Alternative check: Pick x = 12, y = 3. Then A gives 12 : 9 = 4 : 3 → 4/3 = 4/3 (true). B gives 9/12 = 4/3 → 3/4 = 4/3 (false). C gives 12/17 = 3/7 (false). D gives 12/6 = 3/6 → 2 = 0.5 (false). E gives 12/4 = 9/3 → 3 = 3 (true but redundant with A; choose A). Why Other Options Are Wrong: They imply extra constraints beyond xy = 36 (equality of x and y, or a specific value of x), or are simply inconsistent with many valid pairs (x, y) satisfying the product. Common Pitfalls: Not translating proportions to fractional equalities and neglecting to cross multiply; always test with a generic pair satisfying the product. Final Answer: x : 9 = 4 : y

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