Question 1

Ratio equivalence expanded: If 2A = 3B = 4C (all three expressions equal the same positive quantity), determine the simplified ratio A : B : C. Show clearly how you scale each term to obtain integers.

Accepted Answer

Introduction / Context: Problems that present chained equalities such as 2A = 3B = 4C ask you to express each variable in terms of a single common value and then reduce to a clean integer ratio. This checks fluency with proportional scaling and the use of least common multiples to clear denominators neatly. Given Data / Assumptions: 2A = 3B = 4C = k, where k is some positive common value. We seek A : B : C in simplest integer terms. Concept / Approach: Express each variable in terms of k, then remove fractions by multiplying all terms with the least common multiple of denominators. Finally, simplify to a primitive ratio (no common factor). Step-by-Step Solution: From 2A = k ⇒ A = k/2.From 3B = k ⇒ B = k/3.From 4C = k ⇒ C = k/4.So A : B : C = k/2 : k/3 : k/4.Multiply each term by LCM(2,3,4) = 12 to clear denominators.12*(k/2) : 12*(k/3) : 12*(k/4) = 6k : 4k : 3k = 6 : 4 : 3. Verification / Alternative check: Pick k = 12. Then A = 6, B = 4, C = 3. Indeed, 2A = 12, 3B = 12, 4C = 12—consistent. Why Other Options Are Wrong: 2 : 3 : 4 and 3 : 4 : 6 reverse the required scaling; 4 : 3 : 2 inverts the correct order; 12 : 8 : 6 is an unsimplified multiple of the correct ratio. Common Pitfalls: Forgetting to clear denominators, or assuming A : B : C equals 2 : 3 : 4 directly from 2A = 3B = 4C, which is incorrect because the coefficients multiply the variables. Final Answer: 6 : 4 : 3

Question 2

Chain of ratios across four variables: Given A : B = 2 : 3, B : C = 4 : 5, and C : D = 6 : 7 for the same set of quantities, determine A : D. Show the alignment of the repeated term to combine the ratios consistently.

Accepted Answer

Introduction / Context: Chained ratios must be combined by matching the repeated term. Here, B connects A to C, and C connects to D. The task builds ratio-composition skills and careful scaling to get an exact A : D relationship. Given Data / Assumptions: A : B = 2 : 3. B : C = 4 : 5. C : D = 6 : 7. Concept / Approach: To combine A : B and B : C, equalize the value of B in both ratios. Then include C : D similarly by equalizing C. Scale to integers to avoid fractions and then read off A : D. Step-by-Step Solution: Let A : B = 2x : 3x and B : C = 4y : 5y. Match B: 3x = 4y ⇒ choose x = 4, y = 3.Then A = 8, B = 12, C = 15.Next, C : D = 6 : 7 ⇒ if C = 15, set 6k = 15 ⇒ k = 2.5 ⇒ D = 7k = 17.5.Scale all by 2 to remove the decimal: A = 16, B = 24, C = 30, D = 35.Hence A : D = 16 : 35. Verification / Alternative check: Confirm intermediate ratios: A : B = 16 : 24 = 2 : 3; B : C = 24 : 30 = 4 : 5; C : D = 30 : 35 = 6 : 7. Why Other Options Are Wrong: 2 : 7 and 4 : 13 do not preserve the combined scaling; 7 : 8 reverses order; only 16 : 35 satisfies all three conditions. Common Pitfalls: Adding or multiplying ratios directly without aligning the repeated variable; forgetting to scale consistently when introducing the third ratio. Final Answer: 16 : 35

Question 3

Build a three-term ratio from two equations: If A = B/3 and B = C/2 (all positive), express the simplified ratio A : B : C, showing the substitution path from C to A.

Accepted Answer

Introduction / Context: Expressing all variables in terms of one helps create a clean ratio. Here, A is related to B, and B is related to C. Substituting stepwise leads to a direct representation of A and B in terms of C, which can then be scaled. Given Data / Assumptions: A = B/3. B = C/2. All quantities are positive so ratios are meaningful. Concept / Approach: Substitute B from the second equation into the first to express A in terms of C. Then write A : B : C and clear any fractions using a common multiple to get integer terms. Step-by-Step Solution: B = C/2.A = B/3 = (C/2)/3 = C/6.Therefore A : B : C = C/6 : C/2 : C.Cancel C (common factor) to get 1/6 : 1/2 : 1.Multiply all by 6 to clear denominators ⇒ 1 : 3 : 6. Verification / Alternative check: Take C = 6. Then B = 3 and A = 1. Check: A = B/3 = 1, B = C/2 = 3—consistent. Why Other Options Are Wrong: 2 : 3 : 6 doubles only A incorrectly; 3 : 2 : 6 swaps terms; 3 : 1 : 2 and 1 : 2 : 3 do not satisfy the defining equations. Common Pitfalls: Forgetting to divide by 3 after substituting for B, or mixing the order of A, B, C when forming the ratio. Final Answer: 1 : 3 : 6

Question 4

Combine two ratios that share a middle term: If A : B = 5 : 7 and B : C = 6 : 11, determine the three-term ratio A : B : C by correctly aligning the common term B.

Accepted Answer

Introduction / Context: When two ratios share a variable, align the shared term to the same value to combine them into a single three-term ratio. This technique underlies many proportion problems in quantitative aptitude. Given Data / Assumptions: A : B = 5 : 7. B : C = 6 : 11. Concept / Approach: Let A : B = 5x : 7x and B : C = 6y : 11y. Set 7x = 6y to make B consistent, then compute A and C accordingly and write A : B : C in integer form. Step-by-Step Solution: Choose 7x = 6y ⇒ take x = 6 and y = 7 for integer scaling.Then A = 5x = 30; B = 7x = 42; C = 11y = 77.Thus A : B : C = 30 : 42 : 77 (already simplest). Verification / Alternative check: Check subratios: A : B = 30 : 42 = 5 : 7; B : C = 42 : 77 = 6 : 11. Why Other Options Are Wrong: 55 : 77 : 66 does not preserve B : C = 6 : 11; 35 : 49 : 42 mismatches both links; “None of these” is invalid since a correct option exists. Common Pitfalls: Adding numerators/denominators directly, or attempting to average ratios. Only matching the shared term yields a correct combination. Final Answer: 30 : 42 : 77

Question 5

Cascade two ratios to a three-term result: If A : B = 7 : 9 and B : C = 3 : 5, determine A : B : C by aligning the common term B and simplifying.

Accepted Answer

Introduction / Context: Combining chained ratios requires equalizing the shared variable. Here, B links the two given ratios. After setting B equal in both, read off the values for A and C to form a combined three-term ratio. Given Data / Assumptions: A : B = 7 : 9. B : C = 3 : 5. Concept / Approach: Write A : B = 7x : 9x and B : C = 3y : 5y. Match B so 9x = 3y; choose convenient integers to maintain integrality, then compute A, B, C and simplify if needed. Step-by-Step Solution: Set 9x = 3y ⇒ y = 3x.Then A = 7x, B = 9x, C = 5y = 15x.Hence A : B : C = 7 : 9 : 15 (already in lowest terms). Verification / Alternative check: Check subratios: A : B = 7 : 9 and B : C = 9 : 15 = 3 : 5, both satisfied. Why Other Options Are Wrong: 7 : 9 : 5 uses wrong C; 21 : 35 : 45 is not consistent with B : C; 7 : 3 : 15 breaks A : B; 14 : 18 : 30 reduces to 7 : 9 : 15 but is an unsimplified multiple (not minimal form usually expected). Common Pitfalls: Multiplying corresponding terms across ratios or mixing up the order; always align the shared term first. Final Answer: 7 : 9 : 15

Question 6

Interpret decimal proportion precisely: 0.6 of a first number equals 0.09 of a second number. Find the ratio of the first number to the second number (first : second).

Accepted Answer

Introduction / Context: Converting verbal proportional statements with decimals into a clean ratio is a core aptitude skill. The key is to isolate the ratio x : y from an equation of the form ax = by, then reduce to the smallest integer terms. Given Data / Assumptions: 0.6 * (first number) = 0.09 * (second number). Both numbers are positive. Concept / Approach: If 0.6x = 0.09y, then x / y = 0.09 / 0.6. Simplify the fraction carefully by eliminating decimals (e.g., multiply numerator and denominator by 100) and then reduce by common factors to a clean ratio. Step-by-Step Solution: 0.6x = 0.09y ⇒ x / y = 0.09 / 0.6.Multiply top and bottom by 100 ⇒ 9 / 60.Reduce by GCD 3 ⇒ 3 / 20.Therefore, (first : second) = 3 : 20. Verification / Alternative check: Choose y = 20 and x = 3. Then 0.6x = 1.8 and 0.09y = 1.8, satisfying the condition. Why Other Options Are Wrong: 2 : 3, 1 : 15, 20 : 3, and 5 : 18 do not satisfy 0.6x = 0.09y when tested; only 3 : 20 balances the equality. Common Pitfalls: Inverting the ratio (20 : 3) by mistake or dropping zeros incorrectly when clearing decimals. Always double-check by substituting simple values. Final Answer: 3 : 20

Question 7

Percent-to-ratio conversion: If 10% of A equals 20% of B, express the simplified ratio A : B and explain the proportional reasoning behind the result.

Accepted Answer

Introduction / Context: Converting a statement about percentages into a direct ratio is a common aptitude task. The goal is to turn a% of A = b% of B into A : B by isolating the variables and reducing to simplest terms. Given Data / Assumptions: 0.10A = 0.20B. A, B > 0. Concept / Approach: Divide both sides by 0.10 (or equivalently compare coefficients) to get A in terms of B, then write the ratio A : B and reduce if necessary. Step-by-Step Solution: 0.10A = 0.20B ⇒ A = (0.20/0.10)B = 2B.Therefore A : B = 2 : 1. Verification / Alternative check: Test with B = 1 ⇒ A = 2. Then 10% of A = 0.2; 20% of B = 0.2. Equality holds. Why Other Options Are Wrong: 1 : 2 reverses the ratio; 5 : 1 and 10 : 1 overstate A relative to B; 4 : 1 is not supported by the given equality. Common Pitfalls: Mixing up which side to divide by, or comparing 10 and 20 as if they alone define the ratio without accounting for the variables. Final Answer: 2 : 1

Question 8

Reverse ratio required: Given A : B = 2 : 3 and B : C = 4 : 5, compute C : A by first forming A : B : C and then inverting the order to obtain the required pair.

Accepted Answer

Introduction / Context: Often a problem asks for C : A rather than A : C. The proper method is still to combine the given ratios by matching the common term (here, B) and then read off the required order. Given Data / Assumptions: A : B = 2 : 3. B : C = 4 : 5. Concept / Approach: Make B the same in both ratios, derive A : B : C, and then express C : A directly from the obtained triple. Step-by-Step Solution: Let A : B = 2x : 3x and B : C = 4y : 5y.Set 3x = 4y ⇒ choose x = 4 and y = 3.Then A = 8, B = 12, C = 15 ⇒ A : B : C = 8 : 12 : 15.Therefore C : A = 15 : 8. Verification / Alternative check: Subratios check: A : B = 8 : 12 = 2 : 3; B : C = 12 : 15 = 4 : 5. Why Other Options Are Wrong: 8 : 15 reverses the order; 12 : 10, 8 : 5, and 5 : 8 do not match the combined ratio. Common Pitfalls: Answering A : C instead of C : A, or failing to align B properly before concluding. Final Answer: 15 : 8

Question 9

Alloy composition application: In a brass piece, the proportion of Zinc to Copper is 13 : 7 by weight. For a 100 kg piece of this brass, how many kilograms of Zinc are present?

Accepted Answer

Introduction / Context: Ratio problems in mixtures and alloys translate directly into part-to-total calculations. If the composition by weight is known, the mass of a component is its ratio share over the total ratio sum times the total mass. Given Data / Assumptions: Zinc : Copper = 13 : 7. Total mass of brass piece = 100 kg. Concept / Approach: Total ratio parts = 13 + 7 = 20. Zinc share = (13/20) of the total mass. Multiply by 100 kg to get kilograms of Zinc. Step-by-Step Solution: Total parts = 13 + 7 = 20.Zinc mass = (13/20) * 100 kg = 65 kg. Verification / Alternative check: Copper mass = (7/20) * 100 = 35 kg. 65 + 35 = 100 kg, confirming the partition. Why Other Options Are Wrong: 20 kg and 55 kg understate the Zinc share; 70 kg would imply 14 : 6 ratio; 35 kg is actually the Copper portion. Common Pitfalls: Using 13% instead of the ratio fraction (13/20 = 65%); swapping component labels when interpreting the given order. Final Answer: 65 kg

Question 10

Aggregating participation fractions: In a school, one-fourth of the boys and three-eighths of the girls participated in the annual sports. Without the ratio of boys to girls, can we find what fraction of the total student population participated?

Accepted Answer

Introduction / Context: When different fractions apply to subgroups (boys and girls), the overall fraction of participation depends on how large each subgroup is. Without the relative sizes (or ratio) of the subgroups, the combined fraction cannot be uniquely determined. Given Data / Assumptions: Participating boys = 1/4 of boys. Participating girls = 3/8 of girls. No information on the ratio (or counts) of boys to girls. Concept / Approach: Overall participation fraction = (participating boys + participating girls) / total students. This equals ( (1/4)*B + (3/8)*G ) / (B + G ). The expression depends on B and G separately, not just their sum, so B : G is required to get a single number. Step-by-Step Solution: Let B be number of boys, G be number of girls.Overall fraction f = ( (1/4)B + (3/8)G ) / (B + G ).This simplifies only if B and G (or B : G) are known; otherwise f varies with the mix of B and G. Verification / Alternative check: Example 1: If B = G, f = ( (1/4) + (3/8) ) / 2 per capita = (1/4 + 3/8)/(1 + 1) in ratio terms, leading to one value. Example 2: If B is much larger than G, the overall fraction shifts toward 1/4; if G dominates, it shifts toward 3/8. Hence no unique answer. Why Other Options Are Wrong: 4/12 and 8/12 assume equal subgroup sizes without stating it; 5/8 is impossible as it exceeds both contributing subgroup fractions unless weights are extreme and still cannot be justified uniquely; 11/24 is a specific number requiring a specific B : G not given. Common Pitfalls: Averaging 1/4 and 3/8 directly (arithmetic mean) or adding numerators/denominators—both are invalid without weights. Final Answer: Data inadequate

Question 11

Share based on a conditional relation: ₹ 1050 is to be divided among P, Q, and R such that P’s share equals 2/5 of the combined share of Q and R. Find P’s amount received.

Accepted Answer

Introduction / Context: Distribution problems often provide a relationship between one person’s share and the combined shares of others. Translating this relationship into an equation with the total allows solving for the individual amounts cleanly. Given Data / Assumptions: Total amount = ₹ 1050. P = (2/5) * (Q + R). Concept / Approach: Let S = Q + R. Then P = (2/5)S and total is P + S = 1050. Solve for S, then compute P. This method avoids dealing with Q and R separately. Step-by-Step Solution: Let S = Q + R. Given P = (2/5)S.Total: P + S = (2/5)S + S = (7/5)S = 1050.So S = 1050 * (5/7) = 750.P = (2/5) * 750 = ₹ 300. Verification / Alternative check: Then Q + R = 750, P = 300; total 300 + 750 = 1050—consistent with the given total. Why Other Options Are Wrong: ₹ 200, ₹ 320, ₹ 420, and ₹ 350 do not satisfy the key relation P = (2/5)(1050 − P). Common Pitfalls: Misreading the condition as P = 2/5 of the total or as 2/5 of each of Q and R separately; the phrase “combined share of Q and R” means Q + R together. Final Answer: Rs. 300

Question 12

Proportional division with a combined constraint: ₹ 385 is divided among A, B, and C such that A receives 2/9 of what B and C receive together. Determine A’s share accurately.

Accepted Answer

Introduction / Context: This is a classic share-splitting question with a condition tying one person’s amount to the sum of the others. Reducing the condition to one unknown for the combined share makes the arithmetic straightforward. Given Data / Assumptions: Total amount = ₹ 385. A = (2/9) * (B + C). Concept / Approach: Let T = B + C. Then A = (2/9)T and total is A + T = 385. Solve for T and then find A directly from the relation. Step-by-Step Solution: Let T = B + C ⇒ A = (2/9)T.Total: A + T = (2/9)T + T = (11/9)T = 385.Hence T = 385 * (9/11) = 35 * 9 = 315.A = (2/9) * 315 = ₹ 70. Verification / Alternative check: Check: A + T = 70 + 315 = 385; condition holds because 2/9 of 315 is 70. Why Other Options Are Wrong: ₹ 77, ₹ 82.50, and ₹ 85 do not satisfy A = (2/9)(385 − A); ₹ 90 is also inconsistent with the required fraction. Common Pitfalls: Treating 2/9 as a part of the whole instead of specifically of (B + C); ensure you form A + (B + C) for the total. Final Answer: Rs. 70

Question 13

Speed–time inverse relationship: The speeds of three cars are in the ratio 3 : 4 : 5. For the same distance, what is the ratio of the times taken by them? Provide the simplified integer ratio.

Accepted Answer

Introduction / Context: For a fixed distance, time is inversely proportional to speed. Thus, if speeds are in one ratio, times will be in the reciprocal ratio. Converting these reciprocals into integers makes the answer clear and comparable. Given Data / Assumptions: Speed ratio = 3 : 4 : 5. All travel the same distance. Concept / Approach: If v ∝ 3 : 4 : 5, then time t ∝ 1/3 : 1/4 : 1/5. Multiply by the least common multiple of denominators (here 60) to eliminate fractions and obtain an integer ratio. Step-by-Step Solution: t ∝ 1/3 : 1/4 : 1/5.Multiply each by 60 ⇒ 20 : 15 : 12.Therefore the time ratio is 20 : 15 : 12 (slowest car takes the longest time). Verification / Alternative check: Check consistency: times multiplied by speeds should be equal (distance). 20*3 = 60, 15*4 = 60, 12*5 = 60—valid. Why Other Options Are Wrong: 3 : 4 : 5 is the speed ratio, not time; 5 : 4 : 3 is not the correct inverse; 12 : 15 : 20 reverses from fastest to slowest. Common Pitfalls: Forgetting that time and speed are inversely related; copying the speed ratio directly for time. Final Answer: 20 : 15 : 12

Question 14

Total cost from a part and a ratio split: The cost of making an article is split between materials, labour, and overheads in the ratio 3 : 4 : 1. If the materials cost ₹ 67.50, find the total manufacturing cost of the article.

Accepted Answer

Introduction / Context: When a specific component of a total is given alongside the ratio of all components, you can find the “value per part” and then multiply by total parts. This is a standard proportional allocation technique. Given Data / Assumptions: Materials : Labour : Overheads = 3 : 4 : 1. Materials cost = ₹ 67.50 (corresponds to 3 parts). Concept / Approach: Compute one-part value as materials cost divided by its ratio part (3). Sum all ratio parts to get total parts, then multiply by one-part value to obtain the total cost. Step-by-Step Solution: One-part value = 67.50 / 3 = ₹ 22.50.Total parts = 3 + 4 + 1 = 8.Total cost = 8 * ₹ 22.50 = ₹ 180. Verification / Alternative check: Labour = 4 parts = ₹ 90; Overheads = 1 part = ₹ 22.50; Sum: 67.50 + 90 + 22.50 = 180—checks out. Why Other Options Are Wrong: ₹ 122.50 and ₹ 200 misapply the parts; ₹ 380 and ₹ 540 are far from the proportional sum. Common Pitfalls: Taking ₹ 67.50 as 3% or assuming equal splits; always map actual rupees to the ratio parts given. Final Answer: Rs. 180

Question 15

Equal radius, equal volume — cylinder vs cone: A right circular cylinder and a right circular cone have the same radius and the same volume. Find the ratio of the height of the cylinder to the height of the cone.

Accepted Answer

Introduction / Context: For solids with equal radius, comparing volumes quickly leads to a proportional relation between their heights. This question leverages the standard formulas for cylinder and cone volumes, focusing on their height relationship at equal volumes. Given Data / Assumptions: Common radius r for both solids. Volumes are equal. Concept / Approach: Volume of cylinder = π r^2 * h_cyl. Volume of cone = (1/3) * π r^2 * h_cone. Setting these equal and cancelling common terms yields a simple ratio between heights. Step-by-Step Solution: π r^2 * h_cyl = (1/3) * π r^2 * h_cone.Cancel π r^2 (nonzero) ⇒ h_cyl = (1/3) * h_cone.Therefore, h_cyl : h_cone = 1 : 3. Verification / Alternative check: If h_cone = 30, then h_cyl = 10. Substituting gives equal volumes, confirming the ratio. Why Other Options Are Wrong: 3 : 1 inverts the relation; 3 : 5 and 2 : 5 suggest mismatched formulas; 1 : 2 does not satisfy the equality of volumes for equal radii. Common Pitfalls: Forgetting the 1/3 factor in the cone’s volume; mixing radius equality with diameter or slant height. Final Answer: 1 : 3

Question 16

Translate a complement statement to a ratio: If 3/4 of the employees in a supermarket are not college graduates, what is the ratio of college graduates to non-graduates?

Accepted Answer

Introduction / Context: Statements about who is not in a category can be flipped to find proportions that are in the category. Here, knowing three-fourths are not graduates determines the remaining one-fourth who are, enabling a direct ratio. Given Data / Assumptions: Non-graduates fraction = 3/4 of total employees. Graduates fraction = 1 − 3/4 = 1/4. Concept / Approach: Ratio of graduates to non-graduates is simply (1/4) : (3/4). Cancel the common 1/4 factor to simplify the ratio to smallest integers. Step-by-Step Solution: Graduates : Non-graduates = (1/4) : (3/4).Divide both by 1/4 ⇒ 1 : 3. Verification / Alternative check: For a hypothetical 100 employees, 25 graduates and 75 non-graduates give 25 : 75 = 1 : 3. Why Other Options Are Wrong: 1 : 4 suggests only one-fifth are graduates; 3 : 1 reverses roles; 3 : 4 mixes categories; 4 : 1 implies 80% graduates, contradicting the premise. Common Pitfalls: Confusing the complement; ensure you subtract from 1 to get the remaining fraction for graduates. Final Answer: 1 : 3

Question 17

Apply a given ratio to a known total: A school has 840 students. The ratio of students taking English to those not taking English is 4 : 3. How many students take English?

Accepted Answer

Introduction / Context: Ratios split a whole into proportional parts. With a total and a two-part ratio, compute the sum of ratio parts to find one-part value, then scale to get the required subgroup size. Given Data / Assumptions: Total students = 840. English : Not English = 4 : 3. Concept / Approach: Total ratio parts = 4 + 3 = 7. One part is 840 / 7. Multiply by the number of parts for the English group to get its headcount. Step-by-Step Solution: One-part value = 840 / 7 = 120.English students = 4 parts = 4 * 120 = 480. Verification / Alternative check: Not-English students = 3 * 120 = 360; sum 480 + 360 = 840, which matches the total. Why Other Options Are Wrong: 360 is the non-English count; 270 and 420 misuse the ratio; 540 exceeds the 4/7 share of 840. Common Pitfalls: Dividing 840 by 4 or 3 directly; always divide by the sum of ratio parts before scaling. Final Answer: 480

Question 18

Coins in a fixed ratio with a specified total value: A bag contains 25p, 10p, and 5p coins in the ratio 1 : 2 : 3. If the total value is ₹ 60, how many 5p coins are there? (1 rupee = 100 paise.)

Accepted Answer

Introduction / Context: When coin counts are in a ratio but denomination values differ, total value equals the sum of (count * denomination) across types. Express counts using a common multiplier, compute value per “x”, and solve for x from the total value. Then extract the required count. Given Data / Assumptions: Coin count ratio (25p : 10p : 5p) = 1 : 2 : 3. Total value = ₹ 60 = 6000 paise. Denominations: 25p, 10p, 5p. Concept / Approach: Let counts be x, 2x, 3x. Total value V (in paise) = 25x + 10*(2x) + 5*(3x). Simplify, equate to 6000, then solve for x. Finally, 5p coins = 3x. Step-by-Step Solution: Let counts = x (25p), 2x (10p), 3x (5p).Total value (paise) = 25x + 20x + 15x = 60x.Given value = ₹ 60 = 6000 paise ⇒ 60x = 6000 ⇒ x = 100.Number of 5p coins = 3x = 300. Verification / Alternative check: Actual value check: 25*100 + 10*200 + 5*300 = 2500 + 2000 + 1500 = 6000 paise = ₹ 60—consistent. Why Other Options Are Wrong: 200, 180, 240, and 150 do not equal 3x with x = 100; they would produce a total value different from ₹ 60. Common Pitfalls: Forgetting to convert rupees to paise; mixing value ratio with count ratio; or setting counts directly to 1, 2, 3 instead of x, 2x, 3x. Final Answer: 300

Question 19

Equalized multiples define shares: Some money is divided among A, B, and C such that 5 times A’s share equals 12 times B’s share, which also equals 6 times C’s share. Determine the ratio A : B : C.

Accepted Answer

Introduction / Context: When several scaled versions of different shares are equal, set each equal to a common constant. Then express each share in terms of that constant and clear denominators to obtain an integer ratio. Given Data / Assumptions: 5A = 12B = 6C = k, for some k > 0. Concept / Approach: From 5A = k ⇒ A = k/5; from 12B = k ⇒ B = k/12; from 6C = k ⇒ C = k/6. Combine these into A : B : C and multiply by the LCM of denominators to get whole numbers. Step-by-Step Solution: A : B : C = k/5 : k/12 : k/6.Cancel k and multiply by LCM(5, 12, 6) = 60.60*(1/5) : 60*(1/12) : 60*(1/6) = 12 : 5 : 10. Verification / Alternative check: Using A : B : C = 12 : 5 : 10, confirm 5A : 12B : 6C = 60 : 60 : 60, which are equal as required. Why Other Options Are Wrong: 5 : 10 : 12 and 10 : 12 : 5 permute the correct order improperly; 5 : 12 : 10 misplaces A and B; 6 : 5 : 10 comes from picking an arbitrary scale rather than the exact LCM-based simplification. Common Pitfalls: Forgetting to use a common constant or not clearing denominators, which leads to fractional ratios or misordered terms. Final Answer: 12 : 5 : 10

Question 20

Two linked ratios with a fixed total: ₹ 750 is distributed among A, B, and C such that A : B = 5 : 2 and B : C = 7 : 13. Determine A’s share in rupees.

Accepted Answer

Introduction / Context: When two pairwise ratios are given with a common person in both, you can build a three-term ratio by aligning the shared term. After obtaining A : B : C, distribute the known total accordingly to find an individual’s share. Given Data / Assumptions: A : B = 5 : 2. B : C = 7 : 13. Total amount = ₹ 750. Concept / Approach: Let A : B = 5x : 2x and B : C = 7y : 13y. Equalize B (2x = 7y) to combine ratios. Then compute A : B : C, sum the parts, find one-part value, and scale for A’s rupees. Step-by-Step Solution: Set 2x = 7y ⇒ choose x = 7, y = 2 (for integers).Then A = 5x = 35, B = 2x = 14, C = 13y = 26.A : B : C = 35 : 14 : 26; total parts = 35 + 14 + 26 = 75.One part = ₹ 750 / 75 = ₹ 10.A’s share = 35 parts = 35 * ₹ 10 = ₹ 350. Verification / Alternative check: B’s share = 14 * 10 = ₹ 140; C’s = 26 * 10 = ₹ 260; sums to ₹ 750 and matches the given linked ratios. Why Other Options Are Wrong: ₹ 140 is B’s share; ₹ 250 and ₹ 260 belong to neither A nor the computed parts for A; ₹ 300 does not fit the derived three-term split. Common Pitfalls: Adding ratios directly without aligning the common term B; forgetting to divide the total by the sum of all parts before scaling. Final Answer: ₹ 350

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