Question 1

A 729 ml mixture contains milk and water in the ratio 7 : 2. How much additional water (in millilitres) must be added so that the new milk : water ratio becomes 7 : 3?

Accepted Answer

Introduction / Context: This is a classic mixture-and-ratio question. We know the initial volumes and composition (milk : water = 7 : 2) and we are asked to add only water so that the final mixture has milk : water = 7 : 3. Because only water is added, the milk quantity remains constant throughout the process. Given Data / Assumptions: Total initial mixture = 729 ml. Initial ratio milk : water = 7 : 2 (9 parts total). Only water is added; milk stays unchanged. Target ratio milk : water = 7 : 3 (10 parts total). Concept / Approach: Convert the given ratio into actual quantities using parts. Since there are 9 parts initially, each part = 729 / 9 = 81 ml. Milk is 7 parts and does not change. We then enforce the target ratio using the unchanged milk as the reference to compute the new total and hence the water to add. Step-by-Step Solution: Initial part size = 729 / 9 = 81 mlInitial milk = 7 * 81 = 567 ml (fixed)For ratio 7 : 3, total parts = 10; milk 7 parts ⇒ total = (567 / 7) * 10 = 81 * 10 = 810 mlWater to add = final total − initial total = 810 − 729 = 81 ml Verification / Alternative check: After adding 81 ml, water becomes initial water 162 ml + 81 ml = 243 ml, and milk is 567 ml. Ratio 567 : 243 = 7 : 3 (both divisible by 81). Correct. Why Other Options Are Wrong: 60, 70, 72, and 90 ml do not yield the exact 7 : 3 ratio when recalculated; the simplified ratio will deviate from 7 : 3. Common Pitfalls: Changing milk as well, or averaging ratios instead of working with actual quantities. Always lock the unchanged component and scale the other. Final Answer: 81 ml

Question 2

Two alloys of gold and copper are prepared in the ratios 7 : 2 and 7 : 11 respectively. Equal masses of these alloys are melted to form a third alloy C. What is the ratio of gold to copper in alloy C?

Accepted Answer

Introduction / Context: When mixing alloys, you must combine actual fractions of constituents, not the ratios directly. Since equal quantities (equal masses) of the two alloys are mixed, the overall fraction of a metal in the mixture is the simple average of the two alloys’ fractions for that metal. Given Data / Assumptions: Alloy A: gold : copper = 7 : 2 ⇒ gold fraction = 7/9. Alloy B: gold : copper = 7 : 11 ⇒ gold fraction = 7/18. Equal masses of A and B are mixed to form C. Concept / Approach: For equal masses, the gold fraction in C equals the average of the gold fractions of A and B. The copper fraction is then 1 − (gold fraction). Finally, convert the fractions into a simplest whole-number ratio. Step-by-Step Solution: Gold fraction in A = 7/9Gold fraction in B = 7/18Gold fraction in C = [(7/9) + (7/18)] / 2 = (14/18 + 7/18) / 2 = (21/18) / 2 = 21/36 = 7/12Copper fraction in C = 1 − 7/12 = 5/12Hence, gold : copper = (7/12) : (5/12) = 7 : 5 Verification / Alternative check: Assume 1 kg of A and 1 kg of B. Gold mass in C = 1*(7/9) + 1*(7/18) = 21/18 = 7/6 kg out of 2 kg total ⇒ fraction 7/12. Copper becomes 5/12. Ratio 7:5 confirmed. Why Other Options Are Wrong: They either invert the ratio or reflect averaging the ratios (incorrect) rather than the underlying fractions by mass. Common Pitfalls: Taking the mean of 7:2 and 7:11 directly; you must convert ratios to fractions first and then average because mixing is by mass. Final Answer: 7 : 5

Question 3

Some money is divided among A, B, and C such that 5 times A’s share, 3 times B’s share, and 2 times C’s share are all equal. What is the ratio A : B : C?

Accepted Answer

Introduction / Context: The statement “5A, 3B, and 2C are equal” defines a proportional relationship. To extract the basic ratio A : B : C, express A, B, and C in terms of the same common constant and then reduce to whole-number parts. Given Data / Assumptions: 5A = 3B = 2C = k (some positive constant). Find A : B : C. Concept / Approach: From 5A = k ⇒ A = k/5; from 3B = k ⇒ B = k/3; from 2C = k ⇒ C = k/2. The ratio A : B : C is then (1/5) : (1/3) : (1/2). Convert to a common denominator to write whole-number parts. Step-by-Step Solution: A = k/5, B = k/3, C = k/2A : B : C = (1/5) : (1/3) : (1/2)Using denominator 30: (1/5, 1/3, 1/2) = (6/30, 10/30, 15/30) ⇒ 6 : 10 : 15 Verification / Alternative check: Multiply 6 : 10 : 15 by 5, 3, 2 respectively to see equality: 5*6 = 30; 3*10 = 30; 2*15 = 30. Condition satisfied. Why Other Options Are Wrong: They are permutations or scaled versions that do not respect 5A = 3B = 2C when checked. Common Pitfalls: Thinking A : B : C equals 5 : 3 : 2 directly; remember the multiples of the shares are equal, not the shares themselves. Final Answer: 6 : 10 : 15

Question 4

Split 94 into two parts such that one-fifth of the first part and one-eighth of the second part are in the ratio 3 : 4. What is the first part?

Accepted Answer

Introduction / Context: This problem turns a verbal ratio condition on fractional parts into equations in two variables. Once the ratio is translated, cross-multiplication and the sum constraint lead directly to the required parts. Given Data / Assumptions: Total = 94 split into x (first part) and y (second part). (x/5) : (y/8) = 3 : 4. x + y = 94. Concept / Approach: From (x/5)/(y/8) = 3/4 ⇒ (8x)/(5y) = 3/4 ⇒ 32x = 15y. Express y in terms of x and use x + y = 94 to solve for x. Step-by-Step Solution: 32x = 15y ⇒ y = (32/15)xx + (32/15)x = 94 ⇒ (47/15)x = 94x = 94 * 15 / 47 = 30Hence, first part = 30 Verification / Alternative check: Compute parts: First = 30, Second = 64. Then x/5 = 6 and y/8 = 8; ratio 6 : 8 reduces to 3 : 4, as required. Why Other Options Are Wrong: 27, 36, 48, and 24 do not satisfy both the ratio condition and the sum constraint simultaneously. Common Pitfalls: Setting 1/5 of the first equal to 1/8 of the second (equality) instead of the specified 3 : 4 ratio; forgetting to apply the total-sum constraint after forming the ratio equation. Final Answer: 30

Question 5

Rs. 680 is divided among A, B, and C. It is known that A gets two-thirds of B’s share, and B gets one-fourth of C’s share. What is B’s share?

Accepted Answer

Introduction / Context: The statement links each person’s share multiplicatively. The cleanest method is to express all shares in terms of a single variable, typically C, then enforce the total sum to get numerical values. Given Data / Assumptions: A = (2/3)B B = (1/4)C A + B + C = 680 Concept / Approach: Write everything in terms of C. Then compute the total as a multiple of C, solve for C, and back-calculate B. This avoids fractional tangle and ensures consistency. Step-by-Step Solution: B = C/4A = (2/3) * (C/4) = C/6Total = C/6 + C/4 + C = (2/12 + 3/12 + 12/12)C = (17/12)C(17/12)C = 680 ⇒ C = 680 * 12 / 17 = 480B = C/4 = 480/4 = 120 Verification / Alternative check: Compute A = C/6 = 80, then A + B + C = 80 + 120 + 480 = 680 matches. Also A = (2/3)B = 80 (correct) and B = (1/4)C = 120 (correct). Why Other Options Are Wrong: 60, 80, 160, and 140 do not satisfy all three conditions simultaneously when checked. Common Pitfalls: Misreading the statement as A equals two-thirds of (B plus something), or treating “one-fourth of C” as C minus one-fourth. Keep it multiplicative. Final Answer: Rs. 120

Question 6

Rs. 2430 is divided among A, B, and C. If their shares are reduced by Rs. 5, Rs. 10, and Rs. 15 respectively, the resulting amounts are in the ratio 3 : 4 : 5. What is A’s original share?

Accepted Answer

Introduction / Context: Adjusted ratios often point to “let the new amounts be 3k, 4k, 5k,” then add back the adjustments to retrieve the original shares. The sum of original shares must equal the given total. Given Data / Assumptions: A − 5 : B − 10 : C − 15 = 3 : 4 : 5 A + B + C = 2430 Concept / Approach: Let A − 5 = 3k, B − 10 = 4k, C − 15 = 5k. Then A = 3k + 5, B = 4k + 10, C = 5k + 15. Summing and equating to 2430 yields k; substitute back to get A. Step-by-Step Solution: A = 3k + 5, B = 4k + 10, C = 5k + 15A + B + C = (3k + 5) + (4k + 10) + (5k + 15) = 12k + 30 = 243012k = 2400 ⇒ k = 200A = 3k + 5 = 600 + 5 = 605 Verification / Alternative check: New amounts: A − 5 = 600, B − 10 = 810, C − 15 = 985. Their ratio is 600 : 810 : 985 = 3 : 4 : 5 after simplifying by a common factor of 200 for the first two and checking the third consistency (constructed via 5k). Why Other Options Are Wrong: Other choices do not fit the 3k, 4k, 5k structure when rechecked with the original conditions. Common Pitfalls: Forgetting to add back the reductions or summing the reduced amounts to 2430 (which would be incorrect). Final Answer: Rs. 605

Question 7

The sides of a triangle are in the ratio 1/3 : 1/4 : 1/5 and its perimeter is 94 cm. What is the length of the smallest side?

Accepted Answer

Introduction / Context: Ratios with fractional terms can be cleared by multiplying all terms by the least common multiple (LCM) of the denominators. This produces integer parts that sum to a total number of parts matching the perimeter scale. Given Data / Assumptions: Sides ∝ 1/3 : 1/4 : 1/5 Perimeter = 94 cm Concept / Approach: Multiply each term by LCM(3,4,5) = 60 to obtain equivalent integer parts: 20 : 15 : 12. Sum these parts; each side is then proportional to its part times a common multiplier derived from the perimeter. Step-by-Step Solution: Convert ratio: (1/3 : 1/4 : 1/5) × 60 = 20 : 15 : 12Total parts = 20 + 15 + 12 = 47Common multiplier = perimeter / total parts = 94 / 47 = 2Smallest side corresponds to part 12 ⇒ 12 * 2 = 24 cm Verification / Alternative check: Other sides: 20*2 = 40 cm, 15*2 = 30 cm; sum 40 + 30 + 24 = 94 cm, confirming correctness. Why Other Options Are Wrong: They do not match the required scaling once the fractional ratio is converted to integers and matched to the perimeter. Common Pitfalls: Adding fractions directly or confusing which side is “smallest” after clearing denominators; the smallest is the 1/5 term (becoming 12 parts). Final Answer: 24 cm

Question 8

Rs. 53 is divided among A, B, and C such that A gets Rs. 7 more than B, and B gets Rs. 8 more than C. What is the ratio A : B : C?

Accepted Answer

Introduction / Context: When shares differ by fixed amounts, it’s efficient to set the smallest as a variable and express others relative to it. Then apply the total to find actual amounts and convert to a ratio. Given Data / Assumptions: A = B + 7 B = C + 8 A + B + C = 53 Concept / Approach: Let C = x ⇒ B = x + 8 ⇒ A = x + 15. Sum them to equal 53, solve for x, and then form the ratio from the actual amounts. Step-by-Step Solution: C = x, B = x + 8, A = x + 15Sum = (x + 15) + (x + 8) + x = 533x + 23 = 53 ⇒ 3x = 30 ⇒ x = 10Hence (A, B, C) = (25, 18, 10)Ratio A : B : C = 25 : 18 : 10 Verification / Alternative check: Check differences: A − B = 7 and B − C = 8. Sum 25 + 18 + 10 = 53. All conditions satisfied. Why Other Options Are Wrong: They fail either the fixed-difference conditions or the total sum when examined. Common Pitfalls: Accidentally using 7 and 8 as ratios rather than differences; not preserving the order A, then B, then C when forming the ratio. Final Answer: 25 : 18 : 10

Question 9

A sum of money is divided among A, B, and C so that to each rupee A gets, B gets 65 paise and C gets 35 paise. If C’s share is Rs. 28, what is the total sum?

Accepted Answer

Introduction / Context: This is a direct ratio problem with mixed units (rupees and paise). Convert all to a consistent unit or scale up by 100 to avoid decimals, then apply the part-to-total logic using the known share of C. Given Data / Assumptions: A : B : C = 1.00 : 0.65 : 0.35 C’s amount = Rs. 28 Concept / Approach: Scale the ratio by 100 to eliminate decimals: 100 : 65 : 35, which simplifies by 5 to 20 : 13 : 7. With C corresponding to 7 parts, find the value of one part and then the total. Step-by-Step Solution: Scaled ratio = 100 : 65 : 35 = 20 : 13 : 7C corresponds to 7 parts = Rs. 28 ⇒ 1 part = 28 / 7 = Rs. 4Total parts = 20 + 13 + 7 = 40Total sum = 40 * Rs. 4 = Rs. 160 Verification / Alternative check: Compute A = 20 * 4 = Rs. 80, B = 13 * 4 = Rs. 52, C = 7 * 4 = Rs. 28; sum = 160 and proportions 1 : 0.65 : 0.35 hold when normalized. Why Other Options Are Wrong: 120, 140, 180, 200 do not align with C = 28 at 7 parts when the ratio is correctly simplified. Common Pitfalls: Forgetting to convert paise to rupees or not simplifying the ratio, leading to incorrect part values. Final Answer: Rs. 160

Question 10

The incomes of A and B are in the ratio 3 : 2 and their expenditures are in the ratio 5 : 3. If each saves Rs. 1,000 per month, what is A’s income?

Accepted Answer

Introduction / Context: This is a paired-ratio savings problem. Define incomes and expenditures with separate multipliers and use “saving = income − expenditure.” Two linear equations arise and can be solved quickly for the income scale. Given Data / Assumptions: Income ratio A : B = 3 : 2 ⇒ incomes = 3x and 2x. Expenditure ratio A : B = 5 : 3 ⇒ expenditures = 5y and 3y. Savings: 3x − 5y = 1000 and 2x − 3y = 1000. Concept / Approach: Solve the two simultaneous linear equations for x and y. Once x is known, A’s income is 3x. Elimination is quick by aligning one variable’s coefficients. Step-by-Step Solution: 3x − 5y = 1000 … (1)2x − 3y = 1000 … (2)Multiply (2) by 5: 10x − 15y = 5000Multiply (1) by 3: 9x − 15y = 3000Subtract: x = 2000 ⇒ A’s income = 3x = Rs. 6000 Verification / Alternative check: Compute y from (2): 2*2000 − 3y = 1000 ⇒ y = 1000. Expenditures become 5y = 5000 (A) and 3y = 3000 (B). Savings: 6000 − 5000 = 1000 and 4000 − 3000 = 1000, correct. Why Other Options Are Wrong: Other choices do not satisfy both savings equations simultaneously when checked. Common Pitfalls: Assuming income and expenditure share the same multiplier or directly subtracting ratios; keep separate multipliers for each ratio. Final Answer: Rs. 6000

Question 11

Between two stations, first-class and second-class fares are in the ratio 4 : 1, while the number of first-class to second-class passengers is 1 : 40. If the total fare collected is Rs. 1100, how much is collected from first-class passengers?

Accepted Answer

Introduction / Context: Total fare depends on both the fare per passenger and the number of passengers. Multiply the fare ratio by the passenger ratio to get revenue “parts” from each class, then proportion the actual total accordingly. Given Data / Assumptions: Fare ratio (first : second) = 4 : 1 Passenger ratio (first : second) = 1 : 40 Total revenue = Rs. 1100 Concept / Approach: Revenue parts: First-class contributes 4 * 1 = 4 parts; second-class contributes 1 * 40 = 40 parts. Total revenue parts = 44. The first-class share is 4/44 of the total. Step-by-Step Solution: Revenue parts: First = 4*1 = 4; Second = 1*40 = 40Total parts = 4 + 40 = 44First-class revenue = (4/44) * 1100 = Rs. 100 Verification / Alternative check: Second-class revenue = (40/44)*1100 = Rs. 1000; adding to 100 gives Rs. 1100, matching the total. Why Other Options Are Wrong: Other amounts do not equal 4/44 of 1100; they would misallocate the proportional shares. Common Pitfalls: Adding ratios or averaging instead of multiplying fare and passenger ratios; forgetting that revenue is fare × count. Final Answer: Rs. 100

Question 12

The students in three classes are in the ratio 2 : 3 : 5. After 20 students are added to each class, the ratio becomes 4 : 5 : 7. What was the total number of students before the increase?

Accepted Answer

Introduction / Context: When equal numbers are added to each term of a ratio trio, the overall ratio changes in a predictable way. Setting the original classes as multiples of a common factor allows solving with simple equations using two of the ratio comparisons. Given Data / Assumptions: Original classes = 2k, 3k, 5k After adding 20 each: (2k + 20) : (3k + 20) : (5k + 20) = 4 : 5 : 7 Concept / Approach: Form a proportion from any two adjacent classes and solve for k. Then compute the original total as 2k + 3k + 5k = 10k. Step-by-Step Solution: (2k + 20)/(3k + 20) = 4/55(2k + 20) = 4(3k + 20) ⇒ 10k + 100 = 12k + 80 ⇒ 2k = 20 ⇒ k = 10Original total = 10k = 100 Verification / Alternative check: Check second proportion: (3k + 20)/(5k + 20) with k = 10 ⇒ (50)/(70) = 5/7, confirming the final ratio. Why Other Options Are Wrong: 10 and 90 are too small; “None of these” is invalid because 100 satisfies both ratio conditions. Common Pitfalls: Adding 20 to the ratio numbers directly; ratios are not absolute counts. Always return to the k-based actual numbers. Final Answer: 100

Question 13

One year ago, Laxman’s and Gopal’s salaries were in the ratio 3 : 4. Their individual salary growth ratios (last year : this year) are 4 : 5 for Laxman and 2 : 3 for Gopal. If their current total salary is Rs. 4,160, what is Laxman’s current salary?

Accepted Answer

Introduction / Context: When salaries grow by specified ratios, treat those as multiplicative factors on last year’s amounts. Start from a base ratio for last year and apply growth factors to obtain current figures, then use the current total to solve. Given Data / Assumptions: Last year L : G = 3 : 4 ⇒ L0 = 3t, G0 = 4t Growth: L current = (5/4)L0, G current = (3/2)G0 Current total = 4160 Concept / Approach: Compute current salaries in terms of t and set their sum to 4160. Solve for t, then find Laxman’s current salary. Step-by-Step Solution: Lnow = (5/4)*3t = 15t/4Gnow = (3/2)*4t = 6tTotal = 15t/4 + 6t = (15t + 24t)/4 = 39t/4 = 4160t = 4160*4/39 = 16640/39Laxman now = 15t/4 = 15*(16640/39)/4 = 20800/13 = Rs. 1600 Verification / Alternative check: Gopal now = 6t = 6*(16640/39) = 99840/39 = Rs. 2560; total 1600 + 2560 = 4160 as given. Why Other Options Are Wrong: 1040, 3120, 2080, and 2560 do not represent Laxman’s current salary under the given growth factors and total. Common Pitfalls: Adding growth ratios to last-year salaries instead of multiplying; mixing up whose growth factor is which. Final Answer: Rs. 1600

Question 14

Three students A, B, and C hire a computer for a month. Usage is proportional to (number of floppies run) × (number of days). A runs 27 floppies for 19 days, B runs 21 for 17 days, and C runs 24 for 23 days. If the rent is ₹ 23,700, how much should …

Accepted Answer

Introduction / Context: Cost-sharing by usage typically uses a proportional metric. Here, the metric is (floppies run) × (days). Each student’s share equals total rent multiplied by his usage fraction over total usage. Given Data / Assumptions: A-usage = 27 × 19 B-usage = 21 × 17 C-usage = 24 × 23 Total rent = ₹ 23,700 Concept / Approach: Compute each usage, sum them, form A’s fraction, and multiply by the total rent. Simplify ratios where possible to minimize arithmetic. Step-by-Step Solution: A = 27*19 = 513B = 21*17 = 357C = 24*23 = 552Total usage = 513 + 357 + 552 = 1422A’s share = 23700 * (513 / 1422)Reduce: 23700/1422 = 3950/237 and 513/237 = 171/79Hence A’s share = (171/79)*3950 = 50*171 = ₹ 8550 Verification / Alternative check: Compute B and C similarly; the three shares will sum to ₹ 23,700, confirming proportional division. Why Other Options Are Wrong: Only ₹ 8550 matches the exact proportional calculation; others deviate from the correct fraction 513/1422. Common Pitfalls: Adding floppies and days instead of multiplying; forgetting to use exact arithmetic leading to rounding errors in large numbers. Final Answer: ₹ 8550

Question 15

The work done by (x + 2) men in (x − 3) days equals the work done by x men in (x − 2) days (equal efficiencies). Find the value of x.

Accepted Answer

Introduction / Context: With equal efficiencies, work is proportional to (number of men) × (number of days). Equating two such products yields a quadratic that simplifies to a linear equation once expanded and common terms cancel. Given Data / Assumptions: (x + 2)(x − 3) = x(x − 2) Men have identical work rates. Concept / Approach: Expand both sides, cancel matching x^2 terms, and solve the resulting linear equation for x. Check that the resulting days are positive. Step-by-Step Solution: Left: (x + 2)(x − 3) = x^2 − x − 6Right: x(x − 2) = x^2 − 2xEquate: x^2 − x − 6 = x^2 − 2x ⇒ −x − 6 = −2x ⇒ x = 6 Verification / Alternative check: Plug back: (6 + 2)(6 − 3) = 8*3 = 24 and 6*(6 − 2) = 6*4 = 24. Equality holds and days are positive. Why Other Options Are Wrong: 4, 8, and 9 do not satisfy the equation; they create unequal work totals. Common Pitfalls: Arithmetic errors on expansion; ignoring feasibility (days must be positive). Final Answer: 6

Question 16

One year ago, the ratio of mother’s age to daughter’s age was 4 : 1. Four years from now, the ratio will be 3 : 1. What was the ratio of their ages five years ago?

Accepted Answer

Introduction / Context: Age-ratio problems rely on expressing ages at different reference times from a common base. Start with last year’s ratio, solve for the base parameter, then compute ages at any other time point as required. Given Data / Assumptions: One year ago: Mother : Daughter = 4 : 1 ⇒ (M − 1) = 4k, (D − 1) = k Four years from now: (M + 4) : (D + 4) = 3 : 1 Concept / Approach: Express M and D in terms of k from the first condition, substitute into the future ratio, and solve for k. Then compute ages five years ago and form the ratio. Step-by-Step Solution: From (M − 1, D − 1) = (4k, k) ⇒ M = 4k + 1, D = k + 1Future ratio: (M + 4)/(D + 4) = 3/1(4k + 1 + 4)/(k + 1 + 4) = 3 ⇒ (4k + 5)/(k + 5) = 34k + 5 = 3k + 15 ⇒ k = 10Ages 5 years ago: M − 5 = (4k + 1 − 5) = 4k − 4 = 36; D − 5 = (k + 1 − 5) = k − 4 = 6 ⇒ ratio 36 : 6 = 6 : 1 Verification / Alternative check: Current ages: M = 41, D = 11. Last year 40 : 10 = 4 : 1; in 4 years 45 : 15 = 3 : 1. Both match. Why Other Options Are Wrong: They don’t match the back-calculated ages five years ago for k = 10. Common Pitfalls: Forgetting to shift the ages correctly when moving between “ago” and “after” time frames; mixing which ratio applies to which time. Final Answer: 6 : 1

Question 17

The ratio of expenditure to saving of a person is 26 : 3. If the monthly income is ₹ 7250, what is the monthly saving?

Accepted Answer

Introduction / Context: Income equals expenditure plus saving. Given the ratio expenditure : saving, we can treat income as the sum of parts, then compute the value of one part from the total income to get the saving. Given Data / Assumptions: Expenditure : Saving = 26 : 3 Income = ₹ 7250 Income = (26 + 3) parts = 29 parts Concept / Approach: Find one-part value by dividing income by 29. Multiply the one-part value by 3 to get savings. This is a direct part-to-total conversion. Step-by-Step Solution: Total parts = 26 + 3 = 29One part = 7250 / 29 = 250Saving = 3 parts = 3 * 250 = ₹ 750 Verification / Alternative check: Expenditure = 26 parts = 26 * 250 = ₹ 6500. Income check: 6500 + 750 = 7250. Why Other Options Are Wrong: 290, 350, 780, 680 are not 3 out of 29 parts of 7250; they break the ratio when recomputed. Common Pitfalls: Interchanging the order to “saving : expenditure”; always align parts with the given order. Final Answer: ₹ 750

Question 18

Three hikers A, B, and C each start with ₹ 50 and agree to share trip expenses equally. At the end, A has ₹ 20 left, B ₹ 30, and C ₹ 40. By how much must they settle their accounts?

Accepted Answer

Introduction / Context: With equal expense sharing, each person should end up having spent the same amount. Compute each person’s actual spending and compare with the fair share. Transfers should move money from those who underpaid to those who overpaid. Given Data / Assumptions: Initial cash: ₹ 50 each (total ₹ 150) Final cash: A ₹ 20, B ₹ 30, C ₹ 40 Total spent = initial total − final total Concept / Approach: Actual spent by a person = initial − final. The fair share is total spent divided by 3. Compare each person’s actual spend to the fair share and settle the differences with a single minimal transfer. Step-by-Step Solution: Total spent = 150 − (20 + 30 + 40) = 150 − 90 = ₹ 60Fair share = 60 / 3 = ₹ 20 eachA spent = 50 − 20 = ₹ 30 (overpaid ₹ 10)B spent = 50 − 30 = ₹ 20 (fair)C spent = 50 − 40 = ₹ 10 (underpaid ₹ 10)Settlement: C must pay ₹ 10 to A Verification / Alternative check: After transfer, each effectively spends ₹ 20. Totals remain consistent and no further transfers are needed. Why Other Options Are Wrong: They either move money in the wrong direction or between the wrong pair, failing to equalize actual spending to the fair share. Common Pitfalls: Confusing final balances with spending; always compute spending as initial − final, not the other way around. Final Answer: C must pay ₹ 10 to A

Question 19

₹ 1290 is divided between A, B, and C such that A’s share is 1½ times B’s share and B’s share is 1¾ times C’s share. What is C’s share?

Accepted Answer

Introduction / Context: This chained multiplicative relationship is most easily solved by expressing all shares in terms of C. Then enforce the total to find C numerically and back out other shares if desired. Given Data / Assumptions: A = 1.5 B = (3/2)B B = 1.75 C = (7/4)C Total A + B + C = ₹ 1290 Concept / Approach: Substitute B in terms of C into A = (3/2)B to obtain A in terms of C. Sum A + B + C in terms of C to match the total, then solve for C. Step-by-Step Solution: B = (7/4)CA = (3/2)B = (3/2)*(7/4)C = (21/8)CSum = (21/8 + 7/4 + 1)C = (21/8 + 14/8 + 8/8)C = (43/8)C(43/8)C = 1290 ⇒ C = 1290 * 8 / 43 = 240 Verification / Alternative check: B = (7/4)*240 = 420; A = (21/8)*240 = 630; total 630 + 420 + 240 = 1290. Why Other Options Are Wrong: 200, 300, 400, 420 do not satisfy the given multiplicative relationships when summed to 1290. Common Pitfalls: Adding 1.5 and 1.75 to get a direct ratio; always multiply through the chain correctly. Final Answer: ₹ 240

Question 20

A starts a business with ₹ 10,000. After 4 months, B joins with ₹ 5,000. At the end of 12 months the total profit is ₹ 2,000. What is A’s share of the profit?

Accepted Answer

Introduction / Context: In partnerships, profit shares are proportional to “capital × time.” Compute each partner’s capital-months, form the profit-share ratio, and apply it to the total profit. Given Data / Assumptions: A invests ₹ 10,000 for 12 months. B invests ₹ 5,000 for 8 months (joins after 4 months). Total profit = ₹ 2,000. Concept / Approach: Capital-months: A = 10000 * 12; B = 5000 * 8. The ratio A : B determines the split of the profit. Step-by-Step Solution: A’s capital-months = 10000 * 12 = 120000B’s capital-months = 5000 * 8 = 40000A : B = 120000 : 40000 = 3 : 1A’s profit = (3/4) * 2000 = ₹ 1500 Verification / Alternative check: B’s profit = (1/4) * 2000 = ₹ 500. Sum 1500 + 500 = 2000, matching the total profit. Why Other Options Are Wrong: 800, 1000, 1200, and 1800 do not correspond to the 3 : 1 profit division based on capital-months. Common Pitfalls: Using months since start for both partners without adjusting for joining time; ignoring the time component leads to wrong shares. Final Answer: ₹ 1500

Ratio and Proportion Questions