Question 1

Recover ages from past ratio and present sum: The present total of ages of A, B, and C is 90. Ten years ago their ratio was 1 : 2 : 3. What is B’s present age?

Accepted Answer

Introduction / Context:Backdating ratios is common in age problems. Convert the past ratio into algebraic ages “then,” add the elapsed time to each, and match the present total to find the scale factor. Given Data / Assumptions: Present sum = 90 10 years ago ratio = 1 : 2 : 3 Concept / Approach:Let ages 10 years ago be x, 2x, 3x. Present ages are x+10, 2x+10, 3x+10. Sum them to 90 and solve for x, then compute B = 2x + 10. Step-by-Step Solution: (x + 10) + (2x + 10) + (3x + 10) = 906x + 30 = 90 ⇒ 6x = 60 ⇒ x = 10B now = 2x + 10 = 30 years Verification / Alternative check:Present ages are 20, 30, 40 which indeed sum to 90; roll back 10 years gives 10, 20, 30 = 1 : 2 : 3. Why Other Options Are Wrong:20 and 40 mismatch the computed 30; “None of these” is unnecessary. Common Pitfalls:Applying the 10-year shift to the sum only; each person’s age must be adjusted individually before summing. Final Answer:30 years

Question 2

Parent–children sum relationships: A father’s present age is four times the sum of his three children’s present ages. Six years hence, his age will be only double the sum of their ages. What is the father’s present age?

Accepted Answer

Introduction / Context:This is a two-equation problem involving a parent and the combined ages of three children, now and 6 years later. Because three children each gain 6 years, their total increases by 18. Given Data / Assumptions: Let S be the sum of the three children’s present ages Father now F = 4S After 6 years: F + 6 = 2 * (S + 18) Concept / Approach:Use the two linear relations to eliminate S. First relation gives F in terms of S; second relates F and S after 6 years. Solve for S, then get F immediately. Step-by-Step Solution: F = 4SF + 6 = 2(S + 18) ⇒ F + 6 = 2S + 36 ⇒ F = 2S + 30Equate: 4S = 2S + 30 ⇒ 2S = 30 ⇒ S = 15 ⇒ F = 4S = 60 Verification / Alternative check:Check future condition: F + 6 = 66 and S + 18 = 33; 2 * 33 = 66 ✓. Why Other Options Are Wrong:30, 40, 45 do not satisfy both the “now” and “after 6 years” constraints simultaneously. Common Pitfalls:Forgetting that three children together gain 3 * 6 = 18 in six years; using 6 instead of 18 leads to incorrect results. Final Answer:60 years

Question 3

Linked age statements: In ten years, A will be twice as old as B was 10 years ago. If A is now 9 years older than B, find B’s present age.

Accepted Answer

Introduction / Context:Time-shifted age relations are best handled by setting up simultaneous equations. Translate each sentence carefully into algebra and solve the linear system for present ages. Given Data / Assumptions: Let A and B be present ages A + 10 = 2 * (B − 10) A = B + 9 Concept / Approach:Substitute A = B + 9 into the first equation to get B directly; then retrieve A if needed. Ensure sign handling is correct when shifting time forward/backward. Step-by-Step Solution: (B + 9) + 10 = 2(B − 10)B + 19 = 2B − 20 ⇒ 19 + 20 = 2B − B ⇒ B = 39Hence A = B + 9 = 48 (optional) Verification / Alternative check:10 years from now, A = 58; 10 years ago, B = 29; twice 29 is 58 ✓. Why Other Options Are Wrong:40, 36, 49 do not satisfy both the time-shift constraint and the current age difference of 9 years. Common Pitfalls:Reversing the 10-year shifts or forgetting parentheses around (B − 10) when doubling. Final Answer:39 year

Question 4

Expression for present age from a past value: Rahul is 8 years younger than Ritu at present. If Ritu’s age was x three years ago, which expression represents Rahul’s present age?

Accepted Answer

Introduction / Context:We must convert a past value into a present value and then incorporate a fixed difference. Ritu’s present age is derived from “x three years ago,” and Rahul’s is 8 less than that. Given Data / Assumptions: Ritu 3 years ago = x Rahul is 8 years less than Ritu now Concept / Approach:Ritu now = x + 3. Therefore Rahul now = (x + 3) − 8 = x − 5. Choose the expression that matches x − 5 exactly. Step-by-Step Solution: Ritu now = x + 3Rahul now = (x + 3) − 8 = x − 5 Verification / Alternative check:Plug a sample x (e.g., x = 20) to confirm the relationship is preserved correctly through time shifting and difference subtraction. Why Other Options Are Wrong:x + 3 is Ritu now, not Rahul; x − 3 + 8 simplifies to x + 5, and x + 3 + 8 = x + 11—both contradict “8 less than Ritu now.” Common Pitfalls:Subtracting 8 from x (past) instead of from Ritu’s present age (x + 3). Final Answer:x - 5

Question 5

Recover present ages from ratio and past difference: Ramesh : Jayesh = 3 : 2 (present). Four years ago, Ramesh was 6 years older than Jayesh. Find Jayesh’s present age.

Accepted Answer

Introduction / Context:Combine a present-time ratio with a past-time difference. Express both present ages by a common factor and use the past condition to solve for the factor. Given Data / Assumptions: R : J = 3 : 2 ⇒ R = 3k, J = 2k Four years ago, R − 4 = (J − 4) + 6 Concept / Approach:Translate the past statement into an equation in k and solve. Then compute the requested present age of Jayesh. Step-by-Step Solution: 3k − 4 = 2k − 4 + 6 ⇒ 3k − 4 = 2k + 2k = 6 ⇒ Jayesh now = 2k = 12 Verification / Alternative check:Four years ago: R = 18−4=14, J = 12−4=8; difference = 6 ✓. Why Other Options Are Wrong:18 and 6 don’t satisfy both constraints together; “Data is inadequate” is false because the system is solvable and determinate. Common Pitfalls:Forgetting to apply the 4-year subtraction to both ages in the past condition; mixing present and past variables. Final Answer:12 years

Question 6

Past multiple and present sum: Four years ago, a father was 5 times as old as his son. The sum of their present ages is 44. What is the son’s present age?

Accepted Answer

Introduction / Context:This blends a past-time multiplicative relation with a present-time sum. Translate the past statement to an equation in present ages, then combine with the present sum to solve the system. Given Data / Assumptions: Let present ages be F (father) and S (son) F − 4 = 5 * (S − 4) F + S = 44 Concept / Approach:From the past relation express F in terms of S. Substitute into the present sum to compute S directly. Step-by-Step Solution: F − 4 = 5S − 20 ⇒ F = 5S − 16(5S − 16) + S = 44 ⇒ 6S = 60 ⇒ S = 10 Verification / Alternative check:Then F = 34. Four years ago: F = 30, S = 6, and 30 = 5 * 6 ✓. Why Other Options Are Wrong:6 and 8 are past values or misreads; 4 is inconsistent with the equations. Common Pitfalls:Applying the “times” relationship to present ages instead of the correctly shifted past ages. Final Answer:10 years

Question 7

Two-time-ratio system: One year ago, A : B = 3 : 4. One year hence, A : B = 5 : 6. What is B’s present age?

Accepted Answer

Introduction / Context:Two ratios at different times generate a 2×2 linear system in present ages. Solve systematically and check whether the resulting integer matches any listed options; if not, select “None of these.” Given Data / Assumptions: (A − 1) : (B − 1) = 3 : 4 (A + 1) : (B + 1) = 5 : 6 Concept / Approach:Convert ratios to linear equations by cross-multiplication and then solve simultaneously for A and B. Step-by-Step Solution: 4(A − 1) = 3(B − 1) ⇒ 4A − 3B = 16(A + 1) = 5(B + 1) ⇒ 6A − 5B = −1Solve: A = 4, then 4A − 3B = 1 ⇒ 16 − 3B = 1 ⇒ B = 5 Verification / Alternative check:Check both ratios: (4−1):(5−1)=3:4 and (4+1):(5+1)=5:6 ✓. Why Other Options Are Wrong:8, 4, and 6 do not equal the computed B = 5; hence “None of these.” Common Pitfalls:Sign errors when shifting ages by ±1; misapplying ratios to present instead of past/future ages. Final Answer:None of these (B’s present age is 5 years).

Question 8

Sum now and ratio 10 years ago: The sum of the present ages of Yogesh, Prakash, and Sameer is 93. Ten years ago their ages were in the ratio 2 : 3 : 4. What is Sameer’s present age?

Accepted Answer

Introduction / Context:Translate a past ratio into algebraic ages then, and shift forward by 10 years to reach present ages. Sum must match 93; solve for the scale factor and extract Sameer’s present age. Given Data / Assumptions: Present sum = 93 10 years ago ratio = 2 : 3 : 4 Concept / Approach:Let past ages be 2x, 3x, 4x. Present ages are 2x+10, 3x+10, 4x+10. Sum equals 93; solve for x and then compute Sameer = 4x + 10. Step-by-Step Solution: (2x + 10) + (3x + 10) + (4x + 10) = 939x + 30 = 93 ⇒ 9x = 63 ⇒ x = 7Sameer now = 4x + 10 = 28 + 10 = 38 Verification / Alternative check:Present ages sum to 93: (24 + 10) + (21 + 10) + (28 + 10) = 93 ✓. Why Other Options Are Wrong:32, 24, 34 do not match the computed 38; hence “None of these.” Common Pitfalls:Adding 10 only once to the total; each person’s age increases by 10. Final Answer:None of these (Sameer’s present age is 38 years).

Question 9

Father–son linear system: A man is five times as old as his son. Four years later, the sum of their ages will be 56. Find the son’s present age.

Accepted Answer

Introduction / Context:We combine a present multiplicative relation with a future sum. Express the father’s age in terms of the son’s age, then apply the future sum to solve for the son directly. Given Data / Assumptions: Let present ages be M and S M = 5S (M + 4) + (S + 4) = 56 Concept / Approach:Substitute M = 5S into the future-sum equation and solve for S. This yields the son’s current age immediately. Step-by-Step Solution: 5S + S + 8 = 56 ⇒ 6S = 48 ⇒ S = 8 Verification / Alternative check:Father now = 40. After 4 years: 44 and 12; sum = 56 ✓. Why Other Options Are Wrong:12, 5, and 6 do not satisfy both constraints simultaneously. Common Pitfalls:Adding 4 to only one of them; both ages advance by 4 years. Final Answer:8 years

Question 10

Father–son age total after 5 years: A son is 25 years younger than his father. The father was 45 four years ago. What will be the total of their ages after 5 years?

Accepted Answer

Introduction / Context:Determine the present ages first, then project forward by 5 years and add. Translate “was 45 four years ago” correctly into “is 49 now.” Apply the 25-year difference to obtain the son’s current age. Given Data / Assumptions: Father 4 years ago = 45 ⇒ Father now = 49 Son now = Father now − 25 = 24 Find (Father + Son) after 5 years Concept / Approach:After 5 years, add 5 to each current age and sum them. Alternatively, add 10 to the present total since two people each gain 5 years. Step-by-Step Solution: Father in 5 years = 49 + 5 = 54Son in 5 years = 24 + 5 = 29Total = 54 + 29 = 83 Verification / Alternative check:Present total is 73; adding 10 (5 each) gives 83, consistent. Why Other Options Are Wrong:82, 88, 78 do not match the correctly computed sum after 5 years. Common Pitfalls:Using “was 45 four years ago” as the present age or subtracting 25 from 45 instead of from 49. Final Answer:83 years

Question 11

Two trains, same direction crossing time: Two trains of lengths 50 m and 65 m move in the same direction at 18 m/s and 17 m/s. Find the time taken by the faster train to completely cross the slower train.

Accepted Answer

Introduction / Context:When two trains move in the same direction, the effective relative speed equals the difference of their speeds. To fully cross, the faster must cover the sum of lengths at this relative speed. Given Data / Assumptions: Lengths: 50 m and 65 m Speeds: 18 m/s (faster) and 17 m/s (slower) Same-direction motion Concept / Approach:Time = (total length to be cleared) / (relative speed). For same direction, relative speed = 18 − 17 = 1 m/s. Step-by-Step Solution: Total length = 50 + 65 = 115 mRelative speed = 18 − 17 = 1 m/sTime = 115 / 1 = 115 s Verification / Alternative check:Units consistent: metres / (metres per second) gives seconds. No conversion needed as speeds are already in m/s. Why Other Options Are Wrong:100, 114, 95 are not equal to 115 computed from the standard crossing formula. Common Pitfalls:Using sum of speeds (which applies to opposite directions) instead of difference for same-direction motion. Final Answer:115 s

Question 12

Timeline planning to marriage: Raju was 17 at class 12. He finished graduation in 3 years and PG in 2 years. He got a job exactly 1 year after PG, and will marry 3 years after getting the job. At what age will Raju get married?

Accepted Answer

Introduction / Context:This is a chained timeline arithmetic problem. We move step-by-step from a known age point, adding durations to reach the event age (marriage). Given Data / Assumptions: Age at class 12 = 17 Graduation = 3 years PG = 2 years Job = 1 year after PG Marriage = 3 years after job Concept / Approach:Add durations sequentially to the base age (17) to get ages at each milestone. The sum at the marriage milestone is the answer. Step-by-Step Solution: After graduation: 17 + 3 = 20After PG: 20 + 2 = 22Job starts: 22 + 1 = 23Marriage: 23 + 3 = 26 Verification / Alternative check:Equivalently: 17 + (3 + 2 + 1 + 3) = 17 + 9 = 26 ✓. Why Other Options Are Wrong:27 and 28 overcount; 23 is the job-start age, not the marriage age. Common Pitfalls:Skipping the 1-year gap before the job or adding it twice; mixing years and ages improperly. Final Answer:26 yr

Question 13

Mother–daughter present ages from future ratio: Meena is currently 8 times her daughter’s age. In 8 years, their ages will be in the ratio 10 : 3. What is Meena’s present age?

Accepted Answer

Introduction / Context:Two conditions—one multiplicative now, one ratio in the future—allow direct solution. Assign the daughter’s current age a variable and solve from the future ratio to find both ages. Given Data / Assumptions: Meena now = 8x, Daughter now = x (8x + 8) : (x + 8) = 10 : 3 Concept / Approach:Cross-multiply the future ratio and solve for x. Then compute Meena’s present age 8x. Step-by-Step Solution: 3(8x + 8) = 10(x + 8)24x + 24 = 10x + 80 ⇒ 14x = 56 ⇒ x = 4Meena now = 8x = 32 Verification / Alternative check:In 8 years: Meena = 40, Daughter = 12 ⇒ 40 : 12 = 10 : 3 ✓. Why Other Options Are Wrong:40 and 36 are not Meena’s present age under the given constraints. “Cannot be determined” is false since the system is determinate. Common Pitfalls:Switching which age is multiplied by 8; forgetting both increase by 8 years in the future ratio. Final Answer:32 yr

Question 14

Two-time ratios to present ages: Seven years ago, A : B = 3 : 4. Nine years hence, A : B = 7 : 8. What is B’s present age?

Accepted Answer

Introduction / Context:Again, two ratios at different times produce a solvable linear system for present ages. Translate carefully and solve systematically to avoid arithmetic slips. Given Data / Assumptions: (A − 7) : (B − 7) = 3 : 4 (A + 9) : (B + 9) = 7 : 8 Concept / Approach:Cross-multiply each, reduce to standard linear form, and solve by elimination or substitution. Step-by-Step Solution: 4(A − 7) = 3(B − 7) ⇒ 4A − 3B = 78(A + 9) = 7(B + 9) ⇒ 8A − 7B = −9Solve ⇒ A = 19, then 4A − 3B = 7 ⇒ 76 − 3B = 7 ⇒ B = 23 Verification / Alternative check:Seven years ago: 12 : 16 = 3 : 4; in nine years: 28 : 32 = 7 : 8 ✓. Why Other Options Are Wrong:16, 19, 28 are not B’s present age under the constraints; only 23 fits. Common Pitfalls:Arithmetic slips when distributing and moving constants; ensure both persons’ ages shift by the same amount in each time frame. Final Answer:23 yr

Question 15

Recover past age from present ratio and a future value: Tarun : Varun = 3 : 7 (present). Four years from now, Varun will be 39. What was Tarun’s age 4 years ago?

Accepted Answer

Introduction / Context:One person’s future age determines both present ages through the given present ratio. Once present ages are known, subtract 4 to get Tarun’s age 4 years ago. Given Data / Assumptions: Present ratio Tarun : Varun = 3 : 7 Varun in 4 years = 39 ⇒ Varun now = 35 Concept / Approach:Use the ratio to derive Tarun’s present age from Varun’s present age. Then shift back 4 years to answer the question. Step-by-Step Solution: Varun now = 35Tarun now = (3/7) * 35 = 15Tarun 4 years ago = 15 − 4 = 11 Verification / Alternative check:Check ratio now: 15 : 35 = 3 : 7 ✓. Future value for Varun matches 39 after 4 years. Why Other Options Are Wrong:12, 13, 19 do not equal the calculated 11 years. Common Pitfalls:Applying the ratio to future ages; the ratio given is explicitly for present ages. Final Answer:11 yr

Question 16

Present ratio from past and future ratios: Ten years ago, Ram : Rahim = 1 : 3. Five years hence, Ram : Rahim = 2 : 3. Find their present age ratio.

Accepted Answer

Introduction / Context:Two distinct time-based ratios can be reconciled to find a unique present ratio. Translate each ratio into a linear equation in present ages and solve the system carefully. Given Data / Assumptions: (R − 10) : (H − 10) = 1 : 3 (R + 5) : (H + 5) = 2 : 3 Concept / Approach:Cross-multiply to form two linear equations in R (Ram) and H (Rahim). Solve to obtain their present ages; finally reduce to a ratio in simplest form. Step-by-Step Solution: 3(R − 10) = 1(H − 10) ⇒ 3R − H = 203(R + 5) = 2(H + 5) ⇒ 3R − 2H = −5Subtract second from first: (3R − H) − (3R − 2H) = 20 − (−5) ⇒ H = 25Then 3R − 25 = 20 ⇒ 3R = 45 ⇒ R = 15Present ratio = 15 : 25 = 3 : 5 Verification / Alternative check:Ten years ago: 5 : 15 = 1 : 3, and in five years: 20 : 30 = 2 : 3 ✓. Why Other Options Are Wrong:1 : 2 and 2 : 5 do not match the solved values; 3 : 4 is also inconsistent with both time-based constraints. Common Pitfalls:Arithmetic slips when moving constants; mixing up which person corresponds to which ratio term across times. Final Answer:3 : 5

Question 17

Age ratio back in time: Harsh is 40 years old and Sumit is 60 years old at present. How many years ago were their ages in the ratio 3 : 5?

Accepted Answer

Introduction / Context:This problem tests reverse-age ratio reasoning. We are given two present ages and asked to find how many years back their ages had a specified ratio. The method is to write a linear equation in a single variable representing “x years ago” and use the definition of a ratio (cross-multiplication) to solve for x. Given Data / Assumptions: Harsh's present age = 40 years. Sumit's present age = 60 years. Some x > 0 years ago, (40 − x) : (60 − x) = 3 : 5. Ages increase uniformly with time; x must be less than both current ages. Concept / Approach:For a ratio a : b, we use a/b. So, set (40 − x)/(60 − x) = 3/5 and solve. Cross-multiplication is the quickest consistent method for such linear age questions. Step-by-Step Solution: (40 − x)/(60 − x) = 3/55(40 − x) = 3(60 − x)200 − 5x = 180 − 3x20 = 2x ⇒ x = 10 Verification / Alternative check:10 years ago, ages would be 30 and 50, which indeed are in the ratio 3 : 5 (since 30/50 = 3/5). Why Other Options Are Wrong: 15 yr, 20 yr, 37 yr produce age pairs that do not simplify to 3 : 5. None of these is invalid because 10 years works exactly. Common Pitfalls:Reversing the ratio order or forgetting that the same x is subtracted from both ages. Always check feasibility: x must be less than 40 and 60, which 10 is. Final Answer:10 yr

Question 18

Father four times son now; seven times five years ago: Ravi’s father is four times Ravi’s current age. Five years back, the father was seven times Ravi’s age then. What is the father’s present age?

Accepted Answer

Introduction / Context:This is a standard two-equation age problem. The current multiplicative relation (father is 4 times the son) is paired with an earlier-time multiplicative relation (7 times five years ago). Solving the simultaneous linear equations gives the exact ages. Given Data / Assumptions: Let Ravi’s present age be R, father’s present age be F. F = 4R (now). Five years ago: F − 5 = 7(R − 5). Concept / Approach:Substitute F from the first relation into the second and solve for R, then compute F. This avoids introducing extra variables and minimizes algebraic errors. Step-by-Step Solution: F = 4RF − 5 = 7(R − 5)4R − 5 = 7R − 35 ⇒ 30 = 3R ⇒ R = 10F = 4R = 40 Verification / Alternative check:Five years ago: father 35, Ravi 5; 35 is indeed 7 times 5. The data is fully consistent. Why Other Options Are Wrong:84/70/35 are inconsistent with both relations; they fail either the current 4x relation or the 7x past relation. Common Pitfalls:Reversing who is a multiple of whom, or applying the 5-year shift to only one person. Always subtract the same number of years from both ages when going backwards. Final Answer:40 yr

Question 19

Average age unchanged after a birth: Three years ago, the average age of a family of 5 was 17 years. A new child was born since then, and today the average remains 17. What is the child’s present age?

Accepted Answer

Introduction / Context:Average age questions hinge on total ages. When the average is unchanged after several years and an extra family member is added, comparing totals before and after gives the newcomer’s current age directly. This problem tests careful accounting of the total age “carry-forward”. Given Data / Assumptions: 3 years ago: 5 members, average 17 ⇒ total then = 5 * 17 = 85. Today: 6 members (one newborn added sometime in the last 3 years), average still 17 ⇒ total now = 6 * 17 = 102. The initial 5 members each aged 3 years over the period. Concept / Approach:Total of the original 5 today = 85 + (5 * 3) = 100. Since the current family total is 102, the difference 102 − 100 is precisely the present age of the child. Step-by-Step Solution: Past total = 85Added aging of original 5 = 5 * 3 = 15Current total of original 5 = 100Current family total = 102 ⇒ child’s age = 102 − 100 = 2 years Verification / Alternative check:If the child were born exactly t years ago (0 ≤ t ≤ 3), the child’s age now is t. The calculation shows t = 2, which is feasible within 0–3 years. Why Other Options Are Wrong:1, 3, or 2.5 fail to reconcile both totals and the fixed average condition. Common Pitfalls:Multiplying the average by the wrong headcount or forgetting that the five elders gain 3 years each. Final Answer:2 yr

Question 20

Age ratios then and later: Two brothers currently have ages in the ratio 1 : 2. Five years ago the ratio was 1 : 3. What will be their age ratio after 5 years from now?

Accepted Answer

Introduction / Context:Age-ratio problems are solved by translating ratios into linear equations for actual ages. Once present ages are known, future ratios are straightforward. This checks comfort with setting up ratio equations at different time points. Given Data / Assumptions: Let present ages be x and 2x (older is 2x). Five years ago: (x − 5) : (2x − 5) = 1 : 3. We seek (x + 5) : (2x + 5). Concept / Approach:Solve the past-ratio equation for x, then compute the ages after 5 years and simplify the ratio to lowest terms. Step-by-Step Solution: 3(x − 5) = (2x − 5)3x − 15 = 2x − 5 ⇒ x = 10Present ages: 10 and 20After 5 years: 15 and 25 ⇒ ratio = 15 : 25 = 3 : 5 Verification / Alternative check:Check the given past ratio: 5 years ago they were 5 and 15, which is indeed 1 : 3. Why Other Options Are Wrong:1:4, 2:3, 5:6 are not the simplified ratio of 15:25. Common Pitfalls:Using the same ratio across time points without solving for actual ages first. Ratios typically change as equal years are added or subtracted. Final Answer:3 : 5

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