Question 1

Age puzzle across perfect powers: Ram's age was a perfect square last year and it will be a perfect cube next year. Starting from his current age, how many years must he wait until his age is again (after that) a perfect cube?

Accepted Answer

Introduction / Context:This classic ages problem mixes number theory with linear reasoning. We are told that the present age x satisfies two near-neighbor conditions: last year (x − 1) was a perfect square, and next year (x + 1) will be a perfect cube. We then ask: counting forward from today, after next year's cube, how long until his age is once again a perfect cube? Given Data / Assumptions: Current age: x (a positive integer). x − 1 = n^2 for some integer n ≥ 1. x + 1 = m^3 for some integer m ≥ 1. We seek the smallest t > 1 so that x + t is a perfect cube again (i.e., the next cube after x + 1). Concept / Approach:Subtract the two conditions: (x + 1) − (x − 1) = m^3 − n^2 ⇒ 2 = m^3 − n^2. Test small cubes until the right-hand side equals 2. With m = 3, m^3 = 27 and n^2 = 25, satisfying 27 − 25 = 2. Hence x + 1 = 27 ⇒ x = 26 and x − 1 = 25 = 5^2 (consistent). Step-by-Step Solution:1) Solve m^3 − n^2 = 2 ⇒ m = 3, n = 5.2) Current age x = 26; next year x + 1 = 27 = 3^3 (first cube).3) The next cube after 27 is 64 = 4^3.4) Waiting time from today to reach 64 is 64 − 26 = 38 years. Verification / Alternative check:Check neighbor conditions around x = 26: last year 25 is 5^2; next year 27 is 3^3. The next cube after 27 is indeed 64, so 38 years is correct. Why Other Options Are Wrong: 39 years / 10 years / 64 years: Do not match the actual gap to the next cube after 27; 64 years is not a waiting time but a value. Common Pitfalls:Answering “1 year” by misreading “again” as the first cube (next year). The problem asks for the next cube after the immediate one. Final Answer:38 years

Question 2

Linear ages equation: A father is twice his son's present age. If the son is now 20 years old, how many years ago was the father three times the son's age?

Accepted Answer

Introduction / Context:Ages problems typically translate English statements into linear equations in one variable, then solve for the elapsed time. Here the relationship “twice now, thrice some years ago” becomes a simple one-unknown equation. Given Data / Assumptions: Son's present age S = 20. Father's present age F = 2S = 40. Find t > 0 such that (F − t) = 3(S − t). Concept / Approach:Use one unknown (t) for “years ago” and set up the linear equality that matches the ratio condition at that time. Step-by-Step Solution:1) Equation: 40 − t = 3(20 − t).2) Expand: 40 − t = 60 − 3t.3) Rearrange: −t + 3t = 60 − 40 ⇒ 2t = 20 ⇒ t = 10. Verification / Alternative check:Ten years ago: father 30, son 10; indeed 30 = 3 × 10. Why Other Options Are Wrong:They do not satisfy the equation (plug them back to see the mismatch between 3× and the actual ages then). Common Pitfalls:Confusing “twice” and “three times,” or subtracting t from only one age. Both ages change with time. Final Answer:10 years

Question 3

Find today's age from a forward–backward relation: Today is Vishal's birthday. One year from today he will be twice as old as he was 12 years ago. How old is Vishal today?

Accepted Answer

Introduction / Context:Statements tying a future age to a past age convert neatly into linear equations. The key is to represent today's age with x, then express the future and past in terms of x ± k. Given Data / Assumptions: Let current age be x. One year from now: x + 1. Twelve years ago: x − 12. Condition: x + 1 = 2(x − 12). Concept / Approach:Translate the English sentence literally into an equation, then solve for x. Step-by-Step Solution:1) x + 1 = 2x − 24.2) Bring terms together: −x = −25 ⇒ x = 25.3) Hence, Vishal is 25 years old today. Verification / Alternative check:Check: One year from now, 26; twelve years ago, 13; 26 is twice 13. The condition holds exactly. Why Other Options Are Wrong:They fail the equality x + 1 = 2(x − 12) when substituted. Common Pitfalls:Misplacing the offsets (using 12 instead of 1 or vice versa), or forgetting that “one year from today” is x + 1, not x. Final Answer:25 years

Question 4

Two-time comparisons, one solution: A father was 12 times as old as his son 20 years ago. Now he is exactly twice as old as his son. What are their present ages?

Accepted Answer

Introduction / Context:Age problems with two different time references yield a solvable linear system. We align both statements to today's variables and solve simultaneously. Given Data / Assumptions: Let present ages be F (father) and S (son). Now: F = 2S. Twenty years ago: F − 20 = 12(S − 20). Concept / Approach:Substitute the present-day relation F = 2S into the past relation to eliminate F, leaving a single equation in S. Step-by-Step Solution:1) 2S − 20 = 12(S − 20) = 12S − 240.2) Rearrange: 2S − 20 = 12S − 240 ⇒ 220 = 10S ⇒ S = 22.3) Then F = 2S = 44. Verification / Alternative check:Twenty years ago: F=24, S=2, and indeed 24 = 12 × 2. Now 44 = 2 × 22, consistent. Why Other Options Are Wrong:Other pairs do not satisfy both the “20 years ago” and “now” constraints simultaneously. Common Pitfalls:Placing 12 outside the parentheses incorrectly (e.g., 12S − 20) or forgetting to subtract 20 from both ages. Final Answer:Father 44 and son 22

Question 5

Sum in the future, ratio in the present: A father is currently twice his son's age. After 5 years the sum of their ages will be 85. What are their present ages?

Accepted Answer

Introduction / Context:We combine a present-day multiplicative relation with a future sum condition. One variable (the son) is enough, since the father's age is expressed in terms of it. Given Data / Assumptions: Let son's present age be s; father's present age is 2s. After 5 years: (s + 5) + (2s + 5) = 85. Concept / Approach:Translate the future sum to a linear equation in s and solve directly. Step-by-Step Solution:1) 3s + 10 = 85.2) 3s = 75 ⇒ s = 25.3) Father's present age = 2s = 50. Verification / Alternative check:After 5 years: 30 + 55 = 85, which matches the condition. Why Other Options Are Wrong:Other pairs do not produce a future sum of 85 under the present “twice” relation. Common Pitfalls:Forgetting that both ages increase by 5 when forming the future sum. Final Answer:50, 25

Question 6

Two-time conditions, find a ratio today: Two years ago a mother was four times as old as her son. Six years from now her age will exceed twice her son's age by 10 years. What is their present age ratio (mother : son)?

Accepted Answer

Introduction / Context:We are given two timed conditions to pin down two unknown present ages, then asked for their ratio. As usual, convert each sentence into an equation in present-day variables. Given Data / Assumptions: M = mother's current age; S = son's current age. Two years ago: M − 2 = 4(S − 2). Six years from now: M + 6 = 2(S + 6) + 10. Concept / Approach:Solve the system. From the first, M = 4S − 6. Substitute into the second to find S, then M, and finally reduce M : S. Step-by-Step Solution:1) Substitute: (4S − 6) + 6 = 2S + 22 ⇒ 4S = 2S + 22.2) 2S = 22 ⇒ S = 11.3) M = 4S − 6 = 44 − 6 = 38.4) Ratio M : S = 38 : 11. Verification / Alternative check:Two years ago: 36 vs 9 → 36 = 4 × 9. In six years: 44 vs 17 → 44 = 2 × 17 + 10, OK. Why Other Options Are Wrong:They do not match both time-shifted conditions simultaneously. Common Pitfalls:Dropping the +10 in the future condition or mishandling the −2 on both sides in the past condition. Final Answer:38 : 11

Question 7

Daughter’s present age from two conditions: Five years ago a mother was three times her daughter’s age. Five years from now the mother will be twice her daughter’s age. How old is the daughter today?

Accepted Answer

Introduction / Context:Two time-shifted multiplicative relationships determine present ages uniquely. We solve a pair of linear equations derived from the English statements. Given Data / Assumptions: M = mother's present age; D = daughter's present age. Five years ago: M − 5 = 3(D − 5). Five years from now: M + 5 = 2(D + 5). Concept / Approach:Convert each to an equation in M and D, then eliminate M. Step-by-Step Solution:1) From the first: M = 3D − 10.2) From the second: M = 2D + 5.3) Equate: 3D − 10 = 2D + 5 ⇒ D = 15. Verification / Alternative check:Then M = 2D + 5 = 35. Check: Five years ago 30 vs 10 (3×); in five years 40 vs 20 (2×). Both hold. Why Other Options Are Wrong:Other values fail one of the two conditions when verified. Common Pitfalls:Arithmetic slips when moving constants: note −5 and +5 belong to both ages at the respective times. Final Answer:15 years

Question 8

Proportional ages then and now: Neela is now three times as old as her daughter Leela. Ten years ago, Neela was five times as old as Leela. What is Leela’s present age?

Accepted Answer

Introduction / Context:Present and past multiplicative relations between ages form a solvable system. Let present ages be in a simple ratio, then use the past condition to fix the scale. Given Data / Assumptions: N = 3L (now). Ten years ago: N − 10 = 5(L − 10). Concept / Approach:Substitute N = 3L into the ten-years-ago equation and solve for L. Step-by-Step Solution:1) 3L − 10 = 5L − 50.2) Move terms: 40 = 2L ⇒ L = 20.3) Therefore Leela is 20 years old now (Neela is 60). Verification / Alternative check:Ten years ago: 50 vs 10 → 5×. Now: 60 vs 20 → 3×. Consistent. Why Other Options Are Wrong:They break either the present 3× or the past 5× condition. Common Pitfalls:Forgetting both ages decrease by 10 simultaneously when moving to “ten years ago.” Final Answer:20

Question 9

Chain of differences among three people: Amit was born 5 years before Rakesh, and Rakesh is 3 years younger than Anil. If Amit is now 17, how old is Anil?

Accepted Answer

Introduction / Context:Age chains are solved by stepping differences forward or backward. Here we link three people via fixed gaps and use one absolute age to anchor the rest. Given Data / Assumptions: Amit (A) is 5 years older than Rakesh (R): A = R + 5. Rakesh is 3 years younger than Anil (N): R = N − 3. A = 17 (now). Concept / Approach:Express R and N in terms of A or vice versa, then compute. Step-by-Step Solution:1) From A = R + 5 and A = 17 ⇒ R = 12.2) From R = N − 3 ⇒ N = R + 3 = 15. Verification / Alternative check:Check gaps: A − R = 5 (17 − 12); N − R = 3 (15 − 12). All consistent. Why Other Options Are Wrong:They violate one of the fixed differences when back-checked. Common Pitfalls:Reversing “younger than” vs “older than.” Carefully track sign and direction. Final Answer:15 years

Question 10

Three-person sum with given ratios: Ram's age is double Shyam's and half Suresh's. If the sum of their ages is 70, what is Ram's age?

Accepted Answer

Introduction / Context:Sum with linked ratios is reduced to one variable by expressing all ages in terms of the same unknown. Then a single equation in that unknown yields the answer. Given Data / Assumptions: Let Ram = R. Shyam = R / 2 (since R is double Shyam). Suresh = 2R (since R is half Suresh). R + R/2 + 2R = 70. Concept / Approach:Combine like terms; solve for R. Step-by-Step Solution:1) Total = 3.5R = 70.2) R = 70 / 3.5 = 20. Verification / Alternative check:Check: Shyam = 10, Suresh = 40; sum = 20 + 10 + 40 = 70; all relations hold. Why Other Options Are Wrong:They do not satisfy both ratio statements and the total simultaneously. Common Pitfalls:Misreading “half” and “double,” or forgetting to include all three in the sum. Final Answer:20

Question 11

Nested relations with a total: X is four years older than Y, and Y is twice as old as Z. If X + Y + Z = 34, how old is X?

Accepted Answer

Introduction / Context:Express all ages in terms of Z, use the total to solve Z, then recover X. This is a standard linear reduction technique. Given Data / Assumptions: Y = 2Z. X = Y + 4 = 2Z + 4. (2Z + 4) + 2Z + Z = 34. Concept / Approach:Collapse to a single variable Z; compute backwards to X. Step-by-Step Solution:1) 5Z + 4 = 34 ⇒ 5Z = 30 ⇒ Z = 6.2) Y = 12; X = 16. Verification / Alternative check:Sum check: 16 + 12 + 6 = 34; difference check: X − Y = 4; Y = 2Z. All consistent. Why Other Options Are Wrong:They fail either the total or one of the given relations. Common Pitfalls:Arithmetic slips adding three terms, or forgetting the +4 offset for X. Final Answer:16

Question 12

Direct proportion with given absolute age: Mani is twice Prabhu's age, and Ramona is half of Prabhu's age. If Mani is 60 now, what is Ramona's present age?

Accepted Answer

Introduction / Context:When one absolute age is known, proportional links determine the others immediately by scaling up/down. Given Data / Assumptions: Mani = 2 × Prabhu. Mani = 60 ⇒ Prabhu = 30. Ramona = (1/2) × Prabhu. Concept / Approach:Compute Prabhu's age from Mani's, then halve it for Ramona. Step-by-Step Solution:1) Prabhu = 60 / 2 = 30.2) Ramona = 30 / 2 = 15. Verification / Alternative check:Ratios hold: Mani:Prabhu:Ramona = 60:30:15 = 4:2:1. Why Other Options Are Wrong:They contradict the given 2× and 1/2 relations with Mani = 60. Common Pitfalls:Halving the wrong person or misreading the direction of each ratio. Final Answer:15

Question 13

Future sum shift: The current sum of ages of a mother, daughter, and son is 87. What will their combined age be after 8 years?

Accepted Answer

Introduction / Context:When every person in a group ages by the same amount, the total increases by (number of people) × (years). No individual ages are required. Given Data / Assumptions: Present total = 87. Number of people = 3. Time shift = +8 years for each person. Concept / Approach:Total increase = 3 × 8 = 24; add to the current total. Step-by-Step Solution:1) Increment = 24.2) Future total = 87 + 24 = 111. Verification / Alternative check:This logic holds regardless of the individual ages as long as the group size remains three. Why Other Options Are Wrong:They use an incorrect increment (not 24) or arithmetic error. Common Pitfalls:Adding 8 only once rather than for each of the three people. Final Answer:111

Question 14

Past ratio with current difference: Two years ago A was twice as old as B. If their current age difference is 2 years, what is A's present age?

Accepted Answer

Introduction / Context:One condition at a past time and one present-day difference allow a quick linear solution. Given Data / Assumptions: Let present ages be A and B with A − B = 2. Two years ago: A − 2 = 2(B − 2). Concept / Approach:Express A in terms of B from the past condition, then use the present difference to solve B and A. Step-by-Step Solution:1) From A − 2 = 2B − 4 ⇒ A = 2B − 2.2) Plug into A − B = 2 ⇒ (2B − 2) − B = 2 ⇒ B − 2 = 2 ⇒ B = 4.3) A = 2B − 2 = 6. Verification / Alternative check:Two years ago: A=4, B=2 ⇒ A was twice B; difference now is 2. Checks out. Why Other Options Are Wrong:They fail either the past doubling or the present difference. Common Pitfalls:Subtracting 2 from only one age when shifting to “two years ago.” Final Answer:6

Question 15

Becoming twice in the future: Twelve-year-old Sami is three times as old as his brother Vinay today. After how many years will Sami be exactly twice as old as Vinay, and what will Sami’s age be then?

Accepted Answer

Introduction / Context:We are given a current multiplicative relation and asked when a different multiplicative relation will hold in the future. Set up a single “t years from now” equation. Given Data / Assumptions: Sami now = 12. Sami is 3 × Vinay ⇒ Vinay now = 4. Find t such that 12 + t = 2(4 + t). Concept / Approach:Both ages grow by t; impose the “twice” relation at that future instant and solve for t, then compute Sami's age then. Step-by-Step Solution:1) 12 + t = 8 + 2t.2) 4 = t.3) Sami's future age = 12 + 4 = 16. Verification / Alternative check:Vinay will be 8 then; 16 = 2 × 8, confirming the condition. Why Other Options Are Wrong:They are not the age at which the “twice” relation becomes true. Common Pitfalls:Using 12 × 2 directly without shifting Vinay's age forward by t as well. Final Answer:16 years

Question 16

Equal gaps around a middle age: Anil is as much younger than Vivek as he is older than Tarun. If (Vivek + Tarun) = 48, how old is Anil?

Accepted Answer

Introduction / Context:The phrase “as much younger than V as older than T” means Anil is exactly midway between Vivek and Tarun in age. That converts directly to a simple average relation. Given Data / Assumptions: V − A = A − T ⇒ 2A = V + T. V + T = 48. Concept / Approach:Use the midpoint identity to compute A as the average of V and T. Step-by-Step Solution:1) 2A = 48 ⇒ A = 24. Verification / Alternative check:The differences on both sides are equal: (V − 24) = (24 − T) if V + T = 48. Why Other Options Are Wrong:They are not the average of 48/2. Common Pitfalls:Overcomplicating with multiple unknowns; the wording already encodes the midpoint relation. Final Answer:24

Question 17

From 8× in the past to 4× now: Four years ago a father's age was eight times his son's age. At present the father's age is four times the son's age. What is the son's current age?

Accepted Answer

Introduction / Context:Two time-spaced multiplicative relations in ages yield a tidy linear system. Translate both statements into equations in today's variables and solve. Given Data / Assumptions: Let present ages be F (father) and S (son). Now: F = 4S. Four years ago: F − 4 = 8(S − 4). Concept / Approach:Substitute F = 4S into the past condition and solve for S. Step-by-Step Solution:1) 4S − 4 = 8S − 32.2) Move terms: 28 = 4S ⇒ S = 7. Verification / Alternative check:Check: Now F = 28. Four years ago: F=24, S=3 → 24 = 8 × 3; now 28 = 4 × 7. Consistent. Why Other Options Are Wrong:They fail one of the multiplicative checks when validated. Common Pitfalls:Forgetting both ages decrease by 4 in the “four years ago” equation. Final Answer:7 years

Question 18

Three-member family with linked differences: A man is 3 years older than his wife and four times as old as his son. If the son will be 15 after 3 years, what is the wife's present age?

Accepted Answer

Introduction / Context:Combine a known future age of the son with proportional and difference relations to deduce the wife's present age. Given Data / Assumptions: Son in 3 years = 15 ⇒ Son now = 12. Man = 4 × Son ⇒ Man now = 48. Man is 3 years older than wife ⇒ Wife now = 48 − 3. Concept / Approach:Anchor absolute ages using the son's future age, then step through the chained relations. Step-by-Step Solution:1) Son now = 15 − 3 = 12.2) Man now = 4 × 12 = 48.3) Wife now = 48 − 3 = 45. Verification / Alternative check:Relations hold: (48, 45, 12) satisfy “man 3 older than wife” and “man 4× son.” Why Other Options Are Wrong:They break one of the given constraints when checked. Common Pitfalls:Adding 3 instead of subtracting when moving from the husband's age to the wife's age. Final Answer:45 years

Question 19

From 4× now to 2× in 10 years: Dr. Pandey is four times his son's age at present. After 10 years, he will be twice his son's age. What is the son's current age?

Accepted Answer

Introduction / Context:Two time-separated multiplicative relations allow a linear solution. Express the father's present age in terms of the son's age and apply the 10-year condition. Given Data / Assumptions: Let son's present age be S; Dr. Pandey's present age is 4S. After 10 years: 4S + 10 = 2(S + 10). Concept / Approach:Solve the single linear equation for S. Step-by-Step Solution:1) 4S + 10 = 2S + 20.2) 2S = 10 ⇒ S = 5. Verification / Alternative check:Now: father 20, son 5 (4×). In 10 years: father 30, son 15 (2×). Works perfectly. Why Other Options Are Wrong:They do not satisfy both multiplicative relations. Common Pitfalls:Forgetting to add 10 to both ages in the future condition. Final Answer:5 years

Question 20

Past 3× and future 2× around today: Five years ago Mayank's mother was three times as old as Mayank. Five years from now she will be twice his age. What is Mayank's present age?

Accepted Answer

Introduction / Context:Two symmetric time conditions (−5 and +5) precisely fix the present ages. Translate to equations and solve. Given Data / Assumptions: M = mother's present age; m = Mayank's present age. Five years ago: M − 5 = 3(m − 5). Five years from now: M + 5 = 2(m + 5). Concept / Approach:Solve the pair by expressing M from both equations and equating them. Step-by-Step Solution:1) From past: M = 3m − 10.2) From future: M = 2m + 5.3) Equate: 3m − 10 = 2m + 5 ⇒ m = 15.4) Then M = 2m + 5 = 35. Verification / Alternative check:Five years ago: 30 vs 10 → 3×. In five years: 40 vs 20 → 2×. Both conditions hold. Why Other Options Are Wrong:They fail one of the two multiplicative checks upon substitution. Common Pitfalls:Mishandling the ±5 shift on both ages; both change simultaneously. Final Answer:15 years

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