Question 1

Comparing parents’ ages via child’s milestone: A man was 32 years old when he had his first son. When the son turned 7, the wife was 35 years old. What is the age difference between the man and his wife?

Accepted Answer

Introduction / Context: This problem uses ages at specific events to infer the parents’ relative ages. Align both parents’ ages at the same reference time (when the son is 7) to compute their difference. Given Data / Assumptions: Father’s age when son born = 32. When son is 7, wife is 35. Concept / Approach: At the time the son is 7, the father is 32 + 7 = 39. Compare this to the wife’s 35 at the same instant to get the difference. Step-by-Step Solution: 1) Father’s age when son is 7: 32 + 7 = 39.2) Wife’s age at that time: 35.3) Difference = |39 − 35| = 4 years. Verification / Alternative check: Because both ages are referenced to the same moment, the computed difference remains valid at all times. Why Other Options Are Wrong: 7/3/5 years do not match the aligned-time comparison; 4 years is uniquely correct. Common Pitfalls: Comparing ages at different times or overlooking which parent’s age is specified at which event leads to errors. Final Answer: 4 years

Question 2

Past double condition with a future age given: Hema will be 40 years old 10 years from now. Ten years ago, Hema was twice as old as Geetha. How old is Geetha today?

Accepted Answer

Introduction / Context: We combine a known future age for Hema with a past-time multiplicative relation involving Geetha. First find Hema’s present age, then move 10 years back to apply the “twice” condition and step forward to the present for Geetha. Given Data / Assumptions: Hema in 10 years = 40 ⇒ Hema now = 30. Ten years ago: Hema = 2 × Geetha. Concept / Approach: Ten years ago, Hema was 20 (since 30 − 10). If that was twice Geetha's age then, Geetha was 10 at that time. Therefore, now Geetha is 20. Step-by-Step Solution: 1) Hema now = 40 − 10 = 30.2) Ten years ago Hema = 20.3) 20 = 2 × (Geetha then) ⇒ Geetha then = 10.4) Geetha now = 10 + 10 = 20. Verification / Alternative check: Check the future statement: Hema in 10 years = 40 (true). Past statement: 20 and 10 satisfy the 2× relation (true). Why Other Options Are Wrong: 25/15/35/30 do not satisfy both the past multiple and the future age constraint together. Common Pitfalls: Confusing “10 years hence” with “10 years ago” or failing to compute Hema’s present age correctly before applying the past condition. Final Answer: 20 years

Question 3

Translate a verbal age condition into an equation: Naresh’s age is 4 years less than twice the age of his brother (let brother’s age be P). Which expression correctly represents Naresh’s age?

Accepted Answer

Introduction / Context: Algebraic translation questions require careful reading. The phrase “4 years less than twice” indicates a “twice” operation followed by a subtraction of 4 from that result. Given Data / Assumptions: Brother’s present age = P. Naresh’s present age = ? Phrase: “4 years less than twice the age of his brother.” Concept / Approach: “Twice the age of his brother” → 2P. “4 years less than that” → subtract 4 from 2P. Hence the correct expression is 2P − 4. Step-by-Step Solution: 1) Identify the base quantity: brother’s age P.2) Apply the multiplier: twice ⇒ 2P.3) Apply the difference: less by 4 ⇒ 2P − 4. Verification / Alternative check: Test with a sample: If P = 10, “twice” = 20; “4 less” = 16. The formula 2P − 4 gives 16 (consistent). Why Other Options Are Wrong: 2P + 4 adds instead of subtracting. 4P + 2 misreads both multiplier and difference. P − 4 omits the “twice.” P + 4 ignores both “twice” and the subtraction. Common Pitfalls: Students frequently reverse “less than” into 4 − 2P or add 4 instead of subtracting. Read left to right but apply operations to the base quantity correctly. Final Answer: 2P – 4

Question 4

Parent–child ages with a future double condition: My father is 21 years older than me. In 12 years, his age will be twice mine. How old am I now?

Accepted Answer

Introduction / Context: Future multiple conditions with a known present difference translate into an equation in one variable. Represent your present age as x; father's present age is x + 21. Use the future-time doubling relation to solve for x. Given Data / Assumptions: Father now = x + 21. In 12 years: father = 2 × (my age then). Concept / Approach: Set up (x + 21 + 12) = 2(x + 12) and solve for x directly. Step-by-Step Solution: 1) x + 33 = 2x + 24.2) 33 − 24 = 2x − x ⇒ 9 = x.3) Therefore, my present age is 9 years. Verification / Alternative check: In 12 years: me = 21, father = 30; indeed, 30 is twice 15? Wait—recompute correctly: me then = 9 + 12 = 21; father then = (9 + 21) + 12 = 42; 42 = 2 × 21 (true). Condition satisfied. Why Other Options Are Wrong: 8/10/11/12 years do not satisfy the future double relation with a 21-year current difference. Common Pitfalls: Adding 12 to the difference (which must remain 21) or misapplying the doubling to present ages instead of future ages leads to errors. Final Answer: 9 years

Question 5

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 6

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 7

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 8

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 9

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 10

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 11

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 12

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 13

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 14

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 15

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 16

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 17

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 18

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 19

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 20

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

Problems on Ages Questions