Question 1

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 2

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 3

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 4

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

Question 5

Ages (sum and past product condition): The sum of the ages of a father and his son is 45 years. Five years ago, the product of their ages was four times the father's age at that time. Find their present ages.

Accepted Answer

Introduction / Context: This item blends a present-time linear relation (the sum of ages) with a past-time multiplicative relation. Such problems are standard in algebraic age reasoning and test the ability to translate verbal statements into equations and apply factor reasoning efficiently. Given Data / Assumptions: Father's present age = F years; Son's present age = S years. F + S = 45. Five years ago: (F − 5)(S − 5) = 4(F − 5). Ages are non-negative integers and F > S in typical father–son contexts. Concept / Approach: Use the product condition from five years ago. If (F − 5)(S − 5) = 4(F − 5), then either (F − 5) = 0 or (S − 5) = 4. The first case would give F = 5, which cannot fit a realistic father–son pair with sum 45. Hence use the second case to determine S, then use the sum to obtain F. Step-by-Step Solution: From (S − 5) = 4 ⇒ S = 9. Use F + S = 45 ⇒ F = 45 − 9 = 36. Check the product condition: five years ago ages were 31 and 4; product = 31 * 4 = 124; four times father's then-age = 4 * 31 = 124 (✓). Verification / Alternative check: Consider (F − 5) = 0 ⇒ F = 5. Then S = 40, inconsistent for a father–son pair and contradicts typical expectations, so discard. The chosen pair satisfies both constraints exactly. Why Other Options Are Wrong: Each alternative pair either violates the sum 45 or fails the past product relation when tested numerically. Common Pitfalls: Missing the factor trick and expanding fully (creating unnecessary algebra), or accepting the unrealistic F = 5 branch without checking plausibility. Final Answer: 36 years, 9 years

Question 6

Ages (difference now; past ratio given): The difference between the present ages of Arun and Barun is 14 years. Seven years ago, the ratio of their ages was 5 : 7. Find Barun's present age.

Accepted Answer

Introduction / Context: This problem mixes a present absolute difference with a past ratio, a frequent pattern in age questions. The key is to express present ages from the ratio's time-slice and then align them with today's difference. Given Data / Assumptions: Let present ages be A (Arun) and B (Barun), with B − A = 14 (Barun older, consistent with the 5 : 7 past ratio). Seven years ago: (A − 7) : (B − 7) = 5 : 7. Concept / Approach: Convert the past ratio into an equation and combine with the present difference. Solving the two linear equations yields today's ages directly. Step-by-Step Solution: From the ratio: 7(A − 7) = 5(B − 7) ⇒ 7A − 49 = 5B − 35 ⇒ 7A = 5B + 14. Use B = A + 14 in the previous: 7A = 5(A + 14) + 14 = 5A + 70 + 14 ⇒ 2A = 84 ⇒ A = 42. Then B = A + 14 = 56. Verification / Alternative check: Seven years ago: A = 35, B = 49 ⇒ 35 : 49 = 5 : 7 (✓). Difference now 14 (✓). Why Other Options Are Wrong: All other listed numbers fail either the difference constraint or the reconstructed past ratio when checked. Common Pitfalls: Applying the 14-year difference to the past snapshot or reversing the elder–younger roles; the ratio 5 : 7 indicates Barun is older. Final Answer: 56 years

Question 7

Ages (chained ratios summing to a total): The ratio Ram : Mohan is 4 : 5, and Mohan : Anil is 5 : 6. If the sum of their ages is 90 years, find Mohan's age.

Accepted Answer

Introduction / Context: Linking ratios across three people can be handled by expressing all ages with one common multiplier. The question is a straightforward ratio-sum application aimed at quick, accurate computation. Given Data / Assumptions: Ram : Mohan = 4 : 5. Mohan : Anil = 5 : 6. R + M + A = 90 years. Concept / Approach: Use Mohan as the bridge between the two ratios to put all three in the same scale. With M = 5k, we have R = 4k and A = 6k, so the sum becomes 15k, from which k follows immediately. Step-by-Step Solution: Let M = 5k, R = 4k, A = 6k ⇒ R + M + A = 4k + 5k + 6k = 15k. Given 15k = 90 ⇒ k = 6 ⇒ M = 5k = 30 years. Verification / Alternative check: Check other ages: R = 24, A = 36. Ratios 24 : 30 = 4 : 5 and 30 : 36 = 5 : 6 both hold (✓). Why Other Options Are Wrong: They do not yield the correct total of 90 when combined with compatible R and A values preserving both ratios. Common Pitfalls: Trying to solve two ratios independently without anchoring on the shared person (Mohan), which leads to inconsistent scaling. Final Answer: 30 years

Question 8

Ages (past half relation and present ratio): Ten years ago, Divya's age was half of Namrata's. Today their ages are in the ratio 3 : 4. Find the sum of their present ages.

Accepted Answer

Introduction / Context: This problem links a past proportional statement with a present ratio. Expressing present ages using a single parameter lets us translate the “ten years ago” condition into a solvable equation. Given Data / Assumptions: Present ratio Divya : Namrata = 3 : 4 ⇒ D = 3x, N = 4x. Ten years ago: D − 10 = (1/2)(N − 10). Concept / Approach: Substitute D = 3x and N = 4x into the past relation and solve for x. Then compute present ages and sum them to answer the question. Step-by-Step Solution: 3x − 10 = (1/2)(4x − 10) = 2x − 5. Thus x = 5 ⇒ D = 15, N = 20. Sum of present ages = 15 + 20 = 35 years. Verification / Alternative check: Ten years ago they were 5 and 10 respectively, so Divya was indeed half of Namrata (✓). The present ratio 15 : 20 reduces to 3 : 4 (✓). Why Other Options Are Wrong: Each alternative sum implies a value of x that breaks either the past “half” relation or the present ratio. Common Pitfalls: Misapplying the half relation to the present instead of to the past snapshot, or mixing the ten-year adjustment asymmetrically. Final Answer: 35 years

Question 9

Ages (present ratio with past total): Three persons have present ages in the ratio 4 : 7 : 9. Eight years ago, the sum of their ages was 56. Find their present ages.

Accepted Answer

Introduction / Context: Combining a present-day ratio with a past sum is a typical constraint-matching task. Represent present ages by a common multiplier and adjust backwards to match the past total, then return to present values. Given Data / Assumptions: Present ages: 4x, 7x, 9x. Eight years ago: (4x − 8) + (7x − 8) + (9x − 8) = 56. Concept / Approach: Compute the past sum in terms of x and solve. The linearity of age changes ensures each person reduces by the same eight years, simplifying the algebra. Step-by-Step Solution: Past sum = (4x + 7x + 9x) − 24 = 20x − 24 = 56. Hence 20x = 80 ⇒ x = 4. Present ages = 16, 28, and 36 years. Verification / Alternative check: Eight years ago they were 8, 20, and 28; the sum is 8 + 20 + 28 = 56 (✓). The present ratio 16 : 28 : 36 reduces to 4 : 7 : 9 (✓). Why Other Options Are Wrong: Other triplets either do not align to 4 : 7 : 9 or fail the past-sum condition after subtracting eight from each age. Common Pitfalls: Subtracting the total of 24 only once from one person rather than distributing −8 across all three individuals. Final Answer: 16, 28, 36

Question 10

Ages (two linked snapshot ratios): The present ratio of A : B is 5 : 3. Also, A's age 4 years ago equals B's age 4 years hence. Find the ratio of A's age 4 years hence to B's age 4 years ago.

Accepted Answer

Introduction / Context: The problem binds two people's ages across different time points. Once the present ratio is parameterized, the cross-time equality determines the scale, and the requested ratio follows with simple arithmetic. Given Data / Assumptions: Present: A : B = 5 : 3 ⇒ A = 5k, B = 3k. Constraint: A − 4 = B + 4 ⇒ A = B + 8. Concept / Approach: Combine A = 5k and B = 3k with A = B + 8 to find k, then compute A + 4 and B − 4 and take the ratio requested. Step-by-Step Solution: 5k = 3k + 8 ⇒ 2k = 8 ⇒ k = 4. A = 20, B = 12. A + 4 = 24; B − 4 = 8 ⇒ (A + 4) : (B − 4) = 24 : 8 = 3 : 1. Verification / Alternative check: Check the cross-time equality: A − 4 = 16 and B + 4 = 16 (✓). Why Other Options Are Wrong: They contradict either the present 5 : 3 ratio or the A − 4 = B + 4 condition. Common Pitfalls: Applying the +4 and −4 to the wrong person, which inverts the intended equality. Final Answer: 3 : 1

Question 11

Ages (husband–wife ratio now and after 4 years; marriage-time ratio): The present ratio of a man and his wife is 4 : 3. After 4 years it will be 9 : 7. At the time of marriage their ratio was 5 : 3. How many years ago were they married?

Accepted Answer

Introduction / Context: This multi-snapshot problem provides two present/future ratios and an earlier marriage-time ratio. First determine the actual present ages from the now/after-4-years relation; then walk back to the marriage-time ratio to compute the elapsed years since marriage. Given Data / Assumptions: Present: Man : Wife = 4 : 3 ⇒ M = 4k, W = 3k. After 4 years: (4k + 4) : (3k + 4) = 9 : 7. At marriage (t years ago): (M − t) : (W − t) = 5 : 3. Concept / Approach: First solve for k using the “after 4 years” ratio. Then substitute present values into the marriage-time ratio to solve for t, the time elapsed since marriage. Step-by-Step Solution: From after-4-years: 7(4k + 4) = 9(3k + 4) ⇒ 28k + 28 = 27k + 36 ⇒ k = 8. Thus M = 32 and W = 24. Marriage-time condition: (32 − t) : (24 − t) = 5 : 3 ⇒ 3(32 − t) = 5(24 − t) ⇒ 96 − 3t = 120 − 5t ⇒ 2t = 24 ⇒ t = 12. Verification / Alternative check: After 4 years: 36 : 28 reduces to 9 : 7 (✓). Marriage-time ratio 20 : 12 is 5 : 3 (✓). Why Other Options Are Wrong: Any other t fails at least one of the ratio checks when ages are rolled back or forward appropriately. Common Pitfalls: Using the same k at marriage time (ages do not scale; they shift by t), or forgetting to subtract t from both spouses. Final Answer: 12 years

Question 12

Ages (linear relation with total): A is two years older than B, and B is twice as old as C. If A + B + C = 27, find B's present age.

Accepted Answer

Introduction / Context: Linear relations among three ages combined with a total allow quick variable substitution. This problem is a straightforward arithmetic exercise intended to reinforce systematic setup and solution. Given Data / Assumptions: A = B + 2. B = 2C. A + B + C = 27. Concept / Approach: Eliminate A and B in favor of C (or any single variable). Solve for C from the sum, then compute B directly from B = 2C. Step-by-Step Solution: Substitute into total: (B + 2) + B + C = 27. With B = 2C ⇒ (2C + 2) + 2C + C = 27 ⇒ 5C + 2 = 27 ⇒ C = 5. Hence B = 2C = 10; A = B + 2 = 12. Verification / Alternative check: Sum 12 + 10 + 5 = 27 (✓); relations A = B + 2 and B = 2C both hold (✓). Why Other Options Are Wrong: They correspond to incorrect solutions for C that do not satisfy the total or the given relations. Common Pitfalls: Adding instead of substituting, or mixing which person is twice as old as whom, which flips the relation. Final Answer: 10

Question 13

Ages (two-time multiples; ratio asked 8 years ago): Sixteen years ago Sudha's grandfather was “eight times older” than her, and eight years from now he will be three times her age. What was the ratio Sudha : Grandfather eight years ago?

Accepted Answer

Introduction / Context: Statements like “eight times older” are often intended as “grandfather's age equals 9 times Sudha's age” at that snapshot (older by 8 times Sudha's age). Interpreting English phrasing correctly is vital. Here, two time-separated multiplicative constraints uniquely determine the present ages and allow the requested ratio eight years ago. Given Data / Assumptions: Let present ages be S (Sudha) and G (Grandfather). Sixteen years ago: G − 16 = 9(S − 16) (interpreting “8 times older” as 9 times as old). Eight years hence: G + 8 = 3(S + 8). Concept / Approach: Solve the two linear equations for S and G. Then compute their ages eight years ago and simplify the ratio S : G at that time. Step-by-Step Solution: From future: G = 3S + 16. From past: G = 9S − 128. Equate: 9S − 128 = 3S + 16 ⇒ 6S = 144 ⇒ S = 24; then G = 3*24 + 16 = 88. Eight years ago: S = 16, G = 80 ⇒ ratio = 16 : 80 = 1 : 5. Verification / Alternative check: Check the 16-years-ago relation: 88 − 16 = 72 and 9(24 − 16) = 72 (✓). Check the 8-years-hence relation: 88 + 8 = 96 and 3(24 + 8) = 96 (✓). Why Other Options Are Wrong: Other ratios do not simultaneously satisfy both time-based multiplicative conditions when reverse-engineered. Common Pitfalls: Treating “8 times older” as “8 times as old,” which gives inconsistent (non-integer) ages here; or forming a ratio at the wrong time slice. Final Answer: 1 : 5

Question 14

Ages (sum shift comparison): The sum of A and B is 12 more than the sum of B and C. How many years younger is C than A?

Accepted Answer

Introduction / Context: Comparing two sums that share one common term is a quick route to a direct difference between the other two terms. No time-shift is involved; it is a pure algebraic rearrangement task. Given Data / Assumptions: A + B = (B + C) + 12. Concept / Approach: Cancel the common B from both sides to isolate A and C. The resulting equation immediately reveals how much older A is than C. Step-by-Step Solution: A + B = B + C + 12 ⇒ A = C + 12. Therefore, C is 12 years younger than A. Verification / Alternative check: Pick any C and add 12 to obtain A; then sums differ by 12 for all values of B, confirming generality. Why Other Options Are Wrong: 10 and 24 contradict the derived identity; “Data inadequate” is incorrect because the relation uniquely fixes the difference A − C. Common Pitfalls: Forgetting to cancel B or overcomplicating with unnecessary assumptions about actual ages. Final Answer: 12

Question 15

Ages (marriage-time multiplier and sibling offset): Ram married 8 years ago. His present age is 6/5 times his age at marriage. Ram's sister was 10 years younger than him at his marriage. Find the sister's present age.

Accepted Answer

Introduction / Context: Here, a multiplicative relation connects Ram's present age to his marriage age, and a sibling offset is anchored to the same marriage time. Walking through the timeline consistently yields the sister's present age. Given Data / Assumptions: Let Ram's marriage age be x; present age = (6/5)x. Since marriage was 8 years ago, present age also equals x + 8. At marriage, sister's age = x − 10. Concept / Approach: First find x by equating x + 8 with (6/5)x. Then add 8 years to the sister's marriage-time age to get her age today. Step-by-Step Solution: x + 8 = (6/5)x ⇒ 8 = (1/5)x ⇒ x = 40. Sister at marriage: 40 − 10 = 30; today: 30 + 8 = 38 years. Verification / Alternative check: Ram today is (6/5)*40 = 48, and was 40 at marriage, exactly 8 years earlier (✓). Sister's present age follows consistently. Why Other Options Are Wrong: They result from algebraic slips (e.g., treating 6/5 as 5/6) or misapplying the 10-year offset and 8-year shift. Common Pitfalls: Confusing who is 10 years younger at marriage (sister vs. Ram) and mixing up the 8-year advancement. Final Answer: 38 years

Question 16

Ages (midpoint relation; information sufficiency): Q is as much younger than R as he is older than T. If R + T = 50 years, what is the difference between R and Q's ages?

Accepted Answer

Introduction / Context: Information sufficiency is crucial in age problems. The statement “Q is as much younger than R as he is older than T” implies Q is exactly midway between R and T on the number line of ages. The sum R + T is provided, but no further separation is given. Given Data / Assumptions: R − Q = Q − T ⇒ 2Q = R + T ⇒ Q = (R + T)/2. R + T = 50. Concept / Approach: Compute Q from the midpoint relation: Q = 25. However, without either R − T or an additional ratio, we cannot pin down R and T individually, hence we cannot determine R − Q uniquely. Step-by-Step Reasoning: From 2Q = R + T and R + T = 50 ⇒ Q = 25. But R − Q = (R − 25) can vary while keeping R + T fixed at 50 (e.g., R = 30 gives R − Q = 5; R = 28 gives 3). Verification / Alternative check: Multiple (R, T) pairs satisfy the constraints and yield different R − Q values, confirming insufficiency. Why Other Options Are Wrong: Any specific numeric difference (1, 2, 2.5, 5) asserts extra information not provided by the stem. Common Pitfalls: Assuming symmetry further forces R − T = 0, which the problem never states. Final Answer: Data inadequate

Question 17

Ages in arithmetic progression: Five children are born at intervals of 3 years each. The sum of their ages is 50. Find the age of the youngest child.

Accepted Answer

Introduction / Context: Ages at equal intervals form an arithmetic progression (AP). The sum of an AP can be written in terms of the first term and common difference, enabling a direct solution without enumerating all ages explicitly. Given Data / Assumptions: Let the youngest be x. Then ages are x, x+3, x+6, x+9, x+12, and their sum is 50. Concept / Approach: Use the AP sum formula specialized to these five terms with common difference 3. Solve for x and interpret as the youngest child's current age. Step-by-Step Solution: Sum = 5x + (0+3+6+9+12) = 5x + 30 = 50. Hence 5x = 20 ⇒ x = 4 years. Verification / Alternative check: Ages are 4, 7, 10, 13, 16; sum = 50 (✓). Why Other Options Are Wrong: Other numbers do not satisfy the fixed sum when the AP structure is honored. Common Pitfalls: Using 3 as the average interval incorrectly or miscounting the total increment (it is 30 across five terms). Final Answer: 4 years

Question 18

Ages (birth-time equality statement): A father says, “I was as old as you are now at the time of your birth.” If the father is 38 now, how old was the son five years ago?

Accepted Answer

Introduction / Context: The statement implies that the current difference between father and son equals the son's current age: at the son's birth, the father's age equaled today's son's age. This unlocks the ages immediately without heavy algebra. Given Data / Assumptions: Let father's present age be F = 38 and son's present age be S. The statement gives F − S = S ⇒ F = 2S. Concept / Approach: Solve for S from F = 2S, then roll back five years to answer the question asked. Step-by-Step Solution: From F = 38 = 2S ⇒ S = 19. Five years ago: S − 5 = 14 years. Verification / Alternative check: At the time of birth (19 years ago), father's age would have been 38 − 19 = 19, which equals the son's current age—matching the statement (✓). Why Other Options Are Wrong: They contradict F = 2S or fail the five-years-ago rollback. Common Pitfalls: Interpreting “was as old as you are now” as “twice as old” or some other multiplier; it is an equality with the son's present age. Final Answer: 14 years

Question 19

Ages (present multiple plus future-sum constraint): A mother is five times as old as her son. After 4 years, the sum of their ages will be 44. Find the son's present age.

Accepted Answer

Introduction / Context: Combining a present-time multiplicative relation with a future-time sum creates a compact system solvable in one or two steps. Ensure both people advance by the same 4 years in the future constraint. Given Data / Assumptions: Let son's present age be s; mother's present age is 5s. After 4 years, (s + 4) + (5s + 4) = 44. Concept / Approach: Translate the future sum into a present equation in s and solve directly. The linearity of age changes keeps algebra simple. Step-by-Step Solution: (s + 4) + (5s + 4) = 44 ⇒ 6s + 8 = 44 ⇒ 6s = 36 ⇒ s = 6. Verification / Alternative check: Mother is 30 now. In four years: 10 and 34 ⇒ sum 44 (✓). Why Other Options Are Wrong: Plugging them into 6s + 8 = 44 yields sums different from 44. Common Pitfalls: Forgetting to advance both ages by +4, or misreading “five times older” for “five years older.” Final Answer: 6 years

Question 20

Ages (algebraic identity puzzle): “Take my age three years hence, multiply it by 3, then subtract 3 times my age three years ago; you will get my present age.” Find my age.

Accepted Answer

Introduction / Context: This is a neat identity-based age puzzle. The arithmetic structure collapses to a constant regardless of the unknown age, allowing a one-line solution when the algebra is set up correctly. Given Data / Assumptions: Let present age be x. The statement translates to 3(x + 3) − 3(x − 3) = x. Concept / Approach: Compute the left-hand expression and equate to x. Simplification reveals a fixed value for x. Step-by-Step Solution: 3(x + 3) − 3(x − 3) = 3x + 9 − 3x + 9 = 18. Set equal to x ⇒ 18 = x ⇒ x = 18. Verification / Alternative check: Plug back: in 3 years, 21; 3*21 = 63. Three years ago, 15; 3*15 = 45. Difference 63 − 45 = 18, which equals the present age (✓). Why Other Options Are Wrong: They do not satisfy the identity when substituted. Common Pitfalls: Expanding incorrectly or forgetting that subtracting 3(x − 3) changes signs for both terms. Final Answer: 18

Problems on Ages Questions