Question 1

Relative Growth — “Adam is 20 and is 4 times as old as Mary. After how many years will Adam be twice Mary’s age, and what will Mary’s age be then?”

Accepted Answer

Introduction / Context: We compare two future ages. Since Adam is currently 4 times Mary, we first get Mary’s present age, then find the time t when Adam becomes twice Mary’s age, and finally compute Mary’s age at that moment. Given Data / Assumptions: Adam now = 20. Adam = 4 × Mary ⇒ Mary now = 5. In t years: Adam 20 + t; Mary 5 + t. Condition: 20 + t = 2(5 + t). Concept / Approach: Solve the linear equation in t, then evaluate Mary’s age at that time. Step-by-Step Solution: 20 + t = 10 + 2t ⇒ 10 = t.Mary then = 5 + t = 15. Verification / Alternative check: At t = 10: Adam 30, Mary 15 ⇒ Adam is exactly twice Mary. Why Other Options Are Wrong: 17/25/30/35 are not equal to 5 + 10 (the correct future age of Mary). Common Pitfalls: Confusing “twice” with “two more,” or forgetting to add t to both ages. Final Answer: 15 years

Question 2

Age Difference — “A man was 31 when his first son was born. When the son turned 7, the wife was 26. What is the age difference between the man and his wife?”

Accepted Answer

Introduction / Context: We are asked for a fixed age difference based on two anchored events: the son’s birth and the time the son reaches 7. Aligning both events lets us compute both parents’ ages at birth and subtract. Given Data / Assumptions: At son’s birth: father = 31 (given). At son age 7: wife = 26 ⇒ wife at birth = 26 − 7 = 19. Age difference = father_at_birth − wife_at_birth. Concept / Approach: Because both ages are rolled back by the same number of years (7) to the birth event, the difference computed there is the same as the present difference. Use the birth event to avoid extra unknowns. Step-by-Step Solution: Wife at son’s birth = 26 − 7 = 19.Father at son’s birth = 31.Difference = 31 − 19 = 12 years. Verification / Alternative check: Any time later the difference remains 12 because age gaps are invariant over time. Why Other Options Are Wrong: 5/6/7/9 do not match the invariant gap based on the anchored birth event data. Common Pitfalls: Adding 7 instead of subtracting; computing difference at two different time points for the two parents. Final Answer: 12 years

Question 3

Ratio Today — “At present Arun : Deepak = 4 : 3. After 6 years, Arun will be 26. What is Deepak’s present age?”

Accepted Answer

Introduction / Context: We first recover Arun’s present age from the future statement, then use the 4:3 ratio to scale down to Deepak’s present age proportionally. Given Data / Assumptions: Arun in 6 years = 26 ⇒ Arun now = 20. Present ratio A:D = 4:3. Concept / Approach: If 4 parts correspond to 20 (Arun), then 1 part is 20/4 = 5; Deepak has 3 parts, so 3 * 5. Step-by-Step Solution: Arun now = 26 − 6 = 20.Deepak now = (3/4) * 20 = 15. Verification / Alternative check: In 6 years: Deepak 21; ratio becomes 26:21 which simplifies close to 4:3 given the fixed initial ratio used for present ages. The present ages are consistent with the given data. Why Other Options Are Wrong: 12/19/25/13 do not equal (3/4) of 20. Common Pitfalls: Applying the 4:3 ratio to the future ages instead of to the present ages; or forgetting to first compute Arun’s present age. Final Answer: 15

Question 4

Present ages from future ratio and given sum — The sum of the present ages of a father and his son is 70 years. Exactly 10 years from now, the son’s age will be one-half of the father’s age then. What are their present ages (Father, Son)?

Accepted Answer

Introduction / Context: This is a classic linear-ages setup that mixes a present-time sum with a future-time ratio. We are told the combined present ages equal 70, and in 10 years the son will be exactly half the father. Translating both statements into equations in the present variables is the cleanest approach. Given Data / Assumptions: Let F = father’s present age; S = son’s present age. F + S = 70 (present sum). After 10 years: S + 10 = (F + 10) / 2. All ages are whole years; algebra works regardless of integrality. Concept / Approach: Convert the future relation to a present equation and combine with the present sum. From S + 10 = (F + 10)/2, multiply by 2 to avoid fractions and isolate F in terms of S; then substitute into F + S = 70 to solve systematically. Step-by-Step Solution: 2(S + 10) = F + 10 ⇒ 2S + 20 = F + 10 ⇒ F = 2S + 10.Sum: (2S + 10) + S = 70 ⇒ 3S + 10 = 70 ⇒ 3S = 60 ⇒ S = 20.Therefore F = 2(20) + 10 = 50. Verification / Alternative check: In 10 years: son = 30, father = 60; indeed 30 is half of 60. Present sum 50 + 20 = 70, as required. Why Other Options Are Wrong: Pairs like 50 & 25 or 47 & 23 do not satisfy both the present sum and the specified future half relation simultaneously. Common Pitfalls: Mistaking “son will be half the father in 10 years” for “now,” or misapplying the +10 to only one person. Both ages increase by the same amount when moving forward in time. Final Answer: 50 years, 20 years

Problems on Ages Questions