Question 1

Five years ago Raj was one third as old as Rahim. If Raj is 17 years old now, how old is Rahim at present?

Accepted Answer

Introduction / Context:This age problem connects the present age of one person with the past ages of two people. It tests the ability to back-calculate ages using simple ratios and then move forward to find the current age of the second person. Such questions are typical in aptitude exams and can be solved efficiently with basic algebra. Given Data / Assumptions: Raj is 17 years old at present. Five years ago, Raj was one third as old as Rahim. We need to find the present age of Rahim. All ages are measured in years and are positive. Concept / Approach:The problem gives a relationship between their ages at a past time. We first compute Raj age 5 years ago and then use the given ratio to find Rahim age 5 years ago. After that, we add 5 years to Rahim past age to get his current age. This method uses the idea that ages increase linearly with time and that age differences remain constant. Step-by-Step Solution:Present age of Raj = 17 years.Five years ago, Raj age was 17 - 5 = 12 years.At that time, Raj was one third as old as Rahim, so Rahim age 5 years ago was 3 times 12.Rahim age 5 years ago = 3 * 12 = 36 years.To find Rahim present age, add 5 years: 36 + 5 = 41 years.Therefore Rahim is currently 41 years old. Verification / Alternative check:Check the relationship 5 years ago: Raj was 12, Rahim was 36.Compute the ratio: 12 : 36 = 1 : 3, so Raj was indeed one third as old as Rahim.Check the present ages: Raj is 17 and Rahim is 41, so the age difference is 24 years.The age difference 5 years ago would also be 24 years, confirming consistency. Why Other Options Are Wrong:40 years: If Rahim were 40 now, then 5 years ago he would have been 35, and the ratio 12 : 35 is not 1 : 3.36 years: This is Rahim age 5 years ago, not his current age.48 years: If Rahim were 48 now, then 5 years ago he would have been 43, which does not satisfy the one third relationship with Raj age 12. Common Pitfalls:Some learners directly apply the ratio to current ages instead of correctly going back 5 years first.Another mistake is forgetting to add 5 years back after computing Rahim age 5 years ago.Misinterpreting one third as Raj being three times older than Rahim also leads to wrong equations. Final Answer:The present age of Rahim is 41 years.

Question 2

Ten years ago A was half as old as B. The ratio of their present ages is 3 : 4. What is the sum of their present ages in years?

Accepted Answer

Introduction / Context:This question involves both a past age relationship and a present age ratio between two people, A and B. It checks your ability to use linear equations to connect these conditions and then find the required sum of their present ages. Such problems are common in competitive exams under the topic problems on ages. Given Data / Assumptions: Ten years ago, A was half as old as B. The ratio of their present ages is 3 : 4. We need to find the sum of their present ages. All ages are in years and taken as positive. Concept / Approach:The problem combines information from two different times. We start by expressing the present ages of A and B in terms of a common variable using the given ratio. Then we relate these present ages back to the condition that held ten years ago. From this, we obtain a simple equation in one variable and solve it to find the ages and their sum. Step-by-Step Solution:Let present age of A be 3 * x years.Let present age of B be 4 * x years.Ten years ago, A age was 3 * x - 10 and B age was 4 * x - 10.The condition says that ten years ago A was half as old as B, so 3 * x - 10 = (1 / 2) * (4 * x - 10).Multiply both sides by 2 to clear the fraction: 2 * (3 * x - 10) = 4 * x - 10.This gives 6 * x - 20 = 4 * x - 10.Rearrange: 6 * x - 4 * x = -10 + 20, so 2 * x = 10.Thus x = 5.Present age of A = 3 * 5 = 15 years, present age of B = 4 * 5 = 20 years.Sum of their present ages = 15 + 20 = 35 years. Verification / Alternative check:Check the past condition: Ten years ago, A was 15 - 10 = 5 years old, B was 20 - 10 = 10 years old.Now verify the relationship: 5 is half of 10, so the condition is satisfied.Check the present ratio: 15 : 20 simplifies to 3 : 4, confirming consistency with the given ratio. Why Other Options Are Wrong:15 years: This is only the age of A, not the sum of ages.25 years: Does not match the sum of any pair of ages that satisfies the given conditions.45 years: This would imply significantly larger ages and would not satisfy the past relationship. Common Pitfalls:Some students forget to subtract 10 from both ages when considering the situation ten years ago.Another mistake is treating A was half of B as A = 2 * B instead of A = B / 2.Algebraic mistakes during rearrangement of 6 * x - 20 = 4 * x - 10 can also lead to incorrect values of x. Final Answer:The sum of the present ages of A and B is 35 years.

Question 3

The ratio of the present age of Yuvaraj to the present age of Ganguly is 4 : 3. Yuvaraj will be 26 years old after 6 years. What is the present age of Ganguly?

Accepted Answer

Introduction / Context:This age problem combines a present age ratio with information about the future age of one person. The task is to determine the current age of the second person. It checks your ability to handle ratios and to move forward or backward in time when working with ages. Given Data / Assumptions: Present age ratio of Yuvaraj to Ganguly is 4 : 3. Yuvaraj will be 26 years old after 6 years. We need the present age of Ganguly. All ages are expressed in years and are positive. Concept / Approach:When a future age is given, we subtract the time difference to find the current age. Using the ratio, we then express both ages in terms of a common variable and match it to the known current age of one person. This directly gives the variable and the age of the other person. Step-by-Step Solution:Let present age of Yuvaraj be 4 * x years.Let present age of Ganguly be 3 * x years.We know that after 6 years, Yuvaraj will be 26 years old.So 4 * x + 6 = 26.Rearrange: 4 * x = 26 - 6 = 20.Therefore x = 20 / 4 = 5.Present age of Yuvaraj = 4 * 5 = 20 years.Present age of Ganguly = 3 * 5 = 15 years. Verification / Alternative check:Check future age: After 6 years, Yuvaraj age will be 20 + 6 = 26 years, which matches the given statement.Check the present ratio: 20 : 15 simplifies to 4 : 3 by dividing both numbers by 5.Hence the solution for Ganguly age as 15 years is consistent with both pieces of information. Why Other Options Are Wrong:17 years: This would correspond to a ratio that does not match 4 : 3 when paired with Yuvaraj correct age.19 years: This is too large and cannot maintain the 4 : 3 ratio given the future condition.12 years: This is too small and also fails to maintain the correct ratio and future information. Common Pitfalls:Some students mistakenly set 4 * x equal to 26 instead of 4 * x + 6, forgetting the 6 years to be added.Others misinterpret the ratio and use 3 : 4 rather than 4 : 3, which reverses the roles of Yuvaraj and Ganguly.Mixing present and future values without adjusting correctly often leads to wrong equations. Final Answer:The present age of Ganguly is 15 years.

Question 4

Rajeev age after 15 years will be 5 times his age 5 years ago. What is the present age of Rajeev in years?

Accepted Answer

Introduction / Context:This problems-on-ages question relates Rajeev age at two different points in time: 5 years ago and 15 years in the future. The relation given is multiplicative, meaning the future age is a multiple of the past age. The goal is to translate this into an equation and compute Rajeev present age. Given Data / Assumptions: Rajeev age after 15 years will be 5 times his age 5 years ago. We are asked to find his current age. All ages are in years and considered positive. Concept / Approach:When a relationship involves a person age at two different times, we first express both of those ages in terms of the present age. Then we use the given multiplicative relationship to write a single equation. Solving that equation yields the present age. This is a basic application of linear equations in one variable. Step-by-Step Solution:Let the present age of Rajeev be A years.His age 5 years ago was A - 5 years.His age after 15 years will be A + 15 years.The condition says: age after 15 years = 5 times age 5 years ago.So we write: A + 15 = 5 * (A - 5).Expand the right side: A + 15 = 5 * A - 25.Rearrange terms: Move A to the right and -25 to the left.15 + 25 = 5 * A - A.40 = 4 * A.So A = 40 / 4 = 10 years. Verification / Alternative check:If Rajeev is 10 now, then 5 years ago he was 10 - 5 = 5 years old.15 years from now he will be 10 + 15 = 25 years old.Check the relationship: 25 is indeed 5 times 5.Thus, the condition is perfectly satisfied for A = 10 years. Why Other Options Are Wrong:12 years: If A = 12, then A - 5 = 7 and A + 15 = 27. Here 27 is not 5 times 7.14 years: If A = 14, then A - 5 = 9 and A + 15 = 29, and 29 is not 5 times 9.22 years: If A = 22, then A - 5 = 17 and A + 15 = 37, which again does not satisfy the relationship. Common Pitfalls:Students sometimes interchange the time intervals and write A - 15 and A + 5, which mixes up past and future ages.Misplacing terms while rearranging the equation can lead to incorrect values of A.Occasionally, learners confuse five times with increasing by 5, which is a completely different relationship. Final Answer:The present age of Rajeev is 10 years.

Question 5

The present age of a man is three times the total of the present ages of his two daughters. Five years from now, his age will be twice the total of the ages of his daughters at that time. What is the current age of the father?

Accepted Answer

Introduction / Context:This is a standard age puzzle involving a parent and two children. The father age is related to the combined ages of his two daughters at two different points in time: now and five years later. The problem tests understanding of linear equations and relationships involving sums of ages rather than individual ages. Given Data / Assumptions: The present age of the father is three times the sum of the present ages of his two daughters. After 0.5 decades, that is after 5 years, the father age will be twice the sum of the daughters ages at that time. We need to find the present age of the father. All ages are in years and positive. Concept / Approach:Instead of tracking each daughter separately, we can work with the combined age of the daughters. This simplifies the algebra. We represent the current sum of daughters ages by a single variable and then express the father age in terms of this variable. Using the future condition, we set up a second equation and solve for the daughters total and then the father age. Step-by-Step Solution:Let the present total age of the two daughters be D years.Then the present age of the father is 3 * D years, since he is three times their combined age.After 5 years, each daughter will be 5 years older, so the total age of the daughters will be D + 10 years.After 5 years, the father age will be 3 * D + 5 years.The problem states that after 5 years, the father age will be twice the total age of the daughters at that time.So we write the equation: 3 * D + 5 = 2 * (D + 10).Expand the right side: 3 * D + 5 = 2 * D + 20.Rearrange: 3 * D - 2 * D = 20 - 5.So D = 15.Therefore the father present age is 3 * D = 3 * 15 = 45 years. Verification / Alternative check:Present ages: father = 45 years, total daughters ages = 15 years.After 5 years: father = 45 + 5 = 50 years, daughters total = 15 + 10 = 25 years.Check the future condition: 50 is indeed twice 25.This confirms that the father present age of 45 years is correct. Why Other Options Are Wrong:35 years: With this age, it is not possible to satisfy both the three times and twice conditions using positive daughter ages.40 years: Fails the future condition when we compute the daughters total age.47 years: This does not fit neatly into the required multiplicative relationships with an integer daughters sum. Common Pitfalls:Some students mistakenly treat three times the age of each daughter separately rather than the sum.Others forget that both daughters grow older, and therefore the daughters total increases by 10, not 5, after 5 years.Carelessness in forming the equation 3 * D + 5 = 2 * (D + 10) can lead to wrong algebraic rearrangements. Final Answer:The present age of the father is 45 years.

Question 6

Eight years ago Rama mother was five times as old as Rama. After 8 years from now, the mother will be twice as old as Rama. What is the present age of Rama?

Accepted Answer

Introduction / Context:This problem involves the ages of Rama and her mother at two different times, 8 years in the past and 8 years in the future. The relationships given are multiplicative, making it a classic two-variable linear equation question under the topic of problems on ages. Given Data / Assumptions: Eight years ago, the mother was five times as old as Rama. After 8 years from now, the mother will be twice as old as Rama. We are asked to find the present age of Rama. Ages are measured in years and may be fractional. Concept / Approach:The strategy is to define variables for their present ages and then translate both time-based statements into equations. Each equation links the same two unknown ages but at different times, which allows us to solve the system. Even if the result is fractional, that is mathematically acceptable, and the answer choices include fractional ages. Step-by-Step Solution:Let the present age of Rama be R years.Let the present age of the mother be M years.Eight years ago, Rama age was R - 8 and mother age was M - 8.At that time, the mother was five times as old as Rama: M - 8 = 5 * (R - 8).After 8 years, Rama age will be R + 8 and mother age will be M + 8.At that future time, the mother will be twice as old as Rama: M + 8 = 2 * (R + 8).Now expand both equations.From the first equation: M - 8 = 5 * R - 40, so M = 5 * R - 32.From the second equation: M + 8 = 2 * R + 16, so M = 2 * R + 8.Set the two expressions for M equal: 5 * R - 32 = 2 * R + 8.Rearrange: 5 * R - 2 * R = 8 + 32.This gives 3 * R = 40, so R = 40 / 3 years. Verification / Alternative check:Rama present age = 40 / 3 years, which is approximately 13.33 years.Mother present age M = 2 * R + 8 = 2 * (40 / 3) + 8 = 80 / 3 + 24 / 3 = 104 / 3 years.Eight years ago: Rama was 40 / 3 - 8 = 40 / 3 - 24 / 3 = 16 / 3 years, mother was 104 / 3 - 8 = 104 / 3 - 24 / 3 = 80 / 3 years.Check first condition: 5 times Rama age then is 5 * 16 / 3 = 80 / 3, equal to the mother age then.After 8 years: Rama will be 40 / 3 + 8 = 64 / 3 years, mother will be 104 / 3 + 8 = 128 / 3 years.Check second condition: Twice Rama age then is 2 * 64 / 3 = 128 / 3, equal to mother age then. Why Other Options Are Wrong:10 years: Substitution gives inconsistent values for the mother age that do not satisfy the future condition.20/3 years: This smaller age does not simultaneously satisfy both relationships.50/3 years: This larger age for Rama makes the mother younger than expected based on the given ratios. Common Pitfalls:Errors often occur when shifting ages by 8 years in the wrong direction for past or future conditions.Some learners incorrectly assume ages must be integers and reject correct fractional results.Algebra mistakes when equating 5 * R - 32 and 2 * R + 8 can easily lead to the wrong value of R. Final Answer:The present age of Rama is 40/3 years.

Question 7

When I was 24 years old, my mother age was twice my age. Now I am 44 years old. How old is my mother at present?

Accepted Answer

Introduction / Context:This is a simple age-difference problem. The key relationship is that at a certain point in the past, the mother was exactly twice as old as the child. We use this to fix the age difference, and then use the current age of the child to determine the present age of the mother. Given Data / Assumptions: When the child was 24 years old, the mother age was twice that, that is 48 years. Now the child is 44 years old. We need to find the present age of the mother. All ages are in years. Concept / Approach:In problems like this, the age difference between parent and child remains constant over time. Once we know both ages at one point in time, we can compute their age difference. Then, as time passes, we simply add the same number of years to both while keeping the difference fixed. This allows us to calculate the mother present age easily. Step-by-Step Solution:At the time when the child was 24 years old, the mother age was twice 24, which is 2 * 24 = 48 years.So at that moment, the age difference between mother and child was 48 - 24 = 24 years.This age difference remains constant throughout their lives.Now the child is 44 years old.Therefore the present age of the mother is 44 + 24 = 68 years. Verification / Alternative check:When the child was 24, the mother was 48. There is a 24-year gap.Now 20 years have passed since the child went from 24 to 44.The mother age would also increase by 20 years, from 48 to 68.This confirms that the calculation is consistent with both the age difference and the passage of time. Why Other Options Are Wrong:84 years: This would imply an age difference of 40 years instead of 24, which contradicts the original information.48 years: This is the mother age when the child was 24, not the present age.88 years: This would create a 44-year age difference, which is inconsistent with the original twice-as-old statement. Common Pitfalls:Some learners mistakenly double the current age of the child or use the wrong age for the twice relationship.Others forget that the age difference is fixed and instead try to set up unnecessary equations.Confusing the time gap between the two events can also cause arithmetic mistakes. Final Answer:The mother present age is 68 years.

Question 8

The present age of a man is two fifths of the age of his father. After 8 years, the man will be one half of the age of his father. What is the present age of the father?

Accepted Answer

Introduction / Context:This age problem uses fractional relationships between the ages of a man and his father at different times. At present, the man age is a fixed fraction of his father age, and after some years, the man age will be another fraction of his father age. This leads to a pair of equations that can be solved to find the father present age. Given Data / Assumptions: The present age of the man is two fifths of the age of his father. After 8 years, the man will be one half of the age of his father. We are asked to find the current age of the father. All ages are in years and positive. Concept / Approach:We introduce variables for the present ages of the man and the father. Using the fraction relationships, we derive two equations: one for the present and one for the future. Solving these equations simultaneously yields the exact ages. This technique is standard for problems involving fractional comparisons at different times. Step-by-Step Solution:Let the father present age be F years.Then the man present age is (2 / 5) * F years.After 8 years, the father age will be F + 8 years.After 8 years, the man age will be (2 / 5) * F + 8 years.The condition says that after 8 years, the man will be one half of his father age, so:(2 / 5) * F + 8 = (1 / 2) * (F + 8).Multiply every term by 10 to clear the fractions: 10 * ((2 / 5) * F + 8) = 10 * (1 / 2) * (F + 8).This gives 4 * F + 80 = 5 * (F + 8).Expand the right side: 4 * F + 80 = 5 * F + 40.Rearrange: 80 - 40 = 5 * F - 4 * F, so 40 = F.Therefore the present age of the father is 40 years. Verification / Alternative check:If F = 40, man present age is (2 / 5) * 40 = 16 years.After 8 years, father age will be 40 + 8 = 48 years, man age will be 16 + 8 = 24 years.Check the future condition: 24 is exactly one half of 48.Both the present and future fractional relationships are satisfied, confirming that F = 40 years is correct. Why Other Options Are Wrong:50 years: If the father were 50, the man would be 20 now and 28 after 8 years, while the father would be 58, and 28 is not half of 58.33 years: This leads to non-integer or inconsistent values when the future half relationship is applied.30 years: This would give a present man age of 12 and future ages of 20 and 38, which do not satisfy the one half condition. Common Pitfalls:Misinterpreting the fractions as F = (2 / 5) * man or swapping numerator and denominator is a common error.Some students forget to add 8 years to both ages for the future condition.Algebraic mistakes when clearing fractions, such as incorrect multiplication, often result in wrong values. Final Answer:The present age of the father is 40 years.

Question 9

One year ago, Vijaya was four times as old as her daughter Simran. Six years from now, Vijaya age will exceed Simran age by 9 years. What is the ratio of their present ages?

Accepted Answer

Introduction / Context:This question presents a mother daughter age situation at two different times: one year in the past and six years in the future. The relationships describe a multiple of ages at one time and a fixed difference at another time. The task is to find the ratio of their present ages, which is a typical type of problem in the ages topic. Given Data / Assumptions: One year ago, Vijaya was four times as old as her daughter Simran. Six years from now, Vijaya age will exceed Simran age by 9 years. We need the ratio of their present ages. All ages are in years and positive. Concept / Approach:We assign variables to represent the current ages of Vijaya and Simran. Then we use each time-based statement to write an equation involving those variables. Solving the pair of equations gives the exact ages, from which we derive the required ratio. The key is correctly shifting ages backward and forward in time and translating the relationships accurately. Step-by-Step Solution:Let Vijaya present age be V years.Let Simran present age be S years.One year ago, their ages were V - 1 and S - 1 respectively.Given that one year ago Vijaya was four times as old as Simran: V - 1 = 4 * (S - 1).Simplify: V - 1 = 4 * S - 4, so V = 4 * S - 3.Six years from now, their ages will be V + 6 and S + 6.At that time, Vijaya age will exceed Simran age by 9 years, so V + 6 = S + 6 + 9.This simplifies to V + 6 = S + 15, hence V = S + 9.Now we have two expressions for V: V = 4 * S - 3 and V = S + 9.Set them equal: 4 * S - 3 = S + 9.Rearrange: 4 * S - S = 9 + 3, so 3 * S = 12.Thus S = 4 years, and V = S + 9 = 4 + 9 = 13 years.Therefore the ratio of their present ages is V : S = 13 : 4. Verification / Alternative check:One year ago, Vijaya was 13 - 1 = 12 years old, Simran was 4 - 1 = 3 years old.Check first condition: 12 is four times 3, so it holds.Six years from now, Vijaya will be 13 + 6 = 19, Simran will be 4 + 6 = 10.Check second condition: 19 - 10 = 9, so the future difference is 9 years, as given. Why Other Options Are Wrong:8 : 1: This would imply a much larger age gap and fails both given conditions.10 : 2: This simplifies to 5 : 1, which is inconsistent with being four times at one time and a difference of 9 at another time.11 : 4: Close in appearance but does not satisfy either the past four times condition or the future difference condition. Common Pitfalls:Some students misinterpret four times as V - 1 = 4 * S instead of 4 * (S - 1).Forgetting to adjust both ages when moving backward or forward in time is another frequent source of error.Errors in forming or solving the simultaneous equations can easily lead to a wrong ratio. Final Answer:The ratio of the present ages of Vijaya and Simran is 13 : 4.

Question 10

The sum of the present ages of Rajesh, Suresh, Mahesh and Dinesh is 76 years. Seven years from now, the ratio of their ages will be 7 : 6 : 5 : 8 respectively. What is the present age of Mahesh?

Accepted Answer

Introduction / Context:This problem deals with four people whose combined present ages are known. We are given the ratio of their ages seven years into the future and asked to determine the present age of one of them. The question is a good example of using ratios along with total sum information to solve for individual ages. Given Data / Assumptions: Present total of the ages of Rajesh, Suresh, Mahesh and Dinesh is 76 years. After 7 years, their ages will be in the ratio 7 : 6 : 5 : 8 respectively. We must find the present age of Mahesh. All ages are in years and positive. Concept / Approach:The ratio provided applies to their ages in the future, not now. We start by representing their ages after 7 years as multiples of a common factor. The sum of those future ages can be related to the sum of their present ages. Once we find the common factor, we can determine each person future age and then subtract 7 to obtain the present ages. From there, Mahesh present age is read off directly. Step-by-Step Solution:Let the ages of Rajesh, Suresh, Mahesh and Dinesh after 7 years be 7 * k, 6 * k, 5 * k and 8 * k respectively.Then the sum of their ages after 7 years is (7 + 6 + 5 + 8) * k = 26 * k.The sum of their present ages is 76 years.In 7 years, each person will be 7 years older, so the total increase in sum of ages will be 4 * 7 = 28 years.Therefore, sum of their ages after 7 years is 76 + 28 = 104 years.So we have 26 * k = 104.Solve for k: k = 104 / 26 = 4.Thus, ages after 7 years are:Rajesh: 7 * 4 = 28 years.Suresh: 6 * 4 = 24 years.Mahesh: 5 * 4 = 20 years.Dinesh: 8 * 4 = 32 years.Present ages are each of those minus 7:Mahesh present age = 20 - 7 = 13 years. Verification / Alternative check:Present ages: Rajesh = 28 - 7 = 21, Suresh = 24 - 7 = 17, Mahesh = 13, Dinesh = 32 - 7 = 25.Check sum: 21 + 17 + 13 + 25 = 76 years, which matches the given total.After 7 years, their ages are 28, 24, 20 and 32, and the ratio 28 : 24 : 20 : 32 simplifies to 7 : 6 : 5 : 8 by dividing each term by 4. Why Other Options Are Wrong:17 years: This would correspond to a different value of k and would not maintain both the future ratio and the total sum.14 years or 16 years: These also lead to inconsistencies in either the total sum or the required future ratio. Common Pitfalls:A common error is to apply the ratio directly to the present ages rather than to the future ages as specified.Some students forget that with four people, the total increase in sum over 7 years is 28 years, not just 7 years.There can also be mistakes in simplifying or computing the sum of the ratio parts. Final Answer:The present age of Mahesh is 13 years.

Question 11

Four persons A, B, C and D have present ages whose sum is 176 years. Four years ago, the ratio of their ages was 11 : 9 : 6 : 14 respectively. Who is the eldest and who is the youngest among them at present?

Accepted Answer

Introduction / Context:This question involves four people with a known sum of present ages and a given ratio of ages at a time in the past. The task is to determine which person is currently the eldest and which is the youngest. The problem tests the ability to move between past and present ages using ratios and simple algebra. Given Data / Assumptions: The sum of the present ages of A, B, C and D is 176 years. Four years ago, the ratio of their ages was 11 : 9 : 6 : 14 for A : B : C : D. We need to identify who is eldest and who is youngest at present. All ages are in years and positive. Concept / Approach:We represent their ages four years ago as 11 * k, 9 * k, 6 * k and 14 * k. From those we obtain their present ages by adding 4 years to each. The total sum of present ages is given, allowing us to solve for k. Once we know k, we can compute each present age and easily compare them to determine the eldest and youngest. Step-by-Step Solution:Let the ages four years ago be:A = 11 * k, B = 9 * k, C = 6 * k, D = 14 * k.Present ages will then be:A now = 11 * k + 4.B now = 9 * k + 4.C now = 6 * k + 4.D now = 14 * k + 4.Given that the sum of present ages is 176 years, we write:(11 * k + 4) + (9 * k + 4) + (6 * k + 4) + (14 * k + 4) = 176.Combine like terms: (11 + 9 + 6 + 14) * k + 4 * 4 = 176.This is (40 * k) + 16 = 176.Subtract 16 from both sides: 40 * k = 160.So k = 160 / 40 = 4.Now compute present ages:A now = 11 * 4 + 4 = 44 + 4 = 48 years.B now = 9 * 4 + 4 = 36 + 4 = 40 years.C now = 6 * 4 + 4 = 24 + 4 = 28 years.D now = 14 * 4 + 4 = 56 + 4 = 60 years.Comparing these, D is the eldest at 60 years, and C is the youngest at 28 years. Verification / Alternative check:Check the sum: 48 + 40 + 28 + 60 = 176 years, matching the given total.Check the ages four years ago: 44, 36, 24 and 56.Their ratio four years ago is 44 : 36 : 24 : 56. Dividing each by 4 gives 11 : 9 : 6 : 14, confirming the given ratio. Why Other Options Are Wrong:A is eldest and C is youngest: A is not the eldest, as D is older.C is eldest and A is youngest: This is the opposite of the actual ordering and does not match the calculated ages.A is eldest and B is youngest: Neither A nor B holds those extreme positions according to the computed ages. Common Pitfalls:Some students mistakenly treat the given ratio as present ages rather than ages four years ago.Others forget to add 4 years to each age when converting from past to present.Algebraic errors when summing the ages and solving for k can lead to incorrect age values. Final Answer:The correct statement is that D is eldest and C is youngest.

Question 12

Raju got married 8 years ago. His present age is six fifths of his age at the time of marriage. Raju sister was 10 years younger than him at the time of his marriage. What is the present age of Raju sister?

Accepted Answer

Introduction / Context:This problem involves age relationships at the time of marriage and in the present. Raju present age is given as a fraction of his age at marriage, and his sister age is related to his age at that same time. The question finally asks for the sister present age, a common format in problems on ages. Given Data / Assumptions: Raju got married 8 years ago. His present age is six fifths of his age at the time of marriage. At the time of marriage, Raju sister was 10 years younger than him. We must find the present age of Raju sister. All ages are in years and positive. Concept / Approach:We first calculate Raju age at the time of marriage by introducing a variable for that age. Using the fractional relationship, we convert to his present age. Once Raju age at marriage is known, we subtract 10 years to get his sister age at marriage. Finally, we add 8 years to find the sister present age. The approach uses stepwise reasoning across time points. Step-by-Step Solution:Let Raju age at the time of marriage be M years.Eight years have passed since his marriage, so his present age is M + 8 years.We are told that his present age is six fifths of his age at marriage.So M + 8 = (6 / 5) * M.Multiply both sides by 5 to clear the fraction: 5 * (M + 8) = 6 * M.This gives 5 * M + 40 = 6 * M.Rearrange: 40 = 6 * M - 5 * M = M.So Raju age at marriage was 40 years.His present age is M + 8 = 40 + 8 = 48 years.Raju sister was 10 years younger than him at the time of marriage, so her age then was 40 - 10 = 30 years.Since 8 years have passed, her present age is 30 + 8 = 38 years. Verification / Alternative check:Raju at marriage: 40 years, sister at marriage: 30 years.Present ages: Raju = 48 years, sister = 38 years.Check the fractional relationship: six fifths of 40 is (6 / 5) * 40 = 48, which matches Raju present age.Check the age difference between Raju and his sister: 40 - 30 = 10 years at marriage and 48 - 38 = 10 years now, consistent across time. Why Other Options Are Wrong:30 years: This is the sister age at the time of Raju marriage, not her present age.32 years: This does not maintain the 10-year age difference with the correctly computed present age of Raju.None of these: This is incorrect because 38 years is a valid option and satisfies all the conditions. Common Pitfalls:Learners sometimes misinterpret the fraction and write present age as five sixths of the marriage age instead of six fifths.Another mistake is to forget that 8 years have also passed for the sister, leading them to give her age at marriage rather than her present age.Errors in solving M + 8 = (6 / 5) * M can lead to a wrong value of M. Final Answer:The present age of Raju sister is 38 years.

Question 13

The ratio of the present ages of Mahesh and Nilesh is 5 : x. Mahesh is 18 years younger than Ramesh. After 9 years, Ramesh will be 47 years old. If the absolute difference between the present ages of Mahesh and Nilesh is equal to the present age of …

Accepted Answer

Introduction / Context:This problem combines age ratios, fixed age differences, and a condition involving the difference between the ages of two people being equal to the age of a third person. It is a multi-step algebraic question typical of higher level problems on ages in aptitude tests. Given Data / Assumptions: The present age ratio of Mahesh to Nilesh is 5 : x. Mahesh is 18 years younger than Ramesh. After 9 years, Ramesh will be 47 years old. The absolute difference between the present ages of Mahesh and Nilesh is equal to the present age of Ramesh. We need to find the value of x. Concept / Approach:The first step is to determine Ramesh present age using the future information. Then, using the relationship between Mahesh and Ramesh, we compute Mahesh age. Using the given ratio 5 : x, we express Nilesh age in terms of x. The condition that the absolute difference between Mahesh and Nilesh ages equals Ramesh age gives an equation in x, which we solve. Finally, we match the numeric value of x to the options. Step-by-Step Solution:After 9 years, Ramesh will be 47 years old.Therefore, Ramesh present age is 47 - 9 = 38 years.Mahesh is 18 years younger than Ramesh.So Mahesh present age = 38 - 18 = 20 years.The ratio of the ages of Mahesh and Nilesh is 5 : x.Let Mahesh age be 5 * k and Nilesh age be x * k.Since Mahesh age is 20, 5 * k = 20, so k = 4.Then Nilesh present age = x * k = 4 * x years.We are told that the absolute difference between Mahesh and Nilesh ages equals Ramesh present age.So |20 - 4 * x| = 38.This gives two possible equations: 20 - 4 * x = 38 or 4 * x - 20 = 38.The first equation 20 - 4 * x = 38 leads to -4 * x = 18, so x = -4.5, which is not a valid ratio value.The second equation is 4 * x - 20 = 38, so 4 * x = 58 and x = 58 / 4 = 14.5. Verification / Alternative check:If x = 14.5, then Nilesh present age is 4 * 14.5 = 58 years.Mahesh present age is 20 years and Ramesh present age is 38 years.The absolute difference between Mahesh and Nilesh ages is |20 - 58| = 38 years.This exactly matches Ramesh present age, confirming that x = 14.5 satisfies all conditions. Why Other Options Are Wrong:11.8, 12.9, 13.7: Each of these values for x produces an age for Nilesh that does not yield a difference of 38 years when compared with Mahesh age of 20 years.Furthermore, negative or illogical ratios would contradict the idea of meaningful age comparisons. Common Pitfalls:Some students forget that the difference is an absolute value and only consider one algebraic case.Others make sign errors while solving 20 - 4 * x = 38 or 4 * x - 20 = 38.It is also easy to mis-handle the ratio and incorrectly relate Mahesh age to x rather than to 5. Final Answer:The required value of x in the ratio of the ages of Mahesh and Nilesh is 14.5.

Question 14

The ages of X and Y are in the ratio 6 : 5 and the sum of their present ages is 44 years. What will be the ratio of their ages exactly 8 years from now?

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Introduction / Context: This problem on ages uses a simple ratio along with the total of ages to form linear equations. Once the present ages are known, we project them into the future and compute the new ratio. Questions like this are very common in aptitude tests and are a good example of how proportional reasoning and basic algebra work together. Given Data / Assumptions: The present ages of X and Y are in the ratio 6 : 5. The sum of their present ages is 44 years. We are asked to find the ratio of their ages after 8 years. Ages increase linearly with time and there is no other change in the relationship. Concept / Approach: We represent ages in terms of a common multiplier using the ratio. Then we use the given total age to find the value of this multiplier. Once we know the actual current ages of X and Y, we add 8 years to each and simplify the resulting ratio. The key algebra ideas are ratios, linear equations and simplification of fractions. Step-by-Step Solution: Let the present ages of X and Y be 6k and 5k years respectively. Given that 6k + 5k = 44, so 11k = 44. Therefore k = 44 / 11 = 4. So the present age of X = 6 * 4 = 24 years. The present age of Y = 5 * 4 = 20 years. After 8 years, the age of X will be 24 + 8 = 32 years. After 8 years, the age of Y will be 20 + 8 = 28 years. Required ratio after 8 years = 32 : 28. Divide both terms by 4 to simplify: 32 : 28 = 8 : 7. Verification / Alternative check: We can quickly check the total present age: 24 + 20 = 44, which matches the given sum. The ratio 24 : 20 simplifies to 6 : 5, which matches the given ratio. After 8 years, the ages 32 and 28 differ by 4 years, the same difference as now, which is consistent. Simplifying 32 : 28 by 4 correctly gives 8 : 7, confirming the result. Why Other Options Are Wrong: The ratio 6 : 5 corresponds to the present ages, not the future ages. The ratio 7 : 6 does not come from any correct computation. The ratio 9 : 5 would require one person to age faster than the other, which does not happen. The ratio 5 : 4 is unrelated to the given data and does not match the calculations. Common Pitfalls: A common error is to take 6 and 5 directly as ages instead of 6k and 5k. Another mistake is to forget to add 8 years to both ages before taking the ratio or to add 8 only to one of them. Some students also simplify the ratio incorrectly or mix up the order of X and Y. Carefully writing each algebraic step helps avoid these issues. Final Answer: Thus, the ratio of the ages of X and Y after 8 years will be 8 : 7.

Question 15

Eight years ago Ajay's age was four third of Vijay's age, and eight years from now Ajay's age will be six fifth of Vijay's age. What is Ajay's present age in years?

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Introduction / Context: This is a classic two person age problem where ratios of ages are given at two different times, one in the past and one in the future. By translating the statements into equations, we can solve for the present ages. Such questions train you to handle linear equations and time shifts in aptitude exams. Given Data / Assumptions: Eight years ago, Ajay's age was 4/3 times Vijay's age at that time. Eight years from now, Ajay's age will be 6/5 times Vijay's age at that time. We need to find Ajay's present age. Ages increase uniformly and there are no other hidden conditions. Concept / Approach: Let the present ages of Ajay and Vijay be A and V years. Use the condition from eight years ago to express A and V in one equation. Then use the condition from eight years in the future for a second equation. Solving these two linear equations simultaneously gives us both ages. We finally report Ajay's present age. Step-by-Step Solution: Let Ajay's present age be A years and Vijay's present age be V years. Eight years ago: A - 8 = (4/3) * (V - 8). Eight years hence: A + 8 = (6/5) * (V + 8). From the second equation, cross multiply: 5(A + 8) = 6(V + 8). This gives 5A + 40 = 6V + 48 so 5A = 6V + 8. From the first equation: 3(A - 8) = 4(V - 8). So 3A - 24 = 4V - 32 which simplifies to 3A = 4V - 8. Now solve 5A = 6V + 8 and 3A = 4V - 8 simultaneously. From 3A = 4V - 8, we get A = (4V - 8) / 3. Substitute into 5A = 6V + 8. 5 * (4V - 8) / 3 = 6V + 8 gives 20V - 40 = 18V + 24 so 2V = 64 and V = 32. Then A = (4 * 32 - 8) / 3 = (128 - 8) / 3 = 120 / 3 = 40. Verification / Alternative check: Check eight years ago: Ajay was 40 - 8 = 32 and Vijay was 32 - 8 = 24. Ratio 32 : 24 simplifies to 4 : 3, which matches. Eight years ahead: Ajay will be 48 and Vijay 40, so 48 / 40 = 6 / 5, which again matches. Hence the solution is correct. Why Other Options Are Wrong: If Ajay were 41, the ratios would not match exactly when checked in past and future. Ages 37 or 33 also fail either the 4/3 or the 6/5 relation. Only 40 years satisfies both time based ratios simultaneously, so the other choices are not consistent with the given conditions. Common Pitfalls: Typical errors include mixing up the direction of time shifts, forgetting to subtract or add 8 correctly, or mismanaging fractions when forming equations. Another common issue is solving only one of the equations and guessing the ages instead of solving the system. Careful algebra and checking both conditions prevent these mistakes. Final Answer: Therefore, Ajay's present age is 40 years.

Question 16

If 6 years are subtracted from the present age of Arun and the result is divided by 18, we obtain the present age of his grandson Gokul. Gokul is 2 years younger than Madan, whose present age is 5 years. What is the present age of Arun?

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Introduction / Context: This age problem links three generations using a simple linear relation. The present age of Arun is connected to the age of his grandson through subtraction and division, and the grandson's age is related to another child, Madan. Translating the language into algebraic expressions makes the solution straightforward. Given Data / Assumptions: If 6 years are subtracted from Arun's present age and the result is divided by 18, we get Gokul's present age. Gokul is 2 years younger than Madan. Madan's present age is 5 years. We assume all ages are in whole years and consistent. Concept / Approach: We first compute Gokul's present age using the information about Madan. Then we use the equation involving Arun's age and Gokul's age to solve for Arun. The problem becomes a simple one variable linear equation once we substitute the known values. Step-by-Step Solution: Let Arun's present age be A years. We are told that (A - 6) / 18 equals Gokul's present age. Madan's present age is given as 5 years. Gokul is 2 years younger than Madan, so Gokul's age = 5 - 2 = 3 years. Therefore, (A - 6) / 18 = 3. Multiply both sides by 18 to clear the denominator: A - 6 = 3 * 18. Compute 3 * 18 = 54, so A - 6 = 54. Add 6 to both sides to get A = 54 + 6. Thus, A = 60 years. Verification / Alternative check: Check the result: Arun is 60 years old now. Subtracting 6 gives 54. Dividing 54 by 18 gives 3, which matches Gokul's age. Gokul being 3 years old is indeed 2 years younger than Madan, who is 5. All given conditions are satisfied, so the solution is confirmed. Why Other Options Are Wrong: If Arun were 72 years old, (72 - 6) / 18 = 66 / 18 which is not equal to 3. If he were 54, (54 - 6) / 18 = 48 / 18 is also not 3. For 47 years, (47 - 6) / 18 is not an integer and does not give 3 either. Only 60 years makes the equation match the known age of Gokul. Common Pitfalls: A frequent mistake is to mix the operations, for example dividing by 6 instead of subtracting 6, or dividing by 18 at the wrong time. Another error is miscomputing Gokul's age from Madan's age. Writing down the relationship clearly before substituting values helps avoid confusion and calculation slips. Final Answer: Therefore, Arun's present age is 60 years.

Question 17

The present average age of a family of five members is 26 years. If the present age of the youngest member is 10 years, what was the average age of the family at the time of the youngest member's birth, assuming nobody has died in the family since t…

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Introduction / Context: This problem tests understanding of average ages over time and how the birth of a new member affects the family average. We compare the present situation with the situation 10 years ago when the youngest member was just born. The key idea is to track total ages rather than only averages. Given Data / Assumptions: There are 5 family members at present. The present average age of the family is 26 years. The youngest member is currently 10 years old. No death has occurred in the family since the birth of the youngest member. We need the average age of the family at the time of the youngest member's birth. Concept / Approach: First find the total present age of all 5 members using the average. Then subtract the years that have passed for each member to move back in time to the year of birth of the youngest child. At that time there were only 4 members, because the youngest was just born. We then compute the average from the total age of those 4 members at that earlier time. Step-by-Step Solution: Present average age of 5 members = 26 years. Total present age of family = 26 * 5 = 130 years. Present age of youngest member = 10 years. The sum of the present ages of the other 4 members = 130 - 10 = 120 years. The youngest member was born 10 years ago, so move 10 years back. Ten years ago, each of the older 4 was 10 years younger, so their total then = 120 - 4 * 10 = 120 - 40 = 80 years. At the time of birth, the youngest member was 0 years old, so total age of family = 80 years with 4 members. Average age at that time = 80 / 4 = 20 years. Verification / Alternative check: Check by moving forward: Ten years ago total = 80. After 10 years, each of 4 older members gains 10 years, adding 40, and the youngest reaches 10, adding 10 more. New total = 80 + 40 + 10 = 130, giving present average 130 / 5 = 26 as stated. This confirms that the earlier average 20 years is consistent. Why Other Options Are Wrong: An average of 19 or 18 years would not produce the correct present total when progressed by 10 years. A value like 16 years is far too low and leads to a wrong total when ages are advanced. The option 21 years also fails when the same forward check is done. Only 20 years yields the correct present situation. Common Pitfalls: Many learners mistakenly average only the remaining four members directly or think that the average decreases by 10, which is incorrect. Another error is forgetting that the youngest was 0 years at birth, not 10. Working with totals and carefully counting the number of family members at each time avoids these misconceptions. Final Answer: Hence, the average age of the family at the time of the youngest member's birth was 20 years.

Question 18

Mohan was married 5 years ago. At present his age is six fifth of his age at the time of marriage. His wife is 3 years younger than Mohan. What is his wife's present age?

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Introduction / Context: Here we are given a relationship between Mohan's age now and his age at the time of marriage, along with the fact that his wife is 3 years younger than him. Using these details, we convert the verbal description into equations and determine first Mohan's current age and then his wife's age. Given Data / Assumptions: Mohan was married 5 years ago. His present age is 6/5 times his age at the time of marriage. His wife is 3 years younger than Mohan at present. We must find his wife's present age. Concept / Approach: Let M be Mohan's present age. Since he married 5 years ago, his age at that time was M - 5. The relation says M = (6/5) * (M - 5). We solve this linear equation to get M. Then subtract 3 to obtain his wife's current age. This is a straightforward proportional age problem. Step-by-Step Solution: Let Mohan's present age be M years. Five years ago, at the time of marriage, his age was M - 5 years. Given that M = (6/5) * (M - 5). Multiply both sides by 5 to remove the denominator: 5M = 6(M - 5). So 5M = 6M - 30. Rearrange to get -M = -30, so M = 30 years. His wife is 3 years younger, so wife's present age = 30 - 3 = 27 years. Verification / Alternative check: At the time of marriage, Mohan was 30 - 5 = 25 years old. According to the given relation, his present age should be 6/5 of 25, which equals 30. This matches our computed value. His wife, at 27 years, is indeed 3 years younger than him, so all conditions are satisfied. Why Other Options Are Wrong: If his wife were 21 or 24 years old, Mohan's age would not match the derived 30 years under the 6/5 relation. An age of 30 years for the wife would make her the same age as Mohan, contradicting the statement that she is younger. The value 25 years also fails the condition of being exactly 3 years younger than Mohan at 30 years. The only consistent option is 27 years. Common Pitfalls: A common mistake is to take 6/5 of the present age instead of the marriage age, or to add 5 instead of subtracting 5 when finding the marriage age. Some learners also forget to toggle the equation correctly and miscalculate the linear step. Clearly identifying which age corresponds to which time helps prevent errors. Final Answer: Thus, Mohan's wife is presently 27 years old.

Question 19

The present ages of Deepa and Hyma are in the ratio 5 : 6. After 4 years, the ratio of their ages will become 6 : 7. What is the difference between their present ages?

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Introduction / Context: This age problem uses two different ratios of the same two people taken at two different times. By expressing each person's age in terms of a common variable, we can form an equation from the future ratio and solve for their current ages. Then we find the difference between those ages. Given Data / Assumptions: The ratio of the present ages of Deepa and Hyma is 5 : 6. After 4 years, the ratio of their ages will become 6 : 7. We need to determine the difference between their present ages. Ages increase linearly and there are no other conditions. Concept / Approach: Let the present ages be 5k and 6k. After 4 years, their ages will be 5k + 4 and 6k + 4. The ratio 6 : 7 in the future gives an equation that we solve for k. Once k is known, actual ages and their difference are straightforward to obtain. Step-by-Step Solution: Let Deepa's present age be 5k years and Hyma's present age be 6k years. After 4 years, Deepa's age will be 5k + 4 years. After 4 years, Hyma's age will be 6k + 4 years. Given that (5k + 4) / (6k + 4) = 6 / 7. Cross multiply: 7(5k + 4) = 6(6k + 4). So 35k + 28 = 36k + 24. Rearrange to get 28 - 24 = 36k - 35k, so 4 = k. Deepa's present age = 5k = 5 * 4 = 20 years. Hyma's present age = 6k = 6 * 4 = 24 years. Difference between present ages = 24 - 20 = 4 years. Verification / Alternative check: Check the ratio after 4 years. Deepa will be 24 and Hyma will be 28. The ratio 24 : 28 simplifies to 6 : 7 when both terms are divided by 4. This matches the condition given in the question. The present ratio 20 : 24 also simplifies to 5 : 6, confirming the setup and the solution. Why Other Options Are Wrong: A difference of 6 or 8 years would not preserve the ratio changes as given. A difference of 2 years is too small and results in ratios that do not match 5 : 6 and 6 : 7 at the required times. The option 5 years also fails the ratio check. Only a 4 year difference satisfies all conditions. Common Pitfalls: Some students try to handle the ratios by guessing rather than setting up an equation, which often leads to errors. Another common mistake is to forget to add 4 to both ages when forming the future ratio. Writing a clear equation based on the ratio definition avoids such mistakes. Final Answer: Therefore, the difference between their present ages is 4 years.

Question 20

Two years ago, the average age of A and B was 26 years. The age of A five years from now will be 40 years, and B is 5 years younger than C. What is the difference between the present ages of A and C?

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Introduction / Context: This age problem combines an average age condition with individual future age information and an age difference involving a third person. By turning each sentence into an algebraic equation, we can systematically solve for the present ages of all three people and then compute the required difference. Given Data / Assumptions: Two years ago, the average age of A and B was 26 years. The age of A after 5 years will be 40 years. B is 5 years younger than C at present. We need the difference between the present ages of A and C. Concept / Approach: First determine A's present age using the future age information. Then use the past average to find B's present age. Finally, relate B and C using the given difference and compute C's age. The answer is the absolute difference between the present ages of A and C. Linear equations and average formula are used. Step-by-Step Solution: Let A, B and C be their present ages in years. We are told that A + 5 = 40, so A = 40 - 5 = 35 years. Two years ago, A's age was 35 - 2 = 33 years and B's age was B - 2. The average age two years ago was 26, so (33 + (B - 2)) / 2 = 26. This gives 33 + B - 2 = 52 so B + 31 = 52. Thus B = 52 - 31 = 21 years. B is 5 years younger than C, so C = B + 5 = 21 + 5 = 26 years. Difference between present ages of A and C = 35 - 26 = 9 years. Verification / Alternative check: Check the average condition: two years ago A was 33 and B was 19. Their total was 52, and the average was 52 / 2 = 26, which matches. Check B and C: 21 and 26 differ by 5 years, as stated. Check A's future age: 35 + 5 = 40 years, as given. All conditions are satisfied, confirming that a 9 year difference is correct. Why Other Options Are Wrong: Differences such as 11, 7 or 13 years do not match when you reconstruct the ages and test the given statements. For example, if C were 24 or 28 years old, some of the equations would fail. The option 5 years would mean B and C are almost the same age as A, contradicting the calculations. Only a 9 year difference fits all the data properly. Common Pitfalls: Learners sometimes forget that the average two years ago involves the ages at that past time, not present ages. Another common mistake is to misinterpret "5 years younger" and subtract in the wrong direction. Carefully translating each sentence into a stepwise equation avoids these issues. Final Answer: So, the difference between the present ages of A and C is 9 years.

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