Question 1

In a group of 36 students, the average age is 14 years. When the age of the teacher is also included in this group, the average age increases to 15 years. What is the age of the teacher in years?

Accepted Answer

Introduction / Context: This aptitude question is based on the concept of average age and how it changes when a new person is added to an existing group. Such problems are very common in competitive exams because they test your understanding of averages, total sum, and simple linear reasoning in one short statement. By understanding how to move between average and total sum, you can solve a wide variety of age and marks problems quickly in the exam. Given Data / Assumptions: There are 36 students in a group. The average age of these 36 students is 14 years. When the teacher is added, there are 37 persons in total. The new average age of all 37 persons is 15 years. All ages are measured in complete years and we assume simple arithmetic average. Concept / Approach: The key idea is that average = total sum / number of terms. Therefore, total sum = average * number of terms. First we find the total age of the 36 students using their average. Then we find the total age of the 37 persons using the new average. The difference between these two totals must be the age of the teacher, because only the teacher has been added to the group. Step-by-Step Solution: Step 1: Compute total age of 36 students. Total age of 36 students = average age * number of students = 14 * 36 = 504 years. Step 2: Compute total age of 37 persons including the teacher. Total age of 37 persons = new average age * total number of persons = 15 * 37 = 555 years. Step 3: Find the age of the teacher. Age of teacher = total age of 37 persons - total age of 36 students = 555 - 504 = 51 years. Verification / Alternative Check: If the teacher is 51 years old, then new total age = 504 + 51 = 555 years. Dividing this by 37 persons gives 555 / 37 = 15 years exactly. This matches the given average after including the teacher, so the calculation is consistent and confirms that 51 years is correct. Why Other Options Are Wrong: Option 31 years would give new total 504 + 31 = 535, and 535 / 37 is not 15, so it is incorrect. Option 36 years would give total 540, and 540 / 37 is also not equal to 15, so it is wrong. Option 54 years would give total 504 + 54 = 558, and 558 / 37 does not give an average of 15, so this option is also wrong. Common Pitfalls: A common mistake is to simply average 14 and 15, which has no direct meaning here. Another mistake is to forget that when the average changes, the total sum must be recalculated with the new number of persons. Some students also try to reason by guesswork instead of using the clear formula total = average * number of terms, which can lead to calculation errors. Final Answer: The age of the teacher is 51 years.

Question 2

In a class of 80 boys, the average age is 15 years. The average age of one group of 15 boys is 16 years, and the average age of another group of 25 boys is 14 years. What is the average age of the remaining boys in the class?

Accepted Answer

Introduction / Context: This is an average age problem where a large class is divided into several groups with different average ages. You know the overall average of the entire class and the averages of two subgroups, and you are asked to find the average of the remaining boys. It is a good test of your ability to break down averages into totals and then recombine them correctly. Given Data / Assumptions: Total number of boys in the class = 80. Average age of the whole class = 15 years. One group has 15 boys with average age 16 years. Another group has 25 boys with average age 14 years. The remaining boys form a third group whose average age we need to find. Concept / Approach: The basic principle is that total age = average age * number of boys. We first calculate the total age of the whole class. Then we find the total ages of the two known groups. Subtracting these totals from the overall total age gives the combined total age of the remaining boys. Dividing that by the number of remaining boys yields their average age. Step-by-Step Solution: Step 1: Compute total age of all 80 boys. Total age of class = 15 * 80 = 1200 years. Step 2: Compute total age of the first group of 15 boys. Total age of first group = 16 * 15 = 240 years. Step 3: Compute total age of the second group of 25 boys. Total age of second group = 14 * 25 = 350 years. Step 4: Compute total age of the remaining boys. Total age of known groups = 240 + 350 = 590 years. Total age of remaining boys = 1200 - 590 = 610 years. Step 5: Compute the number of remaining boys. Remaining boys = 80 - (15 + 25) = 80 - 40 = 40. Step 6: Compute average age of remaining boys. Average age = 610 / 40 = 15.25 years. Verification / Alternative Check: We can check reasonableness: one group is slightly older than the overall average (16 vs 15), another is younger (14), and the remaining group has an average of 15.25, which is just above the overall average. Because the sizes of the groups differ, this slightly higher average for the remaining group balances the two other groups to produce the overall average of 15 years over 80 boys. The arithmetic also confirms this correctly. Why Other Options Are Wrong: 12.24 years and 13.25 years are far too low, and would make the overall average of 80 boys much lower than 15 years. 16 years would make the remaining group too old on average, pushing the overall class average above 15 years when combined with the other two groups. Common Pitfalls: Some students try to average the three group averages directly without taking into account the number of boys in each group, which leads to incorrect results. Others forget to compute or subtract the total ages of the known groups before working on the remaining group. Always handle such problems by converting averages into totals, performing additions or subtractions on totals, and then returning to averages. Final Answer: The average age of the remaining boys in the class is 15.25 yrs.

Question 3

A father says to his son, "I was as old as you are now at the time of your birth." If the father is 38 years old at present, what was the age of the son 5 years ago?

Accepted Answer

Introduction / Context: This is a classic age puzzle that involves relating present ages to ages at a past event, namely the sons birth. It tests your ability to translate words into algebraic equations and then compute the required age. Given Data / Assumptions: Present age of the father = 38 years. Present age of the son = x years (unknown). At the time of the sons birth, the father was as old as the son is now. We are asked to find the sons age 5 years ago. Concept / Approach: If the son is x years old now, then the sons birth occurred x years ago. At that time, the father was 38 - x years old. According to the statement, that age of the father equals the son s current age x. This gives an equation which we can solve for x. After finding x, we subtract 5 years to get the sons age five years ago. Step-by-Step Solution: Let the sons present age = x years. The son was born x years ago. At that time, the father s age = 38 - x years. Given: 38 - x = x. So 38 = 2x. Therefore x = 19. Sons present age = 19 years. Sons age 5 years ago = 19 - 5 = 14 years. So the son was 14 years old five years ago. Verification / Alternative check: If the son is 19 now, his birth was 19 years ago. At that time, the father s age would have been 38 - 19 = 19 years, which matches the sons present age. This fits the father s statement perfectly, confirming that the calculation is correct. Why Other Options Are Wrong: Option 19 is the sons present age, not his age five years ago. Option 33 would imply an impossible negative age at birth for the father. Option 38 is the father s present age and does not relate to the sons age five years ago. Common Pitfalls: Students sometimes equate the wrong ages, for example taking the son s present age as the father s present age at birth, which makes no sense. Another error is forgetting to subtract the 5 years at the end and stopping at the present age of the son. Carefully track the time shifts and which age is being discussed at each point in time. Final Answer: The sons age five years ago was 14 years.

Question 4

In 10 years, A will be twice as old as B was 10 years ago. At present, A is 9 years older than B. What is the present age of B?

Accepted Answer

Introduction / Context: This age problem relates future and past ages of two people, A and B. You must form simultaneous equations by carefully translating the statements into algebra and then solve for the present age of B. Given Data / Assumptions: Let present age of B = b years. Present age of A = b + 9 years (A is 9 years older than B). In 10 years, A will be twice as old as B was 10 years ago. We are asked to find b, the current age of B. Concept / Approach: Translate the sentence about ages into an equation. B was b - 10 years old ten years ago. A will be (b + 9) + 10 years old ten years from now. The condition is that this future age of A equals two times that past age of B. Form a linear equation, solve it, and interpret the result. Step-by-Step Solution: Let B s present age = b. Then A s present age = b + 9. Ten years from now, A s age = (b + 9) + 10 = b + 19. Ten years ago, B s age = b - 10. Given: A s age in 10 years = 2 times B s age 10 years ago. So b + 19 = 2(b - 10). b + 19 = 2b - 20. Bring terms together: 19 + 20 = 2b - b. 39 = b. Hence B s present age is 39 years. Verification / Alternative check: Check the condition with b = 39. B was 39 - 10 = 29 years old 10 years ago. In 10 years, A will be (39 + 9) + 10 = 58 years old. Twice B s age 10 years ago is 2 * 29 = 58 years, which matches exactly. Thus the solution is correct. Why Other Options Are Wrong: If b = 19 or 29 or 49, substituting in the condition will not make the future age of A equal to twice B s past age. For example, with b = 29, A s future age would be 29 + 9 + 10 = 48 while twice B s age 10 years ago would be 2 * 19 = 38, which does not match. Common Pitfalls: The main difficulty is handling the time shifts correctly. Many learners mistakenly use b - 10 instead of b + 10 for future ages or mix up who is older by 9 years. Carefully label each age with its time reference, and build the equation step by step to avoid confusion. Final Answer: The present age of B is 39 years.

Question 5

The sum of the present ages of A and B is 12 years more than the sum of the present ages of B and C. By how many years is C younger than A?

Accepted Answer

Introduction / Context: This is an algebraic age relation question. It checks whether you can manipulate simple equations involving sums of ages and interpret the difference between two individuals ages from those sums. Given Data / Assumptions: Let present ages be A, B and C years. Given that A + B is 12 years more than B + C. We must find how many years younger C is than A. Concept / Approach: The key is to write the relation between the sums of ages as an equation. Once we remove B from both sides, we get a direct relationship between A and C. That relationship will immediately reveal the difference in their ages, which is exactly what the question is asking. Step-by-Step Solution: Given: A + B is 12 years more than B + C. So we write: A + B = (B + C) + 12. Simplify by subtracting B from both sides. A = C + 12. This means that the age of A is 12 years more than the age of C. Verification / Alternative check: We can cross-check with a simple example. Suppose C is 10 years old. Then A should be 22 years old since A = C + 12. The sum A + B would be 22 + B, while B + C would be B + 10. The difference (A + B) - (B + C) = 12, which fits the original statement. This confirms our interpretation is consistent. Why Other Options Are Wrong: Options 13, 14 and 15 would mean that A is 13, 14 or 15 years older than C. But the equation derived from the statement clearly produces a difference of exactly 12 years, not any other value. Common Pitfalls: A common mistake is to overcomplicate the situation, creating multiple equations when one simple linear equation is enough. Another pitfall is misreading the word more as less and reversing the direction of the inequality or equation. Always translate such phrases carefully before doing algebraic manipulations. Final Answer: C is 12 years younger than A.

Question 6

The present age of a man is four times the present age of his son. Five years ago, the man was nine times as old as his son was at that time. What is the present age of the man?

Accepted Answer

Introduction / Context: This age problem uses a present age ratio combined with a past age ratio. The goal is to form simultaneous equations from the verbal description and then solve for the present ages, specifically for the age of the man. Given Data / Assumptions: Let the present age of the son be S years. The present age of the man (father) is four times the son, so it is 4S years. Five years ago, the man was nine times as old as the son was at that time. We must find the present age of the man. Concept / Approach: We convert the past relationship into an equation. Five years ago, the father s age was 4S - 5 and the son s age was S - 5. The given condition is that the father s age at that time was nine times the son s age at that time. This provides a linear equation in S, which we can solve and then use to get the man s age. Step-by-Step Solution: Present son age = S. Present man age = 4S. Five years ago, ages were: son = S - 5, man = 4S - 5. Given: 4S - 5 = 9(S - 5). 4S - 5 = 9S - 45. Rearrange: 4S - 9S = -45 + 5. -5S = -40. S = 8. Therefore present man age = 4S = 4 * 8 = 32 years. So the man is 32 years old now. Verification / Alternative check: Check the condition with these ages. Five years ago the son was 8 - 5 = 3 years old. The man was 32 - 5 = 27 years old. Nine times the son s age then is 9 * 3 = 27, which matches the man s age at that time. This confirms the solution. Why Other Options Are Wrong: If the man were 24, 40 or 44 years old now, and his age is four times the son s age, the resulting past ages would not satisfy the nine-times relationship. Those values do not simultaneously satisfy both conditions. Common Pitfalls: Frequent errors include mixing the time reference, such as using present ages directly instead of subtracting five years, or misreading the phrase four times as four more. Always distinguish clearly between multiplicative relationships and additive differences and keep track of the time shifts. Final Answer: The present age of the man is 32 years.

Question 7

The ratio of the present ages of P and Q is 3 : 4. Five years ago, the ratio of their ages was 5 : 7. What are the present ages of P and Q?

Accepted Answer

Introduction / Context: This question involves present and past age ratios. We know the current ratio of ages of P and Q and their ratio five years ago. Using these two ratio conditions, we can set up a simple algebraic equation and find their exact ages. Given Data / Assumptions: Present ratio P : Q = 3 : 4. Five years ago, the ratio of their ages was 5 : 7. We need to determine the present ages of P and Q. Concept / Approach: Let the present ages be multiples of the ratio numbers. If P : Q = 3 : 4, we can write P = 3k and Q = 4k for some positive constant k. Five years ago, the ages would have been P - 5 and Q - 5, and their ratio at that time is given as 5 : 7. This gives one equation in the unknown k, which we solve. Step-by-Step Solution: Let P = 3k and Q = 4k. Five years ago, ages were P - 5 = 3k - 5 and Q - 5 = 4k - 5. Given that (3k - 5) / (4k - 5) = 5 / 7. Cross multiply: 7(3k - 5) = 5(4k - 5). 21k - 35 = 20k - 25. 21k - 20k = -25 + 35. k = 10. Therefore P = 3k = 30 and Q = 4k = 40. So P is 30 years old and Q is 40 years old. Verification / Alternative check: Check the given ratios. Present ratio: 30 : 40 simplifies to 3 : 4, which matches. Five years ago the ages were 25 and 35. The ratio 25 : 35 simplifies to 5 : 7, which matches the condition. Therefore the solution is fully consistent. Why Other Options Are Wrong: Options with 40 and 30 reversed, or 30 and 50, or 50 and 30 do not match both ratio conditions simultaneously. For example, if P were 40 and Q were 30, the present ratio would be 4 : 3, not 3 : 4, so that cannot be correct. Common Pitfalls: A common mistake is to forget to subtract 5 from both ages for the past ratio, or to treat 3 and 4 as the actual ages instead of ratio units. Another issue is failing to simplify and check both ratios after finding k. Always verify your solution against every condition in the problem. Final Answer: The present ages are 30 years for P and 40 years for Q.

Question 8

The sum of the present ages of a father and his son is 60 years. Five years ago, the father's age was four times the age of the son. What is the present age of the son?

Accepted Answer

Introduction / Context: This is a typical linear age puzzle connecting the sum of present ages with a multiplicative relationship in the past. You must use both pieces of information to form a system of equations to find the present age of the son. Given Data / Assumptions: Present age of father = F years. Present age of son = S years. F + S = 60. Five years ago the father was four times as old as the son. We need to find S. Concept / Approach: Five years ago, their ages were F - 5 and S - 5. The condition can be written as: F - 5 = 4(S - 5). We then use the equation F + S = 60 to express F in terms of S or vice versa and substitute into the second equation. This produces a linear equation in one variable which we solve. Step-by-Step Solution: From the sum condition: F + S = 60. So F = 60 - S. Five years ago: F - 5 = 4(S - 5). Substitute F = 60 - S into the second equation. (60 - S) - 5 = 4(S - 5). 55 - S = 4S - 20. Bring terms together: 55 + 20 = 4S + S. 75 = 5S. S = 15. Therefore, the son is 15 years old at present. Verification / Alternative check: If S = 15, then F = 60 - 15 = 45. Five years ago, the son was 10 and the father was 40. The father s age was 40, which is four times 10, matching the given condition precisely. This confirms that S = 15 is correct. Why Other Options Are Wrong: If S were 5, 10 or 20, substituting into the sum equation would give father ages of 55, 50 or 40 respectively. Then the past relationship F - 5 = 4(S - 5) would not hold. Only S = 15 simultaneously satisfies both conditions. Common Pitfalls: Learners sometimes forget to apply the past condition to both ages (subtracting 5 from only one of them). Another common mistake is to mis-handle the algebra when substituting and rearranging. Writing each step clearly and simplifying slowly reduces these errors. Final Answer: The present age of the son is 15 years.

Question 9

The average age of a group of 10 students is 15 years. When 5 more students join the group, the average age increases by 1 year. What is the average age of the 5 new students?

Accepted Answer

Introduction / Context: This problem tests understanding of averages and how adding new members with a different average age affects the overall group average. We relate the total ages before and after new students join. Given Data / Assumptions: Average age of 10 students = 15 years. Therefore total age of the 10 students = 10 * 15 years. 5 more students join the group. New average age of all 15 students = 16 years. We must find the average age of these 5 new students. Concept / Approach: Average age = total age / number of students. From the given averages, we first compute the total age of the original 10 students. Then we compute the total age of all 15 students after the new ones join. The difference between these totals gives the combined age of the 5 new students, from which we can find their average. Step-by-Step Solution: Total age of original 10 students = 10 * 15 = 150 years. After 5 more students join, total students = 15. New average age = 16 years. Total age of 15 students = 15 * 16 = 240 years. Combined age of 5 new students = 240 - 150 = 90 years. Average age of new students = 90 / 5 = 18 years. Thus the required average is 18 years. Verification / Alternative check: We can also reason that the overall average increased by 1 year for all 15 students, which adds 15 years more total age than if the original 10 had just grown. The original group had a total of 150 years. If the new 5 students had also been 15 on average, the new total would be 225. Instead, it is 240, which is 15 more, meaning those 5 students contribute an extra 3 years each on average, giving 15 + 3 = 18 as their average. This matches the earlier calculation. Why Other Options Are Wrong: Options 19, 20 and 21 years would give total ages for the new students of 95, 100 and 105 respectively, which would lead to different overall totals and averages that do not match the given increase to 16 years. Common Pitfalls: Learners sometimes mistakenly add 1 to the original average for the new group only or misinterpret how the average changes. Another common issue is not using total sums consistently when dealing with averages. Always convert average information into total sums, do the arithmetic, and then convert back. Final Answer: The average age of the 5 new students is 18 years.

Question 10

A man says to his son: "I am five times as old as you were when I was as old as you are now." The sum of their present ages is 64 years. What are their present ages?

Accepted Answer

Introduction / Context: This is a classic age puzzle involving a layered time statement. The man compares his present age to the age of his son at a specific time in the past, when the man was as old as the son is now. The puzzle also gives the sum of their current ages. We need to translate the verbal description into equations and solve. Given Data / Assumptions: Let the present age of the father be F years. Let the present age of the son be S years. F + S = 64. The father says he is five times as old as the son was when the father was as old as the son is now. Concept / Approach: When the father was as old as the son is now, his age was S. To reach that moment from the present, we go back in time by F - S years, because F - (F - S) = S. At that time in the past, the sons age would have been S - (F - S) = 2S - F. According to the statement, the father s current age F is five times that past age of the son: F = 5(2S - F). Together with F + S = 64, we can solve for F and S. Step-by-Step Solution: From the sum: F + S = 64. When the father was as old as the son is now, time difference = F - S years ago. At that time, son s age = S - (F - S) = 2S - F. Given: F is five times that past age of the son. So F = 5(2S - F). F = 10S - 5F. Bring terms together: 6F = 10S. So 3F = 5S, hence F = (5/3)S. Substitute into F + S = 64: (5/3)S + S = 64. (8/3)S = 64. S = 64 * 3 / 8 = 24. Then F = 64 - 24 = 40. Thus, the father is 40 years old and the son is 24 years old. Verification / Alternative check: Check the key condition. When the father was as old as the son is now, we go back F - S = 40 - 24 = 16 years. Sixteen years ago, father was 24 and the son was 24 - 16 = 8. The father now is 40. Five times the son s age then is 5 * 8 = 40, exactly the father s current age. This matches the statement perfectly. Why Other Options Are Wrong: Other pairs like 50 and 14 or 48 and 16 do not satisfy both the sum of ages equal to 64 and the five times relationship when checked carefully. Only the pair 40 and 24 satisfies both conditions simultaneously. Common Pitfalls: Many learners mis-handle the phrase when I was as old as you are now and either use the wrong time difference or apply the five times condition at the wrong moment. Another common error is to misinterpret the direction of time, subtracting from the wrong age. Drawing a simple time line and labelling ages clearly usually helps to avoid such confusion. Final Answer: The present ages are Father = 40 years and Son = 24 years.

Question 11

Six years ago, Anita was P times as old as Ben was. If Anita is now 17 years old, how old is Ben now in terms of P?

Accepted Answer

Introduction / Context: This age problem introduces a parameter P that represents how many times older Anita was than Ben six years ago. We are asked to express Ben s present age in algebraic form in terms of P, using the fixed current age of Anita. Given Data / Assumptions: Present age of Anita = 17 years. Let present age of Ben = B years. Six years ago, Anita s age was P times Ben s age at that time. We must express B in terms of P. Concept / Approach: Six years ago, Anita s age was 17 - 6 = 11 years, and Ben s age was B - 6 years. The condition that Anita was P times as old as Ben can be written as: 11 = P(B - 6). We then solve this linear equation for B in terms of P. No numerical value of P is given, so the answer must stay in symbolic form. Step-by-Step Solution: Anita s present age = 17 years. Six years ago, Anita s age = 17 - 6 = 11 years. Let Ben s present age = B years. Six years ago, Ben s age = B - 6 years. Given: Anita s age six years ago was P times Ben s age then. So 11 = P(B - 6). Divide both sides by P: B - 6 = 11 / P. Therefore B = 11 / P + 6. So Ben s present age is 11/P + 6 years. Verification / Alternative check: We can test with a simple hypothetical value of P. Suppose P = 1, meaning Anita and Ben were the same age six years ago. Then Ben s present age formula gives B = 11/1 + 6 = 17, the same as Anita s present age, which matches the interpretation that they had the same age six years ago and have both aged the same amount since then. Why Other Options Are Wrong: Option P/11 + 6 incorrectly inverts the fraction. Option 17 - P/6 mixes units (years) with a variable divisor, which does not follow from the equation 11 = P(B - 6). Option 17/P ignores the six-year difference entirely and does not incorporate the fixed difference between present and past ages. Common Pitfalls: A common mistake is to write P = 11(B - 6) or to misread P times as 11/P. Another error is forgetting to subtract 6 from both ages when going back in time. Always carefully reconstruct the situation at the earlier time before writing your equation. Final Answer: Ben s present age, in terms of P, is 11/P + 6 years.

Question 12

The ratio of the father's age to the son's age is 3 : 1. The product of their present ages is 147. What will be the ratio of their ages after 5 years?

Accepted Answer

Introduction / Context: This age question combines a present ratio and a product of ages. We use these together to find the actual ages of the father and son and then compute their age ratio five years in the future. Given Data / Assumptions: Ratio of father s present age to son s present age = 3 : 1. Product of their present ages = 147. We must find the ratio of their ages after 5 years. Concept / Approach: If the present ages follow a 3 : 1 ratio, we can write the father s age as 3k and the son s age as k. Their product is 3k * k = 3k^2, which is given as 147. Solving for k gives the actual ages. After that, we add 5 years to each age and form the new ratio. Step-by-Step Solution: Let father s present age = 3k. Let son s present age = k. Product of ages: 3k * k = 147. So 3k^2 = 147. Divide both sides by 3: k^2 = 49. Thus k = 7 (age must be positive). Father s present age = 3k = 21 years. Son s present age = k = 7 years. After 5 years, father s age = 21 + 5 = 26 years. After 5 years, son s age = 7 + 5 = 12 years. Future ratio father : son = 26 : 12. Simplify 26 : 12 by dividing by 2 to get 13 : 6. Therefore, the required ratio is 13 : 6. Verification / Alternative check: We can verify the original conditions. The present ages 21 and 7 give a ratio of 21 : 7 = 3 : 1, and their product 21 * 7 = 147 matches the given product. After 5 years they become 26 and 12, whose greatest common divisor is 2, giving the simplified ratio 13 : 6. Everything is consistent. Why Other Options Are Wrong: Options 6 : 13 and 6 : 15 reverse or misrepresent the relationship. Option 15 : 6 would require ages like 25 and 10 after 5 years, which do not fit the original product and ratio conditions. Only 13 : 6 is compatible with all the given information. Common Pitfalls: Some learners misinterpret the ratio and set the ages as 3 and 1 directly instead of 3k and k. Others forget to take the positive square root when solving k^2 = 49. Another error is to add 5 only to one of the ages instead of both when finding the future ratio. Final Answer: The ratio of their ages after 5 years will be 13 : 6.

Question 13

Sachin is younger than Rahul by 7 years. If the ratio of their present ages is 7 : 9, what is the present age of Sachin?

Accepted Answer

Introduction / Context: This question combines a ratio of ages with an absolute difference between two people, Sachin and Rahul. We use these two conditions together to find Sachin s exact age. Given Data / Assumptions: Sachin is 7 years younger than Rahul. The ratio of their present ages Sachin : Rahul is 7 : 9. We must determine Sachin s present age. Concept / Approach: Let Sachin s age be 7k and Rahul s age be 9k to match the ratio 7 : 9. The difference between their ages is then 9k - 7k = 2k. The statement that Sachin is 7 years younger than Rahul means this difference must be 7. Hence 2k = 7, which allows us to determine k and then compute Sachin s age. Step-by-Step Solution: Let Sachin s present age = 7k. Let Rahul s present age = 9k. Age difference Rahul - Sachin = 9k - 7k = 2k. Given that Sachin is 7 years younger, so 2k = 7. Thus k = 7 / 2 = 3.5. Sachin s age = 7k = 7 * 3.5 = 24.5 years. Rahul s age = 9k = 9 * 3.5 = 31.5 years. The key answer required is that Sachin is 24.5 years old. Verification / Alternative check: Check the conditions. The ratio Sachin : Rahul is 24.5 : 31.5. Dividing both by 3.5 gives 7 : 9, matching the given ratio. The difference Rahul - Sachin is 31.5 - 24.5 = 7 years, which matches the statement that Sachin is 7 years younger than Rahul. Therefore the solution is consistent. Why Other Options Are Wrong: If Sachin were 25.5, 26.5 or 27.5 years old, the corresponding Rahul ages required to maintain the 7 : 9 ratio would not give a fixed difference of exactly 7 years. These values do not satisfy both the ratio and the difference conditions simultaneously. Common Pitfalls: Some students try to guess by trial rather than setting up the algebra, which can be slow and error prone. Others incorrectly interpret the ratio as Rahul : Sachin, or forget that the difference must be exactly 7. Using the standard approach of expressing ages as multiples of the ratio numbers helps keep the reasoning clear. Final Answer: Sachin s present age is 24.5 years.

Question 14

The sum of the present ages of two persons A and B is 60 years. If the present age of A is twice that of B, what will be the sum of their ages 5 years from now?

Accepted Answer

Introduction / Context: This age problem is simpler and primarily checks whether you can recognise how sums of ages change over time. When both people age by the same amount, the sum of their ages increases in a straightforward way. Given Data / Assumptions: A and B have present ages that sum to 60 years. A is presently twice as old as B. We need the sum of their ages 5 years from now. Concept / Approach: While we can find the individual ages of A and B, there is a simpler method. After 5 years, A s age increases by 5 and B s age also increases by 5. So the sum of their ages increases by 5 + 5 = 10. Therefore, the new sum is simply 60 + 10. We can still compute their specific ages to check consistency. Step-by-Step Solution: Let B s present age = x. Then A s present age = 2x. Given that A + B = 60. So 2x + x = 60. 3x = 60, hence x = 20. Therefore, B is 20 years old and A is 40 years old. After 5 years, A s age = 40 + 5 = 45. After 5 years, B s age = 20 + 5 = 25. Sum after 5 years = 45 + 25 = 70 years. Alternatively, we can simply add 10 to the original sum of 60 to get 70. Verification / Alternative check: From the computed ages, the present sum is 40 + 20 = 60, and after 5 years the sum is 70. This matches the logical shortcut that adding 5 years to both people adds 10 years to the total, confirming that 70 is correct. Why Other Options Are Wrong: Option 60 would be the present sum only and ignores the passage of time. Options 65 and 75 correspond to adding 5 or 15 years total, which would require asymmetric changes in ages or a different time interval. Common Pitfalls: Some learners mistakenly think that because A is twice as old as B, the change in their sum over time is more complicated. In fact, regardless of the ratio, if both individuals age by the same number of years, the total increase in the sum is just two times that number. Final Answer: The sum of their ages 5 years from now will be 70 years.

Question 15

When I was married 10 years ago, my wife became the sixth member of the family. Today my father has died and a baby has been born to me. The average age of the family at the time of my marriage is the same as the average age today. What was my fathe…

Accepted Answer

Introduction / Context: This is a well known average age puzzle that mixes changes in family composition with the concept of constant average. When the average age of a group stays the same over a period in which everyone grows older, there must be compensating changes such as deaths and births. We use this idea to find the father s age at death. Given Data / Assumptions: At the time of marriage (10 years ago), the wife became the sixth member of the family, so there were 6 members at that time. Ten years have passed since the marriage. Today, the father has died and a baby has been born, so the number of family members is still 6. The average age of the family at the time of marriage is equal to the average age of the family today. We must find the father s age at the time of his death (which is today in the story). Concept / Approach: Let the total age of the 6 family members at the time of marriage be S years. The average then is S / 6. After 10 years, if all 6 original members were still alive, each would be 10 years older, so the total age would be S + 6 * 10 = S + 60. However, the father dies and a newborn baby of age 0 joins. We then use the fact that the average today is still S / 6, which means the total age today is still S. From this, we can solve for the father s age 10 years ago and then find his age at death. Step-by-Step Solution: Let S be the total age of the 6 family members at the time of marriage. Average age at marriage = S / 6. Ten years later, if no one left or joined, each of the 6 would be 10 years older. Total age then would be S + 6 * 10 = S + 60. At this time, the father dies. Suppose the father s age at marriage was F years. So his age today would have been F + 10 if he had not died. After his death, we remove F + 10 from the total. A newborn baby of age 0 joins, so nothing is added to the total for age. Thus total age today = (S + 60) - (F + 10) = S + 50 - F. Given that the average age today equals the average at marriage, the total today must equal S. So we set S + 50 - F = S. This simplifies to 50 - F = 0, hence F = 50. Thus the father was 50 years old at the time of the marriage (10 years ago). His age at death today is F + 10 = 60 years. Verification / Alternative check: At marriage, the father was 50 and there were 5 other family members with some ages adding to S - 50. Ten years later, if all were alive, their total age would be S + 60. Removing the father age of 60 leaves S. Adding a newborn with age 0 keeps the total S, so the average remains S / 6, exactly as at the time of marriage. This confirms the internal consistency of the solution. Why Other Options Are Wrong: If the father had died at 50, 65 or 70, the total today would not match the original total S, so the average would change. Only 60 satisfies the condition that the total and hence the average age remains unchanged despite ten years passing and changes in family membership. Common Pitfalls: A common mistake is to assume that the family size changed, when in fact it remains at 6 members throughout. Others mis-handle the ten year increment or forget to remove the father s age and add the baby s age correctly. Working explicitly with total sums and keeping a clear timeline usually prevents these errors. Final Answer: The father was 60 years old when he died.

Question 16

Two times the age of a daughter is added to the age of her mother and the total becomes 70 years. If instead two times the age of the mother is added to the age of the daughter and the total becomes 95 years, what is the present age of the mother in…

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Introduction / Context: This is a classic problems-on-ages question where the age of the mother and the daughter are related by two linear equations. Such questions test your ability to translate language statements into algebraic expressions and then solve simultaneous equations to find the unknown ages. Given Data / Assumptions: Let the present age of the mother be M years. Let the present age of the daughter be D years. Two times the age of the daughter added to the age of the mother gives a total of 70 years. Two times the age of the mother added to the age of the daughter gives a total of 95 years. All ages are assumed to be positive real numbers and represent present ages. Concept / Approach: The key idea is to convert each English statement into a linear equation in terms of M and D. Once we have two linear equations with two unknowns, we can solve them using substitution or elimination. Problems on ages almost always reduce to solving such small systems of equations. Step-by-Step Solution: From the first statement: 2 times daughter age + mother age = 70.So the first equation is: 2 * D + M = 70.From the second statement: 2 times mother age + daughter age = 95.So the second equation is: 2 * M + D = 95.Rewrite the first equation as: M = 70 - 2 * D.Substitute this into the second equation: 2 * (70 - 2 * D) + D = 95.Simplify: 140 - 4 * D + D = 95.Combine like terms: 140 - 3 * D = 95.Rearrange: 140 - 95 = 3 * D, so 45 = 3 * D.Therefore D = 45 / 3 = 15 years.Now substitute back to find M: M = 70 - 2 * 15 = 70 - 30 = 40 years. Verification / Alternative check: First condition: 2 * D + M = 2 * 15 + 40 = 30 + 40 = 70, which matches the given information.Second condition: 2 * M + D = 2 * 40 + 15 = 80 + 15 = 95, which also matches the given information.Both conditions are satisfied, so the solution M = 40 years and D = 15 years is correct. Why Other Options Are Wrong: 30 years: If M = 30, the equations do not hold, and the computed daughter age becomes inconsistent with the second equation.38 years: If M = 38, substituting into either equation gives a non-integer or inconsistent daughter age.41 years: If M = 41, 2 * M + D cannot be exactly 95 while also satisfying the first equation. Common Pitfalls: Students sometimes reverse the roles of the mother and daughter while forming equations, leading to swapped variables and incorrect answers.Another frequent mistake is mishandling the coefficients when using substitution, especially when expanding expressions like 2 * (70 - 2 * D).Some learners try to guess-and-check ages mentally without forming equations, which is error-prone when the numbers are not very small. Final Answer: The present age of the mother that satisfies both conditions is 40 years.

Question 17

The sum of the present ages of five children who were born at intervals of 3 years each is 50 years. What is the present age of the youngest child in years?

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Introduction / Context: This problem involves ages of several children born at equal time intervals. It tests understanding of arithmetic progressions and how to model such situations using simple algebra. The key is to represent the ages in terms of the youngest child and then use the given total to solve for that youngest age. Given Data / Assumptions: There are 5 children. They were born at intervals of 3 years each. The sum of their present ages is 50 years. We are asked to find the present age of the youngest child. All ages are assumed to be non-negative and measured in years. Concept / Approach: The ages of the children form an arithmetic progression because the difference between consecutive ages is constant, equal to 3 years. If the youngest child has age x years, the others will have ages x + 3, x + 6, x + 9 and x + 12 years. The sum of these five expressions must equal the total given in the problem. We can then solve the resulting linear equation for x. Step-by-Step Solution: Let the age of the youngest child be x years.Then the ages of the other four children are x + 3, x + 6, x + 9 and x + 12 years.Write the equation for the sum of the ages: x + (x + 3) + (x + 6) + (x + 9) + (x + 12) = 50.Combine like terms: 5 * x + (3 + 6 + 9 + 12) = 50.Compute the constant sum: 3 + 6 + 9 + 12 = 30.So we have: 5 * x + 30 = 50.Subtract 30 from both sides: 5 * x = 20.Divide by 5: x = 20 / 5 = 4 years. Verification / Alternative check: Youngest child: 4 years.Other children: 7 years, 10 years, 13 years and 16 years.Check the total: 4 + 7 + 10 + 13 + 16 = 50 years, which matches the given sum.Therefore the age of the youngest child is correctly found as 4 years. Why Other Options Are Wrong: 8 years: If the youngest age is 8, the ages would be 8, 11, 14, 17 and 20, whose sum is 70, not 50.10 years: If the youngest is 10, the sequence becomes 10, 13, 16, 19 and 22, summing to 80, not 50.None of these: This is incorrect because we have found a valid option, 4 years, that satisfies all conditions. Common Pitfalls: Some learners mistakenly take the oldest child as x and then add intervals backward, which complicates the algebra.Others may forget to multiply x by 5 when combining like terms, leading to an incorrect equation.A few students attempt guess-and-check, which can work here, but it is less systematic than forming an equation. Final Answer: The present age of the youngest child is 4 years.

Question 18

A man is 24 years older than his son. In two years, the age of the man will be twice the age of his son at that time. What is the present age of the son?

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Introduction / Context: This is a straightforward problems-on-ages question involving a fixed age difference between a father and his son. The situation also provides a future condition relating their ages. The goal is to form a simple linear equation and solve it to find the present age of the son. Given Data / Assumptions: The man is currently 24 years older than his son. After 2 years, the man will be twice as old as his son. We are asked to find the present age of the son. All ages are measured in years and are assumed to be positive. Concept / Approach: The key idea is to express both ages using a single variable. The difference between their ages remains constant over time. A condition involving their ages after 2 years leads to a linear equation that can be solved for the son age. Once we know the son age, we can confirm the father age and verify the condition. Step-by-Step Solution: Let the present age of the son be S years.Then the present age of the man (father) is S + 24 years.After 2 years, the son age will be S + 2, and the father age will be S + 24 + 2 = S + 26.The problem states that after 2 years, the father age will be twice the son age: S + 26 = 2 * (S + 2).Expand the right side: S + 26 = 2 * S + 4.Rearrange: S + 26 = 2 * S + 4 implies 26 - 4 = 2 * S - S.So 22 = S.Therefore the present age of the son is 22 years. Verification / Alternative check: If the son is 22 years old now, the father is 22 + 24 = 46 years old now.After 2 years, the son will be 24 and the father will be 48.Check the condition: 48 is indeed 2 times 24.Hence the solution S = 22 years satisfies all the given conditions. Why Other Options Are Wrong: 20 years: If the son is 20, the father is 44, and after 2 years they will be 22 and 46; 46 is not twice 22.21 years: If the son is 21, the father is 45, and after 2 years they will be 23 and 47; 47 is not twice 23.23 years: If the son is 23, the father is 47, and after 2 years they will be 25 and 49; 49 is not twice 25. Common Pitfalls: Some students mistakenly set up the equation using current ages instead of future ages, which gives a wrong relationship.Another error is misreading the phrase twice the age and incorrectly writing 2 * S + 2 instead of 2 * (S + 2).Forgetting that the age difference remains 24 years at all times can also lead to confusion. Final Answer: The present age of the son is 22 years.

Question 19

The ages of Krish and Vaibhav are in the ratio 3 : 5 at present. After 9 years, the ratio of their ages will become 3 : 4. What is the current age of Vaibhav in years?

Accepted Answer

Introduction / Context: This question is about age ratios that change over time. It tests your ability to translate ratio information into algebraic expressions and use future conditions to find the actual ages. Such problems are very common in quantitative aptitude exams. Given Data / Assumptions: Present ratio of the ages of Krish and Vaibhav is 3 : 5. After 9 years, the ratio of their ages will be 3 : 4. We need to find the current age of Vaibhav. All ages are in years and assumed to be positive integers. Concept / Approach: When an age ratio is given, we usually represent the actual ages as multiples of a common variable. Then, when we are told about a ratio at a future time, we translate that into another equation and solve for the variable. Once the value of the variable is known, we can compute the individual ages. This approach is systematic and avoids guesswork. Step-by-Step Solution: Let the present age of Krish be 3 * x years.Let the present age of Vaibhav be 5 * x years.After 9 years, Krish age will be 3 * x + 9, and Vaibhav age will be 5 * x + 9.The ratio after 9 years is given as 3 : 4, so (3 * x + 9) / (5 * x + 9) = 3 / 4.Cross-multiply to remove the fraction: 4 * (3 * x + 9) = 3 * (5 * x + 9).Simplify left side: 12 * x + 36.Simplify right side: 15 * x + 27.Set them equal: 12 * x + 36 = 15 * x + 27.Rearrange: 36 - 27 = 15 * x - 12 * x, so 9 = 3 * x.Therefore x = 3.Present age of Vaibhav = 5 * x = 5 * 3 = 15 years. Verification / Alternative check: Krish current age = 3 * 3 = 9 years, Vaibhav current age = 15 years.After 9 years: Krish will be 18, Vaibhav will be 24.Check ratio: 18 : 24 simplifies to 3 : 4 by dividing both terms by 6.The future ratio condition is perfectly satisfied, confirming that Vaibhav current age is 15 years. Why Other Options Are Wrong: 10 years: Using this as Vaibhav age would break the 3 : 5 ratio between Krish and Vaibhav and will not satisfy the future ratio condition.13 years: This does not maintain the required future ratio of 3 : 4 after 9 years.18 years: This also fails either the present ratio or the future ratio condition or both. Common Pitfalls: Students sometimes confuse the ratios and incorrectly use the same ratio at both times without shifting ages by the number of years.Another mistake is to forget to add 9 years to both ages when forming the future ratio equation.Careless arithmetic in cross-multiplication can also lead to wrong values of x. Final Answer: The current age of Vaibhav is 15 years.

Question 20

The ratio of the present ages of Maala and Kala is 4 : 3. The sum of their present ages is 2.8 decades, that is 28 years. What will be the ratio of their ages after 0.8 decades, that is after 8 years?

Accepted Answer

Introduction / Context: This problem deals with age ratios and their change over time. We are given a present ratio, the total of the ages now, and asked to find the ratio after a certain number of years. It tests understanding of ratios, basic algebra, and the idea that each person age increases by the same fixed amount over time. Given Data / Assumptions: Present ratio of ages of Maala and Kala is 4 : 3. The sum of their present ages is 2.8 decades, which equals 28 years. We need the ratio of their ages after 0.8 decades, that is after 8 years. All ages are in years and non-negative. Concept / Approach: When two ages are in a given ratio, we express them as multiples of a common variable. Using the sum of the ages, we solve for that variable and get the actual ages. Then we add the given future time to each age and compute the new ratio. Age ratios change when equal years are added to each person, but the age difference remains constant. Step-by-Step Solution: Let the present age of Maala be 4 * x years and the present age of Kala be 3 * x years.Given that their total age is 28 years, so 4 * x + 3 * x = 28.This simplifies to 7 * x = 28.So x = 28 / 7 = 4.Present age of Maala = 4 * 4 = 16 years.Present age of Kala = 3 * 4 = 12 years.After 8 years, Maala age will be 16 + 8 = 24 years.After 8 years, Kala age will be 12 + 8 = 20 years.The required ratio after 8 years is 24 : 20.Simplify by dividing by 4: 24 : 20 = 6 : 5. Verification / Alternative check: Check the sum now: 16 + 12 = 28 years, which matches the given total.Check the ratio now: 16 : 12 simplifies to 4 : 3, which matches the given ratio.After 8 years, 24 : 20 simplifies to 6 : 5, confirming that 6 : 5 is the correct future ratio. Why Other Options Are Wrong: 4 : 3: This is the present ratio, not the ratio after 8 years. The ratio changes when equal years are added to two different ages.12 : 11: There is no combination of 16 and 12 increased by the same amount that gives this ratio.7 : 4: This does not match 24 : 20 or any reasonable simplification from the correct ages. Common Pitfalls: Forgetting to convert decades to years can lead to wrong calculations. Here 0.8 decades means 8 years.Some students incorrectly keep the ratio constant over time without adding the extra years to each age.Errors sometimes occur when simplifying the final ratio, such as dividing by different numbers. Final Answer: The ratio of their ages after 8 years will be 6 : 5.

Problems on Ages Questions