Question 1

Today is Varun's birthday. One year from today, he will be twice as old as he was 12 years ago. How old is Varun today in years?

Accepted Answer

Introduction / Context: This is a straight numerical age puzzle that connects Varun's age one year from now with his age 12 years ago using a simple multiplicative relation. It is a classic example of forming and solving a single linear equation in one variable based on time shift statements. Given Data / Assumptions: Let Varun's present age be V years. One year from today, his age will be V + 1. Twelve years ago, his age was V - 12. We are told that V + 1 is twice (V - 12). Concept / Approach: Translate the verbal statement into an algebraic equation, namely V + 1 = 2 * (V - 12). Then solve this equation for V. This gives Varun's present age directly. It is important to keep the time references consistent when forming the equation. Step-by-Step Solution: Let Varun's present age be V years. One year from today his age will be V + 1. Twelve years ago his age was V - 12. According to the problem, V + 1 = 2 * (V - 12). Expand the right side: V + 1 = 2V - 24. Rearrange terms: bring V to the right and -24 to the left. So 1 + 24 = 2V - V which gives 25 = V. Therefore Varun's present age V is 25 years. Verification / Alternative check: Check the condition: One year from now, Varun will be 26 years old. Twelve years ago, he was 25 - 12 = 13 years old. Twice that past age is 2 * 13 = 26, which matches his age one year from now. Hence the given relation holds exactly, confirming that 25 years is correct. Why Other Options Are Wrong: If Varun were 24 now, then one year from now he would be 25, but his age 12 years ago would be 12, and 2 * 12 = 24, not 25. Similar checks for 23 or 22 quickly show contradictions with the required condition. The option 26 fails as well, since the numbers no longer satisfy the equation V + 1 = 2 * (V - 12). Only 25 works perfectly. Common Pitfalls: A common error is to misread "one year from today" as "one year ago" or to incorrectly handle the difference between 12 years ago and 1 year ahead. Some learners also make algebraic mistakes when moving terms across the equality. Writing the equation slowly and checking it with the final value helps avoid these issues. Final Answer: So, Varun is 25 years old today.

Question 2

Rajeev is 4 years older than Shilpa. Shilpa's present age can be obtained by subtracting four times her age 4 years ago from four times her age 4 years from now. What was Rajeev's age 4 years ago?

Accepted Answer

Introduction / Context: This puzzle uses an unusual wording to define Shilpa's present age from her past and future ages. We also know that Rajeev is 4 years older than Shilpa. Translating the description into algebra provides a neat way to compute both their ages and the required past age of Rajeev. Given Data / Assumptions: Rajeev is 4 years older than Shilpa at present. Shilpa's present age equals four times her age 4 years from now minus four times her age 4 years ago. We must find Rajeev's age 4 years ago. All ages are in whole years and time flows uniformly. Concept / Approach: Let Shilpa's present age be S years. Then her age 4 years from now is S + 4, and 4 years ago it was S - 4. The statement says S = 4(S + 4) - 4(S - 4). We solve this equation for S. Then Rajeev's present age is S + 4, and his age 4 years ago is (S + 4) - 4. Step-by-Step Solution: Let Shilpa's present age be S years. Her age 4 years from now will be S + 4. Her age 4 years ago was S - 4. Given that S = 4(S + 4) - 4(S - 4). Compute the right side: 4(S + 4) = 4S + 16 and 4(S - 4) = 4S - 16. So S = (4S + 16) - (4S - 16) = 4S + 16 - 4S + 16 = 32. Therefore Shilpa's present age S is 32 years. Rajeev is 4 years older, so his present age is 32 + 4 = 36 years. Rajeev's age 4 years ago was 36 - 4 = 32 years. Verification / Alternative check: Verify the condition for Shilpa. Four years ago she was 28 and in four years she will be 36. Then 4 times her future age is 4 * 36 = 144, and 4 times her past age is 4 * 28 = 112. The difference 144 - 112 = 32, which equals her present age. Rajeev at 36 is indeed 4 years older, and 4 years ago he was 32, matching our answer. Why Other Options Are Wrong: If Rajeev were 36 years old four years ago, his present age would be 40, which changes Shilpa's present age and breaks the relation. Ages 28 or 30 also fail to satisfy the special equation for Shilpa. The option 40 years is similarly inconsistent. Only 32 years as Rajeev's age four years ago fits all given data. Common Pitfalls: The unusual sentence structure may cause confusion. Some learners misread it and form S = 4(S - 4) - 4(S + 4) or reverse the subtraction order. Others forget that Rajeev is older and incorrectly add or subtract 4. Carefully writing equalities for ages at each time point avoids these problems. Final Answer: Hence, Rajeev was 32 years old four years ago.

Question 3

The average age of a couple and their son at the time of the son's marriage was 40 years. The son got married, and a child was born 2 years after the marriage. When the child turned 10 years old, the average age of the family of five became 38 years…

Accepted Answer

Introduction / Context: This is a multi step family age and average problem. The situation starts with three people (a couple and their son), then expands to five members after marriage and childbirth. We are given averages at two different times and the timing of the childbirth relative to marriage. The objective is to find the daughter in law's age at marriage by tracking totals and time intervals carefully. Given Data / Assumptions: At the time of the son's marriage, the average age of the couple and their son was 40 years. So at that time, the total age of the couple and their son was 3 * 40 = 120 years. The son got married then, and a child was born 2 years after the marriage. When the child turned 10 years old, the average age of the family of 5 (couple, son, daughter in law, child) became 38 years. We must find the daughter in law's age at the time of marriage. Concept / Approach: Take the time of marriage as the starting point. At that moment, three people have a known total age of 120 years, and the daughter in law has some unknown age D. Twelve years later (2 years until birth plus 10 years until the child is 10), everyone except the unborn child becomes 12 years older, and the child is 10 years old. We express the total family age at that later time in terms of D and equate it to the given average based total of five people. This provides a simple equation in D. Step-by-Step Solution: At the time of marriage, total age of couple and son = 3 * 40 = 120 years. Let the daughter in law's age at marriage be D years. Two years after marriage, a child is born. Twelve years after marriage (2 years + 10 years), the child is 10 years old. In those 12 years, the couple, the son and the daughter in law each age by 12 years. So, at the time when the child is 10, total age of the couple and son = 120 + 3 * 12 = 120 + 36 = 156 years. The daughter in law's age at that time is D + 12 years. The child's age at that time is 10 years. Hence, total family age when the child is 10 = 156 + (D + 12) + 10 = 178 + D. Given that the average age of the 5 family members is 38 years then, total age = 5 * 38 = 190 years. So 178 + D = 190 which gives D = 190 - 178 = 12 years. Verification / Alternative check: At marriage, the couple and son total 120 and the daughter in law is 12, so the four adults total 132. Twelve years later, their ages increase by 12 each: 4 * 12 = 48, giving a total of 132 + 48 = 180. Adding the 10 year old child gives 190. The average at that time is 190 / 5 = 38, as given. This confirms that the daughter in law's age at marriage must be 12 years. Why Other Options Are Wrong: If the daughter in law had been 10, 13, 14 or 16 at marriage, the total family age when the child is 10 would not be 190 and the resulting average would differ from 38. Each alternative value changes the total by more or less than 190, so only D = 12 is consistent with the given data. Common Pitfalls: A common mistake is to assume the first mentioned average is at present instead of at the time of marriage. Another pitfall is forgetting that the child is born 2 years after marriage, so the time gap from marriage to the child being 10 is 12 years, not just 10 years. Tracking time carefully and working with totals instead of partial averages makes the solution more reliable. Final Answer: Therefore, the daughter in law was 12 years old at the time of marriage.

Question 4

Ishikha got married 9 years ago. Today her age is four third of her age at the time of marriage. At present her daughter's age is one sixth of her age. What was her daughter's age 3 years ago?

Accepted Answer

Introduction / Context: This problem combines proportional age growth over 9 years for a mother with a present age ratio between the mother and her daughter. Once Ishikha's present age is determined, the daughter's age can be obtained easily and then adjusted to find her age 3 years ago. The problem nicely mixes fractions and time shifts. Given Data / Assumptions: Ishikha got married 9 years ago. Her present age is 4/3 times her age at the time of marriage. Her daughter's present age is one sixth of Ishikha's present age. We must find her daughter's age 3 years ago. Concept / Approach: Let I be Ishikha's present age. Her age at marriage was I - 9. The relation tells us I = (4/3) * (I - 9). We solve this equation to find I. Then the daughter's current age is I / 6. Subtracting 3 years from that gives the daughter's age three years ago. The steps use proportion and simple linear equations. Step-by-Step Solution: Let Ishikha's present age be I years. Her age at marriage, 9 years ago, was I - 9 years. The condition says I = (4/3) * (I - 9). Multiply both sides by 3: 3I = 4(I - 9). So 3I = 4I - 36 which simplifies to -I = -36. Hence I = 36 years. Her daughter's present age is one sixth of 36, so daughter's age now = 36 / 6 = 6 years. Three years ago, daughter's age = 6 - 3 = 3 years. Verification / Alternative check: At marriage, Ishikha was 36 - 9 = 27 years old. Four third of 27 is (4/3) * 27 = 36, which matches her present age. The daughter being 6 years old now is indeed one sixth of 36. Three years ago, she would have been 3 years old. All conditions are satisfied, so the result is consistent. Why Other Options Are Wrong: A daughter aged 4 or 5 years three years ago would be 7 or 8 now, which is not one sixth of 36. Values like 1 or 2 years three years ago would make her present age 4 or 5 years, which again does not match the one sixth proportion. Only 3 years ago leads to a present age of 6, maintaining the exact ratio with Ishikha's age. Common Pitfalls: Mistakes often come from misinterpreting the phrase four third and writing the equation incorrectly as (3/4) instead of 4/3. Other errors include forgetting to subtract 9 to obtain the marriage age or mixing up whether to divide or multiply when using one sixth. Keeping clear symbols and carefully translating each condition minimizes such issues. Final Answer: Therefore, Ishikha's daughter was 3 years old three years ago.

Question 5

The average age of a class is 15.8 years. The average age of the boys in the class is 16.4 years and that of the girls is 15.4 years. What is the ratio of the number of boys to the number of girls in the class?

Accepted Answer

Introduction / Context: This problem involves weighted averages. We are given the overall average age of the class along with separate average ages of boys and girls. From this, we need to determine the ratio of the number of boys to the number of girls. It is an example of mixing and alligation in the context of averages. Given Data / Assumptions: The average age of the whole class is 15.8 years. The average age of the boys is 16.4 years. The average age of the girls is 15.4 years. We need the ratio of the count of boys to the count of girls. Concept / Approach: Let the number of boys be B and the number of girls be G. Then the total age of boys is 16.4B and the total age of girls is 15.4G. The overall average 15.8 means (16.4B + 15.4G) / (B + G) = 15.8. This equation simplifies to a relation between B and G from which we find B / G, the required ratio. Step-by-Step Solution: Let the number of boys be B and the number of girls be G. Total age of boys = 16.4 * B. Total age of girls = 15.4 * G. Total number of students = B + G and the average age is 15.8. So (16.4B + 15.4G) / (B + G) = 15.8. Multiply both sides by (B + G): 16.4B + 15.4G = 15.8B + 15.8G. Rearrange terms to get (16.4B - 15.8B) + (15.4G - 15.8G) = 0. This gives 0.6B - 0.4G = 0. Hence 0.6B = 0.4G which implies B / G = 0.4 / 0.6 = 2 / 3. Therefore, the ratio of boys to girls is 2 : 3. Verification / Alternative check: Using alligation, the difference between boys average and class average is 16.4 - 15.8 = 0.6. The difference between class average and girls average is 15.8 - 15.4 = 0.4. The ratio of boys to girls should be the inverse of these differences, that is 0.4 : 0.6 = 2 : 3. This matches the algebraic solution, confirming the ratio. Why Other Options Are Wrong: Ratios like 1 : 3 or 3 : 1 would produce overall averages heavily biased toward one group, not 15.8 as given. Ratios 3 : 2 or 4 : 5 also fail when substituted into the weighted average formula, leading to a different overall average. Only 2 : 3 satisfies the equation exactly. Common Pitfalls: A frequent mistake is to simply average 16.4 and 15.4 without weighting for the different numbers of boys and girls. Others may reverse the differences when using the alligation method. Writing out the equation carefully and checking the direction of differences in alligation helps avoid errors. Final Answer: Thus, the ratio of the number of boys to the number of girls in the class is 2 : 3.

Question 6

The sum of the ages of Anushka, Kajal and Charulatha is 57 years. Kajal is three times as old as Anushka and Charulatha is 12 years older than Anushka. What are the respective present ages of Anushka, Kajal and Charulatha?

Accepted Answer

Introduction / Context: This is a direct linear equations problem involving three people whose ages are linked by simple multiplicative and additive relations. We are given the total of their ages and two separate relationships involving Anushka's age. Solving these relations gives the individual ages exactly. Given Data / Assumptions: The sum of the present ages of Anushka, Kajal and Charulatha is 57 years. Kajal is three times as old as Anushka. Charulatha is 12 years older than Anushka. We must find the present ages of Anushka, Kajal and Charulatha. Concept / Approach: Let Anushka's age be A years. Then Kajal's age is 3A and Charulatha's age is A + 12. Using the total age equation A + 3A + (A + 12) = 57, we solve for A. Once A is known, we immediately obtain Kajal's and Charulatha's ages and match them with the correct option. Step-by-Step Solution: Let Anushka's present age be A years. Then Kajal's present age = 3A years. Charulatha's present age = A + 12 years. The sum of their ages is A + 3A + (A + 12) = 57. Simplify: A + 3A + A + 12 = 57 gives 5A + 12 = 57. Subtract 12 from both sides: 5A = 45. Hence A = 45 / 5 = 9. So Anushka's age is 9 years, Kajal's age = 3 * 9 = 27 years, and Charulatha's age = 9 + 12 = 21 years. Verification / Alternative check: Check the total: 9 + 27 + 21 = 57, which matches the given sum. Check the relations: Kajal is three times as old as Anushka since 27 = 3 * 9, and Charulatha is 12 years older because 21 = 9 + 12. Both conditions are satisfied, confirming that the ages 9, 27 and 21 are correct. Why Other Options Are Wrong: Option 27, 9, 21 does not satisfy the relation that Kajal is three times Anushka's age. Option 9, 21, 27 reverses the roles of Kajal and Charulatha. Option 21, 27, 9 puts Anushka as the oldest, which contradicts the conditions. The set 15, 27, 15 sums to 57 but fails both the three times relation and the 12 year difference. Only 9, 27, 21 satisfies all the given statements simultaneously. Common Pitfalls: A common mistake is to misassign the multiplication factor, for example setting A = 3K instead of K = 3A. Another error is forgetting to include the +12 for Charulatha when writing the total equation. Defining a single variable and expressing all other ages in terms of it is the safest way to approach such problems. Final Answer: Thus, the respective ages of Anushka, Kajal and Charulatha are 9, 27 and 21 years.

Question 7

A mother's present age is three times her daughter's present age. Ten years from now, the mother's age will be exactly double her daughter's age. How old is the daughter today?

Accepted Answer

Introduction / Context: This problem uses two simple relationships between the present ages of a mother and her daughter and their ages after 10 years. Such questions are typical in aptitude tests and are solved using linear equations in one variable, representing the daughter's age or the mother's age. Given Data / Assumptions: The mother's present age is three times her daughter's present age. After 10 years, the mother's age will be twice her daughter's age at that time. We need to find the daughter's present age. Concept / Approach: Let the daughter's present age be D years. Then the mother's present age is 3D. After 10 years, the daughter will be D + 10 and the mother will be 3D + 10. The second condition says 3D + 10 = 2(D + 10). Solving this equation quickly yields D. This is a straightforward linear equation problem. Step-by-Step Solution: Let daughter's present age be D years. Then mother's present age is 3D years. After 10 years, daughter's age = D + 10 and mother's age = 3D + 10. Given that 10 years from now the mother's age is twice the daughter's age: 3D + 10 = 2(D + 10). Expand the right side: 3D + 10 = 2D + 20. Rearrange: 3D - 2D = 20 - 10. So D = 10. Therefore, the daughter's present age is 10 years. Verification / Alternative check: If the daughter is 10 years old now, the mother is 3 * 10 = 30 years old. After 10 years, the daughter will be 20 and the mother will be 40. Indeed, 40 is exactly twice 20, matching the second condition. Both statements in the problem are satisfied, confirming that 10 years is correct. Why Other Options Are Wrong: If the daughter were 8, the mother would be 24 and in 10 years their ages would be 18 and 34, which does not give a 2 : 1 ratio. Similar checks show that daughters of age 9, 11 or 12 lead to future ages that do not satisfy the doubling condition. Only D = 10 meets both the three times and the double conditions simultaneously. Common Pitfalls: Students sometimes reverse the relation and assume daughter's age is three times the mother's, which is impossible here. Another common mistake is to forget to add 10 to both ages or to misplace terms when simplifying the equation. Writing each step clearly and interpreting the words carefully avoids these errors. Final Answer: So, the daughter is 10 years old today.

Question 8

When Raj Tarun was born, his father was 32 years older than his brother and his mother was 25 years older than his sister. At present, Raj Tarun's brother is 6 years older than him and his mother is 3 years younger than his father. How old was Raj T…

Accepted Answer

Introduction / Context: This family age puzzle links the ages of Raj Tarun, his brother, sister, father and mother at the time of his birth using age differences that remain constant over time. We are asked to find the sister's age when Raj Tarun was born, using the current difference between Raj and his brother and the relationships given at birth time. Given Data / Assumptions: At the time of Raj Tarun's birth, his father was 32 years older than his brother. At the same time, his mother was 25 years older than his sister. At present, Raj Tarun's brother is 6 years older than him. At present, his mother is 3 years younger than his father. We must find his sister's age at the time of his birth. Concept / Approach: Age differences between people remain constant over time. At Raj Tarun's birth, his brother was already 6 years old, because he is 6 years older now. So we can compute the father's age then as brother's age plus 32, and the mother's age as father's age minus 3, which also equals sister's age plus 25. From this, we solve for the sister's age at that time. Step-by-Step Solution: At Raj Tarun's birth, his own age was 0 years. His brother is 6 years older than him now, so at the time of his birth the brother was 6 years old. At that time, father's age = brother's age + 32 = 6 + 32 = 38 years. Mother is 3 years younger than father at all times, so at that moment mother's age = 38 - 3 = 35 years. We are told that at that same time mother was 25 years older than the sister. So sister's age at Raj Tarun's birth = mother's age - 25 = 35 - 25 = 10 years. Verification / Alternative check: Confirm the relationships at birth: Brother was 6, father was 38, which is indeed 32 years older than the brother (38 - 6 = 32). Mother was 35, which is 3 years younger than father (38 - 35 = 3). Sister was 10, and mother was 25 years older (35 - 10 = 25). All conditions hold perfectly, verifying our calculation. Why Other Options Are Wrong: If the sister had been 5 years old, mother would be only 30 at that time, which breaks the 3 year age gap with father at 38. Values like 8, 12 or 15 also fail to maintain the stated 25 year difference with the mother once constraints are propagated. Only 10 years for the sister keeps every difference consistent with the data in the question. Common Pitfalls: The main difficulty is keeping track of which age difference applies at which moment and avoiding confusing present ages with ages at birth. Some learners incorrectly try to introduce the present age of Raj Tarun, which is not needed. Focusing on fixed differences and using a timeline from the birth moment makes the logic clear. Final Answer: Therefore, Raj Tarun's sister was 10 years old when he was born.

Question 9

A person's present age is two fifth of the present age of his mother. Eight years from now, he will be one half of his mother's age at that time. How old is the mother at present?

Accepted Answer

Introduction / Context: This age problem gives the current ratio between a person and his mother and another ratio eight years into the future. These two relationships allow us to form and solve a system of equations for their present ages. It is a standard example of age questions using proportion and linear equations. Given Data / Assumptions: The person's present age is two fifth of his mother's present age. After 8 years, the person will be one half of his mother's age at that time. We must determine the mother's present age. Concept / Approach: Let the person's present age be P years and the mother's present age be M years. Then P = (2/5) * M. After 8 years, their ages will be P + 8 and M + 8, and the second condition gives P + 8 = (1/2) * (M + 8). Solving these two equations simultaneously yields M directly. Step-by-Step Solution: Let P be the person's present age and M be the mother's present age. We are given P = (2/5) * M. After 8 years, the person's age will be P + 8 and the mother's age will be M + 8. Given that P + 8 = (1/2) * (M + 8). From P = (2/5) * M, substitute into the second equation: (2/5) * M + 8 = (1/2) * (M + 8). Multiply through by 10 to clear denominators: 4M + 80 = 5(M + 8). Expand the right side: 4M + 80 = 5M + 40. Rearrange to get 80 - 40 = 5M - 4M, so 40 = M. Therefore, the mother's present age M is 40 years. Verification / Alternative check: If the mother is 40, the person's present age is P = (2/5) * 40 = 16 years. After 8 years, the person will be 24 and the mother will be 48. Indeed, 24 is exactly half of 48. Both original conditions are satisfied, confirming the solution is correct. Why Other Options Are Wrong: If the mother were 36, then P would be 14.4 years, which is not an integer and does not fit usual exam expectations. Ages 38, 42 or 44 also lead to future ages that do not satisfy the one half relation after 8 years. Only 40 years makes both the two fifth and half conditions hold simultaneously. Common Pitfalls: Mistakes often arise from incorrectly writing P = (5/2) * M instead of (2/5) * M or forgetting to add 8 to both ages when forming the future relation. Some learners also clear denominators incorrectly and lose track of the coefficients. Careful algebra and checking the final ages in both conditions helps avoid these errors. Final Answer: Hence, the mother's present age is 40 years.

Question 10

Seven years ago, the ratio of the ages of Anirudh and Bhavana was 7 : 9. Chandhu is at present 12 years older than Anirudh and 12 years younger than Bhavana. What is Chandhu's present age?

Accepted Answer

Introduction / Context: This is a challenging age puzzle in which the ages of three people are related by a past ratio and current fixed differences. The ratio from 7 years ago and the two 12 year differences involving Chandhu allow us to construct equations for Anirudh and Bhavana's present ages. With those, Chandhu's present age follows directly. Given Data / Assumptions: Seven years ago, the ratio of Anirudh's age to Bhavana's age was 7 : 9. At present, Chandhu is 12 years older than Anirudh. At present, Chandhu is also 12 years younger than Bhavana. We need to find Chandhu's present age. Concept / Approach: Let the present ages of Anirudh and Bhavana be A and B years. Seven years ago their ages were A - 7 and B - 7, and the given ratio gives (A - 7) / (B - 7) = 7 / 9. Also, Chandhu's present age C satisfies C = A + 12 and C = B - 12, giving B - A = 24. Solving these two equations in A and B yields the present ages, then Chandhu's age is A + 12. Step-by-Step Solution: Let A be Anirudh's present age and B be Bhavana's present age. Seven years ago, Anirudh was A - 7 and Bhavana was B - 7. Given (A - 7) / (B - 7) = 7 / 9, so 9(A - 7) = 7(B - 7). This expands to 9A - 63 = 7B - 49. Rearrange: 9A - 7B = 14. (Equation 1) At present, Chandhu is 12 years older than Anirudh and 12 years younger than Bhavana, so B - A = 24. (Equation 2) From B - A = 24, we have B = A + 24. Substitute into Equation 1: 9A - 7(A + 24) = 14. So 9A - 7A - 168 = 14 which simplifies to 2A - 168 = 14. Thus 2A = 182 and A = 91 years. Then B = A + 24 = 91 + 24 = 115 years. Chandhu's present age C = A + 12 = 91 + 12 = 103 years. Verification / Alternative check: Check the ratio 7 years ago. Anirudh was 91 - 7 = 84 and Bhavana was 115 - 7 = 108. The ratio 84 : 108 simplifies by dividing by 12 to 7 : 9, which matches the problem. At present, Chandhu is 103, which is 12 years more than 91 and 12 years less than 115, confirming both difference conditions. The solution is internally consistent, even though the ages are unusually large, which is acceptable mathematically. Why Other Options Are Wrong: If Chandhu were 91 or 115 years old, he would coincide with Anirudh or Bhavana and not satisfy both 12 year differences. The option "Cannot be determined" is incorrect because the system of equations has a unique solution. The suggestion 87 years does not satisfy both the older and younger relations. Only 103 years fits all the given conditions exactly. Common Pitfalls: Some students reject the numerical result because the ages seem too high and conclude the data is insufficient, which is not correct. Another common error is mishandling the ratio equation or forgetting to subtract 7 from both ages for the past ratio. Carefully formulating the equations and trusting the algebraic outcome is crucial here. Final Answer: Therefore, Chandhu's present age is 103 years.

Question 11

The sum of the present ages of P and Q is 41 years. The age of P 2 years from now will equal the age of R 1 year ago. The age of P 4 years from now will equal the age of Q 1 year ago, and the ratio of the present ages of P and S is 3 : 4. What is th…

Accepted Answer

Introduction / Context: This multi person age problem links four people with several time shifted equalities and a given ratio. We must find the difference between the present ages of R and S. To solve it, we first determine P and Q using the sum of their ages and one time based relation, then find R and S through the remaining equations and the ratio. It is a good exercise in handling multiple constraints consistently. Given Data / Assumptions: P + Q = 41 years at present. P 2 years hence equals R 1 year ago. P 4 years hence equals Q 1 year ago. The ratio of the present ages of P and S is 3 : 4. We must find the difference between the present ages of R and S. Concept / Approach: Let the present ages of P, Q, R and S be P, Q, R and S years. From P + Q = 41 and P 4 years hence equal to Q 1 year ago, we get P + 4 = Q - 1, hence Q in terms of P. Substituting into the sum gives P. From P 2 years hence equals R 1 year ago, we find R. The ratio P : S = 3 : 4 gives S in terms of P. Finally, we compute the difference between R and S. Step-by-Step Solution: Given P + Q = 41. (Equation 1) P 4 years hence equals Q 1 year ago: P + 4 = Q - 1. So Q = P + 5. (Equation 2) Substitute Equation 2 into Equation 1: P + (P + 5) = 41. This gives 2P + 5 = 41, so 2P = 36 and P = 18 years. Then Q = P + 5 = 18 + 5 = 23 years. P 2 years hence equals R 1 year ago: P + 2 = R - 1. So R = P + 3 = 18 + 3 = 21 years. The ratio P : S is 3 : 4, so 18 : S = 3 : 4. From 18 / S = 3 / 4, we get S = 18 * 4 / 3 = 24 years. Difference between present ages of R and S = 24 - 21 = 3 years. Verification / Alternative check: Check each condition. P + Q = 18 + 23 = 41, correct. Four years from now P will be 22, and one year ago Q was 22, matching P + 4 = Q - 1. Two years from now P will be 20, and one year ago R was 20, matching P + 2 = R - 1. The ratio P : S is 18 : 24, which simplifies to 3 : 4. All constraints hold, so the computed ages and the difference are consistent. Why Other Options Are Wrong: Differences of 2, 4, 5 or 6 years would not arise from a consistent set of ages that satisfy all the given equations. Any attempt to adjust one age to change the difference breaks at least one of the time based equalities or the ratio condition. Only a difference of 3 years preserves all relationships simultaneously. Common Pitfalls: It is easy to confuse which age is advanced or reduced in each statement, such as mixing up P 4 years hence with Q 4 years hence. Another frequent error is to misapply the ratio P : S and invert it when solving. Writing each equation explicitly and substituting step by step prevents algebraic and logical confusion. Final Answer: Thus, the difference between the present ages of R and S is 3 years.

Question 12

Hari Ram's present age is three times his son's present age and two fifth of his father's present age. The average of the present ages of all three of them (Hari Ram, his son and his father) is 46 years. What is the difference between the present ag…

Accepted Answer

Introduction / Context: This three generation age problem links Hari Ram with his son and his father using two proportions and an overall average. From these relationships, we can determine each individual's present age and then compute the difference between the youngest and the oldest. It combines proportion and average concepts in a neat way. Given Data / Assumptions: Hari Ram's present age is three times his son's present age. Hari Ram's present age is two fifth of his father's present age. The average of the present ages of Hari Ram, his son and his father is 46 years. We must find the difference between the present ages of his son and his father. Concept / Approach: Let Hari Ram's present age be H years. Then his son's age is H / 3 and his father's age is (5 / 2) * H. The average age 46 means (H + H / 3 + (5 / 2) * H) / 3 = 46. Solving this equation yields H. We then compute the son's and father's ages and find their difference, which is the required answer. Step-by-Step Solution: Let Hari Ram's present age be H years. Then his son's present age is H / 3 years. His father's present age is (5 / 2) * H years because H is two fifth of the father's age. The average of their present ages is 46 years, so (H + H / 3 + (5 / 2) * H) / 3 = 46. First find the total age: H + H / 3 + (5 / 2) * H. Compute the coefficients using a common denominator 6: H = 6H / 6, H / 3 = 2H / 6, (5 / 2) * H = 15H / 6. So total age = (6H + 2H + 15H) / 6 = 23H / 6. The average equation becomes (23H / 6) / 3 = 46, that is 23H / 18 = 46. Multiply both sides by 18: 23H = 46 * 18 = 828. So H = 828 / 23 = 36 years. Then son's present age = H / 3 = 36 / 3 = 12 years. Father's present age = (5 / 2) * 36 = 5 * 18 = 90 years. Difference between father's and son's present ages = 90 - 12 = 78 years. Verification / Alternative check: Check the relationships. Hari Ram is 36, which is three times his son at 12. He is also two fifth of his father, since 2/5 of 90 is 36. The total of their ages is 36 + 12 + 90 = 138, and dividing by 3 gives an average of 46 as stated. All conditions are satisfied, so the calculations are correct and the age difference 78 is confirmed. Why Other Options Are Wrong: Differences such as 44, 56, 67 or 52 years would require different underlying ages that would not satisfy both the three times and two fifth relations at the same time while also giving an average of 46. Only the combination 12, 36 and 90 gives a consistent solution, which produces a 78 year gap. Common Pitfalls: Some students incorrectly take H as the son's age instead of Hari Ram's age, which leads to wrong proportions. Others misinterpret "two fifth of his father's age" and invert the fraction. Calculation mistakes in combining fractions and solving for H are also common. Using a common denominator carefully and double checking the proportion directions helps avoid these issues. Final Answer: Therefore, the difference between the present ages of Hari Ram's son and his father is 78 years.

Question 13

In an aptitude problem on ages, the present ages of Renuka and Sony are in the ratio 5 : 4. Four years ago, Sony was 24 years old. What will be Renuka's age 5 years from now?

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Introduction / Context: This is a standard aptitude question on ages involving a ratio between the present ages of two people and information about one person's age in the past. Our goal is to use this data to find Renuka's present age and then compute her age after 5 more years. Given Data / Assumptions: - Present age ratio of Renuka : Sony = 5 : 4. - Four years ago, Sony was 24 years old. - Ages increase uniformly with time for both Renuka and Sony. - We must find Renuka's age 5 years from now. Concept / Approach: First we convert the past information about Sony's age into her present age. Then we apply the given ratio of present ages to calculate Renuka's current age. Finally, we add 5 years to Renuka's present age to obtain her age in the future. The problem mainly uses ratios and basic linear equations. Step-by-Step Solution: Step 1: Sony's age four years ago was 24 years. Step 2: Sony's present age = 24 + 4 = 28 years. Step 3: Let Renuka's present age be R years. Using the ratio R : 28 = 5 : 4. Step 4: From R / 28 = 5 / 4, we get 4R = 5 × 28 = 140. Step 5: Therefore, R = 140 / 4 = 35 years. Step 6: Renuka's age after 5 years = 35 + 5 = 40 years. Verification / Alternative check: Check the ratio using the found ages: 35 : 28. Dividing both by 7 gives 5 : 4, which matches the question. Adding 5 years to 35 gives 40, confirming the result. Why Other Options Are Wrong: 30 years, 35 years, 38 years, and 45 years do not match Renuka's age 5 years from now once the correct present ages are calculated. Only 40 years is consistent with the ratio and time conditions given. Common Pitfalls: Students often forget to shift from "four years ago" to the present age correctly or stop after finding the present age of Renuka without adding the required 5 years. Always track the time references carefully. Final Answer: Renuka's age after 5 years will be 40 years.

Question 14

Five years ago, the average age of a newly married couple was 24 years, but now the average age of the husband, wife and their child is 20 years. The child was born during this 5-year interval. What is the child's present age?

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Introduction / Context: This question tests your understanding of average age problems and how adding a new member to a family affects the overall average. We are given the average age of a couple at the time of marriage and the present average age after the birth of a child, and we must determine the present age of the child. Given Data / Assumptions: - Five years ago, the average age of the husband and wife was 24 years. - Therefore, the total of their ages at that time was 2 × 24 = 48 years. - The child was born sometime during the last 5 years. - At present, the average age of the husband, wife and child together is 20 years. - We need the child's present age in years. Concept / Approach: Average age is defined as total age divided by the number of persons. The key idea is to first find the present total age of the couple, then add the child's age and equate it to the present total implied by the new average. From this equation, we can solve for the child's age. The time shift of 5 years must be applied correctly to the ages of the husband and wife only, because the child did not exist throughout the entire interval. Step-by-Step Solution: Step 1: Total age of the couple 5 years ago = 2 × 24 = 48 years. Step 2: In 5 years, each of them grows 5 years older, so the present total age of the couple = 48 + (2 × 5) = 48 + 10 = 58 years. Step 3: Let the present age of the child be x years. Step 4: Present total age of the family (husband + wife + child) = 58 + x. Step 5: The present average age of the three is 20 years, so (58 + x) / 3 = 20. Step 6: Multiply both sides by 3 to get 58 + x = 60. Step 7: Therefore, x = 60 − 58 = 2 years. Verification / Alternative check: If the child is 2 years old now, then the present ages of the husband and wife must add to 58 years, and including the child gives a total of 60 years. The average for three people is 60 / 3 = 20 years, which matches the statement in the question. This confirms that the child's present age is indeed 2 years. Why Other Options Are Wrong: 3 years, 4 years, 5 years, and 6 years would lead to present totals of 61, 62, 63, and 64 years respectively, giving averages of more than 20 years. None of these values match the required average of 20 years, so they are incorrect. Common Pitfalls: A common error is to forget that both adults age by the full 5 years before the average including the child is calculated. Another mistake is to assume that the child has been present for the entire 5-year period, which is not necessary. Carefully tracking total ages and averages avoids these mistakes. Final Answer: The present age of the child is 2 years.

Question 15

The present age of X is two-thirds of the present age of Y. After 6 years, X will be 46 years old. What is the present age of Y?

Accepted Answer

Introduction / Context: This question is a simple algebraic problem based on ages and ratios. We are given a relationship between the present ages of X and Y and a future value for X's age. Using this, we must determine Y's present age. Given Data / Assumptions: - X's present age is two-thirds of Y's present age. - After 6 years, X will be 46 years old. - Ages increase linearly by the same number of years over time. - We are asked to find Y's present age in years. Concept / Approach: We first use the information about X's age 6 years in the future to determine X's present age. Then we use the given fractional relationship between X and Y to compute Y's present age. The problem can be solved using a simple linear equation and basic fraction manipulation. Step-by-Step Solution: Step 1: After 6 years, X will be 46 years old. Step 2: Therefore, X's present age = 46 − 6 = 40 years. Step 3: X's present age is two-thirds of Y's present age. Let Y's present age be Y years. Step 4: According to the relation, 40 = (2 / 3) × Y. Step 5: Multiply both sides by 3 to get 120 = 2Y. Step 6: Divide both sides by 2 to get Y = 120 / 2 = 60 years. Verification / Alternative check: If Y is 60 years old now, then X must be two-thirds of 60, which is (2 / 3) × 60 = 40 years. In 6 years, X will be 40 + 6 = 46 years, which matches the given condition. Therefore, the value Y = 60 years is fully consistent with the information in the problem. Why Other Options Are Wrong: 40 years and 56 years do not satisfy the relation that X is two-thirds of Y when X is 40 years old. For Y = 40 years, two-thirds of 40 is not 40. For Y = 56 years, two-thirds is 37.33, not 40. The value 64 years gives two-thirds equal to approximately 42.67, and 100 years gives two-thirds equal to about 66.67. None of these match X's present age of 40 years. Common Pitfalls: Students may incorrectly treat the given ratio as X : Y = 2 : 3 without checking which value corresponds to which person or may forget to convert the future age back to the present age before applying the ratio. Always compute the present age first when future or past ages are given, then apply the ratio relationship. Final Answer: The present age of Y is 60 years.

Question 16

Ram is 20 years younger than Shyam. Five years ago, the ratio of their ages was 3 : 5. What is the sum of their present ages?

Accepted Answer

Introduction / Context: This is a classic age-based word problem where we are given the age difference between two people and a ratio of their ages at a previous time. We must use this information to compute their current ages and then find the sum of those ages. Given Data / Assumptions: - Ram is 20 years younger than Shyam at present. - Five years ago, the ratio of their ages was 3 : 5 (Ram : Shyam). - Ages of both increase by 5 years from five years ago to the present. - We are asked for the sum of their present ages. Concept / Approach: We set up algebraic expressions for their present ages using a variable for Shyam's age. Next we form an equation from the ratio given for their ages five years ago. Solving this equation gives the present ages of both Ram and Shyam, from which we compute their sum. This problem uses linear equations and the idea that age differences remain constant over time. Step-by-Step Solution: Step 1: Let Shyam's present age be S years. Then Ram's present age is S − 20 years. Step 2: Five years ago, Shyam's age was S − 5 and Ram's age was (S − 20) − 5 = S − 25. Step 3: The ratio five years ago was (Ram : Shyam) = 3 : 5, so (S − 25) / (S − 5) = 3 / 5. Step 4: Cross-multiply to get 5(S − 25) = 3(S − 5). Step 5: Expand: 5S − 125 = 3S − 15. Step 6: Rearrange to get 2S = 110, so S = 55 years. Step 7: Ram's present age = S − 20 = 55 − 20 = 35 years. Step 8: Sum of their present ages = 55 + 35 = 90 years. Verification / Alternative check: Check the ages five years ago: Ram = 35 − 5 = 30 years and Shyam = 55 − 5 = 50 years. The ratio 30 : 50 simplifies to 3 : 5, which matches the given ratio. The sum of their current ages is indeed 90 years, confirming our solution. Why Other Options Are Wrong: 75 years, 80 years, 85 years, and 95 years are all sums that do not arise from any integer solution that fits the ratio 3 : 5 five years ago and the 20-year age difference. Only 90 years satisfies both the ratio condition and the fixed difference in their ages. Common Pitfalls: Students sometimes mix up which age is older or misapply the ratio at the wrong time (present instead of five years ago). Another frequent mistake is to ignore the constant age difference information, which is crucial in forming the correct equation. Carefully aligning the time frame and the given ratio avoids these issues. Final Answer: The sum of Ram's and Shyam's present ages is 90 years.

Question 17

The present ages of Manisha and Sudeshna are in the ratio 5 : 6. After 8 years, the ratio of their ages will be 7 : 8. What is the difference between their present ages?

Accepted Answer

Introduction / Context: This question involves present and future age ratios of two people. By using the ratio form at two different times, we can set up an equation to determine their actual ages and then find the difference between them. Such problems test understanding of ratios and linear equations in age-related situations. Given Data / Assumptions: - Present age ratio of Manisha : Sudeshna = 5 : 6. - After 8 years, the ratio of their ages will be 7 : 8. - Both individuals grow older by the same 8 years over the period. - We are required to find the difference between their present ages. Concept / Approach: When present ages are given in a ratio, we can represent them as multiples of a common variable. Then we apply the future ratio condition after adding the given years to both ages. This leads to a solvable linear equation in one variable. Once the variable value is found, we compute the actual ages and take their difference. Step-by-Step Solution: Step 1: Let the present ages of Manisha and Sudeshna be 5x and 6x years respectively. Step 2: After 8 years, their ages will be 5x + 8 and 6x + 8 years. Step 3: The ratio after 8 years is given as (5x + 8) : (6x + 8) = 7 : 8. Step 4: Form the equation (5x + 8) / (6x + 8) = 7 / 8. Step 5: Cross-multiply to get 8(5x + 8) = 7(6x + 8). Step 6: Expand both sides: 40x + 64 = 42x + 56. Step 7: Rearrange to find 2x = 8, so x = 4. Step 8: Present ages are Manisha = 5x = 20 years and Sudeshna = 6x = 24 years. Step 9: Difference between their present ages = 24 − 20 = 4 years. Verification / Alternative check: Check the ratio after 8 years: Manisha will be 20 + 8 = 28 years and Sudeshna will be 24 + 8 = 32 years. Their ratio is 28 : 32, which simplifies to 7 : 8 when both numbers are divided by 4. This matches the given future ratio, confirming that the ages and the difference of 4 years are correct. Why Other Options Are Wrong: Differences of 2 years, 6 years, 8 years, or 10 years do not satisfy both the current ratio 5 : 6 and the future ratio 7 : 8 when actual ages are computed. Only a difference of 4 years leads to consistent ages that match both ratio conditions. Common Pitfalls: A frequent mistake is to treat the ratio numbers as direct ages, ignoring that they must be multiplied by a common factor x. Another mistake is to forget to add the 8 years to both ages when using the future ratio. Carefully setting up the equation with variables and the correct time adjustment is essential. Final Answer: The difference between the present ages of Manisha and Sudeshna is 4 years.

Question 18

In a joint family of 24 members, the average age decreases by 4 months when a girl aged 24 years is replaced by a new boy. What is the age of the new boy?

Accepted Answer

Introduction / Context: This problem is based on the concept of average age and how replacing one member of a group with another affects the overall average. We are told that a 24-year-old girl is replaced by a boy, and the average age of a 24-member joint family decreases by 4 months. Using this change in average, we must find the boy's age. Given Data / Assumptions: - There are 24 members in the joint family. - One girl aged 24 years leaves the family. - A new boy joins in her place. - Due to this replacement, the average age of the 24 members decreases by 4 months. - Four months is equal to 4 / 12 = 1 / 3 year. Concept / Approach: A change in average age multiplied by the number of members gives the change in the total sum of ages. Here, the average decreases by 4 months, so the total age of the group decreases. That total decrease must equal the difference between the age of the girl who left and the age of the boy who joined. We can write this as an equation and solve for the boy's age. Step-by-Step Solution: Step 1: Change in average age = 4 months = 1 / 3 year. Step 2: Number of family members = 24. Step 3: Decrease in total age of the family = change in average × number of members = (1 / 3) × 24 = 8 years. Step 4: The girl who left was 24 years old. Let the boy's age be B years. Step 5: The total age decreases when a 24-year-old is replaced by a B-year-old boy, so decrease in total = 24 − B. Step 6: This must equal 8 years, so 24 − B = 8. Step 7: Solving gives B = 24 − 8 = 16 years. Verification / Alternative check: If the girl is 24 years old and the boy is 16 years old, the family loses 8 years of total age (24 − 16 = 8). Dividing this total decrease of 8 years by 24 members yields 8 / 24 = 1 / 3 year, which is 4 months. This matches the stated decrease in average age, confirming that the boy's age is correctly found as 16 years. Why Other Options Are Wrong: A boy aged 12, 15, 18, or 20 years would produce decreases in total age of 12, 9, 6, or 4 years respectively, leading to average decreases of 12 / 24, 9 / 24, 6 / 24, or 4 / 24 years. None of these correspond to exactly 4 months, so those ages are incorrect. Common Pitfalls: Students sometimes confuse a decrease in average age with a decrease in individual ages or forget to convert months into a fraction of a year correctly. Another common error is to ignore the number of family members when calculating the change in total age. Always remember that total change in age = change in average × number of members. Final Answer: The age of the new boy is 16 years.

Question 19

Sweety is 54 years old and her mother is 80 years old. How many years ago was Sweety's mother three times Sweety's age?

Accepted Answer

Introduction / Context: This age problem involves determining a time in the past when one person's age was a specific multiple of another person's age. We are told the present ages of Sweety and her mother and must find how many years ago the mother was three times as old as Sweety. Given Data / Assumptions: - Sweety's present age is 54 years. - Sweety's mother's present age is 80 years. - We are looking for a number of years in the past, say t years ago, when mother's age = 3 × Sweety's age. Concept / Approach: If we go back t years, both ages are reduced by t. At that time, we apply the condition that the mother's age was three times Sweety's age. This gives a simple linear equation in t, which we can solve to find how many years ago the condition was true. Step-by-Step Solution: Step 1: Let t be the number of years ago when the condition held. Step 2: t years ago, Sweety's age was 54 − t years. Step 3: At the same time, the mother's age was 80 − t years. Step 4: According to the question, 80 − t = 3 × (54 − t). Step 5: Expand the right-hand side: 3 × (54 − t) = 162 − 3t. Step 6: Set up the equation: 80 − t = 162 − 3t. Step 7: Rearranging gives 2t = 162 − 80 = 82, so t = 82 / 2 = 41 years. Verification / Alternative check: 41 years ago, Sweety's age would have been 54 − 41 = 13 years. At the same time, her mother would have been 80 − 41 = 39 years old. The ratio 39 : 13 is equal to 3 : 1, meaning the mother was exactly three times as old as Sweety. This matches the condition given in the question, confirming that t = 41 years is correct. Why Other Options Are Wrong: If we try 20, 26, 33, or 45 years instead of 41, the resulting ages for Sweety and her mother do not satisfy the condition that the mother is exactly three times Sweety's age. The products and ratios do not match 3 : 1, so these options are incorrect. Common Pitfalls: A common mistake is to set up the equation incorrectly, for example using 3 × (80 − t) instead of 3 × (54 − t), or forgetting that both ages must be reduced by the same amount t in the past. Another error is mismanaging the algebra when simplifying the equation. Carefully writing each step helps avoid these issues. Final Answer: Sweety's mother was three times her age 41 years ago.

Question 20

Avanthi's present age is one-sixth of her father's present age. Her father's age will be twice Kapil's age after 10 years. Kapil's eighth birthday was celebrated two years ago. What is Avanthi's present age?

Accepted Answer

Introduction / Context: This is a multi-step age problem involving three people: Avanthi, her father, and Kapil. We are given a fractional relationship between Avanthi's age and her father's age, as well as information about Kapil's age and a future relationship between Kapil and the father. Our aim is to work through these relationships to find Avanthi's present age. Given Data / Assumptions: - Avanthi's present age is one-sixth of her father's present age. - After 10 years, the father's age will be twice Kapil's age at that time. - Kapil's eighth birthday was celebrated two years ago, so Kapil is currently 10 years old. - All ages are measured in complete years. Concept / Approach: We start by determining Kapil's present age from the birthday information. Then we use the future statement about the father's age being twice Kapil's age after 10 years to find the father's current age. Finally, we use the fractional relationship between Avanthi and her father to compute Avanthi's present age. This problem combines time shifts with proportional relationships. Step-by-Step Solution: Step 1: Kapil's eighth birthday was celebrated two years ago, so his present age is 8 + 2 = 10 years. Step 2: Let the father's present age be F years. Step 3: After 10 years, Kapil's age will be 10 + 10 = 20 years. Step 4: After 10 years, the father's age will be F + 10 years. Step 5: The problem states that, after 10 years, the father's age will be twice Kapil's age. So F + 10 = 2 × 20 = 40. Step 6: Solving gives F = 40 − 10 = 30 years. Step 7: Avanthi's present age is one-sixth of her father's age. So Avanthi's age = (1 / 6) × 30 = 5 years. Verification / Alternative check: Check all relationships: If the father is 30 years old now and Kapil is 10, then after 10 years they will be 40 and 20 years old respectively, satisfying the condition that the father is twice Kapil's age. Avanthi's age as one-sixth of 30 is 5 years, consistent with the given fraction. All conditions are satisfied simultaneously. Why Other Options Are Wrong: Ages of 4, 6, 7, or 8 years for Avanthi would not equal one-sixth of a father's age that fits the future condition with Kapil. For example, if Avanthi were 6 years old, her father would be 36, and the future ages would not give the specified 2 : 1 ratio with Kapil. Only 5 years matches both the fractional relationship and the future age condition. Common Pitfalls: Some students misread the birthday information and think Kapil is 8 instead of 10, or they forget to apply the 10-year shift to both the father and Kapil before using the "twice the age" condition. Others attempt to guess Avanthi's age directly without first finding the father's age. A systematic approach avoids these errors. Final Answer: Avanthi's present age is 5 years.

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