Question 1

The present age of Rajan is three times the age of his daughter and equal to nine thirteenths of the present age of his mother. The sum of the present ages of all three of them is 125 years. What is the difference between the present age of Rajan's …

Accepted Answer

Introduction / Context: This question involves three related people: Rajan, his daughter, and his mother. The relations between their ages are given using multiplication and a fractional comparison, and the total of their present ages is known. From this, we must find the difference between the ages of the daughter and the mother. Problems like this are common in aptitude tests on ages and require solving a small system of equations. Given Data / Assumptions: Rajan's present age is three times the age of his daughter. Rajan's present age is also nine thirteenths of his mother's present age. The sum of their present ages (Rajan, daughter, and mother) is 125 years. We must find the difference between the present age of Rajan's daughter and that of his mother. Concept / Approach: We introduce variables for the ages of the daughter and Rajan and express all other ages in terms of those variables. The relation that Rajan's age is three times his daughter's age gives one equation. The relation involving nine thirteenths gives a connection between Rajan's age and his mother's age. Adding these and using the total sum of ages gives a solvable system. Once the ages are found, the required difference can be calculated. Step-by-Step Solution: Step 1: Let the daughter's present age be D years.Step 2: Rajan's present age is three times his daughter's age, so his age is 3D years.Step 3: Rajan's age is nine thirteenths of his mother's age. So 3D = (9 / 13) * M, where M is the mother's age.Step 4: Rearrange to express M in terms of D: multiply both sides by 13 and divide by 9, giving M = (13 / 9) * 3D = (39D) / 9 = (13D) / 3.Step 5: The sum of their present ages is D + 3D + M = 125.Step 6: Substitute M = 13D / 3 into the sum: D + 3D + 13D / 3 = 125.Step 7: Combine terms: D + 3D = 4D. So 4D + 13D / 3 = (12D + 13D) / 3 = 25D / 3.Step 8: Set 25D / 3 = 125. Multiply both sides by 3: 25D = 375, so D = 375 / 25 = 15 years.Step 9: Then Rajan's age is 3D = 45 years, and M = 13D / 3 = 13 * 15 / 3 = 65 years.Step 10: The required difference between the mother's age and the daughter's age is 65 - 15 = 50 years. Verification / Alternative check: Check the sum: 15 (daughter) + 45 (Rajan) + 65 (mother) = 125 years, which matches the problem statement. Check the fractional relation: Rajan's age as nine thirteenths of his mother's age gives (9 / 13) * 65 = 585 / 13 = 45 years, which matches Rajan's age. This confirms that our calculations are correct. Why Other Options Are Wrong: A difference of 32, 48, or 62 years would correspond to different combinations of ages that do not satisfy all the constraints simultaneously. For example, they would either violate the three times relation between Rajan and his daughter or the nine thirteenths relation between Rajan and his mother, or fail to sum to 125 years. Only a difference of 50 years preserves all given relationships. Common Pitfalls: Common mistakes include mishandling the fraction nine thirteenths, incorrectly cross multiplying, or forgetting to express all three ages in terms of a single variable before using the sum. Another pitfall is to treat 9/13 as 13/9 by mistake. Writing each relationship slowly and checking each step prevents such errors. Final Answer: The difference between the present ages of Rajan's daughter and his mother is 50 years.

Question 2

Manish is 12 years old and currently three times as old as his brother Rajeev. How old will Manish be when he becomes exactly twice as old as Rajeev?

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Introduction / Context: This is a straightforward problem on ages involving two brothers. We know Manish's present age and his current relationship to his younger brother Rajeev. We must find Manish's age at the time when he will be exactly twice as old as Rajeev. Such questions are commonly used to test basic algebraic reasoning in aptitude exams. Given Data / Assumptions: Manish is 12 years old at present. He is currently three times as old as his brother Rajeev. We are asked to find Manish's age when he becomes twice as old as Rajeev. Ages are assumed to be in whole years. Concept / Approach: We first compute Rajeev's present age using the three times relation. Then we introduce a variable t for the number of years after which Manish will be twice as old as Rajeev. We then form a simple linear equation using their future ages and solve for t. Finally, we add t to Manish's current age to get the required age. Step-by-Step Solution: Step 1: Let Manish's present age be M = 12 years.Step 2: Since Manish is three times as old as his brother, Rajeev's present age R satisfies M = 3R.Step 3: Substitute M = 12 into 12 = 3R to get R = 12 / 3 = 4 years.Step 4: Let t be the number of years from now when Manish will be twice as old as Rajeev.Step 5: After t years, Manish's age will be 12 + t and Rajeev's age will be 4 + t.Step 6: At that time, 12 + t must equal 2 * (4 + t).Step 7: Expand: 12 + t = 8 + 2t. Rearranging gives 12 - 8 = 2t - t, so 4 = t.Step 8: Therefore, after 4 years Manish will be 12 + 4 = 16 years old. Verification / Alternative check: At present, Manish is 12 years old and Rajeev is 4 years old. After 4 years, Manish will be 16 and Rajeev will be 8. At that time, 16 is exactly twice 8, satisfying the required condition. So the solution is valid. Why Other Options Are Wrong: If Manish's age were 12 or 14 years at the moment he is twice as old as Rajeev, simple backward calculations show that the age relation 2:1 would not hold. For example, at 12 years, Rajeev would have to be 6 years old, which contradicts the given information that he is currently 4. If Manish were 18 at that moment, Rajeev would be 9, which would not arise from the original three times relation at age 12. Therefore, only 16 years satisfies all conditions. Common Pitfalls: A common mistake is to assume that Manish will be twice as old as Rajeev at some fixed age like 24 without actually setting up the equation. Another error is to forget that both ages increase by the same amount t when moving into the future. Treating the problem systematically with variables prevents such errors. Final Answer: Manish will be 16 years old when he is twice as old as Rajeev.

Question 3

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

Accepted Answer

Introduction / Context: This problem involves several children whose ages form an arithmetic sequence, because each child is born 3 years after the previous one. The sum of their present ages is given, and we are asked to determine the age of the youngest child. This is a standard question under the topic problems on ages and arithmetic progressions. Given Data / Assumptions: There are five children. Each successive child is 3 years older than the next younger one, so their ages differ by 3 years. The sum of the present ages of all five children is 50 years. We are required to find the present age of the youngest child. Ages are treated as whole numbers in years. Concept / Approach: If the youngest child is x years old, then the other children are x + 3, x + 6, x + 9, and x + 12 years old, since they are each 3 years apart. The sum of these five ages can be expressed in terms of x. We then set this sum equal to 50, solve for x, and identify the youngest child's age. Step-by-Step Solution: Step 1: Let the present age of the youngest child be x years.Step 2: Because the children are born at intervals of 3 years, their ages are x, x + 3, x + 6, x + 9, and x + 12 years.Step 3: The sum of their ages is given as 50 years. So x + (x + 3) + (x + 6) + (x + 9) + (x + 12) = 50.Step 4: Combine like terms: x + x + x + x + x = 5x, and 3 + 6 + 9 + 12 = 30. So the equation becomes 5x + 30 = 50.Step 5: Subtract 30 from both sides: 5x = 20.Step 6: Divide both sides by 5: x = 20 / 5 = 4 years.Step 7: Hence, the youngest child is 4 years old at present. Verification / Alternative check: Compute all ages: youngest 4 years, then 7, 10, 13, and 16 years. Their sum is 4 + 7 + 10 + 13 + 16 = 50 years, which matches the total given in the question. This confirms that the solution is correct and consistent with all the information provided. Why Other Options Are Wrong: If the youngest child were 6 years old, the ages would be 6, 9, 12, 15, and 18 years, summing to 60 years, not 50. If the youngest were 8 or 16 years old, the sums would be even larger and would clearly not equal 50 years. Therefore, those options do not satisfy the total sum condition. Common Pitfalls: Some learners may forget that there are five children and mistakenly use fewer terms in the arithmetic sequence. Others may miscalculate the sum of the constant differences (3, 6, 9, 12). Writing the sequence explicitly and adding carefully helps avoid such mistakes. Final Answer: The youngest child is 4 years old at present.

Question 4

At present, the ratio between the ages of Arun and Harish is 4:3. After 6 years, the age of Arun will be 26 years. What is the present age of Harish?

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Introduction / Context: This question involves finding one person's present age from a given ratio and a piece of future age information. The present age ratio of Arun and Harish is known, and Arun's age after 6 years is specified. We must compute Harish's present age. Such questions are common in the topic problems on ages and ratios. Given Data / Assumptions: The present ratio of the ages of Arun and Harish is 4:3. After 6 years, Arun's age will be 26 years. We are required to find Harish's present age in years. Ages are assumed to be whole numbers. Concept / Approach: First, we use the future age of Arun to find his present age by subtracting 6 years. Then we use the ratio 4:3 to find Harish's age. Expressing the ratio as Arun : Harish = 4 : 3 means that Arun's age is 4 parts and Harish's age is 3 parts. Once the value of one part is determined, we can scale the ratio appropriately. Step-by-Step Solution: Step 1: After 6 years, Arun's age will be 26 years. Therefore, Arun's present age is 26 - 6 = 20 years.Step 2: The present age ratio Arun : Harish is 4 : 3. So we can write Arun's age as 4k and Harish's age as 3k for some positive number k.Step 3: Since Arun's present age is 20, we have 4k = 20.Step 4: Solve for k: k = 20 / 4 = 5.Step 5: Harish's present age is 3k = 3 * 5 = 15 years. Verification / Alternative check: The ages 20 and 15 are in the ratio 20:15, which simplifies to 4:3 when divided by 5, matching the given ratio. After 6 years, Arun will be 26 and Harish will be 21. The question only requires Harish's present age, and the numbers are consistent, confirming the correctness of the solution. Why Other Options Are Wrong: If Harish were 16, 18, or 21 years old at present, the corresponding ratios with Arun's age of 20 would be 20:16, 20:18, or 20:21, which simplify to 5:4, 10:9, and approximately 20:21, none of which equal 4:3. Thus, those options do not satisfy the given ratio and must be rejected. Common Pitfalls: Some learners mistakenly use Arun's future age of 26 directly in the ratio instead of first converting it to the present age of 20. Another error is to set up the ratio in the reverse order. Correctly identifying whose age corresponds to which part of the ratio and adjusting for the time shift is crucial. Final Answer: Harish is 15 years old at present.

Question 5

The present age of Riya is four times the present age of Siya, and the age of Riya is 24 years. What is the difference between the ages of Riya and Siya?

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Introduction / Context: This is a simple age ratio problem involving two people, Riya and Siya. Riya's age is given to be a multiple of Siya's age, and Riya's actual age is specified. We must find the difference between their ages. This type of problem tests basic proportional reasoning and subtraction. Given Data / Assumptions: The present age of Riya is four times the present age of Siya. Riya is 24 years old at present. We need to find the age difference between Riya and Siya. Ages are in years and assumed to be whole numbers. Concept / Approach: When one person's age is a multiple of another's, we can express the smaller age as a variable and the larger as a multiple of that variable. Here, Riya's age equals 4 times Siya's age. Knowing Riya's age allows us to divide by 4 to find Siya's age. The difference between their ages is then easily obtained by subtraction. Step-by-Step Solution: Step 1: Let Siya's present age be S years.Step 2: Riya's present age is four times Siya's age, so Riya's age is 4S years.Step 3: It is given that Riya is 24 years old, so 4S = 24.Step 4: Solve for S: S = 24 / 4 = 6 years.Step 5: The difference between their ages is Riya's age minus Siya's age, which is 24 - 6 = 18 years. Verification / Alternative check: Check that Riya's age is indeed four times Siya's: 4 * 6 = 24, which matches Riya's given age. The computed difference of 18 years is therefore consistent with the problem data. There are no additional conditions to check, so the solution is straightforward and complete. Why Other Options Are Wrong: A difference of 2, 12, or 6 years would imply Siya's age is 22, 12, or 18 respectively if Riya is 24. None of these ages satisfies the condition that Riya is four times Siya's age. Only when Siya is 6 years old does the four times condition hold, giving a difference of 18 years. Common Pitfalls: A common mistake is to interpret "four times the age" incorrectly and either subtract 4 or add 4 instead of multiplying by 4. Another error is to misplace which person is older. Carefully translating the phrase "four times the age" into a multiplication equation prevents such confusion. Final Answer: The difference between the ages of Riya and Siya is 18 years.

Question 6

In an ages-in-a-row problem, seven persons are sitting in a single row facing east. The average age of all 7 persons is 26 years. The average age of the first 3 persons (from the left end) is 19 years, and the average age of the last 3 persons (from…

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Introduction: This question tests the core idea of averages in age problems. When a group average is given, you can convert it into a total sum of ages by multiplying average by number of persons. Then, if partial group averages are given (like first 3 and last 3), you can find their totals and subtract from the overall total to get the missing individual (the middle person). Given Data / Assumptions: Total persons = 7 Average age of all 7 = 26 years Average age of first 3 = 19 years Average age of last 3 = 32 years Middle person is the 4th person Concept / Approach: Total age = average * number of persons. The middle person's age = (total of all 7) - (total of first 3) - (total of last 3). Step-by-Step Solution: Total age of all 7 = 26 * 7 = 182Total age of first 3 = 19 * 3 = 57Total age of last 3 = 32 * 3 = 96Middle person's age = 182 - 57 - 96 = 29 Verification / Alternative check: First 3 + middle + last 3 totals should equal 182. Here 57 + 29 + 96 = 182, so the calculation is consistent. Why Other Options Are Wrong: 27 years: would make the total 180, not 182.24 years: would make the total 177, not 182.32 years: is the last-3 average value, not the middle person's computed age.30 years: would make the total 183, not 182. Common Pitfalls: Using averages directly without converting to totals.Forgetting that the middle person is excluded from both first-3 and last-3 groups.Mixing up position logic in a row (the middle in 7 persons is the 4th person). Final Answer: 29 years

Question 7

In a chain age relation problem, Anu's present age is 9 years more than what Raj's age will be after 5 years. Raj's present age is 7 years more than what Renu's age was 4 years ago. If Renu's present age is 19 years, what will be Anu's age after 5 y…

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Introduction: This question checks your ability to translate age statements into simple arithmetic. Such problems usually involve moving backward or forward in time (like 4 years ago, 5 years after) and linking ages across people. The safest approach is to start from the only fixed value given (Renu's present age), then compute Raj's present age, then compute Anu's present age, and finally move to Anu's age after 5 years. Given Data / Assumptions: Renu's present age = 19 years Raj's present age = (Renu's age 4 years ago) + 7 Anu's present age = (Raj's age after 5 years) + 9 We need Anu's age after 5 years Concept / Approach: Convert each statement into a step: first shift Renu back 4 years, then add 7 to get Raj. Next shift Raj forward 5 years, then add 9 to get Anu. Then shift Anu forward 5 years for the final requirement. Step-by-Step Solution: Renu's age 4 years ago = 19 - 4 = 15Raj's present age = 15 + 7 = 22Raj's age after 5 years = 22 + 5 = 27Anu's present age = 27 + 9 = 36Anu's age after 5 years = 36 + 5 = 41 Verification / Alternative check: If Anu is 36 now, Raj after 5 years is 27, and 27 + 9 = 36 matches the condition. Raj now is 22, Renu 4 years ago was 15, and 15 + 7 = 22 also matches. So the chain is consistent. Why Other Options Are Wrong: 36 years: this is Anu's present age, not after 5 years.39 years and 46 years: come from incorrect time shifting (adding/subtracting years in the wrong place).34 years: would imply Anu's present age 29, breaking the given relations. Common Pitfalls: Confusing “was 4 years ago” with “will be after 4 years”.Applying the +9 years to Raj's present age instead of Raj's age after 5 years.Stopping at Anu's present age and forgetting to add 5 more years. Final Answer: 41 years

Question 8

In an age ratio relationship, Ravi's age is twice Prakash's age and three times Kumar's age. If Prakash is 24 years old at present, what will be Kumar's age after 4 years?

Accepted Answer

Introduction: This is a standard “multiples of ages” question. When one person's age is given as a multiple of another person's age, you can convert these relations into equations. Since Prakash's age is known, we can directly compute Ravi's age, and then use Ravi's age to compute Kumar's present age. Finally, we add 4 years to answer the “after 4 years” requirement. Given Data / Assumptions: Prakash's present age = 24 years Ravi's age = 2 * (Prakash's age) Ravi's age = 3 * (Kumar's age) We need Kumar's age after 4 years Concept / Approach: Use the chain: Prakash -> Ravi -> Kumar. Convert multiples into direct arithmetic and then shift time by adding 4 years to Kumar's present age. Step-by-Step Solution: Ravi's present age = 2 * 24 = 48Since Ravi = 3 * Kumar, Kumar's present age = 48 / 3 = 16Kumar's age after 4 years = 16 + 4 = 20 Verification / Alternative check: Check the multiples: If Kumar is 16, then 3 * 16 = 48 which equals Ravi's age. Also Ravi is twice Prakash: 2 * 24 = 48. Both relations hold, so the base ages are correct. Why Other Options Are Wrong: 16 years: this is Kumar's present age, not after 4 years.18 years: implies adding only 2 years instead of 4.24 years: confuses Kumar's age with Prakash's age.22 years: would require Kumar present to be 18, which breaks Ravi = 3 * Kumar. Common Pitfalls: Forgetting to add 4 years at the end.Dividing by the wrong multiple (using 2 instead of 3).Mixing up which person is a multiple of whom. Final Answer: 20 years

Question 9

The average age of Sushanth, Krish, and Rishi is 43 years. The average age of Sushanth, Rishi, and Rohit is 49 years. If Rohit is 54 years old, what is Krish's present age (in years)?

Accepted Answer

Introduction: This problem tests comparing two averages where two people are common in both groups. When two averages are given, convert each average into a total sum. Because Sushanth and Rishi appear in both groups, subtracting the totals removes the common part and isolates the third person's age difference. Then, using Rohit's known age, we can find Krish's age directly. Given Data / Assumptions: Average(Sushanth, Krish, Rishi) = 43 Average(Sushanth, Rishi, Rohit) = 49 Rohit = 54 All ages are present ages in years Concept / Approach: Total of 3 persons = average * 3. Use the two totals to find (Sushanth + Rishi) and then substitute into the first total to get Krish. Step-by-Step Solution: Sushanth + Krish + Rishi = 43 * 3 = 129Sushanth + Rishi + Rohit = 49 * 3 = 147Sushanth + Rishi = 147 - 54 = 93Krish = 129 - 93 = 36 Verification / Alternative check: If Krish is 36, then Sushanth + Rishi = 93 makes the first group total 93 + 36 = 129, average 129/3 = 43 which matches. Second group total is 93 + 54 = 147, average 147/3 = 49 which also matches. Why Other Options Are Wrong: 34 years and 26 years: come from subtracting averages directly instead of totals.24 years: ignores Rohit's value in the second total.38 years: would change the first average away from 43. Common Pitfalls: Subtracting 49 - 43 and treating it as a direct age difference.Forgetting to multiply the averages by 3.Assuming Rohit is part of the first group by mistake. Final Answer: 36 years

Question 10

Five years ago, Sushma was 5 times as old as her son. Five years from now, Sushma's age will be 8 years less than three times her son's age at that time. What are the present ages of Sushma and her son (in years)?

Accepted Answer

Introduction: This is a classic ages equation problem involving “years ago” and “years hence” with multiplicative relations. The key is to define variables for present ages, then carefully shift them backward or forward in time based on the statement. Once both conditions are converted into equations, solve them like a simple linear system to get present ages. Given Data / Assumptions: Let Sushma's present age = M years Let son's present age = S years 5 years ago: M - 5 = 5 * (S - 5) 5 years hence: M + 5 = 3 * (S + 5) - 8 Concept / Approach: Convert time-shift statements into equations. Solve the two equations to find S and M. Ensure ages remain positive and logically consistent. Step-by-Step Solution: From 5 years ago: M - 5 = 5(S - 5)M - 5 = 5S - 25M = 5S - 20From 5 years hence: M + 5 = 3(S + 5) - 8M + 5 = 3S + 15 - 8 = 3S + 7M = 3S + 2Equate: 5S - 20 = 3S + 22S = 22 => S = 11M = 3*11 + 2 = 35 Verification / Alternative check: 5 years ago: M-5 = 30 and S-5 = 6, and 5*6 = 30, correct. 5 years hence: M+5 = 40 and S+5 = 16, 3*16 - 8 = 48 - 8 = 40, correct. Why Other Options Are Wrong: 33 and 15: fails the “5 times” condition 5 years ago.48 and 24: gives huge ages and does not satisfy the -8 adjustment.24 and 13: makes mother too young to have a son of 13.40 and 12: does not satisfy both equations simultaneously. Common Pitfalls: Applying the multiplier to the present age instead of the age 5 years ago.Forgetting that “8 less than” means subtract 8 after multiplying.Mixing up “years hence” and “years ago” signs. Final Answer: 35 years and 11 years

Question 11

The average age of Sumit, Krishna, and Rishabh is 43 years. The average age of Sumit, Rishabh, and Rohit is 49 years. If Rohit is 54 years old, what is Krishna's present age (in years)?

Accepted Answer

Introduction: This is an averages comparison problem with overlapping members. Two different 3-person averages are given, where two people (Sumit and Rishabh) are common to both groups. By converting averages into total sums, we can isolate the sum of the common members. Since Rohit's age is known, we can compute (Sumit + Rishabh) and then use it to find Krishna's age from the first group's total. Given Data / Assumptions: Average(Sumit, Krishna, Rishabh) = 43 Average(Sumit, Rishabh, Rohit) = 49 Rohit = 54 All ages are present ages in years Concept / Approach: Total for 3 persons = average * 3. Compute totals, remove Rohit to get common sum, then subtract from the first total to get Krishna. Step-by-Step Solution: Sumit + Krishna + Rishabh = 43 * 3 = 129Sumit + Rishabh + Rohit = 49 * 3 = 147Sumit + Rishabh = 147 - 54 = 93Krishna = 129 - 93 = 36 Verification / Alternative check: Common sum 93 and Krishna 36 gives total 129 and average 43. Common sum 93 and Rohit 54 gives total 147 and average 49. Both match the question, so the result is correct. Why Other Options Are Wrong: 48 years and 42 years: do not satisfy the first group average when combined with common sum 93.40 years: comes from using 49 - 43 as a direct difference without totals.34 years: would reduce the first group total below 129. Common Pitfalls: Subtracting averages instead of converting to sums.Forgetting to multiply by 3 for totals.Accidentally solving for Sumit or Rishabh instead of Krishna. Final Answer: 36 years

Question 12

A person says: (1) His son is 5 times as old as his daughter. (2) His wife is 5 times as old as his son. (3) He is twice as old as his wife. The sum of their ages (father + wife + son + daughter) equals 81 years, which is the grandmother's age today…

Accepted Answer

Introduction: This question tests forming and simplifying age relations using multiples, then using a total sum to find actual ages. The key observation is that all ages are linked to the daughter via multiplication. Once everything is written in terms of one variable, the total becomes a single equation. Solving that equation gives all ages immediately, and the son's age is one of them. Given Data / Assumptions: Let daughter's age = D years Son is 5 times daughter: Son = 5D Wife is 5 times son: Wife = 5 * (5D) = 25D Father is twice wife: Father = 2 * 25D = 50D Total of four ages = 81 years Concept / Approach: Express each person's age as a multiple of D, add them, and equate to 81. Then solve for D and compute the son's age = 5D. Step-by-Step Solution: Daughter = DSon = 5DWife = 25DFather = 50DTotal = D + 5D + 25D + 50D = 81DGiven total is 81: 81D = 81So D = 1Son = 5D = 5 * 1 = 5 Verification / Alternative check: With D=1, the ages become: daughter 1, son 5, wife 25, father 50. Their sum is 1 + 5 + 25 + 50 = 81, matching the given total exactly. Why Other Options Are Wrong: 9, 12, 15, 18: would force daughter to be non-integer or break the fixed multiple chain, and the sum would not stay at 81 under the exact 1:5:25:50 structure. Common Pitfalls: Misreading “5 times as old” and writing addition instead of multiplication.Forgetting to include all four people in the total sum.Making “wife is 5 times older” incorrectly as wife = son + 5*son (not needed here). Final Answer: 5 years

Question 13

Eighteen years ago, a mother was 3 times as old as her daughter. At present, the mother is 2 times as old as her daughter. What is the sum of their present ages (mother + daughter) in years?

Accepted Answer

Introduction: This is a standard “mother-daughter ages” question involving two time-based ratio conditions. The main skill tested is forming equations correctly for “years ago” and “at present.” The best method is to use variables for present ages, write both conditions as equations, and solve. Finally, the question asks for the sum of present ages, not the individual ages alone. Given Data / Assumptions: Let mother's present age = M years Let daughter's present age = D years 18 years ago: M - 18 = 3 * (D - 18) Now: M = 2D We need M + D Concept / Approach: Use the present ratio equation M = 2D, substitute into the “18 years ago” equation, solve for D, then compute M and the sum M + D. Step-by-Step Solution: From present condition: M = 2DUse 18 years ago condition: M - 18 = 3(D - 18)Substitute M = 2D: 2D - 18 = 3D - 54Move terms: -18 + 54 = 3D - 2D36 = DSo M = 2D = 72Sum = M + D = 72 + 36 = 108 Verification / Alternative check: 18 years ago: mother 72-18 = 54, daughter 36-18 = 18. 54 is 3 times 18, correct. Present: 72 is 2 times 36, correct. So the solution is consistent. Why Other Options Are Wrong: 84 and 96: result from incorrect subtraction of 18 or mixing the ratio conditions.116: typically comes from adding wrong ages or using 3 times relation in the present instead of 18 years ago.100: does not satisfy both equations together. Common Pitfalls: Writing M - 18 = 3D instead of 3(D - 18).Assuming the same ratio applies at present and in the past.Solving for individual ages but forgetting the question asks for the sum. Final Answer: 108 years

Question 14

Data sufficiency (Problems on Ages): What is Teja's present age? Statements: A) Four years ago, Raju was as old as Teja is at present. B) Sita's present age is 2 times Raju's present age. C) The average of Teja's and Sita's present ages is 19 years.…

Accepted Answer

Introduction: This is a data sufficiency question in ages. The goal is not to compute Teja's age directly from one equation, but to decide which statements provide enough independent information to solve for Teja uniquely. The safest strategy is to define variables for present ages and test each statement (or combination) to see whether it produces a unique value for Teja. Given Data / Assumptions: Let Teja's present age = T Let Raju's present age = R Let Sita's present age = S A: Four years ago, Raju was as old as Teja is now B: S = 2R C: (T + S) / 2 = 19 Concept / Approach: Translate each statement into equations and check if T can be uniquely determined. If not, add more statements until T becomes uniquely solvable. Step-by-Step Solution: From A: R - 4 = T, so R = T + 4From B: S = 2RFrom C: (T + S) / 2 = 19 => T + S = 38Use A and B and C together:R = T + 4 => S = 2(T + 4) = 2T + 8T + S = 38 => T + (2T + 8) = 383T = 30 => T = 10 (unique) Verification / Alternative check: Now test sufficiency of smaller sets: A + C gives R = T + 4 and T + S = 38, but no link between R and S, so infinite solutions. B + C gives S = 2R and T + S = 38, but no link between T and R, so infinite solutions. A + B gives R = T + 4 and S = 2R, but no numeric total, so not unique. Hence all three are required. Why Other Options Are Wrong: “Only C”: gives T + S = 38 but cannot separate T.“A and C only”: still leaves S unknown.“A and B only”: gives relations but no fixed number.“Cannot be determined”: false because A+B+C gives T = 10 uniquely. Common Pitfalls: Assuming an average equation alone can give Teja's age.Forgetting that data sufficiency asks about necessity, not just calculation.Not testing smaller combinations before concluding. Final Answer: All three statements A, B, and C are necessary to determine the answer

Question 15

Four years ago, the ratio of (1/2 of Anita's age at that time) to (4 times Bablu's age at that time) was 5:12. Eight years from now, half of Anita's age will be 2 years less than Bablu's age at that time. What is Bablu's present age (in years)?

Accepted Answer

Introduction: This ages question combines a past ratio condition with a future comparison involving “half of age.” It tests careful time shifting (4 years ago and 8 years hence) and forming equations from ratios. The best method is to let present ages be variables, translate both conditions into equations, and solve systematically. Given Data / Assumptions: Let Anita's present age = A Let Bablu's present age = B 4 years ago: ratio of (0.5 of Anita's age) to (4 times Bablu's age) was 5:12 8 years hence: 0.5(Anita's age) is 2 years less than Bablu's age Concept / Approach: Convert the ratio into an equation by cross-multiplication. Convert the future “less than by 2 years” into a direct equality. Solve the two equations to get B. Step-by-Step Solution: Ages 4 years ago: Anita = A - 4, Bablu = B - 4Ratio condition: (0.5(A - 4)) : (4(B - 4)) = 5 : 12So (0.5(A - 4)) / (4(B - 4)) = 5/12(A - 4) / (8(B - 4)) = 5/12Cross-multiply: 12(A - 4) = 40(B - 4)Simplify: 3(A - 4) = 10(B - 4) => 3A = 10B - 28Future condition (8 years hence): 0.5(A + 8) is 2 less than (B + 8)0.5(A + 8) = (B + 8) - 2 => 0.5A + 4 = B + 6So 0.5A = B + 2 => A = 2B + 4Substitute: 3(2B + 4) = 10B - 286B + 12 = 10B - 28 => 40 = 4B => B = 10 Verification / Alternative check: If B=10, then A=2B+4=24. Check 4 years ago: Anita 20, Bablu 6. 0.5*20=10 and 4*6=24, ratio 10:24 = 5:12 correct. Check 8 years hence: Anita 32, half is 16; Bablu 18; 16 is 2 less than 18, correct. Why Other Options Are Wrong: 9, 12, 15, 24: do not satisfy both the past ratio equation and the future half-age difference simultaneously. Common Pitfalls: Forgetting the multiplier 4 on Bablu's age in the ratio.Misreading “less than by 2” and adding 2 instead of subtracting.Applying the 4-year shift to only one person. Final Answer: 10 years

Question 16

The ratio of the present ages of Ramita and Satyajit is 9:7. After 9 years, the ratio of their ages will become 5:4. What is Ramita's present age (in years)?

Accepted Answer

Introduction: This is a ratio-based ages problem where the ratio changes after a fixed number of years. The concept is that if present ages are in the ratio 9:7, we can represent them as 9x and 7x. After 9 years, we add 9 to both and use the new ratio 5:4 to solve for x. Once x is found, Ramita's present age is 9x. Given Data / Assumptions: Present ratio Ramita : Satyajit = 9 : 7 After 9 years, ratio becomes 5 : 4 Let Ramita = 9x and Satyajit = 7x at present Concept / Approach: Represent ages using a common factor x. Use the future ratio equation: (9x + 9)/(7x + 9) = 5/4 and solve for x. Step-by-Step Solution: Let Ramita = 9x and Satyajit = 7xAfter 9 years: Ramita = 9x + 9, Satyajit = 7x + 9Given (9x + 9)/(7x + 9) = 5/4Cross-multiply: 4(9x + 9) = 5(7x + 9)36x + 36 = 35x + 45x = 9Ramita's present age = 9x = 9 * 9 = 81 Verification / Alternative check: If Ramita is 81 and Satyajit is 63, the present ratio is 81:63 = 9:7. After 9 years, they become 90 and 72, ratio 90:72 = 5:4. Both match, so 81 is correct. Why Other Options Are Wrong: 56 and 63: are based on the initial ratio but do not satisfy the second ratio after adding 9 years.80: close but breaks exact ratio conversion.72: would imply x=8, which fails the future ratio equation. Common Pitfalls: Adding 9 years to only one person.Treating 9:7 as difference rather than a ratio multiplier.Solving for x but forgetting to multiply by 9 to get Ramita's age. Final Answer: 81 years

Question 17

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

Accepted Answer

Introduction: This is a simple algebraic ages comparison question. It checks whether you can translate a sentence about sums into an equation and simplify it. The important insight is that B appears in both sums, so it cancels out when we compare them. After cancellation, the relationship between A and C becomes direct and the difference can be read immediately. Given Data / Assumptions: (A + B) is 12 more than (B + C) All values represent present ages in years We need how many years C is younger than A Concept / Approach: Convert the statement into an equation, cancel common terms, and interpret the remaining difference as an age gap. Step-by-Step Solution: Translate: A + B = (B + C) + 12A + B = B + C + 12Subtract B from both sides: A = C + 12So A is 12 years older than CTherefore, C is 12 years younger than A Verification / Alternative check: If A = C + 12, then A + B = (C + 12) + B = (B + C) + 12, which matches the given statement exactly. So the reasoning is consistent. Why Other Options Are Wrong: 13, 14, 15: would require the difference between A and C to be different from 12, contradicting the simplified equation A = C + 12.11: similarly contradicts the exact “12 more” condition in the question. Common Pitfalls: Trying to assign random numbers to A, B, and C instead of simplifying algebraically.Forgetting that B cancels because it appears on both sides.Answering “A is older than C” but not giving the numeric difference. Final Answer: 12 years

Question 18

The average age of Vishal and Raghavendra is 24 years. When a new person named Veerender is included with them, the average age of the three persons increases by 1.5 years. What is Veerender's age (in years)?

Accepted Answer

Introduction: This problem tests how averages change when a new value is added to a group. The standard technique is to convert averages into totals. First, find the total age of Vishal and Raghavendra using their average. Then, compute the new total age of three people using the increased average. The difference between the new total and the old total gives Veerender's age directly. Given Data / Assumptions: Average of Vishal and Raghavendra = 24 years Number of persons initially = 2 After including Veerender, number of persons = 3 New average increases by 1.5 years, so new average = 24 + 1.5 = 25.5 years Concept / Approach: Total = average * count. Veerender's age = (new total of 3) - (old total of 2). Step-by-Step Solution: Old total age (2 persons) = 24 * 2 = 48New average (3 persons) = 25.5New total age (3 persons) = 25.5 * 3 = 76.5Veerender's age = 76.5 - 48 = 28.5 Verification / Alternative check: If Veerender is 28.5, then total is 48 + 28.5 = 76.5. Divide by 3 gives 25.5, which is exactly 1.5 more than 24. So the result checks out. Why Other Options Are Wrong: 27 years and 26.5 years: would make the new average less than 25.5.29.5 years and 30.5 years: would push the new average higher than 25.5. Common Pitfalls: Adding 1.5 to the total directly instead of to the average.Forgetting to multiply averages by the number of people.Using 2 people instead of 3 while computing the new total. Final Answer: 28.5 years

Question 19

When Swetha got married to Sudeer, Swetha's age was 3/4 of her husband's age. After 12 years, Swetha's age became 5/6 of her husband's age. What was Swetha's age at the time of marriage (in years)?

Accepted Answer

Introduction: This ages problem involves fractional ratios at two different times: at marriage and 12 years later. The skill tested is converting fraction-based age relations into equations and handling the time shift correctly. Since both Swetha and Sudeer age by the same 12 years, we can set up a clean equation using their marriage-age relationship and the later ratio condition. Given Data / Assumptions: Let Swetha's age at marriage = S Let Sudeer's age at marriage = H At marriage: S = (3/4) * H After 12 years: S + 12 = (5/6) * (H + 12) We need S Concept / Approach: Use the first relation to express H in terms of S, then substitute into the second equation and solve for S. Step-by-Step Solution: From S = (3/4)H, we get H = (4/3)SAfter 12 years condition: S + 12 = (5/6)(H + 12)Substitute H: S + 12 = (5/6)((4/3)S + 12)Multiply both sides by 6: 6S + 72 = 5((4/3)S + 12)Expand: 6S + 72 = (20/3)S + 60Multiply by 3: 18S + 216 = 20S + 180216 - 180 = 20S - 18S36 = 2S => S = 18 Verification / Alternative check: If S=18, then H=(4/3)*18=24. After 12 years: Swetha 30, Sudeer 36, and 30/36 = 5/6. At marriage: 18/24 = 3/4. Both match perfectly. Why Other Options Are Wrong: 17, 21, 23, 20: do not satisfy both fraction ratios simultaneously when 12 years are added to both ages. Common Pitfalls: Applying the 12-year shift to only one person.Using 3/4 and 5/6 as differences instead of ratios.Forgetting to rearrange H in terms of S before substitution. Final Answer: 18 years

Question 20

Sushma joins a job at the age of 20 years. For the first 3 years, her salary is ₹10,000 per annum. After that, her salary increases by ₹2,000 every year for the next 10 years, and then the salary becomes constant until retirement. If her average sal…

Accepted Answer

Introduction: This is an “average over a career” problem combining a fixed salary period, an increment period (arithmetic progression), and then a constant salary period. The main idea is: average salary = (total earnings over all years) / (number of years). So we compute total earnings for the known parts, represent the unknown constant-salary years using a variable, and then solve using the given average salary ₹25,000 per annum. Finally, add the total career years to the starting age (20) to get the retirement age. Given Data / Assumptions: Starting age = 20 years Years 1 to 3 salary = ₹10,000 per annum Next 10 years salary increases by ₹2,000 each year After that, salary is constant (equals salary after increments end) Average salary over entire career = ₹25,000 per annum Concept / Approach: Total earnings = sum of salary each year. Use arithmetic series sum for the 10 increment years. Let total career length be N years, then total earnings = 25,000 * N and solve for N. Step-by-Step Solution: First 3 years total = 3 * 10,000 = 30,000Increment period is 10 years: salaries are 12,000, 14,000, ..., up to 30,000This is an arithmetic series with first term 12,000, last term 30,000, number of terms 10Sum = (first + last) / 2 * terms = (12,000 + 30,000) / 2 * 10Sum = 42,000 / 2 * 10 = 21,000 * 10 = 210,000Total for first 13 years = 30,000 + 210,000 = 240,000After 13 years, salary becomes constant at 30,000 per yearLet total career length = N years, remaining years = N - 13Total earnings = 240,000 + 30,000*(N - 13)Average condition: 240,000 + 30,000*(N - 13) = 25,000*N240,000 + 30,000N - 390,000 = 25,000N30,000N - 150,000 = 25,000N5,000N = 150,000 => N = 30 yearsRetirement age = 20 + 30 = 50 years Verification / Alternative check: If N=30, total earnings should be 25,000*30=750,000. Our earnings: 240,000 + 30,000*(17) = 240,000 + 510,000 = 750,000. Matches exactly. Why Other Options Are Wrong: 42, 45, 47: imply shorter career lengths that would produce a lower average because many early years are below 25,000.52: implies a longer career which would push the average closer to 30,000, not stay at 25,000. Common Pitfalls: Not treating the 10 years as an arithmetic series and summing incorrectly.Using increment for more than 10 years or ending at the wrong last salary.Forgetting that retirement age = start age + total career years. Final Answer: 50 years

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