Question 1

The ratio of the present ages of Anil and Aakash is 4 : 5. Three years later, the ratio of their ages will be 7 : 8. What is the present age, in years, of Anil?

Accepted Answer

Introduction / Context: This problem is a typical age ratio question where the ratio of present ages is given along with the ratio of future ages after a fixed number of years. The task is to determine the actual present age of one person. Such questions test your ability to convert ratio information into algebraic equations and handle time shifts correctly. Given Data / Assumptions: At present, the ages of Anil and Aakash are in the ratio 4 : 5. After 3 years, the ratio of their ages will be 7 : 8. We must find Anil present age. Ages are measured in years and are positive integers. Concept / Approach: We express Anil and Aakash present ages in terms of a common multiple using the given ratio. After 3 years, we add 3 to each expression and impose the second ratio as an equation. This leads to an algebraic equation in the common multiple, which we solve to obtain the actual ages. The method illustrates how ratios and linear equations work together in age problems. Step-by-Step Solution: Step 1: Let Anil present age be 4x years and Aakash present age be 5x years. Step 2: After 3 years, Anil age will be 4x + 3 years. Step 3: After 3 years, Aakash age will be 5x + 3 years. Step 4: At that time, the ratio of their ages is given as 7 : 8. Step 5: Therefore, (4x + 3) / (5x + 3) = 7 / 8. Step 6: Cross multiply to obtain 8 * (4x + 3) = 7 * (5x + 3). Step 7: Expand both sides: 32x + 24 = 35x + 21. Step 8: Rearrange: 32x - 35x = 21 - 24, so -3x = -3. Step 9: Divide by -3 to get x = 1. Step 10: Therefore, Anil present age is 4x = 4 * 1 = 4 years. Verification / Alternative check: With x = 1, Aakash present age is 5 years. After 3 years, their ages will be 4 + 3 = 7 years for Anil and 5 + 3 = 8 years for Aakash. The ratio 7 : 8 exactly matches the condition given in the problem. This confirms that our calculated ages are consistent with both the present and future ratios, and thus the answer is correct. Why Other Options Are Wrong: Option A (8 years) would lead to a present ratio that is inconsistent with the ratio 4 : 5 when Aakash age is determined. Option B (6 years) would not produce the correct ratio of 7 : 8 after 3 years when matched with a suitable age for Aakash. Option D (10 years) similarly fails to satisfy both ratio conditions simultaneously, so it cannot be the correct answer. Common Pitfalls: One common mistake is to apply the ratio 7 : 8 directly to present ages instead of to ages after 3 years. Another error is to forget to add 3 to both ages when forming the future ratio equation. It is also easy to commit algebraic mistakes while cross multiplying and simplifying. Careful step by step writing of expressions is the safest way to avoid such issues. Final Answer: The present age of Anil is 4 years.

Question 2

The ratio of the present ages of P and Q is 9 : 4, and the difference between their present ages is 20 years. What will be the sum of their ages, in years, after 10 years from now?

Accepted Answer

Introduction / Context: This problem is a straightforward application of age ratios and differences, followed by a simple time shift. It is commonly used in aptitude tests to check basic comfort with ratio and proportion and with the idea that all people age by the same number of years over any given interval. Given Data / Assumptions: The present age ratio of P and Q is 9 : 4. The difference between their present ages is 20 years. We must find the sum of their ages after 10 years. Ages are in years and positive. Concept / Approach: We first express the present ages of P and Q as 9x and 4x respectively, using the given ratio. The difference between these expressions is then set equal to 20 years, which allows us to solve for x. Once we know x, we can compute the actual present ages of P and Q. To find the sum of their ages after 10 years, we add 10 years to each age and then add them together. This method uses the ideas of ratio based representation and constant age differences over time. Step-by-Step Solution: Step 1: Let the present age of P be 9x years and the present age of Q be 4x years. Step 2: The difference between their present ages is 9x - 4x = 5x. Step 3: According to the question, this difference is 20 years, so 5x = 20. Step 4: Divide both sides by 5 to obtain x = 4. Step 5: Therefore, P present age is 9x = 9 * 4 = 36 years. Step 6: Q present age is 4x = 4 * 4 = 16 years. Step 7: After 10 years, P age will be 36 + 10 = 46 years and Q age will be 16 + 10 = 26 years. Step 8: The sum of their ages after 10 years is 46 + 26 = 72 years. Step 9: Hence, the required sum is 72 years. Verification / Alternative check: We can verify by checking the ratio and difference at the present time. With ages 36 and 16, the ratio 36 : 16 simplifies by dividing both numbers by 4 to get 9 : 4, which matches the given ratio. The difference 36 - 16 equals 20 years, which also matches the statement. Since both conditions hold at the present, and adding the same 10 years to each age does not change the difference, the computed future sum of 72 years is reliable. Why Other Options Are Wrong: Option A (62 years) would correspond to significantly smaller ages after 10 years and does not match the given ratio and difference when traced back to the present. Option B (66 years) also fails to align with the correct present age ratio and difference when checked. Option D (76 years) is too large for the sum after 10 years and would imply present ages that do not satisfy both the 9 : 4 ratio and the 20 year difference. Common Pitfalls: A frequent mistake is to add 10 years to the age difference as well, but the difference between two ages remains constant when the same number is added to both. Another pitfall is incorrectly setting up the equation for the difference, such as writing 9x + 4x instead of 9x - 4x. Keeping a clear picture of ratio based representation and the concept of age difference helps avoid these errors. Final Answer: The sum of their ages after 10 years will be 72 years.

Question 3

In an aptitude problem on ages, the average age of Ram and Rahim is 18 years, the average age of Rahim and Ramesh is 25 years, and the average age of Ram and Ramesh is 29 years. What is the age (in years) of the eldest among these three friends?

Accepted Answer

Introduction / Context: This aptitude question on ages involves three people whose pairwise average ages are given. We are asked to determine the age of the eldest among them. Such questions test your ability to convert average information into total ages and then solve a simple system of linear equations. Given Data / Assumptions: - The average age of Ram and Rahim is 18 years. - The average age of Rahim and Ramesh is 25 years. - The average age of Ram and Ramesh is 29 years. - All ages are in whole years and positive. Concept / Approach: Average age is defined as total age divided by the number of persons. For each pair, the average allows us to compute the sum of their ages. With three unknown ages and three equations coming from these sums, we can solve the system to obtain the individual ages of Ram, Rahim, and Ramesh. Finally, we identify the largest value to find the eldest person's age. Step-by-Step Solution: Step 1: Let the present ages of Ram, Rahim and Ramesh be R1, R2 and R3 years respectively. Step 2: From the first average, (R1 + R2) / 2 = 18 ⇒ R1 + R2 = 36. Step 3: From the second average, (R2 + R3) / 2 = 25 ⇒ R2 + R3 = 50. Step 4: From the third average, (R1 + R3) / 2 = 29 ⇒ R1 + R3 = 58. Step 5: Solve these equations. Subtract the first from the third: (R1 + R3) − (R1 + R2) = 58 − 36 ⇒ R3 − R2 = 22. Step 6: Use R2 + R3 = 50 and R3 − R2 = 22. Adding them gives 2R3 = 72 ⇒ R3 = 36 years. Step 7: Substitute back: R2 + 36 = 50 ⇒ R2 = 14 years. Step 8: From R1 + R2 = 36 ⇒ R1 + 14 = 36 ⇒ R1 = 22 years. Verification / Alternative check: Check each average: Ram and Rahim = (22 + 14) / 2 = 36 / 2 = 18 years, correct. Rahim and Ramesh = (14 + 36) / 2 = 50 / 2 = 25 years, correct. Ram and Ramesh = (22 + 36) / 2 = 58 / 2 = 29 years, correct. All the given conditions are satisfied by the ages 22, 14 and 36 years. Why Other Options Are Wrong: Ages such as 14 years, 22 years, 28 years or 40 years cannot be the eldest age because the system of equations forces Ramesh to be 36 years old. No other value is consistent with all three pairwise averages at the same time. Common Pitfalls: Students sometimes treat the given averages as individual ages or forget to convert them into sums correctly. Another error is attempting to guess and check values rather than solving systematically. Always remember that average × number of persons = total age, and use that to set up clean algebraic equations. Final Answer: The eldest among the three is 36 years old.

Question 4

The sum of the present ages of Rohit and Ankit is 78 years. If Rohit's present age is twice the present age of Ankit, what is Rohit's age (in years)?

Accepted Answer

Introduction / Context: This is a straightforward aptitude question on ages involving a sum and a simple multiple relationship. We are given the total of the ages of Rohit and Ankit, and we know that Rohit is twice as old as Ankit. Our task is to find Rohit's present age. Given Data / Assumptions: - The sum of the present ages of Rohit and Ankit is 78 years. - Rohit's present age is twice Ankit's present age. - All ages are in years and positive. Concept / Approach: We can solve this problem using simple algebra. By representing Ankit's age with a variable and expressing Rohit's age as a multiple of that variable, we can use the given sum to form and solve a linear equation. This method is common in introductory questions on ratios and ages. Step-by-Step Solution: Step 1: Let Ankit's present age be A years. Step 2: Rohit's present age is twice Ankit's age, so Rohit's age = 2A years. Step 3: The sum of their present ages is given as 78 years, so A + 2A = 78. Step 4: Combine like terms to obtain 3A = 78. Step 5: Divide both sides by 3 to get A = 78 / 3 = 26 years. Step 6: Rohit's present age = 2A = 2 × 26 = 52 years. Verification / Alternative check: We can quickly verify the result. If Ankit is 26 years old and Rohit is 52 years old, then the sum of their ages is 26 + 52 = 78 years, exactly as stated in the question. Also, 52 is twice 26, which matches the requirement that Rohit's age is twice Ankit's age. Therefore, the answer is consistent with all given conditions. Why Other Options Are Wrong: Values like 48 years, 54 years or 60 years would not satisfy both conditions simultaneously. For example, if Rohit were 48 years old, Ankit would have to be 30 years old to sum to 78, and 48 is not twice 30. Similar inconsistencies arise for the other incorrect options. Only 52 years satisfies both the sum and the "twice as old" relationship. Common Pitfalls: Some students may mistakenly divide 78 directly by 2, assuming both persons are the same age, or they might misinterpret the phrase "twice the age" and set up a wrong equation. Carefully defining a variable for one person's age and then expressing the other in terms of that variable helps avoid such errors. Final Answer: Rohit's present age is 52 years.

Question 5

The average age of four children is 12 years. The average age of these four children together with their father is 20 years. What is the father's present age?

Accepted Answer

Introduction / Context: This question checks your understanding of average age and how to move between totals and averages. We know the average age of four children and the average age when their father is included. From this, we must determine the father's present age. Given Data / Assumptions: - There are four children, and their average age is 12 years. - When these four children and their father are taken together (five persons), the average age is 20 years. - All ages are in years and positive. Concept / Approach: Average age is equal to the total of all ages divided by the number of people. Using the average of just the children, we can find the total of their ages. Using the average of the children plus the father, we can find the total age of all five people. The father's age is then the difference between these two totals. Step-by-Step Solution: Step 1: Number of children = 4 and their average age = 12 years. Step 2: Total age of the four children = 4 × 12 = 48 years. Step 3: Number of persons including the father = 4 children + 1 father = 5 persons. Step 4: Average age of these five persons = 20 years. Step 5: Total age of the four children and the father together = 5 × 20 = 100 years. Step 6: Father's present age = total age of group − total age of children = 100 − 48 = 52 years. Verification / Alternative check: We can verify by reconstructing the averages. If the children together sum to 48 years and the father is 52 years old, then the total for all five is 48 + 52 = 100 years. Dividing by 5 gives an average of 20 years, which matches the given information. The children's average remains 48 / 4 = 12 years, also consistent. Hence, the father's age must be 52 years. Why Other Options Are Wrong: If the father were 48, 54, 56 or 62 years old, the total ages and resulting averages would not match both the given average of 12 years for the children and 20 years for the combined group. Only 52 years satisfies both averages simultaneously. Common Pitfalls: A common mistake is to average the averages directly, which is not valid because the number of people differs in each group. Another error is to forget to multiply the averages by the number of people when converting to total age. Always use the relationship total age = average age × number of persons. Final Answer: The father's present age is 52 years.

Question 6

A man is 3 years older than his wife and four times as old as his son. If the son will be 15 years old after 3 years, what is the present age of the wife?

Accepted Answer

Introduction / Context: This aptitude problem connects the ages of a man, his wife and his son through age differences and a future value. We are told that the man is older than his wife and that he is four times as old as his son. The son's future age is given, and we must find the wife's present age. Given Data / Assumptions: - The man is 3 years older than his wife. - The man is four times as old as his son at present. - The son will be 15 years old after 3 years. - All ages are measured in years and are positive. Concept / Approach: First we use the son's future age to determine his present age. Then we use the multiple relationship to compute the man's present age as a multiple of the son's age. Finally, we use the age difference between the man and his wife to determine the wife's present age. The problem involves basic linear reasoning and proportional relationships between ages. Step-by-Step Solution: Step 1: The son will be 15 years old after 3 years. Step 2: Therefore, the son's present age = 15 − 3 = 12 years. Step 3: The man is four times as old as his son. So the man's present age = 4 × 12 = 48 years. Step 4: The man is 3 years older than his wife. Let the wife's present age be W years, so 48 = W + 3. Step 5: Solving for W gives W = 48 − 3 = 45 years. Verification / Alternative check: Check all relationships: The son is now 12 years old and will be 15 after 3 years, matching the condition. The man is 48 years old, which is four times 12, so the multiple relationship is correct. The wife is 45 years old, and the man is 3 years older (48 − 45 = 3), confirming the difference. All statements in the problem are satisfied by these ages. Why Other Options Are Wrong: Options such as 48, 51, 54 or 60 years for the wife do not maintain the required 3-year difference from the man's age of 48 years while also keeping the man four times as old as the son and the son becoming 15 in 3 years. Only 45 years fits every condition simultaneously. Common Pitfalls: Some students mistakenly add 3 instead of subtracting when calculating the wife's age from the man's age. Others may forget to convert the son's future age into his present age before applying the "four times" multiplier. Always move step by step: find present ages first, then apply relationships. Final Answer: The present age of the wife is 45 years.

Question 7

The present age of A is three times the present age of B. What is the ratio of the present age of B to the present age of A?

Accepted Answer

Introduction / Context: This is a basic ratio question on ages. The present age of A is given as three times the present age of B. We are asked to express the ratio of B's age to A's age. This type of question checks your understanding of how to set up and interpret ratios correctly. Given Data / Assumptions: - A's present age is three times B's present age. - We must find the ratio (B's age) : (A's age). - Ages are positive real numbers. Concept / Approach: Rather than working with actual ages, we can represent B's age with a variable and then express A's age as a multiple of that variable. Ratios are then formed by comparing the two expressions. Since the actual numeric values are not needed, we only need the relative relationship between the two ages. Step-by-Step Solution: Step 1: Let B's present age be B years. Step 2: It is given that A's present age is three times B's age, so A's age = 3B years. Step 3: We are asked to find the ratio of the age of B to the age of A, i.e., B : A. Step 4: Substitute the expressions: B : A = B : 3B. Step 5: Divide both terms of the ratio by B (B ≠ 0) to simplify: B / B : 3B / B = 1 : 3. Verification / Alternative check: We can verify with a simple example. Suppose B is 5 years old. Then A would be 3 × 5 = 15 years old. The ratio of B's age to A's age is 5 : 15. Dividing both numbers by 5 yields 1 : 3, confirming our simplified ratio. Any other positive value for B will give the same simplified ratio because the relationship is strictly multiplicative. Why Other Options Are Wrong: Ratios like 1 : 2, 1 : 4, 2 : 3 or 3 : 1 do not correctly represent the statement "A is three times as old as B" when expressed as B : A. For instance, 3 : 1 would correspond to B being three times as old as A, which is the opposite of the given relationship. Common Pitfalls: A very common mistake is to invert the ratio and select 3 : 1 by thinking directly of "three times" without paying attention to the order requested in the question. Always read carefully whether the ratio is A : B or B : A before answering. The first term of the ratio must correspond to the first person mentioned in the question. Final Answer: The ratio of the present age of B to the present age of A is 1 : 3.

Question 8

The present ages of Ratnabali and Shaukat are in the ratio 8 : 5. After 22 years, the ratio of their ages will be 10 : 9. What is Ratnabali's present age?

Accepted Answer

Introduction / Context: This problem involves present and future ratios of the ages of two people, Ratnabali and Shaukat. We are given the ratio of their present ages and the ratio of their ages after 22 years. Using these two pieces of information, we must determine Ratnabali's present age. Given Data / Assumptions: - Present age ratio of Ratnabali : Shaukat = 8 : 5. - After 22 years, the ratio of their ages will be 10 : 9. - All ages are in years and positive. Concept / Approach: When present ages are given in a ratio, we can represent them as multiples of a common variable. We then add the same number of years (22) to each age to reflect the future and set up a new ratio equation. Solving this equation for the variable gives us their actual present ages. This is a standard algebraic method for handling ratio-based age problems. Step-by-Step Solution: Step 1: Let the present ages of Ratnabali and Shaukat be 8x years and 5x years respectively. Step 2: After 22 years, Ratnabali's age will be 8x + 22 years and Shaukat's age will be 5x + 22 years. Step 3: After 22 years, the ratio of their ages is given as (8x + 22) : (5x + 22) = 10 : 9. Step 4: Form the equation (8x + 22) / (5x + 22) = 10 / 9. Step 5: Cross-multiply: 9(8x + 22) = 10(5x + 22). Step 6: Expand both sides: 72x + 198 = 50x + 220. Step 7: Rearrange: 72x − 50x = 220 − 198 ⇒ 22x = 22 ⇒ x = 1. Step 8: Ratnabali's present age = 8x = 8 × 1 = 8 years. Verification / Alternative check: With x = 1, Shaukat's present age is 5 × 1 = 5 years. After 22 years, Ratnabali will be 8 + 22 = 30 years old and Shaukat will be 5 + 22 = 27 years old. Their ages will then be in the ratio 30 : 27, which simplifies to 10 : 9 when both numbers are divided by 3. This matches the given future ratio, confirming that Ratnabali's present age is indeed 8 years. Why Other Options Are Wrong: Values such as 5 years, 11 years, 14 years or 81 years for Ratnabali do not yield a pair of present ages that satisfy both the initial ratio of 8 : 5 and the future ratio of 10 : 9 after 22 years. Only 8 years leads to consistent future ages of 30 and 27 with the required ratio. Common Pitfalls: Students often forget to add the same number of years to both ages when forming the future ratio, or they mistakenly set up the initial ratio using actual numbers instead of multiples of x. Another error is to make arithmetic mistakes when cross-multiplying. Carefully writing down each algebraic step helps avoid these issues. Final Answer: Ratnabali's present age is 8 years.

Question 9

The ratio of the present ages of R and S is 11 : 17. Eleven years ago, the ratio of their ages was 11 : 20. What is R's present age (in years)?

Accepted Answer

Introduction / Context: This question deals with present and past ratios of the ages of two people, R and S. We know their current age ratio and their age ratio eleven years ago. Using these two relations, we must determine R's present age. This is a classic linear equation and ratio problem from aptitude tests on ages. Given Data / Assumptions: - Present age ratio of R : S = 11 : 17. - Eleven years ago, the ratio of their ages was 11 : 20. - All ages are positive and measured in years. Concept / Approach: We represent the present ages of R and S as multiples of a common variable using the present ratio. Then we subtract 11 years from each to obtain their ages eleven years ago and form the second ratio equation. Solving the resulting algebraic equation gives the value of the variable and hence R's present age. Step-by-Step Solution: Step 1: Let the present ages of R and S be 11x years and 17x years respectively. Step 2: Eleven years ago, R's age was 11x − 11 years and S's age was 17x − 11 years. Step 3: The ratio eleven years ago is given as (11x − 11) : (17x − 11) = 11 : 20. Step 4: Form the equation (11x − 11) / (17x − 11) = 11 / 20. Step 5: Cross-multiply: 20(11x − 11) = 11(17x − 11). Step 6: Expand: 220x − 220 = 187x − 121. Step 7: Rearrange to get 220x − 187x = 220 − 121 ⇒ 33x = 99 ⇒ x = 3. Step 8: R's present age = 11x = 11 × 3 = 33 years. Verification / Alternative check: With x = 3, S's present age is 17 × 3 = 51 years. Eleven years ago, R was 33 − 11 = 22 years and S was 51 − 11 = 40 years. The ratio 22 : 40 simplifies to 11 : 20, which matches the given past ratio. The present ratio is 33 : 51, which simplifies to 11 : 17 when divided by 3. Both conditions are satisfied. Why Other Options Are Wrong: If R were 22, 40, 51 or 55 years old at present, the resulting ages and ratios eleven years ago would fail to match both the given present ratio 11 : 17 and the past ratio 11 : 20. Only 33 years leads to consistent ages for both time points. Common Pitfalls: A frequent mistake is to treat the ratio numbers directly as ages or to forget to subtract 11 years from both ages when forming the past ratio. Another error involves incorrect cross-multiplication or simplification of equations. Always maintain the structure: present ages as multiples of x, subtract the time shift, then form and solve the ratio equation. Final Answer: R's present age is 33 years.

Question 10

The average age of 40 students in a class is 16 years. The average age of 24 of these students is 15.5 years, and the average age of another 15 students is 16 2/3 years. What is the age of the 40th student?

Accepted Answer

Introduction / Context: This question involves average ages and missing data for one student. We know the overall average age of 40 students and the average ages of two subgroups within the class. Using this information, we must find the age of the remaining (40th) student. This is a common type of problem in aptitude tests involving weighted averages and totals. Given Data / Assumptions: - Total number of students in the class = 40. - Average age of all 40 students = 16 years. - Average age of 24 students = 15.5 years. - Average age of another 15 students = 16 2/3 years (i.e., 16.666... years or 50/3 years). - We must find the age of the 40th student (the one not included in the two described subgroups). Concept / Approach: Average age is total age divided by number of students. We can compute the total age of all 40 students using the overall average. Then, we calculate the total ages of the two specified subgroups from their averages. Subtracting these known totals from the total for 40 students leaves the age of the remaining student. This method is a direct application of the weighted average concept. Step-by-Step Solution: Step 1: Total age of all 40 students = 40 × 16 = 640 years. Step 2: Total age of the first group of 24 students = 24 × 15.5 = 24 × 15.5 = 372 years. Step 3: Convert 16 2/3 years to an improper fraction: 16 2/3 = 50/3 years. Step 4: Total age of the second group of 15 students = 15 × (50/3) = 250 years. Step 5: Total number of students accounted for in the two groups = 24 + 15 = 39 students. Step 6: Total age of these 39 students = 372 + 250 = 622 years. Step 7: Age of the 40th student = total age of 40 students − total age of 39 students = 640 − 622 = 18 years. Verification / Alternative check: To verify, include the 40th student's age back into the sum. The new total is 622 + 18 = 640 years, and dividing by 40 gives an average age of 16 years, which matches the given data. This confirms that the missing student must be 18 years old for all averages to remain consistent. Why Other Options Are Wrong: If the 40th student's age were 16, 16.5, 17 or 19 years, the overall total age would be different and the overall average for the 40 students would not remain exactly 16 years. Only 18 years preserves the overall average once the two subgroup totals are accounted for. Common Pitfalls: Students may forget to convert the mixed fraction 16 2/3 to a proper fractional or decimal form correctly, or they might accidentally average the two subgroup averages directly instead of using totals. Always convert each average to a total by multiplying by the number of students in that group and then work with the totals to avoid such errors. Final Answer: The age of the 40th student is 18 years.

Question 11

Six years ago, Parvez’s age was equal to the present age of Manish. Parvez’s present age is one-fourth more than Manish's present age. In how many years from now will Parvez’s age become exactly double Manish’s present age?

Accepted Answer

Introduction / Context: This age problem links Parvez and Manish with both a past equality and a current percentage relationship. We are told that Parvez's age six years ago was the same as Manish's present age, and Parvez is now one-fourth more than Manish. We must determine after how many years Parvez's age will be exactly double Manish's present age. This tests your ability to translate word statements into algebraic equations and use both past and future time references correctly. Given Data / Assumptions: - Six years ago, Parvez's age was equal to Manish's present age. - Parvez's present age is one-fourth more than Manish's present age (i.e., Parvez = 1.25 × Manish). - We must find the number of years from now when Parvez's age will be double Manish's present age. - Ages are in whole years. Concept / Approach: We introduce variables for the present ages of Parvez and Manish and use the two given statements to set up simultaneous equations. Solving them gives their current ages. Then, we find the time t such that Parvez's age after t years equals twice Manish's present age. The key is careful handling of time shifts and percentage relations (one-fourth more means 5/4 times, not plus 1/4 year). Step-by-Step Solution: Step 1: Let Manish's present age be M years and Parvez's present age be P years. Step 2: Six years ago, Parvez's age was P − 6 years, and it was equal to Manish's present age, so P − 6 = M. Step 3: Parvez's present age is one-fourth more than Manish's, meaning P = M + (1 / 4)M = (5 / 4)M. Step 4: Substitute P from Step 3 into P − 6 = M, giving (5 / 4)M − 6 = M. Step 5: Rearrange: (5 / 4)M − M = 6 ⇒ (1 / 4)M = 6 ⇒ M = 24 years. Step 6: Parvez's present age P = (5 / 4) × 24 = 30 years. Step 7: We want Parvez's age to become double Manish's present age. Double Manish's present age = 2M = 2 × 24 = 48 years. Step 8: Let t years from now be the required time. Then, Parvez's age after t years will be P + t, and we must have P + t = 48. Step 9: Substitute P = 30: 30 + t = 48 ⇒ t = 48 − 30 = 18 years. Verification / Alternative check: Check the given conditions: Six years ago, Parvez's age was 30 − 6 = 24 years, which equals Manish's present age, 24 years, confirming the first condition. Also, Parvez's present age is 30, which is one-fourth more than 24 because 24 + (1 / 4) × 24 = 24 + 6 = 30. After 18 years, Parvez will be 30 + 18 = 48 years old, exactly double Manish's present age of 24 years. All conditions are satisfied. Why Other Options Are Wrong: Values such as 6 years, 12 years, 15 years, or 20 years would not make Parvez's future age equal to twice Manish's present age when we use the correct present ages of 30 and 24 years. Only 18 years gives 48 years for Parvez, which is exactly double 24. Common Pitfalls: Common mistakes include misinterpreting "one-fourth more" as P = M + 1/4 instead of P = (5 / 4)M, or using Manish's future age rather than his present age when forming the "double" condition. Another typical error is to mis-handle the timeline and forget to subtract or add years consistently. Setting up the equations clearly and checking each condition carefully helps avoid these issues. Final Answer: Parvez's age will become double Manish's present age after 18 years.

Question 12

The ratio of the present ages of Simi and Seema is 5 : 4. After 9 years, the ratio of their ages will be 8 : 7. What is Simi's present age (in years)?

Accepted Answer

Introduction / Context: This is a typical age-ratio problem involving two people, Simi and Seema. We know their present age ratio and the ratio that their ages will have after 9 years. From these two statements, we must determine Simi's present age. The problem tests your skill in forming and solving linear equations from ratio information. Given Data / Assumptions: - Present age ratio of Simi : Seema = 5 : 4. - After 9 years, the ratio of their ages will be 8 : 7. - All ages are in positive whole years. Concept / Approach: We represent their present ages as multiples of a common variable using the current ratio. Then, we add 9 years to each to obtain their ages after 9 years and set up a new ratio equation based on the future ratio. Solving this algebraic equation gives the common factor and thus the actual ages. This approach is standard for problems involving time-shifted ratios. Step-by-Step Solution: Step 1: Let the present ages of Simi and Seema be 5x years and 4x years respectively. Step 2: After 9 years, Simi's age will be 5x + 9 years. Step 3: After 9 years, Seema's age will be 4x + 9 years. Step 4: The ratio of their ages after 9 years is given as (5x + 9) : (4x + 9) = 8 : 7. Step 5: Form the equation (5x + 9) / (4x + 9) = 8 / 7. Step 6: Cross-multiply: 7(5x + 9) = 8(4x + 9). Step 7: Expand both sides: 35x + 63 = 32x + 72. Step 8: Rearrange: 35x − 32x = 72 − 63 ⇒ 3x = 9 ⇒ x = 3. Step 9: Simi's present age = 5x = 5 × 3 = 15 years. Verification / Alternative check: With x = 3, Seema's present age is 4 × 3 = 12 years. After 9 years, Simi will be 15 + 9 = 24 years and Seema will be 12 + 9 = 21 years. The ratio 24 : 21 simplifies to 8 : 7 when both numbers are divided by 3, which matches the given future ratio. Therefore, the calculated present age of Simi, 15 years, is consistent with all data. Why Other Options Are Wrong: Values such as 12, 18, 21 or 24 years for Simi would produce different present ages for Seema and would not yield the correct ratio of 8 : 7 after 9 years. Only 15 years, along with Seema's 12 years, satisfies both the present and future ratio conditions. Common Pitfalls: Students may forget to add the full 9 years to both ages when forming the future ratio, or they might incorrectly assign the ratio 5 : 4 to Seema : Simi instead of Simi : Seema. Another common error is mishandling the cross-multiplication. Writing each step clearly and keeping track of who is older helps avoid these mistakes. Final Answer: Simi's present age is 15 years.

Question 13

The sum of the present ages of Ram and his son is 48 years. If Ram's uncle's age is also included, the average of their ages (Ram, his son and his uncle) becomes 40 years. What is the age (in years) of Ram's uncle?

Accepted Answer

Introduction / Context: This question tests your understanding of averages and how adding a new person affects the average age. We know the combined age of Ram and his son, and we are told what happens to the average when Ram's uncle is included. From this information, we must determine the uncle's present age. Given Data / Assumptions: - The sum of the present ages of Ram and his son is 48 years. - When Ram's uncle is also included, the average age of all three (Ram, his son and his uncle) is 40 years. - We must find the uncle's present age in years. Concept / Approach: Average age is total age divided by number of persons. The total age of Ram and his son is known directly. When we include the uncle and use the new average, we can compute the new total age for the three persons. The uncle's age is then obtained by subtracting the combined age of Ram and his son from this new total. This is a simple application of the formula total = average × number of persons. Step-by-Step Solution: Step 1: Let Ram's present age be R years and his son's present age be S years. Step 2: It is given that R + S = 48 years. Step 3: When Ram's uncle is included, there are 3 persons in total. Step 4: The average age of these three persons is 40 years. Step 5: Therefore, total age of Ram, his son and his uncle together = 3 × 40 = 120 years. Step 6: Let the uncle's present age be U years. Then R + S + U = 120. Step 7: Substitute R + S = 48 to get 48 + U = 120. Step 8: Solve for U: U = 120 − 48 = 72 years. Verification / Alternative check: If the uncle is 72 years old, then the total age of the three is 48 + 72 = 120 years. Dividing 120 by 3 gives an average age of 40 years, which matches the condition in the question. This confirms that the uncle's present age is correctly calculated as 72 years. Why Other Options Are Wrong: Ages such as 34, 48, 50 or 60 years for the uncle would lead to total ages of 82, 96, 98 or 108 years respectively when added to 48, giving averages different from 40 years for 3 persons. Only 72 years produces the exact average of 40 years required by the question. Common Pitfalls: Some students wrongly average the known total of 48 with 40, or attempt to treat 40 as another age instead of an average. Others may forget that the new average applies to 3 people, not 2. Always convert averages into totals using total = average × number of persons before comparing or subtracting ages. Final Answer: Ram's uncle is 72 years old.

Question 14

Chandan is 2 years older than Ankit but 1 year younger than Sumit. Ankit is twice as old as Khushboo. If Khushboo's present age is 25 years, what is the present age (in years) of Sumit?

Accepted Answer

Introduction / Context: This question connects the ages of four people: Chandan, Ankit, Sumit and Khushboo. We are given age differences and a multiple relationship, along with Khushboo's present age. Our goal is to determine Sumit's present age. This problem exercises your ability to chain age relationships step by step. Given Data / Assumptions: - Chandan is 2 years older than Ankit. - Chandan is 1 year younger than Sumit. - Ankit is twice as old as Khushboo. - Khushboo's present age is 25 years. - All ages are in years and positive. Concept / Approach: We start by using Khushboo's known age to determine Ankit's age through the "twice as old" relationship. Then we use the differences to find Chandan's age and finally Sumit's age. The key is to move along the chain of relationships carefully, one step at a time, always substituting known values into the next relation. Step-by-Step Solution: Step 1: Khushboo's present age is given as 25 years. Step 2: Ankit is twice as old as Khushboo. Therefore, Ankit's present age = 2 × 25 = 50 years. Step 3: Chandan is 2 years older than Ankit. So Chandan's present age = 50 + 2 = 52 years. Step 4: Chandan is 1 year younger than Sumit, meaning Sumit is 1 year older than Chandan. Step 5: Sumit's present age = 52 + 1 = 53 years. Verification / Alternative check: We can verify by checking all relations. If Khushboo is 25 years old, Ankit is 50 years old, which is indeed twice 25. Chandan is 52, which is 2 years older than Ankit (50 + 2), and Sumit is 53, which is 1 year older than Chandan (52 + 1). Every relationship exactly matches the statements in the question, confirming that Sumit's present age is 53 years. Why Other Options Are Wrong: Ages such as 50, 51, 52 or 55 years for Sumit contradict at least one of the given relationships. For example, if Sumit were 52, Chandan could not be both 1 year younger than Sumit and 2 years older than Ankit when Ankit is 50. Only 53 years keeps all differences and multiples consistent. Common Pitfalls: Students sometimes mix up "older" and "younger" relationships or forget which person is being compared to which. Another frequent mistake is to apply the "twice as old" relationship to the wrong pair. A useful strategy is to start with the person whose age is known (Khushboo), calculate each subsequent age in sequence, and write a small diagram or list to track each step. Final Answer: Sumit's present age is 53 years.

Question 15

The sum of the present ages of a brother and sister is 21 years. Five years ago, the product of their ages was 28. What are their present ages (brother, sister)?

Accepted Answer

Introduction / Context: This is a classic quadratic-type age problem involving the sum and the product of ages at different times. We are told the present sum of the ages of a brother and sister and the product of their ages five years ago. The question asks us to find their present ages. This problem illustrates how to translate real-world conditions into simultaneous equations and solve them logically. Given Data / Assumptions: - Let the brother's present age be B years and the sister's present age be S years. - The sum of their present ages is B + S = 21 years. - Five years ago, their ages were B − 5 and S − 5 years respectively, and the product of those ages was 28, i.e., (B − 5)(S − 5) = 28. - Ages are positive whole numbers and B, S > 5, since we talk about ages 5 years ago. Concept / Approach: We have two unknowns and two equations: one linear equation (sum of ages) and one quadratic equation (product of ages in the past). We can express one variable in terms of the other using the sum equation and substitute into the product equation, or reason by trying integer pairs that sum to 21 and satisfy the product condition. Both methods ultimately identify the correct pair of ages. Step-by-Step Solution: Step 1: From the sum, B + S = 21. Step 2: Five years ago, their ages were B − 5 and S − 5, and their product was 28, so (B − 5)(S − 5) = 28. Step 3: Expand the product: (B − 5)(S − 5) = BS − 5B − 5S + 25 = 28. Step 4: Rearrange to BS − 5B − 5S + 25 − 28 = 0 ⇒ BS − 5B − 5S − 3 = 0. Step 5: From B + S = 21, we can write S = 21 − B and substitute into the product equation. Step 6: Substitute S: B(21 − B) − 5B − 5(21 − B) − 3 = 0. Step 7: Simplify step by step: 21B − B^2 − 5B − 105 + 5B − 3 = 0. Step 8: Combine like terms: −B^2 + 21B − 105 − 3 = −B^2 + 21B − 108 = 0 ⇒ B^2 − 21B + 108 = 0. Step 9: Factor the quadratic: B^2 − 21B + 108 = (B − 9)(B − 12) = 0. Step 10: So B = 9 or B = 12. Accordingly, S = 21 − B gives (B, S) as (9, 12) or (12, 9). Verification / Alternative check: Taking the pair (9, 12): Five years ago, their ages were 4 and 7. The product 4 × 7 = 28, which matches the given condition. Taking the pair (12, 9) simply swaps the roles of brother and sister. The problem asks for ages in the order (brother, sister), and the option 9 years, 12 years matches one consistent assignment. Both the sum and product conditions hold, so these are the correct present ages. Why Other Options Are Wrong: Pairs like (6, 15), (7, 14), (8, 13) or (10, 11) either do not sum to 21 or, when reduced by 5 years each, do not give a product of 28. For example, (6, 15) sums to 21, but five years ago the ages would be 1 and 10, whose product is only 10, not 28. Similar checks rule out the other pairs. Common Pitfalls: Students sometimes forget that both ages must be reduced by 5 when going back in time, or they misinterpret the product condition as applying to present ages. Others may attempt to solve the quadratic but make factoring errors. A quick integer trial of pairs that sum to 21 can also help: (9, 12) gives past ages (4, 7) with product 28, a strong hint that the algebra is consistent. Final Answer: The present ages of the brother and sister are 9 years and 12 years respectively.

Question 16

James's present age is two-sevenths of his father's present age. James's brother is three years older than James. The ratio of the present ages of James's father and James's brother is 14 : 5. What is James's present age?

Accepted Answer

Introduction / Context: This question connects the ages of James, his father and his brother through fractional and ratio relationships. We are told how James's age compares to his father's, how the brother's age compares to James's, and the ratio between the father's and brother's ages. Using these relationships, we must find James's present age. This problem tests your skill in setting up and solving simultaneous equations involving fractions and ratios. Given Data / Assumptions: - James's present age is two-sevenths of his father's present age. - James's brother is three years older than James. - The ratio of the present ages of James's father and James's brother is 14 : 5. - All ages are in years and positive. Concept / Approach: We introduce a variable for the father's age and express James's age as a fraction of that. Next, we express the brother's age in terms of James's age. Finally, we apply the given ratio between the father and the brother to form an equation in a single variable. Solving this equation yields the father's age, from which we deduce James's age. Step-by-Step Solution: Step 1: Let the father's present age be F years. Step 2: James's present age is given as two-sevenths of F, so James's age J = (2 / 7)F. Step 3: James's brother is three years older than James, so brother's age B = J + 3 = (2 / 7)F + 3. Step 4: The ratio of the present ages of James's father and his brother is F : B = 14 : 5. Step 5: Translate the ratio into an equation: F / B = 14 / 5. Step 6: Substitute B = (2 / 7)F + 3 into the ratio equation: F / ((2 / 7)F + 3) = 14 / 5. Step 7: Cross-multiply: 5F = 14((2 / 7)F + 3). Step 8: Simplify the right-hand side: (2 / 7)F × 14 = 4F, so 14((2 / 7)F + 3) = 4F + 42. Step 9: The equation becomes 5F = 4F + 42 ⇒ F = 42 years. Step 10: James's age J = (2 / 7)F = (2 / 7) × 42 = 12 years. Verification / Alternative check: With F = 42 years and J = 12 years, the brother's age B = 12 + 3 = 15 years. The ratio of father's age to brother's age is 42 : 15, which simplifies to 14 : 5 when both numbers are divided by 3. Also, James's age is indeed two-sevenths of 42 because 42 × 2 / 7 = 12. All given conditions are satisfied by these values. Why Other Options Are Wrong: If James were 13, 14, 15 or 16 years old, the fraction and ratio conditions would not hold simultaneously. For example, if James were 14, then his father's age as 7 / 2 times James's age would be 49, and the ratio to his brother's age would no longer be exactly 14 : 5 when the brother is 3 years older than James. Only 12 years fits every relationship correctly. Common Pitfalls: A common mistake is to interpret "two-sevenths of his father's present age" incorrectly or to forget to include the extra 3 years for the brother when forming the ratio. Others may cross-multiply incorrectly when solving the fraction equation. Keeping the relationships clearly written and simplifying step by step helps avoid such errors. Final Answer: James's present age is 12 years.

Question 17

You are given a question and two statements about the ages of two sisters. Question: What will be the total age of the two sisters 10 years from now? Statements: (1) Lina is 20 years old now. (2) Lina's sister's present age is twice Lina's present a…

Accepted Answer

Introduction / Context: This is a data sufficiency question involving the ages of two sisters. Instead of directly asking for a numerical answer, it asks which combination of the given statements is sufficient to determine the total age of the sisters 10 years from now. Understanding how to interpret and combine such statements is essential for data sufficiency problems in aptitude exams. Given Data / Assumptions: - We need the total of the two sisters' ages 10 years from now. - Statement 1: Lina is 20 years old now. - Statement 2: Lina's sister's present age is twice Lina's present age. - We must decide which statement(s) provide enough information to uniquely determine the total age after 10 years. Concept / Approach: For data sufficiency, we are not primarily focused on computing the exact answer, though we may do so to check. Instead, we determine whether each statement alone, or both together, provide enough information to get a unique numerical result. If the value is uniquely determined, the statements used are sufficient; if multiple values are possible, they are not sufficient. Step-by-Step Solution: Step 1: Using Statement 1 alone: Lina is 20 years old now. We know nothing about her sister's age. Therefore, we cannot find their total age in 10 years because the sister's age is unknown. So, Statement 1 alone is not sufficient. Step 2: Using Statement 2 alone: Lina's sister's age is twice Lina's age. Without knowing Lina's actual age, this only tells us that sister = 2 × Lina, but not any concrete values. Hence, we still cannot determine the total age of both sisters in 10 years. So, Statement 2 alone is also not sufficient. Step 3: Using Statements 1 and 2 together: From Statement 1, Lina's present age is 20 years. From Statement 2, sister's present age = 2 × 20 = 40 years. Step 4: Ten years from now, Lina will be 20 + 10 = 30 years old, and her sister will be 40 + 10 = 50 years old. Step 5: Their total age after 10 years will be 30 + 50 = 80 years, which is uniquely determined. Verification / Alternative check: As soon as both statements are used, there is no ambiguity: Lina is fixed at 20 years, the sister at 40 years, and their future ages after 10 years are fixed at 30 and 50 years respectively. No other values can satisfy both statements if Lina is 20. Therefore, the total age after 10 years is uniquely determined, proving that the combination of Statement 1 and Statement 2 is sufficient. Why Other Options Are Wrong: Option a (Statement 1 alone) is wrong because we still do not know the sister's age. Option b (Statement 2 alone) is wrong because Lina's age itself is unknown. Option c claims that both together are not sufficient, which is false as shown. Option e claims that the statements are insufficient even combined, which is also incorrect since we clearly found a unique total age with both statements. Common Pitfalls: In data sufficiency questions, students sometimes try to compute the final numerical answer when the question only asks which statements are sufficient. Others may overlook the fact that statement 2 alone still leaves Lina's age undefined. The safest approach is to test each statement individually and then in combination, checking whether you can derive a unique numerical result for the quantity asked. Final Answer: Only Statements 1 and 2 together are sufficient to determine the total age of the sisters 10 years from now.

Question 18

The average age of 30 boys in a class is 10 years. If the age of their teacher is also included, the average age of all of them increases by 1 year. What is the teacher's present age (in years)?

Accepted Answer

Introduction / Context: This question involves average age and the effect of adding one more person (the teacher) to the group. We know the average age of 30 boys and the new average when the teacher is included. From this information, we must calculate the teacher's present age. This is a standard problem type in aptitude tests on averages and ages. Given Data / Assumptions: - There are 30 boys in the class. - The average age of these 30 boys is 10 years. - When the teacher's age is included, there are 31 persons, and the average age becomes 11 years. - We are asked to find the teacher's present age. Concept / Approach: Average age is total age divided by number of people. We can first calculate the total age of the 30 boys from their average. Then we calculate the total age of the 30 boys plus the teacher from the new average. The difference between these two totals gives the teacher's age. This uses the formula total age = average age × number of persons. Step-by-Step Solution: Step 1: Total age of the 30 boys = 30 × 10 = 300 years. Step 2: When the teacher is included, the total number of persons is 31. Step 3: The new average age is 11 years. Step 4: Total age of the 30 boys plus the teacher = 31 × 11 = 341 years. Step 5: Let the teacher's age be T years. Then 300 + T = 341. Step 6: Solve for T: T = 341 − 300 = 41 years. Verification / Alternative check: If the teacher is 41 years old, then the total age of all 31 persons is 300 + 41 = 341 years. Dividing this by 31 gives an average age of 11 years, which matches the condition in the question. Therefore, the calculation is consistent and confirms that the teacher's age is 41 years. Why Other Options Are Wrong: If the teacher were 30, 38, 40 or 45 years old, the total age of the group would be 330, 338, 340 or 345 years respectively, and the corresponding averages for 31 persons would not equal 11 years. Only 41 years gives the correct new average. Common Pitfalls: Students sometimes treat 11 years as the teacher's age or mistakenly average the numbers 10 and 11. Others may forget to change the count of persons from 30 to 31 when including the teacher. Always remember to convert averages to totals before adding or subtracting ages in such problems. Final Answer: The teacher's present age is 41 years.

Question 19

Each question contains Quantity I and Quantity II. Read the details and compare the two quantities.  Quantity I: Three years ago, the ratio of the ages of A and B was 3 : 4. Two years from now, the sum of their ages will be 45 years. What is the pre…

Accepted Answer

Introduction / Context: This is a quantitative comparison question involving two different age scenarios. We must compute the present age of A in Quantity I and the present age of P in Quantity II, and then compare these two numerical values. Such questions test your ability to solve age problems and then interpret the relative sizes of derived quantities. Given Data / Assumptions: - Quantity I: Three years ago, the ratio of A's age to B's age was 3 : 4. Two years from now, the sum of their ages will be 45 years. - Quantity II: Five years ago, the ratio of P's age to Q's age was 3 : 4. P's age after 6 years will be equal to Q's present age. - Ages are positive whole numbers. Concept / Approach: We treat each quantity as a separate age problem, form the necessary equations, and solve for the required present ages. Once we have a numerical value for Quantity I and a numerical value for Quantity II, we simply compare them to decide whether Quantity I is greater, smaller or equal to Quantity II. The comparison is then matched with the correct option. Step-by-Step Solution: Step 1 (Quantity I): Let the present ages of A and B be A and B years respectively. Step 2 (Quantity I): Three years ago, their ages were A − 3 and B − 3, and the ratio was (A − 3) : (B − 3) = 3 : 4. Step 3 (Quantity I): Two years from now, their ages will be A + 2 and B + 2, and their sum will be 45: (A + 2) + (B + 2) = 45 ⇒ A + B = 41. Step 4 (Quantity I): From the ratio, (A − 3) / (B − 3) = 3 / 4. Step 5 (Quantity I): Also B = 41 − A. Substitute into the ratio: (A − 3) / (41 − A − 3) = 3 / 4 ⇒ (A − 3) / (38 − A) = 3 / 4. Step 6 (Quantity I): Cross-multiply: 4(A − 3) = 3(38 − A) ⇒ 4A − 12 = 114 − 3A ⇒ 7A = 126 ⇒ A = 18. Step 7 (Quantity I): So Quantity I = 18 years. Step 8 (Quantity II): Let the present ages of P and Q be P and Q years respectively. Step 9 (Quantity II): Five years ago, their ages were P − 5 and Q − 5, and (P − 5) : (Q − 5) = 3 : 4. Step 10 (Quantity II): Also, P's age after 6 years equals Q's present age: P + 6 = Q. Step 11 (Quantity II): Use the ratio equation: (P − 5) / (Q − 5) = 3 / 4 and substitute Q = P + 6. Step 12 (Quantity II): Then (P − 5) / (P + 6 − 5) = 3 / 4 ⇒ (P − 5) / (P + 1) = 3 / 4. Step 13 (Quantity II): Cross-multiply: 4(P − 5) = 3(P + 1) ⇒ 4P − 20 = 3P + 3 ⇒ P = 23. Step 14 (Quantity II): So Quantity II = 23 years. Verification / Alternative check: We now compare Quantity I and Quantity II numerically: Quantity I = 18 and Quantity II = 23. Clearly, 18 is less than 23. This matches the verbal statement that Quantity I is smaller than Quantity II. There is no ambiguity because each quantity is determined uniquely by its respective conditions. Why Other Options Are Wrong: Option a claims Quantity I is greater than Quantity II, which is false since 18 < 23. Option c claims they are equal, which is also false. Option d states both quantities are less than 20, but Quantity II is 23, so this cannot be true. Option e claims both are greater than 25, which is also wrong. Only the statement that Quantity I is smaller than Quantity II matches the computed values. Common Pitfalls: Errors can arise from misinterpreting the time phrases like "three years ago" and "after 6 years", or from setting up the ratios incorrectly. Another common mistake is to try to compare the quantities without fully solving them, which can lead to incorrect assumptions. Always write clear equations for each quantity, solve them completely, and only then perform the comparison. Final Answer: We find Quantity I = 18 and Quantity II = 23, so Quantity I is smaller than Quantity II.

Question 20

Seven years from now, Virat will be twice as old as Mohinder. Five years ago, Mohinder's age was one year less than two-fifths of Virat's age at that time. What is Virat's present age?

Accepted Answer

Introduction / Context: This is a multi-step algebraic age problem involving Virat and Mohinder. We are given a future relationship between their ages and a past relationship where Mohinder's age is tied to a fraction of Virat's age at that past time. We must carefully interpret these time-based conditions to determine Virat's present age. Such problems test both algebraic manipulation and careful reading of age statements involving different time points. Given Data / Assumptions: - Let Virat's present age be V years and Mohinder's present age be M years. - Seven years from now, Virat's age will be twice Mohinder's age at that time: V + 7 = 2(M + 7). - Five years ago, Mohinder's age was one year less than two-fifths of Virat's age at that time: M − 5 = (2 / 5)(V − 5) − 1. - All ages are in whole years. Concept / Approach: We translate each time-based statement into an equation involving V and M. The future condition (seven years from now) gives one linear equation, and the past condition (five years ago) gives another linear equation. Solving these two equations simultaneously yields the present ages of Virat and Mohinder. Finally, we select Virat's age as the required answer. Step-by-Step Solution: Step 1: From the future condition, seven years from now Virat will be twice Mohinder's age at that time: V + 7 = 2(M + 7). Step 2: Expand Step 1: V + 7 = 2M + 14 ⇒ V = 2M + 14 − 7 = 2M + 7. Step 3: From the past condition, five years ago Mohinder's age was one year less than two-fifths of Virat's age at that same time: M − 5 = (2 / 5)(V − 5) − 1. Step 4: Substitute V = 2M + 7 from Step 2 into the past condition. Step 5: Compute V − 5 = (2M + 7) − 5 = 2M + 2. Step 6: Then (2 / 5)(V − 5) = (2 / 5)(2M + 2) = (4M + 4) / 5. Step 7: The past condition becomes M − 5 = (4M + 4) / 5 − 1. Step 8: Simplify the right-hand side: (4M + 4) / 5 − 1 = (4M + 4 − 5) / 5 = (4M − 1) / 5. Step 9: Therefore, M − 5 = (4M − 1) / 5. Step 10: Multiply both sides by 5: 5(M − 5) = 4M − 1 ⇒ 5M − 25 = 4M − 1. Step 11: Rearrange: 5M − 4M = −1 + 25 ⇒ M = 24 years. Step 12: Substitute M = 24 back into V = 2M + 7 to find Virat's present age: V = 2 × 24 + 7 = 48 + 7 = 55 years. Verification / Alternative check: Check both conditions with V = 55 and M = 24. Seven years from now, Virat will be 55 + 7 = 62 years old and Mohinder will be 24 + 7 = 31 years old. Twice Mohinder's age at that time is 2 × 31 = 62, which matches Virat's age, satisfying the first condition. Five years ago, Virat was 55 − 5 = 50 years old and Mohinder was 24 − 5 = 19 years old. Two-fifths of Virat's age at that time is (2 / 5) × 50 = 20. One year less than that is 20 − 1 = 19, which matches Mohinder's age, satisfying the second condition. Both conditions hold exactly. Why Other Options Are Wrong: If Virat's present age were 51, 53, 57 or 60 years, then there would be no corresponding integer age for Mohinder that satisfies both time-based equations simultaneously. For instance, changing Virat's present age to 53 would break either the future doubling condition or the past two-fifths condition when we attempt to solve the equations. Only 55 years fits both constraints. Common Pitfalls: Students often misinterpret the phrase "two-fifths of Virat's age" and may apply it to Virat's present age instead of his age five years ago. Another frequent error is in algebraic manipulation, especially when dealing with fractions and subtracting 1. Carefully identifying the correct time reference for each age and simplifying step by step is crucial to avoid mistakes. Final Answer: Virat's present age is 55 years.

Problems on Ages Questions