Question 1

Sony is one-fifth of her mother's age 15 years ago, and Sony's brother is three-fifths of his mother's age 10 years ago. If the sum of Sony's and her brother's present ages is 31 years, how old is their mother now?

Accepted Answer

Introduction / Context: This age problem involves two children and their mother, with information given about fractional relationships between their present ages and the mother's age at different times in the past. We are also given the sum of the children's present ages, and we must determine the mother's present age. Given Data / Assumptions: - Let the mother's present age be M years. - Fifteen years ago, the mother's age was M − 15 years. - Sony's present age is one-fifth of her mother's age 15 years ago, so Sony's age = (1 / 5) × (M − 15). - Ten years ago, the mother's age was M − 10 years. - Sony's brother's present age is three-fifths of his mother's age 10 years ago, so brother's age = (3 / 5) × (M − 10). - The sum of Sony's and her brother's present ages is 31 years. Concept / Approach: We convert the verbal information into algebraic expressions and form an equation using the given sum of the children's ages. Solving this equation gives the mother's present age. Such questions test your ability to handle time shifts in ages and fractional relationships simultaneously. Step-by-Step Solution: Step 1: Express Sony's present age as S = (1 / 5) × (M − 15). Step 2: Express the brother's present age as B = (3 / 5) × (M − 10). Step 3: The sum of their present ages is given as S + B = 31. Step 4: Substitute S and B into the equation: (1 / 5)(M − 15) + (3 / 5)(M − 10) = 31. Step 5: Multiply both sides by 5 to clear the denominator: (M − 15) + 3(M − 10) = 155. Step 6: Expand and simplify: M − 15 + 3M − 30 = 155 ⇒ 4M − 45 = 155. Step 7: Add 45 to both sides: 4M = 200. Step 8: Divide by 4 to get M = 50 years. Verification / Alternative check: If the mother is 50 years old now, then 15 years ago she was 35, so Sony's age is (1 / 5) × 35 = 7 years. Ten years ago she was 40, so the brother's age is (3 / 5) × 40 = 24 years. Their sum is 7 + 24 = 31 years, which matches the given total. Therefore, the mother's present age of 50 years is correct. Why Other Options Are Wrong: If we try 40, 45, 60, or 70 years for the mother, the resulting values for Sony's and her brother's ages will not add up to 31 years. The equations will fail when checked, so none of those options satisfy all the conditions in the problem. Common Pitfalls: Some students misinterpret the phrases "15 years ago" and "10 years ago" and wrongly link the fractions to the mother's current age instead of her past age. Others forget to convert the word statements into a single combined equation. Always pay attention to which time frame each fraction refers to and write the algebra carefully. Final Answer: Sony's mother is currently 50 years old.

Question 2

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

Accepted Answer

Introduction / Context: This question tests your understanding of weighted averages. We know the average age of the entire class and the average ages of two subgroups of boys. Using this information, we need to find the average age of the remaining boys in the class. Given Data / Assumptions: - Total number of boys in the class = 80. - Overall average age = 15 years. - First subgroup: 15 boys with an average age of 16 years. - Second subgroup: 25 boys with an average age of 14 years. - The remaining boys form the third group whose average age we must find. Concept / Approach: Average age is equal to total age divided by the number of boys. We first compute the total age of all 80 boys using the overall average. Then we calculate the total ages of the first and second subgroups separately. Subtracting these from the total age of the class gives the total age of the remaining boys. Finally, we divide this remaining total by the number of remaining boys to get their average age. Step-by-Step Solution: Step 1: Total age of all 80 boys = 80 × 15 = 1200 years. Step 2: Total age of the first subgroup (15 boys) = 15 × 16 = 240 years. Step 3: Total age of the second subgroup (25 boys) = 25 × 14 = 350 years. Step 4: Total number of boys in the first two groups = 15 + 25 = 40 boys. Step 5: Remaining number of boys = 80 − 40 = 40 boys. Step 6: Total age of the remaining 40 boys = 1200 − (240 + 350) = 1200 − 590 = 610 years. Step 7: Average age of the remaining boys = 610 / 40 = 15.25 years. Verification / Alternative check: We can recheck using a weighted average. Combine the subgroups: first group has total age 240, second group 350, and remaining group 610, giving 240 + 350 + 610 = 1200 years total for 80 boys. Dividing 1200 by 80 again gives 15 years, the original overall average. This confirms that the intermediate calculations, including the 15.25 years for the remaining boys, are consistent. Why Other Options Are Wrong: Values such as 12.24 years, 13.25 years, 14.5 years, or 16 years do not satisfy the total age constraint when combined with the averages of the other two groups. Plugging them back into the total will not restore the overall average of 15 years for the 80 boys, so they must be rejected. Common Pitfalls: A common mistake is to treat the average of the remaining group as the average of the two known subgroup averages, which is incorrect because the group sizes differ. Another error is miscalculating the total ages or forgetting to subtract both subgroups from the class total. Carefully using the formula total = average × number avoids these problems. Final Answer: The average age of the remaining boys in the class is 15.25 years.

Question 3

The present ages of Hema and Chinnu are in the ratio 13 : 17. Four years ago, their ages were in the ratio 11 : 15. What will be the ratio of their ages six years from now?

Accepted Answer

Introduction / Context: This question involves present and past ratios of the ages of two people, Hema and Chinnu. From these ratios, we must determine their current ages and then compute the ratio of their ages six years in the future. It is a good test of working with ratios and linear equations in age-related problems. Given Data / Assumptions: - Present age ratio of Hema : Chinnu = 13 : 17. - Four years ago, the age ratio of Hema : Chinnu was 11 : 15. - Both grow older by the same number of years over time. - We must find the ratio of their ages six years from now. Concept / Approach: We first represent the present ages as multiples of a common variable and then use the ratio from four years ago to form an equation. Solving that equation gives the value of the variable and hence the present ages of Hema and Chinnu. After finding their present ages, we add six years to each and form the required future ratio. Step-by-Step Solution: Step 1: Let the present ages of Hema and Chinnu be 13x and 17x years respectively. Step 2: Four years ago, Hema's age was 13x − 4 and Chinnu's age was 17x − 4. Step 3: The ratio four years ago is given as (13x − 4) : (17x − 4) = 11 : 15. Step 4: Form the equation (13x − 4) / (17x − 4) = 11 / 15. Step 5: Cross-multiply to get 15(13x − 4) = 11(17x − 4). Step 6: Expand both sides: 195x − 60 = 187x − 44. Step 7: Rearrange to obtain 8x = 16, so x = 2. Step 8: Present ages: Hema = 13 × 2 = 26 years, Chinnu = 17 × 2 = 34 years. Step 9: Six years from now, Hema will be 26 + 6 = 32 years and Chinnu will be 34 + 6 = 40 years. Step 10: The future ratio is 32 : 40, which simplifies to 4 : 5. Verification / Alternative check: We can recheck the ratio four years ago using the found values. Four years ago, Hema was 22 years and Chinnu was 30 years old. The ratio 22 : 30 simplifies to 11 : 15 when divided by 2, which matches the given past ratio. This consistency confirms that the present ages and the future ratio 4 : 5 are correct. Why Other Options Are Wrong: Ratios such as 3 : 7, 5 : 6, 7 : 9, and 1 : 3 do not match the ages 32 and 40 that we obtain after 6 years. When we simplify 32 : 40, we only get 4 : 5, so all other options must be rejected. Common Pitfalls: Common errors include using the wrong time frame when applying ratios (for example, mixing the present and four-years-ago ratios incorrectly) or not simplifying the final ratio properly. It is important to carefully translate each statement into equations and to verify the ratios at each stage. Final Answer: The ratio of the ages of Hema and Chinnu six years hence will be 4 : 5.

Question 4

A woman says, "If you reverse the digits of my age, the resulting number represents my husband's age. He is older than me, and the difference between our ages is one-eleventh of their sum." What is the husband's age?

Accepted Answer

Introduction / Context: This puzzle-type age question involves a two-digit number whose digits are reversed to obtain another age. We know that the husband's age is obtained by reversing the woman's age and that the husband is older. In addition, the difference between their ages is one-eleventh of the sum of their ages. We must use this information to find the husband's age. Given Data / Assumptions: - Let the woman's present age be a two-digit number with tens digit a and units digit b. - Therefore, the woman's age can be written as 10a + b. - The husband's age is obtained by reversing the digits, so his age is 10b + a. - The husband is older, which implies b > a. - The difference between their ages is one-eleventh of the sum of their ages. Concept / Approach: We convert the word conditions into algebraic expressions involving a and b. Using the relationship between the difference and the sum of their ages, we form an equation and solve it for a and b. Once we know the digits, we can compute the husband's actual age by forming the reversed number 10b + a. Step-by-Step Solution: Step 1: Woman's age = 10a + b, husband's age = 10b + a. Step 2: Difference between their ages = (10b + a) − (10a + b) = 9(b − a). Step 3: Sum of their ages = (10b + a) + (10a + b) = 11(a + b). Step 4: The problem states that the difference is one-eleventh of the sum, so 9(b − a) = (1 / 11) × 11(a + b) = a + b. Step 5: This simplifies to 9b − 9a = a + b ⇒ 8b = 10a ⇒ 4b = 5a. Step 6: Both a and b are digits from 1 to 9 and b > a. The smallest integer solution for 4b = 5a is a = 4, b = 5. Step 7: Woman's age = 10a + b = 10 × 4 + 5 = 45 years. Step 8: Husband's age = 10b + a = 10 × 5 + 4 = 54 years. Verification / Alternative check: Check the condition using these ages. Sum of the ages = 45 + 54 = 99 years. Difference between the ages = 54 − 45 = 9 years. One-eleventh of the sum is 99 / 11 = 9 years, which equals the difference. The husband is older, and his age is 54 years, satisfying all conditions. Why Other Options Are Wrong: Ages such as 24, 42, 45, and 48 years do not satisfy the specific condition that the difference between the ages is one-eleventh of their sum when combined with a reversed two-digit age for the woman. Only 54 years as the husband's age yields a consistent pair of ages (45 and 54) that meets all requirements. Common Pitfalls: Some students may incorrectly set up the difference or the sum, or forget that both ages must be two-digit numbers with reversed digits. Others may attempt random trial and error without using the equation 4b = 5a, which makes the solution straightforward. Careful algebraic translation of the word problem simplifies the puzzle. Final Answer: The husband's age is 54 years.

Question 5

A is three times as old as B. C was twice as old as A four years ago. In four years' time, A will be 31 years old. What are the present ages of B and C?

Accepted Answer

Introduction / Context: This age problem involves three people: A, B, and C. We know a multiple relationship between the ages of A and B, a past relationship between the ages of C and A, and a future value for A's age. Using these facts, we must determine the current ages of B and C. Given Data / Assumptions: - A is currently three times as old as B. - Four years ago, C was twice as old as A was at that time. - In four years' time, A will be 31 years old. - We must find the present ages of B and C. Concept / Approach: We first determine A's present age using the information about A's age in four years. With A's present age known, we can immediately find B's age using the "three times as old" relationship. Then we use the statement about C being twice as old as A four years ago to calculate C's present age. The problem is solved with simple linear arithmetic rather than complicated equations. Step-by-Step Solution: Step 1: A will be 31 years old in four years. Step 2: Therefore, A's present age = 31 − 4 = 27 years. Step 3: A is three times as old as B now. Let B's age be B years. Step 4: From the relationship, 27 = 3 × B, so B = 27 / 3 = 9 years. Step 5: Four years ago, A's age was 27 − 4 = 23 years. Step 6: At that time, C was twice A's age, so C's age four years ago was 2 × 23 = 46 years. Step 7: Therefore, C's present age = 46 + 4 = 50 years. Verification / Alternative check: Check all conditions: A is 27 and B is 9, so A is indeed three times B's age. Four years ago, A was 23 and C was 46; 46 is twice 23, satisfying the second condition. In four years, A will be 27 + 4 = 31 years, matching the given future age. Thus the present ages are B = 9 years and C = 50 years. Why Other Options Are Wrong: Pairs such as 45 years, 10 years or 10 years, 50 years do not satisfy all the given relationships when checked carefully. In particular, A must be three times B now and C must be exactly twice A's age four years ago. Only the pair 9 years, 50 years satisfies every condition simultaneously. Common Pitfalls: Some students misinterpret the time references, for example using A's age four years from now instead of four years ago when relating to C. Others may incorrectly form equations or forget that age differences remain constant over time. Writing down a timeline and labelling ages at each point in time often helps avoid such confusion. Final Answer: The present ages are B = 9 years and C = 50 years.

Question 6

Ayush was born two years after his father's marriage. Ayush's mother is five years younger than his father but 20 years older than Ayush, who is now 10 years old. At what age did the father get married?

Accepted Answer

Introduction / Context: This problem links the ages of Ayush, his father, and his mother with information about the father's age at the time of marriage. By carefully using the given age differences, we can determine how old the father was when he got married. Given Data / Assumptions: - Ayush is currently 10 years old. - Ayush's mother is 20 years older than Ayush. - Ayush's mother is five years younger than his father. - Ayush was born two years after his father's marriage. - We must find the father's age at the time of marriage. Concept / Approach: We first compute the mother's present age using the information about Ayush's age. Then we use the age difference between the mother and father to find the father's present age. Finally, we use the timing between the father's marriage and Ayush's birth to determine how many years ago the marriage took place and subtract this from the father's present age to get his age at marriage. Step-by-Step Solution: Step 1: Ayush's present age is 10 years. Step 2: The mother is 20 years older than Ayush, so mother's present age = 10 + 20 = 30 years. Step 3: The mother is five years younger than the father. Let the father's present age be F years. Then F − 5 = 30. Step 4: Solving gives F = 30 + 5 = 35 years. Step 5: Ayush was born two years after his father's marriage, and Ayush is now 10 years old. Step 6: Therefore, the marriage took place 10 + 2 = 12 years ago. Step 7: Father's age at the time of marriage = present age − years since marriage = 35 − 12 = 23 years. Verification / Alternative check: If the father is 35 now and was 23 at the time of marriage, then the marriage happened 12 years ago. Since Ayush is 10 years old now, he was born 10 years ago, which is 2 years after the marriage, consistent with the problem statement. The mother's present age is 30, which is both 20 years more than Ayush's age and 5 years less than the father's age. All conditions are satisfied. Why Other Options Are Wrong: Options such as 22, 24, 25, or 26 years lead to inconsistencies when we calculate the father's present age and compare it with the given age differences. They do not simultaneously satisfy the facts that the mother is 5 years younger than the father, 20 years older than Ayush, and that Ayush was born 2 years after the marriage. Common Pitfalls: Students sometimes mix up the directions of the age differences or miscalculate the timing between the marriage and Ayush's birth. Another common error is to forget to add the 2-year gap between the marriage and birth when counting back from the father's present age. Drawing a simple timeline is helpful in avoiding such mistakes. Final Answer: The father got married at the age of 23 years.

Question 7

Mary is 12 years younger than Pinky. After 3 years, the ratio of their ages will be 5 : 8. If Sam's present age is equal to the sum of the present ages of Mary and Pinky, what was Sam's age 15 years ago?

Accepted Answer

Introduction / Context: This question combines age differences, future age ratios, and the sum of ages to determine the past age of a third person, Sam. We are told how Mary and Pinky's ages relate to each other now and in the future, and that Sam's present age equals their combined present ages. From this, we must work backwards to find Sam's age 15 years ago. Given Data / Assumptions: - Mary is 12 years younger than Pinky at present. - After 3 years, the ratio of Mary's age to Pinky's age will be 5 : 8. - Sam's present age is equal to the sum of the present ages of Mary and Pinky. - We must find Sam's age 15 years ago. Concept / Approach: We first express the current ages of Mary and Pinky using a variable, using the age difference and the future ratio to form an equation. Once their ages are found, we add them to get Sam's current age. Finally, we subtract 15 years from Sam's present age to determine how old he was 15 years ago. This problem uses both linear equations and the concept of age sums. Step-by-Step Solution: Step 1: Let Pinky's present age be P years. Then Mary's present age is P − 12 years. Step 2: After 3 years, Mary's age will be (P − 12) + 3 = P − 9 years. Step 3: After 3 years, Pinky's age will be P + 3 years. Step 4: The ratio after 3 years is (P − 9) : (P + 3) = 5 : 8. Step 5: Form the equation (P − 9) / (P + 3) = 5 / 8. Step 6: Cross-multiply: 8(P − 9) = 5(P + 3). Step 7: Expand: 8P − 72 = 5P + 15 ⇒ 3P = 87 ⇒ P = 29 years. Step 8: Mary's present age = P − 12 = 29 − 12 = 17 years. Step 9: Sam's present age is the sum of Mary's and Pinky's present ages: 17 + 29 = 46 years. Step 10: Sam's age 15 years ago = 46 − 15 = 31 years. Verification / Alternative check: Check the future ratio: After 3 years, Mary will be 17 + 3 = 20 years old and Pinky will be 29 + 3 = 32 years old. The ratio 20 : 32 simplifies to 5 : 8, which matches the given condition. Sam's age is correctly set as 46, the sum of 17 and 29. Subtracting 15 gives Sam's age 15 years ago as 31 years, confirming the result. Why Other Options Are Wrong: Values such as 29, 33, 37, or 46 years do not match Sam's age 15 years ago when all relationships are checked. Some of them correspond to Sam's present age or other unrelated quantities, but only 31 years is consistent with all the given conditions. Common Pitfalls: Common mistakes include reversing the age difference (taking Mary older than Pinky), misusing the future ratio, or forgetting to subtract 15 years at the final step. It is important to clearly define the variables and carefully handle the time shift and age difference together. Final Answer: Sam's age 15 years ago was 31 years.

Question 8

Twice the sum of the ages of a father and his son is eight times the son's age. If the average age of the father and the son is 30 years, what is the father's present age?

Accepted Answer

Introduction / Context: This question combines a relationship between the sum of two ages and a multiple of one age with information about their average age. By setting up equations for the father and son's ages, we can solve for the father's present age. Given Data / Assumptions: - Let the father's present age be F years and the son's present age be S years. - Twice the sum of their ages is eight times the son's age: 2(F + S) = 8S. - The average age of the father and son is 30 years, so (F + S) / 2 = 30. - We must find the father's age F. Concept / Approach: We translate the verbal statements into algebraic equations and then solve the system. From the equation involving the average, we can find the sum F + S. Substituting this sum into the first equation allows us to solve for S, and then F is obtained using the sum. This method keeps the calculations straightforward and organized. Step-by-Step Solution: Step 1: From the average condition, (F + S) / 2 = 30. Step 2: Multiply both sides by 2 to get F + S = 60. Step 3: From the first condition, 2(F + S) = 8S. Step 4: Substitute F + S = 60 into 2(F + S) = 8S to get 2 × 60 = 8S. Step 5: This simplifies to 120 = 8S, so S = 120 / 8 = 15 years. Step 6: Use F + S = 60 again to find F = 60 − S = 60 − 15 = 45 years. Verification / Alternative check: Check both conditions with F = 45 and S = 15. The average age is (45 + 15) / 2 = 60 / 2 = 30 years, which matches the given average. The sum of the ages is 60, so twice the sum is 2 × 60 = 120. Eight times the son's age is 8 × 15 = 120, which matches the other condition. Both statements are satisfied, confirming that F = 45 years is correct. Why Other Options Are Wrong: Ages such as 36, 38, 42, or 48 years for the father will not simultaneously satisfy both the average condition and the equation 2(F + S) = 8S when the son's age is calculated. Only 45 years leads to a consistent pair of ages (45 and 15) fulfilling both requirements. Common Pitfalls: Some students may misinterpret "twice the sum" as "the sum is twice" or incorrectly form the relation as F + S = 8S. Others may forget to use the average equation to find F + S. The key is to write each condition carefully and solve the system step by step without skipping algebraic steps. Final Answer: The father's present age is 45 years.

Question 9

The sum of the present ages of a father and his son is 70 years. Five years ago, the age of the father was five times the age of the son. What is the present age of the son in years?

Accepted Answer

Introduction / Context: This is a classic aptitude question from the topic of problems on ages. We are given the sum of the present ages of a father and his son and a condition relating their ages five years ago. Using this information, we need to determine the present age of the son in years. Such questions commonly appear in competitive exams and are solved using simple linear equations based on present, past, and future ages. Given Data / Assumptions: The sum of the present ages of the father and the son is 70 years. Five years ago, the age of the father was five times the age of the son. Ages are assumed to be whole numbers in years. We are asked to find the present age of the son. Concept / Approach: Problems on ages are usually solved by expressing all conditions in terms of present ages and then forming linear equations. When the statement says "five years ago", we subtract 5 from the present age. Ratios or multipliers such as "five times" are then used to relate the ages. Solving the resulting equations gives the required present age. Step-by-Step Solution: Step 1: Let the present age of the son be S years and the present age of the father be F years.Step 2: From the sum condition, F + S = 70.Step 3: Five years ago, the father's age was F - 5 and the son's age was S - 5. At that time, F - 5 = 5 * (S - 5).Step 4: Expand the second equation: F - 5 = 5S - 25, so F = 5S - 20.Step 5: Substitute F = 5S - 20 into F + S = 70 to get (5S - 20) + S = 70.Step 6: Simplify: 6S - 20 = 70, so 6S = 90 and S = 15.Step 7: Therefore, the present age of the son is 15 years. Verification / Alternative check: If the son is 15 years old now, the father is 70 - 15 = 55 years old now. Five years ago, the son was 10 years old and the father was 50 years old. At that time, 50 is exactly 5 times 10, so the given condition is satisfied. This confirms that the solution is consistent and correct. Why Other Options Are Wrong: If the son were 12 years old, the father would be 58 years old, and five years ago their ages would be 7 and 53, which does not satisfy a five times relationship. If the son were 14 years old, the father would be 56 years old, and five years ago they would be 9 and 51, which again is not a five times relation. If the son were 19 years old, the father would be 51 years old, and five years ago the ages would be 14 and 46, which also does not give a five times ratio. Thus, these options are incorrect. Common Pitfalls: A common mistake is to apply the "five times" condition to the present ages instead of the ages five years ago. Another error is to forget that both ages are reduced by 5 when going back five years. Some learners also mis-handle the algebra while substituting one equation into another. Writing the equations carefully and checking the arithmetic step by step helps avoid these issues. Final Answer: The present age of the son is 15 years.

Question 10

The difference between the present ages of Ravali and Swarna is 9 years. After 7 years, the age of Ravali will be twice the age of Swarna. What will be the age of Ravali after 4 years from now?

Accepted Answer

Introduction / Context: This problem on ages involves two people, Ravali and Swarna, with a known difference between their present ages and a future condition relating their ages. Such questions test the ability to convert verbal age relationships into algebraic equations and solve them accurately. They are commonly found in aptitude and competitive exams under the topic problems on ages. Given Data / Assumptions: The difference between the present ages of Ravali and Swarna is 9 years. After 7 years, the age of Ravali will be twice the age of Swarna. We assume ages are measured in whole years. We are required to find Ravali's age after 4 years from now. Concept / Approach: We express both unknown ages using variables and then use the given difference and the future age condition to form equations. The idea is to represent future ages using present ages plus the number of years and then apply the specified relationship, here "twice". Solving the resulting equations gives the present ages, from which we can find the required future age of Ravali. Step-by-Step Solution: Step 1: Let the present age of Ravali be R years and the present age of Swarna be S years.Step 2: The difference condition gives R - S = 9.Step 3: After 7 years, Ravali will be R + 7 years old and Swarna will be S + 7 years old.Step 4: At that time, Ravali's age will be twice Swarna's age, so R + 7 = 2 * (S + 7).Step 5: Expand the equation: R + 7 = 2S + 14, so R = 2S + 7.Step 6: Substitute R = 2S + 7 into R - S = 9 to get (2S + 7) - S = 9.Step 7: Simplify: S + 7 = 9, so S = 2. Then R = S + 9 = 11.Step 8: Ravali's age after 4 years will be R + 4 = 11 + 4 = 15 years. Verification / Alternative check: At present, Ravali is 11 years old and Swarna is 2 years old. The difference is 9 years, which matches the given data. After 7 years, Ravali will be 18 years old and Swarna will be 9 years old. At that time, Ravali's age is exactly twice Swarna's age, since 18 = 2 * 9. Therefore, our solution is consistent, and Ravali will indeed be 15 years old after 4 years. Why Other Options Are Wrong: If Ravali were 16 years old after 4 years, her present age would be 12 years, leading to inconsistent values for Swarna and violating the future twice-age condition. If Ravali were 20 or 21 years old after 4 years, her present age would be 16 or 17 years, respectively, which again cannot satisfy both the age difference of 9 years and the twice relation after 7 years. Hence, these options do not fit the given conditions. Common Pitfalls: Students sometimes reverse the age difference and write S - R = 9 instead of R - S = 9, which leads to negative values or wrong answers. Another common error is to forget that both ages increase by the same amount (7 years) when moving into the future. Careful equation setup based on the statements is essential in problems on ages. Final Answer: Ravali will be 15 years old after 4 years.

Question 11

The present ages of Pranay and Sai are in the ratio 15:17. After 6 years, the ratio of their ages will become 9:10. What will be the age of Pranay after 6 years from now?

Accepted Answer

Introduction / Context: This question involves the present ages of two people, Pranay and Sai, expressed as a ratio. A different ratio is given for their ages after 6 years. Using these two ratios, we need to determine Pranay's age after 6 years. Such questions are typical in aptitude tests under problems on ages and ratios, combining basic algebra with proportional reasoning. Given Data / Assumptions: The present age ratio of Pranay to Sai is 15:17. After 6 years, the ratio of their ages will be 9:10. Ages are assumed to be in whole years. We must find Pranay's age after 6 years. Concept / Approach: When ages are given in a ratio, we can assign a common multiplier to convert the ratio into actual ages. If the present ratio is 15:17, we can write the ages as 15k and 17k. A future ratio condition is then expressed using these same expressions plus the number of years being added. Solving the resulting equation for k gives the actual ages. Step-by-Step Solution: Step 1: Let the present age of Pranay be 15k years and the present age of Sai be 17k years.Step 2: After 6 years, Pranay will be 15k + 6 years old and Sai will be 17k + 6 years old.Step 3: At that time, the ratio of their ages is given as 9:10, so (15k + 6) / (17k + 6) = 9 / 10.Step 4: Cross multiply: 10 * (15k + 6) = 9 * (17k + 6).Step 5: Expand both sides: 150k + 60 = 153k + 54.Step 6: Rearrange to get 150k + 60 - 153k - 54 = 0, which simplifies to -3k + 6 = 0, so 3k = 6 and k = 2.Step 7: Therefore, Pranay's present age is 15k = 30 years, and after 6 years he will be 30 + 6 = 36 years. Verification / Alternative check: With k = 2, Sai's present age is 17 * 2 = 34 years. After 6 years, Pranay will be 36 years old and Sai will be 40 years old. The ratio 36:40 simplifies to 9:10 when both numbers are divided by 4. This matches the ratio given in the question, confirming that the calculation is correct. Why Other Options Are Wrong: If Pranay were 42 years old after 6 years, his present age would be 36 years, and it would be impossible to maintain the required 15:17 current ratio and 9:10 future ratio simultaneously. Similarly, an age of 24 years after 6 years would mean a present age of 18 years, which does not fit these ratios. The option 30 years corresponds to Pranay's present age, not his age after 6 years, so it is not the required answer. Only 36 years satisfies all conditions. Common Pitfalls: Some learners mistakenly set up the ratio with incorrect future ages or forget to add 6 to both ages. Others confuse the present ratio with the future ratio or cancel terms incorrectly when cross multiplying. Always write the ratio equation carefully and solve step by step for the common multiplier k. Final Answer: Pranay will be 36 years old after 6 years.

Question 12

Ten years ago, Karan was three times as old as Shiva. Ten years hence, Karan will be only twice as old as Shiva. Find the present age of Shiva in years.

Accepted Answer

Introduction / Context: This question deals with two people, Karan and Shiva, and gives their age relationship 10 years ago and again 10 years in the future. Problems like this require forming equations from time-shifted age statements and solving for the unknown present ages. They are standard in aptitude exams under problems on ages. Given Data / Assumptions: Ten years ago, Karan was three times as old as Shiva. Ten years hence, Karan will be twice as old as Shiva. We are required to find the present age of Shiva in years, assuming whole number ages. Concept / Approach: We use variables to represent present ages and then translate the past and future conditions into equations. When we go 10 years back, we subtract 10 from the present age; when we go 10 years forward, we add 10. Each statement involving "three times" or "twice" turns into a simple linear equation. Solving the pair of linear equations gives the present ages. Step-by-Step Solution: Step 1: Let the present age of Karan be K years and the present age of Shiva be S years.Step 2: Ten years ago, their ages were K - 10 and S - 10. At that time, Karan was three times as old as Shiva, so K - 10 = 3 * (S - 10).Step 3: Ten years hence, their ages will be K + 10 and S + 10. At that time, Karan will be twice as old as Shiva, so K + 10 = 2 * (S + 10).Step 4: From K - 10 = 3S - 30, we get K = 3S - 20.Step 5: Substitute K = 3S - 20 into K + 10 = 2S + 20 to obtain (3S - 20) + 10 = 2S + 20.Step 6: Simplify: 3S - 10 = 2S + 20, so S = 30.Step 7: Therefore, the present age of Shiva is 30 years. Verification / Alternative check: If Shiva is 30 years old now, then from K = 3S - 20 we get Karan's present age as 3 * 30 - 20 = 70 years. Ten years ago, Shiva was 20 years old and Karan was 60 years old, and 60 is exactly three times 20. Ten years from now, Shiva will be 40 years old and Karan will be 80 years old, and 80 is exactly twice 40. Thus, all conditions are satisfied. Why Other Options Are Wrong: If Shiva were 40 or 60 years old at present, the equations would not be satisfied when checked against both the past and future conditions. The option 70 years corresponds to Karan's present age, not Shiva's, so it is not the required answer. Only 30 years matches all the given relationships. Common Pitfalls: Common mistakes include mixing up who is older, misplacing the time shifts (adding instead of subtracting 10, or vice versa), or forming equations from the wrong time frame. Carefully distinguishing between ages 10 years ago, present ages, and ages 10 years hence is crucial to avoid errors. Final Answer: The present age of Shiva is 30 years.

Question 13

The ratio of the present ages of X and Y is 4:5. Which of the following options cannot represent the ratio of their ages 20 years ago?

Accepted Answer

Introduction / Context: This is a conceptual problem on ages and ratios. We are given the ratio of the present ages of two people, X and Y, and we must determine which ratio among the options is impossible for their ages 20 years ago. The question checks understanding of how age differences remain constant over time while ratios change. Given Data / Assumptions: The present age ratio of X to Y is 4:5. We consider their ages 20 years ago and check which of the given ratios could or could not represent that past ratio. Ages are in years and assumed to be positive. Concept / Approach: Let the present ages be 4k and 5k for some positive number k. Twenty years ago, their ages were 4k - 20 and 5k - 20. For any proposed past ratio a:b to be possible, there must exist a positive k such that (4k - 20) : (5k - 20) = a : b. We can test each option by forming an equation and checking whether a valid positive k exists. Step-by-Step Solution: Step 1: Let present ages be 4k and 5k years.Step 2: Twenty years ago, X was 4k - 20 years and Y was 5k - 20 years.Step 3: Test option 2 : 5. Set (4k - 20) / (5k - 20) = 2 / 5. Cross multiply: 5 * (4k - 20) = 2 * (5k - 20). This gives 20k - 100 = 10k - 40, so 10k = 60 and k = 6 (valid). So 2 : 5 is possible.Step 4: Test option 8 : 15. Set (4k - 20) / (5k - 20) = 8 / 15. Cross multiply: 15 * (4k - 20) = 8 * (5k - 20), so 60k - 300 = 40k - 160, giving 20k = 140 and k = 7 (valid). So 8 : 15 is possible.Step 5: Test option 3 : 5. Set (4k - 20) / (5k - 20) = 3 / 5. Cross multiply: 5 * (4k - 20) = 3 * (5k - 20), so 20k - 100 = 15k - 60, giving 5k = 40 and k = 8 (valid). So 3 : 5 is possible.Step 6: Test option 9 : 10. Set (4k - 20) / (5k - 20) = 9 / 10. Cross multiply: 10 * (4k - 20) = 9 * (5k - 20), so 40k - 200 = 45k - 180, giving -5k = 20 and k = -4, which is not valid because k must be positive. Thus, 9 : 10 is impossible. Verification / Alternative check: For the valid k values, we can compute actual ages and see that they are reasonable. For example, with k = 6, present ages are 24 and 30, and 20 years ago they were 4 and 10, which form the ratio 2 : 5. Similar checks work for k = 7 and k = 8. However, for 9 : 10, the required k is negative, which would make the ages negative and impossible in a real context. So option 9 : 10 cannot represent the ratio 20 years ago. Why Other Options Are Wrong: The options 2 : 5, 8 : 15, and 3 : 5 all correspond to valid positive k values and yield meaningful positive ages 20 years ago, so they can be correct ratios in the past. Only 9 : 10 leads to a negative k, which makes it impossible for real ages. Therefore, 9 : 10 is the only option that cannot be the ratio 20 years ago. Common Pitfalls: Some learners may try to guess by inspection without considering the algebraic relationship between present and past ratios. Others may incorrectly assume that any ratio close to 4:5 must be valid in the past. The key is to remember that difference in ages is fixed, so not every arbitrary ratio is possible for past or future ages. Final Answer: The ratio that cannot represent the ages 20 years ago is 9 : 10.

Question 14

When Ram was born, the age of his father was 32 years more than the age of his brother, and the age of his mother was 25 years more than the age of his sister. Ram's brother is currently 6 years older than Ram, and his mother is 3 years younger than…

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Introduction / Context: This is a family ages puzzle involving Ram, his brother, his sister, and his parents. The relationships among their ages are described at the time of Ram's birth and in the present. The goal is to find the age of Ram's sister at the time Ram was born. Such questions test understanding of constant age differences and careful interpretation of relational statements. Given Data / Assumptions: At the time of Ram's birth, the age of his father was 32 years more than the age of his brother. At the same time, the age of his mother was 25 years more than the age of his sister. At present, Ram's brother is 6 years older than Ram. The mother is 3 years younger than the father at present. We need the age of Ram's sister at the time of Ram's birth. Concept / Approach: Age differences between people do not change over time. Therefore, if Ram's brother is 6 years older than Ram now, he was also 6 years older at the time of Ram's birth. Similarly, the difference between the ages of the father and the mother is constant. By working backward to the time of Ram's birth and using the given differences, we can compute the sister's age at that time. Step-by-Step Solution: Step 1: At the time of Ram's birth, Ram's age was 0 years. His brother was older by 6 years, so the brother's age was 6 years.Step 2: At that time, the age of the father was 32 years more than the age of the brother. So father's age at Ram's birth = 6 + 32 = 38 years.Step 3: The mother is always 3 years younger than the father. So at Ram's birth, mother's age was 38 - 3 = 35 years.Step 4: At the time of Ram's birth, the age of the mother was 25 years more than the age of his sister. So sister's age at that time = 35 - 25 = 10 years.Step 5: Therefore, Ram's sister was 10 years old when Ram was born. Verification / Alternative check: We can verify the consistency. At Ram's birth: brother 6 years, sister 10 years, mother 35 years, father 38 years. The father is indeed 32 years older than the brother (38 - 6 = 32), and the mother is 25 years older than the sister (35 - 10 = 25). The mother is also 3 years younger than the father (38 - 35 = 3). Thus, all conditions match perfectly, so the answer is correct. Why Other Options Are Wrong: If the sister were 6 years old, the mother would be only 31 years old at Ram's birth, contradicting the 3-year difference with the father. If the sister were 8 years old or 12 years old, the difference between the mother's age and the sister's age would not be exactly 25 years. Only 10 years satisfies all the simultaneous conditions given in the problem. Common Pitfalls: Many learners mistakenly apply the age differences to the present instead of the time of Ram's birth, or they confuse which difference (32 or 25 years) applies to which pair of people. Another common mistake is forgetting that the difference between the parents' ages is fixed over time. Keeping track of who is older and by how many years at each stage is essential. Final Answer: Ram's sister was 10 years old when Ram was born.

Question 15

Priya's age was equal to the cube of an integer (other than 1) four years ago and will be equal to the square of an integer four years from now. After how many more years will her age be such that one year earlier it was a perfect square and one yea…

Accepted Answer

Introduction / Context: This is a higher level problem on ages combined with number theory. Priya's age is related to perfect cubes and perfect squares at different times. Four years ago her age was a perfect cube, four years from now it will be a perfect square, and we must find when her age will lie exactly between a perfect square and a perfect cube in consecutive years. This type of puzzle tests both algebraic reasoning and familiarity with perfect powers. Given Data / Assumptions: Four years ago, Priya's age was equal to the cube of an integer greater than 1. Four years from now, her age will be equal to the square of an integer. We must find how many more years she should wait so that at that future time her age will have the property that the age one year earlier is a perfect square and the age one year later is a perfect cube. Ages are in whole years. Concept / Approach: We represent Priya's present age by a variable and convert the conditions into equations involving perfect powers. First, we search for an age x such that x - 4 is a perfect cube and x + 4 is a perfect square. Once we find such an x, we then look for a future age A = x + k such that A - 1 is a perfect square and A + 1 is a perfect cube. This can be approached either by systematic trial with small perfect cubes and squares or by logical reasoning with typical age ranges. Step-by-Step Solution: Step 1: Let Priya's present age be x years.Step 2: Four years ago, her age was x - 4 and this is given to be a perfect cube greater than 1. Try small cubes: 2^3 = 8, 3^3 = 27, and so on. So x - 4 could be 8, 27, etc.Step 3: Four years from now, her age will be x + 4 and this must be a perfect square, such as 9, 16, 25, 36, and so on.Step 4: Testing x - 4 = 8 gives x = 12. Then x + 4 = 16, which is 4^2, a perfect square. So x = 12 satisfies both conditions: 4 years ago her age was 8 (2^3) and 4 years from now it will be 16 (4^2).Step 5: Now we must find k such that at age A = x + k, the age one year earlier, A - 1, is a perfect square and the age one year later, A + 1, is a perfect cube.Step 6: With x = 12, test reasonable future ages. Try A = 26: then A - 1 = 25 = 5^2 (square) and A + 1 = 27 = 3^3 (cube). So A = 26 years works.Step 7: The waiting time is k = A - x = 26 - 12 = 14 years. Verification / Alternative check: Check the initial conditions: Priya is now 12 years old. Four years ago, she was 8 years old, which is the cube of 2. Four years from now, she will be 16 years old, which is the square of 4. After waiting 14 more years, she will be 26 years old. At that time, the previous year's age, 25, is 5^2 and the next year's age, 27, is 3^3. All conditions are perfectly satisfied. Why Other Options Are Wrong: If she waited 7 or 12 years, the resulting ages would not simultaneously satisfy the condition that one year earlier is a perfect square and one year later is a perfect cube. For example, at 19 or 24 years, the neighbouring ages do not form the required perfect powers. Similarly, waiting 21 years would lead to an age where neighbouring ages are not one perfect square and one perfect cube. Only 14 years gives the exact combination needed. Common Pitfalls: Many learners try random guesses for her present age without using the combined constraints of a cube and a square at four-year intervals. Another common mistake is misreading the final condition and thinking that her age itself must be both a square and a cube, which is not possible except for special numbers. Here, her age lies between a square and a cube in consecutive years, which is a subtler requirement. Final Answer: Priya should wait 14 years for the described condition to occur.

Question 16

Three years ago, the average age of a husband, his wife, and their child was 24 years. Five years ago, the average age of only the wife and the child was 25 years. What is the present age of the husband?

Accepted Answer

Introduction / Context: This question uses average ages to find the present age of a family member. The averages of Husband, Wife, and Child at one point in the past and of Wife and Child at a different point in the past are given. From these, we must deduce the present age of the husband. Such problems are common in aptitude exams and require careful handling of averages and time shifts. Given Data / Assumptions: Three years ago, the average age of husband, wife, and child was 24 years. Five years ago, the average age of only the wife and the child was 25 years. We must compute the present age of the husband in years. Ages are assumed to be whole numbers. Concept / Approach: The average age of a group is the total of their ages divided by the number of people. From the given averages, we can first find the total ages at those specific times. Then, by accounting for the passage of time, we can find their present total ages. Finally, subtracting the present sum of the wife and child from the present sum of all three gives the present age of the husband. Step-by-Step Solution: Step 1: Three years ago, the average age of husband, wife, and child was 24 years. With three people, total age three years ago = 24 * 3 = 72 years.Step 2: Let the present ages of husband, wife, and child be H, W, and C years respectively. Three years ago, their ages were H - 3, W - 3, and C - 3, with total (H - 3) + (W - 3) + (C - 3) = 72.Step 3: Simplify: H + W + C - 9 = 72, so H + W + C = 81. This is the present total age of all three.Step 4: Five years ago, the average age of wife and child was 25 years, so their total age at that time was 25 * 2 = 50 years.Step 5: Their present ages are W and C, so five years ago they were W - 5 and C - 5. Thus, (W - 5) + (C - 5) = 50.Step 6: Simplify: W + C - 10 = 50, hence W + C = 60.Step 7: Now subtract: (H + W + C) - (W + C) = 81 - 60 = 21. Therefore, H = 21 years. Verification / Alternative check: Check the earlier data. Three years ago, total age of all three was 72 years. If H is 21 now, then three years ago husband's age was 18 years. Similarly, W + C = 60 now, so three years ago their combined age was 60 - 6 = 54 years. Thus, 18 + 54 = 72, which matches the first condition. Five years ago, W + C = 60 - 10 = 50, and 50 divided by 2 gives an average of 25 years, which matches the second condition. So the solution is consistent. Why Other Options Are Wrong: If the husband were 27, 29, or 31 years old presently, the totals would not reconcile with both average conditions simultaneously. Substituting these values into the equations would either disturb the total of 81 years or fail to keep W + C equal to 60. Therefore, those choices are not possible. Common Pitfalls: Common errors include forgetting to multiply the average by the number of people to get total age, or incorrectly adjusting ages when going back 3 or 5 years. Another pitfall is to treat averages as if they applied to the present without properly accounting for the time shift. Writing separate equations for each given average and working step by step avoids these mistakes. Final Answer: The present age of the husband is 21 years.

Question 17

Ten years ago, Kumar was three times as old as Sailesh. Ten years hence, Kumar will be only twice as old as Sailesh. Find the present age of Kumar in years.

Accepted Answer

Introduction / Context: This is another linear age problem involving two time references, 10 years ago and 10 years hence. Kumar and Sailesh have a three times relation in the past and a two times relation in the future. From these relationships, we must determine Kumar's present age. Problems like this are standard in the topic problems on ages in aptitude tests. Given Data / Assumptions: Ten years ago, Kumar was three times as old as Sailesh. Ten years from now, Kumar will be twice as old as Sailesh. We need to find the present age of Kumar in years. Ages are assumed to be positive integers. Concept / Approach: We assign variables to the present ages, then use the past and future conditions to form two linear equations. When an age is described "ten years ago", we subtract 10 from the present age; "ten years hence" means we add 10. Solving the system of equations gives both present ages, and we can then select Kumar's age. Step-by-Step Solution: Step 1: Let Kumar's present age be K years and Sailesh's present age be S years.Step 2: Ten years ago, Kumar's age was K - 10 and Sailesh's age was S - 10. The condition says K - 10 = 3 * (S - 10).Step 3: Ten years from now, their ages will be K + 10 and S + 10. The condition says K + 10 = 2 * (S + 10).Step 4: From K - 10 = 3S - 30, we get K = 3S - 20.Step 5: Substitute K = 3S - 20 into K + 10 = 2S + 20 to obtain (3S - 20) + 10 = 2S + 20.Step 6: Simplify: 3S - 10 = 2S + 20, so S = 30 years.Step 7: Substitute back to find K: K = 3 * 30 - 20 = 70 years. Thus, Kumar is 70 years old at present. Verification / Alternative check: Ten years ago, Kumar was 60 years old and Sailesh was 20 years old. At that time, 60 is three times 20, which satisfies the first condition. Ten years from now, Kumar will be 80 years old and Sailesh will be 40 years old, and 80 is twice 40, which satisfies the second condition. Therefore, the solution is correct and consistent with all the given information. Why Other Options Are Wrong: If Kumar were 50, 60, or 40 years old at present, substituting those values into the equations would fail to satisfy both the past three times relation and the future twice relation. Only a present age of 70 years for Kumar produces valid ages for both people at both time points, so the other options must be rejected. Common Pitfalls: As with similar problems, learners may confuse which person is older, misapply the time shifts, or mis-handle the algebra. It is important to clearly label present ages, carefully adjust for ten years ago and ten years hence, and then solve the equations in an orderly fashion. Final Answer: Kumar's present age is 70 years.

Question 18

The ratio of the present ages of Karthik and Kalyan is 4:5. The difference between the present age of Kalyan and the age that Karthik will have 5 years from now is 3 years. What is the sum of the present ages of Karthik and Kalyan?

Accepted Answer

Introduction / Context: This question combines ratios of present ages with a difference involving a future age. The present age ratio of Karthik and Kalyan is given, and a condition relates Kalyan's present age to Karthik's age 5 years hence. We must find the total of their present ages. Such problems are standard in aptitude topics dealing with ages and ratios. Given Data / Assumptions: The ratio of the present ages of Karthik and Kalyan is 4:5. The difference between the present age of Kalyan and the age Karthik will have 5 years from now is 3 years. We are asked to find the sum of their present ages. All ages are in years. Concept / Approach: When present ages are in a ratio, we can represent them as multiple of a common factor. Here, we let Karthik's age be 4k and Kalyan's age be 5k. Then we translate the difference condition involving Karthik's age 5 years later and set up an equation in k. Solving for k yields actual ages, and we can then compute their sum. Step-by-Step Solution: Step 1: Let the present age of Karthik be 4k years and the present age of Kalyan be 5k years.Step 2: Five years from now, Karthik's age will be 4k + 5 years.Step 3: The problem states that the difference between Kalyan's present age and Karthik's age 5 years hence is 3 years.Step 4: Since Kalyan is older, we write 5k - (4k + 5) = 3.Step 5: Simplify the equation: 5k - 4k - 5 = 3, which gives k - 5 = 3, so k = 8.Step 6: Therefore, Karthik's present age is 4k = 4 * 8 = 32 years, and Kalyan's present age is 5k = 5 * 8 = 40 years.Step 7: Their total present age is 32 + 40 = 72 years. Verification / Alternative check: Check the given condition. The ratio 32:40 simplifies to 4:5, which matches the present age ratio. Five years hence, Karthik will be 37 years old. The difference between Kalyan's present age (40) and 37 is 3 years, exactly as stated. Therefore, all the information fits perfectly, confirming that the sum of their present ages is 72 years. Why Other Options Are Wrong: Totals of 56, 62, or 69 years do not correspond to any combination of ages that both satisfy the 4:5 ratio and the condition involving a 3 year difference with Karthik's future age. Only a total of 72 years yields ages of 32 and 40, which satisfy all the constraints. Common Pitfalls: One common mistake is to reverse the difference and write (4k + 5) - 5k = 3, which would incorrectly assume Karthik is older even 5 years hence. Another error is forgetting to add 5 only to Karthik's age while leaving Kalyan's present age unchanged in the equation. Careful interpretation of the phrase "difference between the present age of Kalyan and the age of Karthik 5 years hence" is essential. Final Answer: The sum of the present ages of Karthik and Kalyan is 72 years.

Question 19

Two boys are playing on a ground, and both are less than 10 years old. The age of the younger boy is equal to the cube root of the product of the ages of the two boys. If we place the digit representing the age of the younger boy to the left of the …

Accepted Answer

Introduction / Context: This is a logical ages puzzle involving two young boys and their parents. The puzzle links the ages of the boys with their parents' ages using a combination of digit manipulation and a cube root condition. Questions like this test both numerical reasoning and the ability to interpret word problems carefully. Given Data / Assumptions: Both boys are less than 10 years old. The age of the younger boy equals the cube root of the product of the ages of the two boys. When the digit of the younger boy's age is placed in front of the digit of the elder boy's age, the resulting two-digit number gives the age of the younger boy's father. When the digit of the elder boy's age is placed in front of the digit of the younger boy's age and the resulting number is divided by 2, we obtain the age of the younger boy's mother. The mother is 3 years younger than the father. We must determine the ages of the elder boy (E) and younger boy (Y). Concept / Approach: Since both boys are less than 10 years old, their ages are single-digit positive integers. The cube root condition says that Y is the cube root of E * Y. This implies Y^3 = E * Y, and because Y is not zero, we can divide both sides by Y to get Y^2 = E. Thus, E must be the square of Y. We then test small integer values of Y to find suitable E values that remain single-digit and check whether the parent age conditions are satisfied. Step-by-Step Solution: Step 1: From Y^3 = E * Y, divide by Y (Y is positive and nonzero) to get Y^2 = E.Step 2: Try small values of Y. If Y = 1, then E = 1^2 = 1, but then there is no meaningful elder and younger distinction, so discard.Step 3: If Y = 2, then E = 2^2 = 4, which is valid and less than 10.Step 4: If Y = 3, then E = 9, which is not among the options and would lead to different parent ages. Also, options explicitly list specific pairs.Step 5: Now test Y = 2 and E = 4 with the digit conditions. Placing the digit of the younger boy (2) before the elder boy (4) gives 24. This is the age of the father.Step 6: Placing the digit of the elder boy (4) before the younger boy (2) gives 42. Dividing by 2 gives 21, which is the age of the mother.Step 7: Check the difference between the father and mother: 24 - 21 = 3 years, which matches the condition that the mother is 3 years younger than the father.Step 8: Therefore, E = 4 years and Y = 2 years satisfy all the puzzle conditions. Verification / Alternative check: Recheck the cube root condition: E * Y = 4 * 2 = 8. The cube root of 8 is 2, which equals Y, so the first condition holds. The father's age as 24 and mother's age as 21 also satisfy the difference of 3 years. Thus, all relationships are consistent and confirm the solution. Why Other Options Are Wrong: The pairs (15, 3), (14, 12), and (40, 22) violate the basic constraints of the puzzle because the boys must both be less than 10 years old and have single-digit ages. Moreover, they do not satisfy the cube root relationship Y^3 = E * Y nor the specific digit-based conditions for the parents' ages. Only E = 4 and Y = 2 meet every requirement. Common Pitfalls: Some learners may ignore the restriction that both boys are less than 10 years old and start considering two-digit ages. Others might misinterpret the digit placement or forget that the cube root condition forces a very specific relationship between E and Y. Staying systematic by translating each statement into a clear algebraic or digit condition greatly simplifies the puzzle. Final Answer: The elder and younger boys are E = 4 years and Y = 2 years old respectively.

Question 20

The present age of Rajitha is three years less than twice her age 12 years ago. The ratio of the present age of Raj to the present age of Rajitha is 4:9. What will be the age of Raj after 5 years?

Accepted Answer

Introduction / Context: This problem relates the present age of Rajitha to her own age 12 years ago and then links her current age to the present age of Raj through a ratio. We must determine how old Raj will be after 5 years. Such problems combine linear equations with simple ratios and are typical in aptitude questions on ages. Given Data / Assumptions: The present age of Rajitha is three years less than twice her age 12 years ago. The ratio of the present age of Raj to the present age of Rajitha is 4:9. We are asked to find the age of Raj after 5 years. Ages are given in years and are assumed to be whole numbers. Concept / Approach: We first use the relation between Rajitha's present age and her age 12 years ago to form a linear equation and solve for her current age. Then, using the given ratio between Raj and Rajitha, we find Raj's present age. Finally, we add 5 to obtain his age after 5 years. This stepwise approach is efficient and clear. Step-by-Step Solution: Step 1: Let Rajitha's present age be Rj years.Step 2: Her age 12 years ago was Rj - 12. Twice that age is 2 * (Rj - 12).Step 3: The problem states that her present age is three years less than twice her age 12 years ago. So Rj = 2 * (Rj - 12) - 3.Step 4: Expand: Rj = 2Rj - 24 - 3, which simplifies to Rj = 2Rj - 27.Step 5: Rearranging gives 2Rj - Rj = 27, so Rj = 27 years.Step 6: The ratio of the present age of Raj to the present age of Rajitha is 4:9, so Raj's present age R satisfies R / 27 = 4 / 9.Step 7: Cross multiply: 9R = 4 * 27 = 108, so R = 108 / 9 = 12 years.Step 8: Raj's age after 5 years will be R + 5 = 12 + 5 = 17 years. Verification / Alternative check: Check Rajitha's age condition: Twelve years ago, her age was 27 - 12 = 15 years. Twice that age is 2 * 15 = 30, and three years less than 30 is 27, which matches her present age. For the ratio, Raj is 12 years old and Rajitha is 27 years old, and 12:27 simplifies to 4:9 when both numbers are divided by 3. Adding 5 years gives Raj's age as 17 years, which is consistent with all the given information. Why Other Options Are Wrong: If Raj were 16, 19, or 21 years old after 5 years, his present age would be 11, 14, or 16 years respectively. None of these values would produce a ratio of 4:9 with Rajitha's age of 27 years. Therefore, those options are not consistent with the given ratio and must be rejected. Common Pitfalls: Some learners misinterpret the phrase "three years less than twice her age 12 years ago" and mistakenly write 2Rj - 12 - 3 instead of 2 * (Rj - 12) - 3. Another common error is to invert the ratio and compute 9:4 instead of 4:9 when relating Raj's age to Rajitha's age. Careful reading of the statement and writing equations step by step avoids these mistakes. Final Answer: Raj will be 17 years old after 5 years.

Problems on Ages Questions