Question 1

Two years ago, the average age of A and B was 26 years. The age of A five years from now will be 40 years, and B is 5 years younger than C. What is the difference between the present ages of A and C?

Accepted Answer

Introduction / Context: This age problem connects the ages of three people: A, B and C. We are given the past average age of A and B, the future age of A, and an age difference between B and C. Using this information, we must find the difference between the present ages of A and C. The question tests your ability to handle average age information and relative age differences across time. Given Data / Assumptions: - Two years ago, the average age of A and B was 26 years. - A's age five years from now will be 40 years. - B is 5 years younger than C at present. - All ages are in years and positive. Concept / Approach: We first use A's future age to compute A's present age. Then, from the past average of A and B, we compute B's present age. Using the information that B is 5 years younger than C, we derive C's present age. Finally, we calculate the absolute difference between A's and C's present ages. This requires careful tracking of time shifts and differences. Step-by-Step Solution: Step 1: Let A's present age be A years and B's present age be B years. Step 2: We are told that A's age five years from now will be 40 years, so A + 5 = 40. Step 3: Solve for A: A = 40 − 5 = 35 years. Step 4: Two years ago, the average age of A and B was 26 years. Step 5: Two years ago, their ages were A − 2 and B − 2, and their average was (A − 2 + B − 2) / 2 = 26. Step 6: Simplify: (A + B − 4) / 2 = 26 ⇒ A + B − 4 = 52 ⇒ A + B = 56. Step 7: Substitute A = 35 into A + B = 56 to find B: 35 + B = 56 ⇒ B = 21 years. Step 8: B is 5 years younger than C, meaning C's present age = B + 5 = 21 + 5 = 26 years. Step 9: Now we have A = 35 years and C = 26 years. The difference between their present ages is |35 − 26| = 9 years. Verification / Alternative check: We can quickly verify all the given conditions. If A is 35 years old now, then in 5 years A will be 40 years old, matching the future age statement. Two years ago, A was 33 and B was 19. Their average then was (33 + 19) / 2 = 52 / 2 = 26 years, which matches the given average. With B at 21 and C at 26 now, B is indeed 5 years younger than C. Therefore, all given conditions are satisfied by these values, and the calculated difference of 9 years is consistent. Why Other Options Are Wrong: Differences such as 6, 12, 15 or 18 years would imply different present ages for C that would either break the two-years-ago average or the "5 years younger" relationship between B and C. Only a 9-year difference keeps all conditions consistent when A is 35 years old and B is 21 years old. Common Pitfalls: Students may misread the time references, for example treating the 26-year average as the current average rather than the average two years ago. Others might forget to adjust both A and B by 2 years when using that average, or misapply the "5 years younger" relationship. Writing down a small timeline and methodically converting averages into total ages helps to avoid such mistakes. Final Answer: The difference between the present ages of A and C is 9 years.

Question 2

At present, the sum of the ages of four people A, B, C, and D is 76 years. After 7 years, their ages will be in the ratio 7:6:5:8 respectively. What is the present age of person C in years?

Accepted Answer

Introduction / Context:This aptitude question on problems related to ages involves four different people whose total present age is known. We are also given a future ratio of their ages after a fixed number of years. The task is to determine the present age of person C. Questions of this type test the ability to convert ratio conditions and total sum information into algebraic equations and then solve them logically. Given Data / Assumptions:- There are four people A, B, C, and D.- The sum of their present ages is 76 years.- After 7 years, their ages will be in the ratio 7:6:5:8 in the order A, B, C, D.- All ages are measured in complete years.- We must find the present age of person C. Concept / Approach:The key idea is that when all four ages are shifted by the same number of years, the ratio of their new ages can be represented using a common multiplying factor. We express each person's future age in terms of this factor and then convert back to present ages. Using the fact that the total present age is known, we set up an equation in the common factor and solve for it. Once this factor is known, the present age of each person can be calculated easily. Step-by-Step Solution:Step 1: Let the common multiplying factor for the ratio after 7 years be k.Step 2: After 7 years, the ages of A, B, C, and D will be 7k, 6k, 5k, and 8k years respectively.Step 3: Therefore, their present ages are A = 7k - 7, B = 6k - 7, C = 5k - 7, and D = 8k - 7.Step 4: The sum of their present ages is given as (7k - 7) + (6k - 7) + (5k - 7) + (8k - 7) = 76.Step 5: Simplify this expression: 7k + 6k + 5k + 8k = 26k and the constants sum to -28, so 26k - 28 = 76.Step 6: Add 28 to both sides to get 26k = 104. Thus k = 104 / 26 = 4.Step 7: Substitute k = 4 in C = 5k - 7 to get C = 5 * 4 - 7 = 20 - 7 = 13 years. Verification / Alternative check:Using k = 4, the present ages are A = 7 * 4 - 7 = 21 years, B = 6 * 4 - 7 = 17 years, C = 13 years, and D = 8 * 4 - 7 = 25 years. The sum 21 + 17 + 13 + 25 equals 76 years, which matches the given total. After 7 years, their ages will be 28, 24, 20, and 32 years. The ratio 28:24:20:32 simplifies by dividing all terms by 4 to 7:6:5:8, which matches the required ratio. This confirms that the value of C is correct. Why Other Options Are Wrong:Values like 11 years, 12 years, 15 years, or 17 years for C do not produce a total present age of 76 when combined with consistent ages for A, B, and D that maintain the future ratio 7:6:5:8. When tested, they either break the total sum of 76 or fail to give the correct ratio after 7 years. Therefore, they cannot be correct answers for this problem. Common Pitfalls:One common mistake is to try to apply the ratio directly to the present ages instead of to the ages after 7 years. Another error is to forget that the same number, 7, is subtracted from each future age to obtain the present ages. Some learners also mismanage the algebra when summing and simplifying the expressions. Working step by step with a clear definition of k helps avoid these errors. Final Answer:The present age of person C is 13 years.

Question 3

The average age of 8 men increases by 4 years when one man aged 30 years is replaced by a new man. What is the age of the new man in years?

Accepted Answer

Introduction / Context:This is a standard average and age replacement problem from aptitude. We have a group of men with a known average age, and when one member of the group leaves and is replaced by another, the average changes. From this change, we can deduce the age of the new person. Such questions are frequently used to test understanding of averages and how they relate to totals. Given Data / Assumptions:- There are 8 men in the group.- The average age increases by 4 years after the replacement.- The man who leaves is 30 years old.- The new man has an unknown age that we must determine.- All ages are in complete years. Concept / Approach:The average age of a group is equal to the total of the ages divided by the number of people. When one person leaves and another joins, the number of people remains the same, but the total age changes. The increase in average multiplied by the number of people gives the increase in total age. Knowing how the total age changed when a 30 year old man left makes it possible to find the age of the new man by simple algebra. Step-by-Step Solution:Step 1: Let the original average age of the 8 men be A years.Step 2: The original total age of the 8 men is 8 * A.Step 3: After the 30 year old man is replaced, the new average becomes A + 4 years.Step 4: The new total age is 8 * (A + 4) = 8A + 32.Step 5: Let the age of the new man be N years. The new total age can also be written as original total minus 30 plus N, that is 8A - 30 + N.Step 6: Equate the two expressions for the new total age: 8A - 30 + N = 8A + 32.Step 7: Cancel 8A from both sides to obtain -30 + N = 32. Therefore N = 32 + 30 = 62 years. Verification / Alternative check:We can check the logic with a simple interpretation. Because there are 8 men and the average age increases by 4 years, the total age of the group must have increased by 8 * 4 = 32 years. When the 30 year old man leaves and the new man joins, the net increase in total age is the age of the new man minus 30. This increase must equal 32. So new age minus 30 equals 32, leading to new age equal to 62 years, which confirms the algebraic solution. Why Other Options Are Wrong:If the new man were 55, 42, 69, or 38 years old, the change in total age would not be exactly 32 years. For example, if he were 55 years old, the increase in total age would be 55 - 30 = 25 years, which would only raise the average by 25 / 8, not by 4 years. Similar checks show that 42, 69, and 38 years do not give the required average increase. Common Pitfalls:Some learners mistakenly add 4 directly to the age of the man who left or divide the change in total age by the wrong number of people. Others confuse the original and new averages. The key is to remember that average times number of people equals total age, and that the increase in average multiplied by the number in the group gives the change in total age. Final Answer:The age of the new man is 62 years.

Question 4

The ratio of the present ages of P and Q is 5:8. Three years from now the ratio of their ages will be 8:11. What is the present age of Q in years?

Accepted Answer

Introduction / Context:This question is a classic example of age problems based on ratios at different times. The present ages of P and Q are in one ratio, while their ages after 3 years are in another ratio. Using these two ratio conditions, we must determine the current age of Q. Such problems test understanding of how ratios and linear equations work together when ages increase uniformly over time. Given Data / Assumptions:- The current ratio of ages of P and Q is 5:8.- After 3 years, the ratio of their ages will be 8:11.- Ages increase by the same number of years over time for each person.- The required quantity is the present age of Q in years. Concept / Approach:When ages are given in a ratio, we represent them using a common multiplier. If the present ratio is 5:8, we can write the ages as 5k and 8k. After 3 years, these ages become 5k + 3 and 8k + 3. The second ratio 8:11 is then used to form an equation, which allows us to find the value of k. Once k is known, Q's present age 8k can be calculated. Step-by-Step Solution:Step 1: Let the present age of P be 5k years and the present age of Q be 8k years.Step 2: After 3 years, P will be 5k + 3 years old and Q will be 8k + 3 years old.Step 3: The ratio after 3 years is given as (5k + 3) : (8k + 3) = 8 : 11.Step 4: Write the equation from the ratio: (5k + 3) / (8k + 3) = 8 / 11.Step 5: Cross multiply to get 11 * (5k + 3) = 8 * (8k + 3).Step 6: Expand both sides: 55k + 33 = 64k + 24.Step 7: Rearrange the equation: 55k + 33 - 64k - 24 = 0, which simplifies to -9k + 9 = 0, so 9k = 9 and k = 1.Step 8: Therefore, the present age of Q is 8k = 8 * 1 = 8 years. Verification / Alternative check:With k equal to 1, P is currently 5 years old and Q is 8 years old. After 3 years, P will be 8 years old and Q will be 11 years old. Their ages after 3 years are 8 and 11, which gives the ratio 8:11 exactly as stated in the problem. This confirms that the value of k and the computed age of Q are both correct. Why Other Options Are Wrong:If Q were 5, 11, 14, or 20 years old at present, then the current ratio with P would no longer be 5:8 or the future ratio after 3 years would not be 8:11. For example, if Q were 11 years old, P would have to be 6.875 years old to keep the ratio 5:8, which is not a natural age. Similar contradictions occur with the other options, so they cannot be correct. Common Pitfalls:Some learners attempt to work directly with the numbers in the ratios without using a common multiplier. This often leads to fractional ages or confusion. Another common mistake is to forget that both P and Q age by the same number of years, 3, before applying the second ratio. Carefully setting up the algebraic equation based on the ratio conditions prevents these errors. Final Answer:The present age of Q is 8 years.

Question 5

Seven years ago the age of Sahil was equal to the present age of Nihal. The sum of the age of Sahil 5 years ago and the age of Nihal 6 years later is 58 years. If Ruchi is 4 years older than Sahil at present, what will be the age of Ruchi after 10 y…

Accepted Answer

Introduction / Context:This is a multi person age problem involving Sahil, Nihal, and Ruchi. The relationships among their ages are described at different points in time, both in the past and in the future. Using these relationships, we first find the present ages of Sahil and Nihal and then determine Ruchi's age in the future. Such questions appear frequently in aptitude tests and require setting up simple linear equations. Given Data / Assumptions:- Seven years ago, the age of Sahil was equal to the present age of Nihal.- The sum of the age of Sahil 5 years ago and the age of Nihal 6 years later is 58 years.- Ruchi is currently 4 years older than Sahil.- All ages are in complete years.- We must find Ruchi's age after 10 years from now. Concept / Approach:We assign variables to the present ages of Sahil and Nihal and then translate each statement into an equation. The first statement connects Sahil's age seven years ago with Nihal's current age. The second statement connects two different times for Sahil and Nihal and gives a sum. Once we solve these equations to find Sahil's age, we can add 4 years to find Ruchi's present age and then add 10 more years to find Ruchi's age after 10 years. Step-by-Step Solution:Step 1: Let the present age of Sahil be S years and the present age of Nihal be N years.Step 2: Seven years ago, Sahil was S - 7 years old. The problem states that S - 7 equals the present age of Nihal, so S - 7 = N.Step 3: Five years ago, Sahil's age was S - 5. Six years from now, Nihal's age will be N + 6. The sum of these two ages is given as 58 years, so S - 5 + N + 6 = 58.Step 4: Simplify the sum equation to S + N + 1 = 58, which gives S + N = 57.Step 5: From S - 7 = N, we have N = S - 7. Substitute this into S + N = 57 to get S + (S - 7) = 57.Step 6: Simplify: 2S - 7 = 57, so 2S = 64 and S = 32 years. Thus Sahil is 32 years old now and Nihal is N = S - 7 = 25 years old.Step 7: Ruchi is 4 years older than Sahil, so her present age is 32 + 4 = 36 years. After 10 years, Ruchi will be 36 + 10 = 46 years old. Verification / Alternative check:Seven years ago, Sahil was 32 - 7 = 25 years old, which matches Nihal's present age of 25 years, so the first condition is satisfied. Five years ago, Sahil was 27 years old, and six years from now Nihal will be 31 years old. Their sum 27 + 31 equals 58, which matches the given sum. Finally, Ruchi being 36 years old now and 46 years old after 10 years is consistent with the information. All conditions are satisfied. Why Other Options Are Wrong:If Ruchi were 38, 42, 36, or 40 years old after 10 years, her present age would be 28, 32, 26, or 30 years respectively, which would lead to inconsistent ages for Sahil and Nihal when the conditions are checked. These inconsistent values would not satisfy both the seven year and the sum conditions. Therefore, those options cannot be correct. Common Pitfalls:Many learners misread the time references in the question and mix up ages 5 years ago, 6 years later, and 7 years ago. Another common mistake is to assume that the difference between Sahil and Nihal's ages changes over time, whereas it actually remains constant. Setting up the equations carefully using the correct time frames is crucial for reaching the right solution. Final Answer:Ruchi will be 46 years old after 10 years.

Question 6

The ratio of the present ages of P and Q is 7:9. Ten years ago the ratio of their ages was 5:7. What is the present age of Q in years?

Accepted Answer

Introduction / Context:This problem is based on the concept of age ratios at two different times. The present ages of P and Q are in one ratio, and their ages ten years ago were in another ratio. Using both ratios together, we can determine their current ages. Here, we are specifically asked to find the present age of Q. Such questions strengthen understanding of how ratios and linear equations are combined in age problems. Given Data / Assumptions:- Present ratio of ages P:Q is 7:9.- Ten years ago, the ratio of their ages was 5:7.- Ages increase uniformly by 10 years over the ten year period.- We need to find the present age of Q in years. Concept / Approach:We represent the present ages of P and Q using a common multiplier based on their present ratio. Then we express their ages ten years ago in terms of this multiplier and set up another ratio equation. Solving this system gives the value of the multiplier and therefore their current ages. This method relies heavily on the idea that both people age by the same fixed amount over the same period of time. Step-by-Step Solution:Step 1: Let the present age of P be 7k years and the present age of Q be 9k years.Step 2: Ten years ago, P was 7k - 10 years old and Q was 9k - 10 years old.Step 3: The ratio ten years ago is given as (7k - 10) : (9k - 10) = 5 : 7.Step 4: Write the equation from the ratio: (7k - 10) / (9k - 10) = 5 / 7.Step 5: Cross multiply to obtain 7 * (7k - 10) = 5 * (9k - 10).Step 6: Expand both sides: 49k - 70 = 45k - 50.Step 7: Rearrange to get 49k - 70 - 45k + 50 = 0, which simplifies to 4k - 20 = 0, so 4k = 20 and k = 5.Step 8: The present age of Q is 9k = 9 * 5 = 45 years. Verification / Alternative check:With k equal to 5, P is 7 * 5 = 35 years old and Q is 45 years old now. Ten years ago, P was 25 years old and Q was 35 years old. The ratio 25:35 simplifies to 5:7 when both numbers are divided by 5. The present ratio of 35:45 simplifies to 7:9 when divided by 5. Both ratios match the given conditions, confirming that Q is 45 years old. Why Other Options Are Wrong:If Q were 35, 25, 55, or 63 years old at present, the corresponding age of P calculated from the ratio 7:9 would not satisfy the second ratio condition from ten years ago. For example, if Q were 35 years old, P would be 35 * 7 / 9, which is not an integer age, and the ten years ago ratio would not match 5:7. Similar contradictions arise for the other options. Common Pitfalls:A frequent error is to attempt to work with ratios by guessing ages instead of using an algebraic approach. Another mistake is to forget to subtract 10 from both ages when forming the earlier ratio. Careful definition of variables and consistent application of the time shift are essential to avoid confusion and miscalculations. Final Answer:The present age of Q is 45 years.

Question 7

The average of the present ages of Rajan and his son is 20 years. Rajan is three times as old as his son at present. What will be the age of Rajan after 5 years?

Accepted Answer

Introduction / Context:This is a straightforward average and ratio based problem on ages. We are told the average age of a father and his son and also how many times the father is older than the son. From this information, we must calculate the father's present age and then determine his age after 5 years. Such questions are useful for practicing basic algebra and understanding how averages relate to sums. Given Data / Assumptions:- The average of the present ages of Rajan and his son is 20 years.- Rajan is currently three times as old as his son.- Ages are in complete years.- We are asked to find Rajan's age 5 years from now. Concept / Approach:The average of two numbers is the sum of the numbers divided by 2. If the average of two ages is known, we can multiply the average by 2 to obtain the total of their ages. The condition that one person is three times as old as the other allows us to express their ages in terms of a single variable and solve a simple system of equations. Step-by-Step Solution:Step 1: Let the present age of the son be S years.Step 2: Rajan is three times as old as his son, so Rajan's present age is 3S years.Step 3: The average of their present ages is given as 20 years, so (S + 3S) / 2 = 20.Step 4: Simplify the numerator to get 4S / 2 = 20, which reduces to 2S = 20.Step 5: Divide both sides by 2 to obtain S = 10 years. Therefore, the son is 10 years old now.Step 6: Rajan's present age is 3S = 3 * 10 = 30 years.Step 7: After 5 years, Rajan's age will be 30 + 5 = 35 years. Verification / Alternative check:We can confirm the result by checking the average and the age multiple. Currently, the son is 10 years old and Rajan is 30 years old. The average age is (10 + 30) / 2 = 40 / 2 = 20 years, which matches the given average. Also, Rajan is 3 times as old as his son, because 30 = 3 * 10. After 5 years, Rajan will indeed be 35 years old. Everything is consistent. Why Other Options Are Wrong:If Rajan were 40, 28, 25, or 30 years old after 5 years, his current age would be 35, 23, 20, or 25 years respectively. None of these current ages, paired with a suitable son age, would satisfy both the given average of 20 years and the condition that the father is three times as old as the son. Only the value of 35 years after 5 years leads to a present age of 30 years and a son age of 10 years that meet all conditions. Common Pitfalls:Some learners mistakenly interpret the average as the age of each person instead of as a mean value. Others forget that both ages are included in the average and divide by the wrong number. Another error is to misapply the three times relation and write S = 3R instead of R = 3S. Writing the equations carefully and solving them step by step avoids these mistakes. Final Answer:Rajan will be 35 years old after 5 years.

Question 8

The average age of 19 boys in a class is 21 years. When the age of the teacher is included, the average increases to 22 years. What is the age of the teacher in years?

Accepted Answer

Introduction / Context:This is a typical average age problem where a teacher is added to a group of students and the average age of the combined group changes. From the change in average, we can compute the teacher's age. Such questions help reinforce the link between average, total sum, and number of individuals in a group. Given Data / Assumptions:- There are 19 boys in a class.- The average age of these 19 boys is 21 years.- When the teacher is included, the average age becomes 22 years.- Ages are given in years.- We must find the age of the teacher. Concept / Approach:The average age is equal to the total of all ages divided by the number of individuals. We first compute the total age of the 19 boys using their average. Then, using the new average when the teacher is included, we compute the total age of the 20 individuals. The difference between these totals gives the teacher's age directly. Step-by-Step Solution:Step 1: Total age of the 19 boys is 19 * 21 years.Step 2: Compute this total: 19 * 21 = 399 years.Step 3: When the teacher is included, there are 20 individuals with average age 22 years, so the new total age is 20 * 22 years.Step 4: Compute this new total: 20 * 22 = 440 years.Step 5: The teacher's age is the difference between the total age of the group including the teacher and the total age of only the boys.Step 6: Teacher's age = 440 - 399 = 41 years. Verification / Alternative check:If we add a teacher aged 41 years to the group of 19 boys whose total age is 399 years, the new total becomes 399 + 41 = 440 years. Dividing by 20 individuals gives an average of 440 / 20 = 22 years, which matches the problem statement. This confirms that the calculated age of the teacher is correct. Why Other Options Are Wrong:If the teacher were 39, 40, 44, or 47 years old, the total age with the teacher would be 399 plus that value. Dividing those totals by 20 would not give an exact average of 22 years. For example, with a 39 year old teacher, the total would be 438 and the average would be 438 / 20 = 21.9 years, which is not equal to 22 years. Therefore, those options are incorrect. Common Pitfalls:Common mistakes include forgetting to multiply the average by the number of individuals to get total age, or using 19 instead of 20 when computing the new total. Some learners also try to adjust the average directly without working with totals, which can cause confusion. The safe approach is always to convert averages to sums and then back to averages as needed. Final Answer:The age of the teacher is 41 years.

Question 9

The sum of the present ages of four children born at intervals of 4 years each is 60 years. What is the present age of the youngest child?

Accepted Answer

Introduction / Context:This problem deals with ages of children born at equal intervals of time. Since the children are born at intervals of 4 years, their ages form an arithmetic sequence. We are given the total of their present ages and asked to compute the age of the youngest child. This is a common type of question in aptitude exams for practicing arithmetic progressions and basic algebra. Given Data / Assumptions:- There are four children in the family.- Each child is 4 years older than the next younger child.- The sum of the present ages of all four children is 60 years.- Ages are in complete years.- We need to find the present age of the youngest child. Concept / Approach:When children are born at equal intervals of time, their ages form an arithmetic sequence. If we let the youngest child's age be x years, then the other children will be x + 4, x + 8, and x + 12 years old. The sum of these four expressions must equal the given total of 60 years. This leads to a simple linear equation in x that can be solved directly. Step-by-Step Solution:Step 1: Let the present age of the youngest child be x years.Step 2: Since the children are born at 4 year intervals, the other children are x + 4, x + 8, and x + 12 years old.Step 3: The sum of their present ages is x + (x + 4) + (x + 8) + (x + 12) = 60.Step 4: Combine like terms: x + x + x + x = 4x and 4 + 8 + 12 = 24, so the equation becomes 4x + 24 = 60.Step 5: Subtract 24 from both sides to obtain 4x = 36.Step 6: Divide both sides by 4 to get x = 36 / 4 = 9 years.Step 7: Therefore, the youngest child is 9 years old at present. Verification / Alternative check:Using x = 9, the ages of the four children are 9, 13, 17, and 21 years. The differences between consecutive ages are 4 years, as required. The sum 9 + 13 + 17 + 21 equals 60 years, matching the total given in the problem. This confirms that the youngest child's age has been calculated correctly. Why Other Options Are Wrong:If the youngest child were 7 years old, the ages would be 7, 11, 15, and 19 years, and their sum would be 52, not 60. If the youngest child were 10, 12, or 5 years old, the sums of the resulting age sequences would also not equal 60. Therefore, those values are inconsistent with the data and cannot be correct answers. Common Pitfalls:A frequent mistake is to misinterpret the phrase "born at intervals of 4 years" and incorrectly assign ages. Another error is to forget that there are exactly four children and write too many or too few terms. Some learners also add the constants incorrectly when forming the sum. Writing the ages in order and then summing them carefully helps avoid these issues. Final Answer:The youngest child is 9 years old at present.

Question 10

The average age of 36 students in a group is 14 years. When the age of the teacher is included, the average increases by 1 year. What is the age of the teacher in years?

Accepted Answer

Introduction / Context:This question is another example of average age calculation where a teacher is added to a group of students. The average age of the students is known, and including the teacher increases the average by a fixed amount. Using the relationship between average, total age, and number of individuals, we can compute the teacher's age. This type of problem is common in quantitative aptitude tests. Given Data / Assumptions:- There are 36 students in the group.- The average age of these 36 students is 14 years.- When the teacher's age is included, the average age increases by 1 year, becoming 15 years.- Ages are in complete years.- We are required to find the age of the teacher. Concept / Approach:The average age equals total age divided by the number of people. We first find the total age of the 36 students using their average. Then we compute the new total age when the teacher is included, using the new average and the new number of individuals, which is 37. The difference between the two totals is the age of the teacher. Step-by-Step Solution:Step 1: Total age of the 36 students is 36 * 14 years.Step 2: Compute this total: 36 * 14 = 504 years.Step 3: After including the teacher, there are 37 individuals with an average age of 15 years.Step 4: The new total age is 37 * 15 years.Step 5: Compute this new total: 37 * 15 = 555 years.Step 6: The teacher's age is equal to the new total age minus the original total age of the students, that is 555 - 504.Step 7: Teacher's age = 555 - 504 = 51 years. Verification / Alternative check:If the teacher is 51 years old, then adding this to the students' total age of 504 years gives 555 years. Dividing by 37 individuals gives an average of 555 / 37 = 15 years, which matches the increased average given in the question. This confirms that the calculated age of the teacher is correct and consistent with the data. Why Other Options Are Wrong:If the teacher's age were 31, 36, 41, or 45 years, the combined total age would be 535, 540, 545, or 549 years respectively. Dividing those totals by 37 would give average ages different from 15 years. For example, 540 / 37 is not equal to 15. Therefore, none of those ages can produce the specified new average and must be rejected. Common Pitfalls:Some learners mistakenly use 36 instead of 37 when calculating the new total from the new average. Others try to add 1 year directly to each student's age, which is not what the average increase means here. The correct approach is always to convert averages into totals by multiplying by the number of individuals, and then compare the totals before and after the change. Final Answer:The age of the teacher is 51 years.

Question 11

The average age of a class of 39 students is 15 years. When the age of the teacher is included, the average increases by 3 months. What is the age of the teacher in years?

Accepted Answer

Introduction / Context:This problem again involves average ages, but here the average increases by a fraction of a year, namely 3 months. The scenario describes a class of students with a known average age, and then the teacher is added. The new average allows us to find the teacher's age. This type of question tests careful handling of fractional years and understanding of averages. Given Data / Assumptions:- There are 39 students in a class.- The average age of the 39 students is 15 years.- When the teacher is included, the average age increases by 3 months, which is 0.25 years, so the new average is 15.25 years.- Ages are in years, but fractions of years are allowed for the average.- We must find the age of the teacher. Concept / Approach:The average age is equal to the total of all ages divided by the number of individuals. We first calculate the total age of the 39 students. Then, using the new average and the fact that there are now 40 individuals including the teacher, we compute the new total age. The teacher's age is obtained by subtracting the original total from the new total. Careful conversion of 3 months into years is essential. Step-by-Step Solution:Step 1: Total age of the 39 students is 39 * 15 years.Step 2: Compute this total: 39 * 15 = 585 years.Step 3: When the teacher is added, there are 40 individuals and the new average age is 15 years + 0.25 years = 15.25 years.Step 4: New total age including the teacher is 40 * 15.25 years.Step 5: Compute this new total: 40 * 15.25 = 610 years.Step 6: The teacher's age is the difference between the new total age and the original total age of the students, that is 610 - 585.Step 7: Teacher's age = 610 - 585 = 25 years. Verification / Alternative check:If the teacher is 25 years old, the combined total age is 585 + 25 = 610 years. Dividing by 40 individuals gives an average of 610 / 40 = 15.25 years, which corresponds to 15 years and 3 months. This matches the condition that the average increases by 3 months when the teacher is included. The solution is therefore consistent. Why Other Options Are Wrong:If the teacher were 30, 35, 40, or 28 years old, the new total age would be 615, 620, 625, or 613 years respectively. Dividing each by 40 would give averages that differ from 15.25 years. For instance, 620 / 40 = 15.5 years, not 15.25 years. Therefore, none of those values fit the given information. Common Pitfalls:Some learners do not convert 3 months into 0.25 years and instead treat it as 3 years or ignore the fractional part. Others mistakenly use 39 instead of 40 when multiplying by the new average. Procedural errors like these lead to incorrect teacher ages. Correctly handling units and remembering that the number of individuals changes from 39 to 40 are crucial steps. Final Answer:The age of the teacher is 25 years.

Question 12

The average age of A and B is 18 years. After 3 years, C joins them and at that time the average age of A, B, and C together becomes 22 years. What is the present age of C in years?

Accepted Answer

Introduction / Context:This question combines the concept of average age with the idea of a new person joining the group after some years. The average age of A and B is given, and three years later C joins them, at which point the new average of all three is known. From this information, we must determine the current age of C. This type of problem is a good test of how well you can handle time shifts in average calculations. Given Data / Assumptions:- The average age of A and B at present is 18 years.- This means their total present age is 2 * 18 years.- After 3 years, C joins A and B.- At that time, the average age of A, B, and C together is 22 years.- Ages are measured in years.- We need the present age of C, not C's age after 3 years. Concept / Approach:First we calculate the total present age of A and B using their average. After 3 years, both A and B will have aged 3 years each, so their combined age increases accordingly. When C joins them after 3 years, we use the new average to find the total age of the three at that time. Subtracting the future total age of A and B from this combined total gives C's age at that time, and then we subtract 3 years to get C's present age. Step-by-Step Solution:Step 1: Total present age of A and B is 2 * 18 = 36 years.Step 2: After 3 years, A and B will both be 3 years older. Their total age at that time will be 36 + 3 + 3 = 42 years.Step 3: When C joins them after 3 years, there are 3 people with an average age of 22 years.Step 4: Total age of A, B, and C at that time is 3 * 22 = 66 years.Step 5: Therefore, C's age at that time is 66 - 42 = 24 years.Step 6: Since this is C's age after 3 years, C's present age is 24 - 3 = 21 years. Verification / Alternative check:Assume C is currently 21 years old. After 3 years, C will be 24 years old. If the total age of A and B at present is 36 years, after 3 years their total will be 42 years. Adding C's age of 24 years at that time gives 42 + 24 = 66 years. Dividing by 3 people gives an average of 22 years, which matches the problem statement. This confirms that C's present age is indeed 21 years. Why Other Options Are Wrong:If C were 20, 23, 24, or 27 years old at present, then C's age after 3 years would be 23, 26, 27, or 30 years. When we add any of these to the future total age of A and B, the combined total age would not be 66, and the average would not be exactly 22 years. Thus, those options violate the conditions given in the question. Common Pitfalls:The most common mistake is to treat 22 years as the present average instead of the average after 3 years. Another error is failing to increase the ages of A and B by 3 years before forming the equation with the new average. Some learners also try to treat C's present age and C's age after 3 years as the same, which is incorrect. Carefully separating present and future ages is essential. Final Answer:The present age of C is 21 years.

Question 13

A man was 32 years old when his first son was born. His wife was 35 years old when the son was 7 years old. What is the difference between the age of the father and the age of the mother in years?

Accepted Answer

Introduction / Context:This question involves a family age relationship among a father, a mother, and their son. The ages of the parents are given at specific times relative to the son's birth and childhood. From these, we have to find the constant age difference between the father and the mother. Problems like this test understanding of how age differences remain constant over time. Given Data / Assumptions:- The father was 32 years old when his first son was born.- When the son was 7 years old, the mother was 35 years old.- Ages are in years.- We need the difference between the age of the father and the age of the mother. Concept / Approach:The key concept is that the difference in age between any two people does not change over time. Instead of tracking ages year by year, we can find the ages of the father and mother at one common time and compute their difference. Here, the natural common reference point is the time when the son is 7 years old, since both parents' ages are easy to express then. Step-by-Step Solution:Step 1: At the time the son was born, the father was 32 years old.Step 2: When the son is 7 years old, 7 years have passed since his birth.Step 3: Therefore, when the son is 7, the father's age is 32 + 7 = 39 years.Step 4: The problem states that when the son is 7 years old, the mother is 35 years old.Step 5: At that same time, the difference between the father's age and the mother's age is 39 - 35.Step 6: Compute this difference: 39 - 35 = 4 years.Step 7: Because age differences do not change over time, the father is always 4 years older than the mother. Verification / Alternative check:We can check consistency at both time points. At the time of the son's birth, if the father is 32 and the mother is 28, their difference is 4 years. Seven years later, the father would be 39 and the mother 35, again a difference of 4 years. This matches the given information that the mother is 35 when the son is 7 and confirms that the age difference is 4 years in every time frame. Why Other Options Are Wrong:A difference of 7, 5, 3, or 2 years would not fit the data. For instance, if the difference were 7 years, the mother would have to be 32 years old when the son was 7, which contradicts the given information that she was 35 at that time. Only a 4 year difference is consistent with the ages provided at the specific milestone when the son is 7 years old. Common Pitfalls:Some learners get confused and try to track the ages from the father's perspective only, without using the second piece of information about the mother. Others mistakenly think that the difference between the parents' ages might change over time. Remember that while ages increase, the difference between them always stays the same. Focusing on a single reference time where both ages are known makes the problem simple. Final Answer:The difference between the age of the father and the age of the mother is 4 years.

Question 14

The sum of the present ages of a father and his son is 60 years. Six years ago, the age of the father was five times the age of the son. What will be the age of the son after 6 years?

Accepted Answer

Introduction / Context:This is a classic problem on the ages of a father and son. We are given their total present age and a relationship between their ages six years ago. Using this information, we can determine their individual present ages and then find the son's age after 6 more years. Such questions are common in aptitude exams and test the ability to form and solve linear equations from word statements. Given Data / Assumptions:- The sum of the present ages of the father and his son is 60 years.- Six years ago, the father's age was five times the son's age.- Ages are in years.- We must find the son's age after 6 years from now. Concept / Approach:We assign variables to the present ages of the father and the son. The given total age at present provides one equation. The condition from six years ago gives another equation relating their ages at that time. Solving this system of two linear equations yields the present ages. Finally, we add 6 years to the son's present age to answer the question. Step-by-Step Solution:Step 1: Let the present age of the father be F years and the present age of the son be S years.Step 2: The sum of their present ages is F + S = 60.Step 3: Six years ago, the father was F - 6 years old and the son was S - 6 years old.Step 4: The problem states that six years ago the father's age was five times the son's age, so F - 6 = 5 * (S - 6).Step 5: Expand the second equation: F - 6 = 5S - 30, which gives F = 5S - 24.Step 6: Substitute F = 5S - 24 into F + S = 60 to obtain (5S - 24) + S = 60.Step 7: Combine terms to get 6S - 24 = 60, so 6S = 84 and S = 84 / 6 = 14 years.Step 8: The son is currently 14 years old, so after 6 years his age will be 14 + 6 = 20 years. Verification / Alternative check:If the son is 14 years old now, then the father's age is F = 60 - 14 = 46 years. Six years ago, the son was 8 years old and the father was 40 years old. The ratio 40:8 equals 5:1, so the father's age was indeed five times the son's age at that time. This confirms that the ages are consistent with the given information and that the son will be 20 years old after 6 years. Why Other Options Are Wrong:If the son's age after 6 years were 25, 33, 45, or 26 years, then his present age would be 19, 27, 39, or 20 years respectively. Substituting these values into the total and ratio conditions either breaks the sum of 60 years or violates the five times relation from six years ago. Therefore, none of those options can satisfy all the constraints simultaneously. Common Pitfalls:Some learners mistakenly set F = 5S directly instead of applying the condition six years ago, which leads to incorrect equations. Others forget to subtract 6 from both ages when working with the past condition. It is important to pay attention to time phrases such as "six years ago" and to adjust both ages accordingly before applying the ratio or multiple condition. Final Answer:The son will be 20 years old after 6 years.

Question 15

Ten years ago, a man was seven times as old as his son. Two years hence, twice his age will be equal to five times the age of his son. What is the present age of the son?

Accepted Answer

### Concept & Formula The core concept is defining present ages in terms of a single variable based on past conditions, and then translating future conditions into a solvable algebraic equation. Let the son's age $10$ years ago be $x$. The father's age then was $7x$. Their present ages can be expressed by adding $10$ to both. ### Step-by-Step Solution * **Given:** * $10$ years ago: Man = $7 	imes$ Son * $2$ years hence (future): $2 	imes$ Man = $5 	imes$ Son * **Calculation / Deduction:** * Let the son's age $10$ years ago be $x$. * Then, the man's age $10$ years ago was $7x$. * Their *present* ages are: * Son = $x + 10$ * Man = $7x + 10$ * Their ages $2$ years hence (from present) will be: * Son = $(x + 10) + 2 = x + 12$ * Man = $(7x + 10) + 2 = 7x + 12$ * The problem states that in this future time, twice the man's age equals five times the son's age: * $$2(7x + 12) = 5(x + 12)$$ * $$14x + 24 = 5x + 60$$ * Group the variables and constants: * $$14x - 5x = 60 - 24$$ * $$9x = 36$$ * $$x = 4$$ * We need the *present* age of the son, which is $x + 10$. * Present age = $4 + 10 = 14$ years. ### Exam Strategy & Shortcut Use Option Elimination with timeline matching. The son's present age minus $10$ years must yield a valid, workable past age. Testing option (c) $14$ years: If present son = $14$, past son ($10$ yrs ago) = $4$. Past man = $7 	imes 4 = 28$. Present man = $28 + 10 = 38$. Future man ($+2$ yrs) = $40$. Future son ($+2$ yrs) = $16$. Check condition: $2 	imes 40 = 80$, and $5 	imes 16 = 80$. They match perfectly! ### Common Pitfall A major trap is successfully calculating $x = 4$ and assuming it is the final answer, or adding only $2$ instead of $10$ to find the present age. Always clearly label what time period your variable $x$ belongs to. ### Final Answer **Therefore, the correct answer is 14 years.**

Question 16

4 years ago, the ratio of 1/2 of A's age at that time and four times of B's age at the time was 5 : 12. Eight years hence, 1/2 of A's age at that time will be less than B's age at that time by 2 years. What is B's present age?

Accepted Answer

### Concept & Strategy This requires setting up simultaneous equations across different timelines (past, present, future). The strategy is to clearly define the variables at a single point in time, build the fractional equations, and solve through substitution. ### Step-by-Step Solution * **Given:** Let the ages of A and B **4 years ago** be $a$ and $b$ respectively. * **Calculation (Part 1: The Ratio 4 years ago):** The ratio of $1/2$ of A's age to $4$ times B's age is $5:12$. $$\frac{a/2}{4b} = \frac{5}{12}$$ $$\frac{a}{8b} = \frac{5}{12}$$ Cross-multiply to find a relationship: $$12a = 40b \implies 3a = 10b \implies a = \frac{10b}{3}$$ * **Calculation (Part 2: Eight years hence):** Since $a$ and $b$ are ages 4 years ago, their ages **8 years hence** (which is 12 years after the "4 years ago" timeline) will be: A's future age = $a + 12$ B's future age = $b + 12$ The problem states $1/2$ of A's future age is less than B's future age by 2 years: $$\frac{a + 12}{2} = (b + 12) - 2$$ $$\frac{a + 12}{2} = b + 10$$ $$a + 12 = 2b + 20$$ $$a - 2b = 8$$ * **Calculation (Part 3: Solving for B):** Substitute $a = \frac{10b}{3}$ into the new equation: $$\frac{10b}{3} - 2b = 8$$ $$\frac{10b - 6b}{3} = 8$$ $$\frac{4b}{3} = 8 \implies 4b = 24 \implies b = 6$$ So, B's age 4 years ago was 6. B's present age = $6 + 4 = 10$ years. ### Exam Strategy & Shortcut Working with fractions can be slow. Instead, use the options to work backwards. We are looking for B's present age. Test Option (a) 10: If B is currently 10, then 4 years ago, B was 6. From the first ratio: $\frac{a}{8(6)} = \frac{5}{12} \implies \frac{a}{48} = \frac{5}{12} \implies a = 20$. So 4 years ago A was 20. A's present age is 24. Let's check the second condition (8 years hence): A will be 32, B will be 18. Is half of A's age (16) less than B's age (18) by 2? Yes! $18 - 16 = 2$. Option (a) is verified instantly without solving heavy simultaneous equations. ### Common Pitfall A major trap is mixing up the timelines. When dealing with "4 years ago" and "8 years hence", students often add 8 to the present age correctly, but fail to realize that relative to the "4 years ago" variables ($a$ and $b$), the time jump is actually $4 + 8 = 12$ years. ### Final Answer **Therefore, the correct answer is 10 years.**

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