Problems on Ages : Quantitative Aptitude

Age problems are a common type of quantitative aptitude problem that appears in various competitive exams in India. These problems involve calculating the ages of different individuals based on certain given information, such as their current age, age at a future date, time gaps, etc.

This is a comprehensive guide that provides a step-by-step approach to solving age problems. This Page covers all types of age problems that can appear in competitive exams and provides detailed explanations of the concepts and formulas involved.

The Page is designed to help readers develop a strong foundation in age problems and improve their problem-solving skills. Each chapter includes numerous solved examples and practice problems with detailed solutions to help readers understand the concepts and apply them in real-world scenarios.

Table of Contents

1. Introduction to Age Problems
2. Basic Concepts and Formulas for Problems on Ages
3. Solving Simple Problems on Ages
4. Problems on Ages Involving Fractions
5. Problems on Ages with Multiple Persons
6. Problems on Ages with Reversed Ages
7. Problems on Ages with Time Gaps
8. Problems on Ages Involving Death of a Person
9. Problems on Ages with Current Age and Age at a Future Date
10. Problems on Ages with Average Ages
11. Problems on Ages with Ages in Geometric Progression
12. Problems on Ages with Ages in Harmonic Progression
13. Problems on Ages with Ages and Income
14. Problems on Ages with Ages and Discounts
15. Problems on Ages with Ages and Partnership
    

Understanding Age Problems

Solving Age Problems

Example 1:

Question: Sarah is currently twice as old as her daughter. In 10 years, Sarah's age will be three times her daughter's age. How old is Sarah currently?

Solution:

Step 1: Known quantities: Sarah's age in 10 years will be three times her daughter's age, and Sarah is currently twice as old as her daughter.

Unknown quantity: Sarah's current age.

Step 2: Let x be the daughter's current age.

Then, Sarah's current age is 2x.

Step 3: Based on the given information, we can set up the following equation:

2x + 10 = 3(x + 10)

Step 4: Simplifying the equation, we get:

2x + 10 = 3x + 30

x = 20

Step 5: Sarah's current age is:

2x = 2(20) = 40

Therefore, Sarah is currently 40 years old.

Step 6: Checking the answer, we can verify that in 10 years, Sarah will be 50 years old, which is three times her daughter's age of 30.

Common Pitfalls

Misinterpreting the Information

One of the most common pitfalls in solving age problems is misinterpreting the information given in the problem. Age problems often contain extraneous information that may distract students from the key details that are relevant to solving the problem. As a result, it is important to read the problem carefully and identify the known quantities and unknown quantities.

For example, consider the following problem:

John is twice as old as Jane. If John is 30 years old, how old is Jane?

The known quantity is John's age (30 years old), and the unknown quantity is Jane's age. The relationship between their ages is that John is twice as old as Jane. By identifying the known and unknown quantities and the relationship between them, we can set up an equation to solve for Jane's age.

Using the Wrong Variable

Another common pitfall is using the wrong variable to represent the unknown quantity. In age problems, it is common to use x to represent the unknown quantity (i.e., the current age of an individual). However, some problems may involve multiple unknown quantities, in which case it may be necessary to use different variables to represent each one.

For example, consider the following problem:

The sum of the ages of three people is 90. The first person is twice as old as the second person, and the third person is 10 years younger than the first person. How old is the second person?

In this problem, we can use x to represent the age of the second person. However, we need to use different variables to represent the ages of the other two people. Let y be the age of the second person, then the age of the first person is 2y, and the age of the third person is 2y - 10. By setting up an equation based on the sum of their ages, we can solve for the value of y.

Not Setting Up Equations or Proportions Correctly

Another common pitfall is not setting up equations or proportions correctly. Age problems often involve ratios and proportions, which can be used to relate the ages of different individuals to each other and to the passage of time. It is important to pay attention to the relationships between the ages of the individuals involved and to use variables appropriately to represent the unknown quantities.

For example, consider the following problem:

The sum of the ages of a mother and daughter is 60. The mother is four times as old as the daughter. How old is the daughter?

In this problem, we can let x be the daughter's age, then the mother's age is 4x. We can set up an equation based on the sum of their ages: x + 4x = 60. However, this equation does not give us enough information to solve for the daughter's age. Instead, we need to set up a proportion based on the relationship between the mother's age and the daughter's age: 4x / x = 4. Solving for x, we get x = 15, so the daughter is 15 years old.

Not Simplifying Equations or Proportions Correctly

Another common pitfall is not simplifying equations or proportions correctly. It is important to combine like terms and apply mathematical operations correctly in order to solve for the unknown quantities.

For example, consider the following problem:

The sum of the ages of two people is 70. One person is 10 years older than the other. How old is each person?

In this problem, we can use x to represent the age of one person, and x + 10 to represent the age of the other person. We can set up an equation based on the sum of their ages: x + (x + 10) = 70. Simplifying this equation, we get 2x + 10 = 70. Solving for x, we get x = 30, so one person is 30 years old, and the other person is 40 years old.

Not Checking the Solution for Reasonableness

Finally, it is important to check the solution to make sure it is reasonable and makes sense in the context of the problem. If the solution implies that an individual's age is negative or greater than the lifespan of a human, it is likely that an error has been made in the solution.

For example, consider the following problem:

The sum of the ages of a father and son is 80. The father is three times as old as the son. How old is each person?

In this problem, we can use x to represent the age of the son, and 3x to represent the age of the father. We can set up an equation based on the sum of their ages: x + 3x = 80. Simplifying this equation, we get 4x = 80. Solving for x, we get x = 20, so the son is 20 years old, and the father is 60 years old.

It is important to check the solution to make sure it makes sense in the context of the problem. In this case, we can check that the father is indeed three times as old as the son (60 = 3 x 20) and that their ages add up to 80 (20 + 60 = 80). Therefore, our solution is reasonable and makes sense.

• Age ratio: This concept involves finding the ratio of the ages of two or more people at a given time. The age ratio can be used to form an equation based on the given condition and solve for the unknowns.

For example, if a father is 40 years old and his son is 10 years old, the age ratio of the father to his son is 4:1. This can be used to solve problems such as "If the father is four times as old as his son, how old is the son?"

Solution: Let x be the age of the son. Then, the age of the father is 4x. We know that the father is 40 years old and the son is 10 years old. So, we can write:

4x = 40

x = 10

Therefore, the age of the son is 10 years old.

• Sum of ages: This concept involves finding the sum of the ages of two or more people at a given time. The sum of ages can be used to form an equation based on the given condition and solve for the unknowns.

For example, The sum of the ages of a father and his son is 50 years. The father is 30 years older than his son. What are their ages?

Solution: Let x be the age of the son. Then, the age of the father is x + 30. We know that the sum of their ages is 50. So, we can write:

x + (x + 30) = 50

2x + 30 = 50

2x = 20

x = 10

Therefore, the age of the son is 10 years old and the age of the father is 40 years old.

• Age difference: This concept involves finding the difference between the ages of two or more people at a given time. The age difference can be used to form an equation based on the given condition and solve for the unknowns.

For example,  A man is 10 years older than his wife. The difference between their ages is 20 years. What are their ages?

Solution: Let x be the age of the wife. Then, the age of the man is x + 10. We know that the difference between their ages is 20. So, we can write:

x + 10 - x = 20

10 = 20

This equation is not possible, as 10 cannot be equal to 20.
Therefore, there is no solution to this problem.

• Age after n years or before n years: This concept involves finding the age of a person after n years or before n years. This can be used to form an equation based on the given condition and solve for the unknowns.

For example,  The present age of a father is 40 years and the age of his son is half his age. What is the age of the son after 10 years?

Solution: Let x be the age of the son. Then, the age of the father is 2x. We know that the present age of the father is 40. So, we can write:

2x = 40

x = 20

Therefore, the age of the son is 20 years old. After 10 years, the age of the son will be:

20 + 10 = 30

Therefore, the age of the son after 10 years will be 30 years old.

• Average age: This concept involves finding the average age of a group of people. The average age can be used to form an equation based on the given condition and solve for the unknowns.

For example,  The average age of a group of 5 people is 30 years. If the age of one person is not known, and the sum of their ages is 150 years, what is the age of the unknown person?

Solution: Let x be the age of the unknown person. We know that the average age of the group is 30, and the sum of their ages is 150. So, we can write:

(4 * 30 + x) / 5 = 30

120 + x = 150

x = 30

Therefore, the age of the unknown person is 30 years old.

• Age and number of children: This concept involves finding the age of a parent based on the number of children they have and their ages. This can be used to form an equation based on the given condition and solve for the unknowns.

For example,  A father has three children whose ages are 2, 4, and 6 years. If the sum of their ages is equal to his age, what is the age of the father?

Solution: The sum of the ages of the children is:

2 + 4 + 6 = 12

Let x be the age of the father. We know that the sum of their ages is equal to his age. So, we can write:

x = 12

x = 24

Therefore, the age of the father is 24 years old.

• Age and their relation with each other: This concept involves finding the relation between the ages of two or more people at a given time. The relation can be used to form an equation based on the given condition and solve for the unknowns.

For example,  The age of a mother is three times the age of her daughter. If the sum of their ages is 36, what are their ages?

Solution: Let x be the age of the daughter. Then, the age of the mother is 3x. We know that the sum of their ages is 36. So, we can write:

x + 3x = 36

4x = 36

x = 9

Therefore, the age of the daughter is 9 years old and the age of the mother is 3 times that, which is:

3 * 9 = 27

Therefore, the age of the mother is 27 years old.