## Part 1: Understanding the Extended Concept of GMAT Age and Digit Problems

The Graduate Management Admission Test (GMAT) is renowned for its challenging quantitative reasoning section. One such topic that often stumps test-takers is the concept of Age and Digit problems. This comprehensive five-part series will demystify these problems and provide you with the tools to tackle them confidently.

### The Concept of GMAT Age and Digit Problems

The core of GMAT Age and Digit problems revolves around understanding algebra, specifically in the context of forming and solving equations. These problems typically involve an element of time and require you to solve for unknowns, often ages or numbers represented by digits.

For instance, you might encounter a problem stating, “Ten years ago, John was twice as old as Jane. Today, their combined age is 60 years. How old are they?” Or a digit problem like, “The digit in the tens place is three times the digit in the ones place. The number is even. What is the number?”

To solve these types of problems, you need to translate the words into algebraic equations and then solve them.

### Formula and Reasoning

The formulaic approach to these problems is to build an equation (or set of equations) that accurately represents the situation. For age problems, it often involves the current age, past or future age, and a relationship between the ages of multiple individuals. For digit problems, it involves understanding the positional value of each digit in a number and the relationship between these digits.

Considering our previous examples, for the age problem, we could say:

1. Let’s denote John’s current age as J and Jane’s as A.
2. According to the problem, John was twice as old ten years ago as Jane. So, we can write the equation as J – 10 = 2(A – 10).
3. The problem also states that John and Jane’s current ages sum up to 60. So, we can write another equation as J + A = 60.

We now have a system of two equations that we can solve simultaneously to find the values of J and A.

Similarly, for the digit problem, we could represent the tens digit as 10T and the one’s digit as O. The problem tells us that the tens digit is three times the one’s digit, or T = 3O and that the number is even.

### Shortcut and Practical Use

The shortcut to solving these problems often lies in understanding the properties of numbers and making intelligent guesses. In our digit problem, for instance, we know that an even number must end in 0, 2, 4, 6, or 8. We can use this knowledge to limit our potential solutions.

As for practical use, these problems help to enhance critical thinking and problem-solving skills. They may seem abstract, but their reasoning applies to many real-world scenarios. Age problems, for example, can be directly related to financial planning or demographic analysis.

Understanding the concept of GMAT Age and Digit problems is the first step towards mastering them. Stay tuned for the next part of this series, where we will delve deeper into the strategies and techniques to solve these problems efficiently.

## Part 2: Mastering the Art of Solving GMAT Age and Digit Problems

Welcome back to our comprehensive series on understanding and mastering GMAT Age and Digit problems. In the first part, we introduced the concept and explained the reasoning behind it. In this installment, we’ll delve into specific strategies and techniques to crack these types of problems.

### A Systematic Approach to Age Problems

Age problems primarily require understanding the time frame and relationships between individuals’ ages. Here’s a step-by-step strategy:

1. Identify the unknowns and assign variables to them. For example, let ‘A’ and ‘B’ denote the current ages of two people.
2. Translate the verbal clues into mathematical expressions or equations. If the problem says, “Person A is twice as old as Person B,” write it as A = 2B.
3. Formulate a system of equations and solve them. If another clue states, “The sum of their ages is 50,” add a second equation, A + B = 50. Solve this system of equations to find the values of A and B.

### Mastering Digit Problems

Digit problems involve understanding the positional value of digits in a number. Here’s a systematic approach to solving them:

1. Assign variables to the unknown digits. If a two-digit number is unknown, let ‘T’ represent the tens and O’ the ones digit.
2. Translate the given conditions into algebraic equations. If the problem states, “The tens digit is three times the one’s digit,” write it as T = 3O.
3. Solve the equations considering additional conditions. If the number is even, remember that the one digit (O) can be 0, 2, 4, 6, or 8.

### Shortcut Techniques

While it’s crucial to understand the systematic approach, knowing a few shortcut techniques can save time:

• For age problems, if there’s a relationship between ages at different times, consider setting your equations at the time when the ages are related to avoid dealing with too many variables.
• For digit problems, remember the properties of numbers. An even number’s one digit must be 0, 2, 4, 6, or 8. A number divisible by 3 must have its digits add up to a multiple of 3. These shortcuts can narrow down your options quickly.

### The Importance of Practice

Mastering GMAT Age and Digit problems is about more than just understanding the concept and formula. It’s about practicing these techniques until they become second nature. The more problems you solve, the more adept you become at identifying the underlying patterns and applying the appropriate techniques.

In the next part of this series, we’ll provide examples of GMAT Age and Digit problems and walk you through the solutions. This will help you apply the strategies and techniques discussed here.

## Part 3: Real-world Examples and Solutions to GMAT Age and Digit Problems

In the previous parts of our series, we’ve explored the concept, reasoning, and strategies for solving GMAT Age and Digit problems. Now, let’s solidify this knowledge by applying our techniques to real-world examples.

Example 1: GMAT Age Problem

Suppose the problem is: “Five years from now, Alice will be twice as old as Bob. Today, their total age is 25 years. How old are they?”

Step 1: Assign variables to the unknowns. Let’s say A represents Alice’s current age, and B represents Bob’s.

Step 2: Translate the problem into algebraic equations. The first part can be written as A + 5 = 2(B + 5). The second part can be written as A + B = 25.

Step 3: Solve the system of equations. Substituting A from the second equation into the first gives us 25 – B + 5 = 2(B + 5). Solving this equation, we find B = 10. Substituting B into the second equation, we find A = 15. So Alice is 15, and Bob is 10.

Example 2: GMAT Digit Problem

Consider this problem: “The digit in the tens place is three times the digit in the ones place, and the two-digit number is even. What is the number?”

Step 1: Assign variables to the unknown digits. Let T be the tens digit and O be the one’s digit.

Step 2: Translate the problem into an algebraic equation. This gives us T = 3O.

Step 3: Solve the equation considering the additional condition. Since the number is even, O can only be 0, 2, 4, 6, or 8. But T must be three times O, and it can’t exceed 9 (as it’s a digit). So, the only solution that fits is O = 2 and T = 6. So, the number is 62.

These examples illustrate how to apply our strategies to solve GMAT Age and Digit problems. As you can see, once you break down the problem into manageable steps, the solution becomes clearer.

In the next part of this series, we’ll discuss common pitfalls to avoid when solving these types of problems and how to overcome them. This will further enhance your problem-solving skills and boost your confidence.

## Part 4: Common Pitfalls and How to Avoid Them in GMAT Age and Digit Problems

Welcome back to the fourth part of our series on GMAT Age and Digit problems. Having discussed the concept, strategies, and solutions to these problems, it’s now crucial to understand common mistakes students make and how to avoid them.

### Pitfall 1: Misinterpretation of the Problem

Age and digit problems are often wordy and can be easily misunderstood. Misinterpretation can lead to incorrect equations and, subsequently, wrong answers.

Solution: Take the time to thoroughly read and understand the problem. Break it down into smaller parts, and translate each part into an equation. Don’t rush this process. A correct interpretation is half the battle won.

### Pitfall 2: Incorrect Assignment of Variables

Incorrectly assigning variables to unknowns can complicate your equations and make them harder to solve.

Solution: Clearly define your variables at the start. If there are multiple individuals or digits involved, ensure each has a distinct variable. Consistency in your variable assignment is key.

### Pitfall 3: Calculation Errors

Even with a correctly formed and simplified equation, calculation errors can lead to incorrect solutions.

Solution: Double-check your calculations at every step. If time allows, consider solving the problem using a different method as a check.

### Pitfall 4: Overlooking Constraints

Some problems have specific constraints, such as the number being even in digit problems. Ignoring these constraints can lead to incorrect answers.

Solution: Always take note of any specific conditions or constraints given in the problem and ensure your solution meets these conditions.

### Pitfall 5: Lack of Practice

Without sufficient practice, you may not be able to recognize patterns or apply strategies efficiently under exam conditions.

Solution: Practice consistently. Solve a variety of age and digit problems to familiarize yourself with different scenarios and improve your problem-solving speed.

The ability to identify and avoid these pitfalls will significantly improve your efficiency in solving GMAT Age and Digit problems. Remember, it’s not just about knowing the right strategies but also about applying them effectively.

In the final part of our series, we’ll discuss some advanced techniques and tips to further enhance your problem-solving skills. Stay tuned!

Keywords: GMAT, Age and Digit problems, pitfalls, solutions, strategies, practice.

(Join us for the final part of this series, where we’ll dive into advanced techniques for mastering GMAT Age and Digit problems.)

Part 5: Advanced Techniques and Final Tips for GMAT Age and Digit Problems

Welcome to the final part of our comprehensive series on GMAT Age and Digit problems. We’ve covered the basics, strategies, solutions, and pitfalls to avoid. Now, let’s elevate your skills further with some advanced techniques and tips.

In some cases, especially when the answer choices are given, you can use back-solving. Start by plugging the answer choices into the problem and see which one fits. This technique can save time, particularly when the algebraic solution seems complex.

Sometimes, exact calculations aren’t necessary. You can estimate the answer, especially when the answer choices are widely spaced. This technique can save precious time during the exam.

### Advanced Technique 3: Logical Reasoning

Certain problems can be solved faster using logical reasoning rather than pure algebra. Look for patterns, consider the constraints of the problem, and use logical deduction to arrive at the solution.

## Final Tips for Mastering GMAT Age and Digit Problems

1. Practice Regularly: Regular practice is the key to mastery. The more problems you solve, the more comfortable you’ll become with these types of questions.
2. Review Mistakes: Don’t just correct your mistakes. Understand why you made them, and learn how to avoid them in the future.
3. Understand the Concepts: Memorizing strategies isn’t enough. You need to understand the underlying concepts, as this will allow you to adapt to different problem types.
4. Manage Your Time: Practice solving problems within a set time. This will help you manage your time effectively during the actual exam.
5. Stay Calm: Don’t panic if you find a problem challenging. Take a deep breath, read the problem carefully, and break it down into manageable parts.

Mastering GMAT Age and Digit problems is a journey. With understanding, practice, and the right strategies, you can confidently solve these problems and score high on your GMAT. Good luck with your preparation!

## Problem-Solving

Question 1:

Adam is currently twice as old as Brian. If the sum of their ages is 36, how old is Brian?

A) 10

B) 12

C) 14

D) 16

E) 18

Solution:

Let’s denote Adam’s age as A and Brian’s as B. We have two equations according to the problem:

1. A = 2B (Adam is twice as old as Brian)
2. A + B = 36 (The sum of their ages is 36)

Substitute the first equation into the second to get 2B + B = 36, simplifying to 3B = 36. Solving for B, we get B = 12. Therefore, Brian is 12 years old.

Question 2:

A two-digit number has a tens digit that is 3 times the one digit. If the number is even, what is the number?

A) 12

B) 24

C) 36

D) 42

E) 62

Solution:

From the problem, we know that the tens digit is three times the ones digit, and the number is even. This means the ones digit must be 2 (an even number), and the tens digit must be 6 (three times 2). Therefore, the number is 62.

Question 3:

Five years ago, Mary was three times as old as her son. Today, the sum of their ages is 50. How old is Mary’s son?

A) 10

B) 12

C) 15

D) 17

E) 20

Solution:

Let’s denote Mary’s current age as M and her son’s age as S. We have two equations:

1. M – 5 = 3(S – 5) (Five years ago, Mary was three times as old as her son)
2. M + S = 50 (The sum of their ages is 50)

Substitute the first equation into the second to get 3S – 10 + S = 50, simplifying to 4S = 60. Solving for S, we get S = 15. Therefore, Mary’s son is 15 years old.

Question 4:

A two-digit number is such that the number obtained by reversing its digits is 27 more than the original number. If the tens digit is 2 more than the ones digit, what is the original number?

A) 14

B) 26

C) 38

D) 45

E) 57

Solution:

The problem shows that the tens digit is two more than the ones digit, and the reversed number is 27 more than the original number. Looking at the answer choices, we can see that the only number that meets these conditions is 38. If we reverse it, we get 83, which is 27 more than 38. Also, the tens digit (3) is 2 more than the ones digit (1).

Question 5:

Tom is twice as old as Jerry. In 6 years, Tom will be 40 years old. How old is Jerry now?

A) 12

B) 14

C) 16

D) 17

E) 18

Solution:

According to the problem, Tom is twice as old as Jerry, and in 6 years, Tom will be 40 years old. This means Tom is currently 40 – 6 = 34 years old. Since Tom is twice as old as Jerry, Jerry must be 34 / 2 = 17 years old.

Question 6:

A two-digit number has a tens digit that is 2 times the one digit. The sum of the digits is 9. What is the number?

A) 36

B) 45

C) 54

D) 63

E) 72

Solution:

From the problem, we know that the tens digit is two times the ones digit, and the sum of the digits is 9. The only number in the choices that meets these conditions is 63. The tens digit (6) is twice the ones digit (3), and their sum is 9.

Question 7:

If John is now twice as old as Sam, and the difference between Sam’s age 5 years from now and John’s age 5 years ago is one, how old is John now?

A) 15

B) 18

C) 20

D) 23

E) 26

Solution:

Let’s denote John’s current age as J and Sam’s as S. We have two equations:

1. J = 2S (John is twice as old as Sam)
2. S + 5 = J – 5 + 1 (The difference between Sam’s age 5 years from now and John’s age 5 years ago is one)

Substitute the first equation into the second to get 2S + 5 = 2S – 5 + 1, simplifying to 2S = 10. Solving for S, we get S = 5. Substituting S into the first equation, we get J = 2*5 = 10.

Question 8:

A two-digit number is such that the sum of the digits is 11, and the number obtained by reversing the digits is 27 less than the original number. What is the original number?

A) 29

B) 38

C) 47

D) 56

E) 65

Solution:

From the problem, we know that the sum of the digits is 11, and the reversed number is 27 less than the original number. The only number in the choices that meets these conditions is 38. If we reverse it, we get 83, which is 27 more than 38. Also, the sum of the digits (3 + 8) equals 11.

Question 9:

Three years from now, Mary’s age will be twice that of her daughter. Five years ago, the sum of their ages was 30. How old is Mary now?

A) 27

B) 31

C) 33

D) 35

E) 37

Solution:

Let’s denote Mary’s current age as M and her daughter’s as D. We have two equations:

1. M + 3 = 2(D + 3) (Three years from now, Mary will be twice as old as her daughter)
2. M – 5 + D – 5 = 30 (Five years ago, the sum of their ages was 30)

Solving these simultaneous equations, we find that Mary’s age is 35.

Question 10:

A two-digit number is such that the number obtained by reversing its digits is 45 more than the original number. If the sum of the digits is 10, what is the original number?

A) 19

B) 28

C) 37

D) 46

E) 55

Solution:

From the problem, we know that the reversed number is 45 more than the original number, and the sum of the digits is 10. The only number in the choices that meets these conditions is 37. If we reverse it, we get 73, which is 45 more than 37. Also, the sum of the digits (3 + 7) equals 10.

Question 11:

If Robert is 4 years younger than his sister Mary and the product of their ages is 320, how old is Mary?

A) 16

B) 18

C) 20

D) 22

E) 24

Solution:

Let’s denote Mary’s age as M and Robert’s as R. We have two equations:

1. R = M – 4 (Robert is 4 years younger than Mary)
2. R * M = 320 (The product of their ages is 320)

Substitute the first equation into the second to get (M – 4) * M = 320. Solving this quadratic equation, we find that Mary’s age is 20.

Question 12:

A two-digit number is such that the number obtained by reversing its digits is 36 less than the original number. If the tens digit is 3 more than the ones digit, what is the original number?

A) 21

B) 32

C) 43

D) 54

E) 65

Solution:

From the problem, we know that the reversed number is 36 less than the original number, and the tens digit is 3 more than the ones digit. The only number in the choices that meets these conditions is 43. If we reverse it, we get 34, which is 36 less than 43. Also, the tens digit (4) is 3 more than the ones digit (1).

Question 13:

The ages of three friends are in the ratio of 3:4:5. If the sum of the ages of the youngest and the oldest friend is 56, how old is the middle friend?

A) 16

B) 20

C) 24

D) 28

E) 32

Solution:

Let the ages of the three friends be 3x, 4x, and 5x. According to the problem, the sum of the youngest and oldest friend’s ages is 56, so 3x + 5x = 56, or 8x = 56. Solving for x gives us x = 7. Therefore, the age of the middle friend is 4x = 4*7 = 28.

Question 14:

A two-digit number is such that the number obtained by reversing its digits is 54 more than the original number, and the sum of the squares of the digits is 85. What is the original number?

A) 47

B) 56

C) 64

D) 73

E) 82

Solution:

From the problem, we know that the reversed number is 54 more than the original number, and the sum of the squares of the digits is 85. The only number in the choices that meets these conditions is 29. If we reverse it, we get 92, which is 54 more than 29. Also, the sum of the squares of the digits (2^2 + 9^2) equals 85.

Question 15:

In five years, the age of David will be three times the age of his son. Three years ago, the sum of their ages was 27. How old is David’s son now?

A) 6

B) 8

C) 10

D) 12

E) 14

Solution:

Let’s denote David’s current age as D and his son’s as S. We have two equations:

1. D + 5 = 3(S + 5) (In five years, David will be three times as old as his son)
2. D – 3 + S – 3 = 27 (Three years ago, the sum of their ages was 27)

Solving these simultaneous equations, we find that David’s son’s age is 8.

Question 16:

A two-digit number is such that the number obtained by reversing its digits is 72 more than the original number, and the tens digit is 5 less than twice the one digit. What is the original number?

A) 18

B) 27

C) 36

D) 45

E) 54

Solution:

From the problem, we know that the reversed number is 72 more than the original number, and the tens digit is 5 less than twice the ones digit. The only number in the choices that meets these conditions is 18. If we reverse it, we get 81, which is 72 more than 18. Also, the tens digit (1) is 5 less than twice the ones digit (2*4 – 5 = 1).

Question 17:

The ages of a father and son are in the ratio of 11:3. In 8 years, the ratio of their ages will be 13:5. How old is the son currently?

A) 12

B) 15

C) 18

D) 21

E) 24

Solution:

Let’s denote the father’s current age as 11x and the son’s as 3x. According to the problem, in 8 years, the ratio of their ages will be 13:5. Therefore, we get this equation: (11x + 8) / (3x + 8) = 13 / 5. Solving this equation for x gives us x = 3. Hence, the son’s current age is 3x = 3 * 3 = 9.

Question 18:

A two-digit number is such that the number obtained by reversing its digits is 63 more than the original number, and the difference between the tens digit and the ones digit is 3. What is the original number?

A) 26

B) 37

C) 48

D) 59

E) 72

Solution:

From the problem, we know that the reversed number is 63 more than the original number, and the difference between the tens digit and the ones digit is 3. The only number in the choices that meets these conditions is 37. If we reverse it, we get 73, which is 63 more than 37. Also, the difference between the tens digit (3) and the ones digit (7) is 3.

Question 19:

John, Paul, and George have ages that are consecutive, even integers. If the sum of twice John’s age, thrice Paul’s age, and four times George’s age is 124, how old is Paul?

A) 8

B) 10

C) 12

D) 14

E) 16

Solution:

Let John’s age be x, Paul’s age be x+2, and George’s age be x+4. We have the equation:

2x + 3(x+2) + 4(x+4) = 124

Solving this equation, we find x=10. Therefore, Paul’s age is x+2 = 12.

Question 20:

A two-digit number is such that the number obtained by reversing its digits is 108 more than the original number. If the product of the digits is 20, what is the original number?

A) 24

B) 35

C) 46

D) 57

E) 68

Solution:

From the problem, we know that the reversed number is 108 more than the original number, and the product of the digits is 20. The only number in the choices that meets these conditions is 68. If we reverse it, we get 86, which is 108 less than 68. Also, the product of the digits (6*8) equals 20.

Question 21:

The sum of the ages of a father and his son is 50 years. Four years ago, the product of their ages was 4 times the son’s age at that time. How old is the son now?

A) 10

B) 12

C) 14

D) 16

E) 18

Solution:

Let the son’s current age be S and the father’s be F. We have two equations:

1. S + F = 50 (The sum of their ages is 50)
2. (S – 4)(F – 4) = 4(S – 4) (Four years ago, the product of their ages was 4 times the son’s age)

From the first equation, we can express F as F = 50 – S. Substituting this into the second equation and solving for S gives S = 14.

Question 22:

A two-digit number is such that the number obtained by reversing its digits is 99 more than the original number, and the sum of the squares of the digits is 65. What is the original number?

A) 26

B) 37

C) 48

D) 59

E) 82

Solution:

From the problem, we know that the reversed number is 99 more than the original number, and the sum of the squares of the digits is 65. The only number in the choices that meets these conditions is 26. If we reverse it, we get 62, which is 99 less than 26. Also, the sum of the squares of the digits (2^2 + 6^2) equals 65.

Question 23:

The ages of a father and son are in the ratio of 7:2. Six years ago, the ratio of their ages was 5:1. How old is the son currently?

A) 12

B) 14

C) 16

D) 18

E) 20

Solution:

Let’s denote the father’s current age as 7x and the son’s as 2x. According to the problem, six years ago, the ratio of their ages was 5:1. Therefore, we get this equation: (7x – 6) / (2x – 6) = 5 / 1. Solving this equation for x gives us x = 4. Hence, the son’s current age is 2x = 2 * 4 = 8.

Question 24:

A two-digit number is such that the number obtained by reversing its digits is 110 more than the original number, and the difference between twice the tens digit and the ones digit is 7. What is the original number?

A) 34

B) 45

C) 56

D) 67

E) 78

Solution:

From the problem, we know that the reversed number is 110 more than the original number, and the difference between twice the tens digit and the ones digit is 7. The only number in the choices that meets these conditions is 45. If we reverse it, we get 54, which is 110 less than 45. Also, the difference between twice the tens digit (2*4) and the ones digit (5) is 7.

Question 25:

A father is three times as old as his son. How old is the son?

Statement 1: The sum of their ages is 40.

Statement 2: The difference in their ages is 20 years.

Solution:

Statement 1: If the sum of their ages is 40, then we can express their ages as son’s age = x, father’s age = 3x. From the equation x + 3x = 40, we find that x = 10, so the son is 10 years old.

Statement 2: If the difference in their ages is 20 years, then the equation becomes 3x – x = 20, from which we find that x = 10, so the son is 10 years old.

Thus, each statement alone is sufficient to answer the question.

Answer: D) Each statement alone is sufficient.

Question 26:

Is the number x an integer?

Statement 1: x is evenly divisible by 2.

Statement 2: x is evenly divisible by 3.

Solution:

Statement 1: If x is evenly divisible by 2, it does not necessarily mean that x is an integer. It could be a fraction or a decimal as well, such as 0.5 or 1.5, which are divisible by 2.

Statement 2: Similarly, if x is evenly divisible by 3, it does not necessarily mean that x is an integer. It could be a fraction or a decimal, such as 0.3 or 1.5, which are divisible by 3.

Even when considering both statements together, we cannot determine whether x is an integer. It could be a decimal or fraction that is divisible by both 2 and 3, such as 1.5.

Answer: E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Question 27:

A man is now three times as old as his son. In six years, the man will be twice as old as his son will be then. How old is the son now?

Statement 1: The man is 36 years old.

Statement 2: The difference in their ages is 24 years.

Solution:

Statement 1: If the man is 36 years old, then the son is 36/3 = 12 years old. In six years, the man will be 42, and the son will be 18, and indeed, 42 is twice 18. Thus, statement 1 alone is sufficient to answer the question.

Statement 2: If the difference in their ages is 24 years, then the son’s age is x, and the father’s age is x + 24. The equation from the problem statement becomes 2(x + 6) = (x + 24 + 6). Solving this equation, we find that x = 12. Thus, statement 2 alone is sufficient to answer the question.

Answer: D) Each statement alone is sufficient.

Question 28:

A two-digit number is such that the sum of its digits is 9, and the number obtained by reversing its digits is 27 more than the original number. What is the original number?

Statement 1: The tens digit is 3 more than the ones digit.

Statement 2: The original number is less than 50.

Solution:

Statement 1: If the tens digit is 3 more than the ones digit, the original number could be 36 or 47. However, only 36 meets the condition that the number obtained by reversing its digits (63) is 27 more than the original number (36). Thus, statement 1 alone is sufficient to answer the question.

Statement 2: If the original number is less than 50, there are several two-digit numbers with digits summing to 9 (e.g., 18, 27, 36, 45). However, only 36 meets the condition that the number obtained by reversing its digits (63) is 27 more than the original number (36). Thus, statement 2 alone is sufficient to answer the question.

Answer: D) Each statement alone is sufficient.

Question 29:

John is now four times as old as his son. In five years, John will be three times as old as his son will be then. How old is the son now?

Statement 1: The difference in their ages is 30 years.

Statement 2: John is 40 years old.

Solution:

Statement 1: If the difference in their ages is 30 years, then the son’s age is x, and John’s age is x + 30. The equation from the problem statement becomes 3(x + 5) = (x + 30 + 5). Solving this equation, we find that x = 10. Thus, statement 1 alone is sufficient to answer the question.

Statement 2: If John is 40 years old, then the son is 40/4 = 10 years old. In five years, John will be 45, and the son will be 15, and indeed, 45 is three times 15. Thus, statement 2 alone is sufficient to answer the question.

Answer: D) Each statement alone is sufficient.

Question 30:

A two-digit number is such that the difference of its digits is 4, and the number obtained by reversing its digits is 36 more than the original number. What is the original number?

Statement 1: The tens digit is 4 more than the ones digit.

Statement 2: The original number is less than 50.

Solution:

Statement 1: If the tens digit is 4 more than the ones digit, the original number could be 14, 25, 36, or 47. However, only 14 meets the condition that the number obtained by reversing its digits (41) is 36 more than the original number (14). Thus, statement 1 alone is sufficient to answer the question.

Statement 2: If the original number is less than 50, there are several two-digit numbers with digit differences of 4 (e.g., 14, 25, 36, 47). However, only 14 meets the condition that the number obtained by reversing its digits (41) is 36 more than the original number (14). Thus, statement 2 alone is sufficient to answer the question.

Answer: D) Each statement alone is sufficient.

Question 31:

Sasha is now twice as old as her daughter. In ten years, Sasha will be 1.5 times as old as her daughter will be then. How old is the daughter now?

Statement 1: The difference in their ages is 20 years.

Statement 2: Sasha is 40 years old.

Solution:

Statement 1: If the difference in their ages is 20 years, then the daughter’s age is x, and Sasha’s age is x + 20. The equation from the problem statement becomes 1.5(x + 10) = (x + 20 + 10). Solving this equation, we find that x = 20. Thus, statement 1 alone is sufficient to answer the question.

Statement 2: If Sasha is 40 years old, then the daughter is 40/2 = 20 years old. In ten years, Sasha will be 50, and the daughter will be 30, and indeed, 50 is 1.5 times 30. Thus, statement 2 alone is sufficient to answer the question.

Answer: D) Each statement alone is sufficient.

Question 32:

A two-digit number is such that the product of its digits is 12, and the number obtained by reversing its digits is 36 more than the original number. What is the original number?

Statement 1: The tens digit is three less than the ones digit.

Statement 2: The original number is less than 40.

Solution:

Statement 1: If the tens digit is three less than the ones digit, the original number could be 14, 25, or 36. However, only 14 meets the condition that the number obtained by reversing its digits (41) is 36 more than the original number (14). Thus, statement 1 alone is sufficient to answer the question.

Statement 2: If the original number is less than 40, there are several two-digit numbers with digit products of 12 (e.g., 14, 21, 34). However, only 14 meets the condition that the number obtained by reversing its digits (41) is 36 more than the original number (14). Thus, statement 2 alone is sufficient to answer the question.

Answer: D) Each statement alone is sufficient.

Question 33:

A father is five times as old as his son. In seven years, the father will be four times as old as his son will be then. How old is the son now?

Statement 1: The difference in their ages is 40 years.

Statement 2: The son will be 16 years old in three years.

Solution:

Statement 1: If the difference in their ages is 40 years, then the son’s age is x, and the father’s age is x + 40. The equation from the problem statement becomes 4(x + 7) = (x + 40 + 7). Solving this equation, we find that x = 12. Thus, statement 1 alone is sufficient to answer the question.

Statement 2: If the son will be 16 years old in three years, then the son is currently 13 years old. If we plug this into the problem statement, we find that the father is currently 65 years old, which satisfies the condition. Thus, statement 2 alone is sufficient to answer the question.

Answer: D) Each statement alone is sufficient.

Question 34:

A two-digit number is such that the product of its digits is 15, and the number obtained by reversing its digits is 27 more than the original number. What is the original number?

Statement 1: The tens digit is two less than the ones digit.

Statement 2: The original number is less than 40.

Solution:

Statement 1: If the tens digit is two less than the ones digit, the original number could be 35 or 24. However, only 35 meets the condition that the number obtained by reversing its digits (53) is 27 more than the original number (35). Thus, statement 1 alone is sufficient to answer the question.

Statement 2: If the original number is less than 40, there are several two-digit numbers with digit products of 15 (e.g., 15, 24, 35, 51). However, only 35 meets the condition that the number obtained by reversing its digits (53) is 27 more than the original number (35). Thus, statement 2 alone is sufficient to answer the question.

Answer: D) Each statement alone is sufficient.