Question 1

The sum of the ages of Jayant, Prem and Paras is 93 years. Ten years ago, the ratio of their ages was 2 : 3 : 4. What is the present age of Paras?

Accepted Answer

Concept & Logic: To use a past ratio, you must adjust the present sum to match the past timeframe. Subtract the elapsed time for *each* person involved before distributing the total according to the ratio. Past Sum = Present Sum - (n × years) Step-by-Step Solution: • Given: Present sum = 93. Ratio 10 years ago = 2:3:4. Number of people = 3. • Calculation: Find the sum of their ages 10 years ago. • Past sum = 93 - (3 × 10) = 93 - 30 = 63 years. • 10 years ago, their ages were 2x, 3x, and 4x. • Equation: 2x + 3x + 4x = 63 \implies 9x = 63 \implies x = 7. • Paras's ratio share 10 years ago was 4x. • Paras's age 10 years ago = 4 × 7 = 28 years. • Paras's present age = 28 + 10 = 38 years. Exam Strategy & Shortcut: Option Alignment: The question asks for the present age of Paras. Since his ratio 10 years ago was 4, his (Present Age - 10) must be a multiple of 4. Check the options: (a) 24 - 10 = 14 (No) (b) 28 - 10 = 18 (No) (c) 32 - 10 = 22 (No) (d) 34 - 10 = 24 (Yes) (e) 38 - 10 = 28 (Yes) We have two candidates. But 34 implies x=6, making past sum 54, which doesn't match 63. So 38 is the only correct answer. Common Pitfall: Finding the age 10 years ago (28) and forgetting to add 10 to get the *present* age. Examiners frequently put 28 in the options as a trap (Option B). Final Answer: Therefore, the correct answer is 38 years.

Question 2

The ratio between the present ages of A and B is 5 : 3 respectively. The ratio between A's age 4 years ago and B's age 4 years hence is 1 : 1. What is the ratio between A's age 4 years hence and B's age 4 years ago?

Accepted Answer

Concept & Strategy: When one person's past age equals another person's future age (a 1 : 1 ratio), it means their present ages are separated by the sum of those two time shifts. We can model this linearly using their present ratio. (A - t_1/B + t_2) = (1/1) \implies A - t_1 = B + t_2 Step-by-Step Solution: • Given: Present ratio A : B = 5 : 3. Condition: (A's age - 4) : (B's age + 4) = 1 : 1. • Calculation: Let A's present age be 5x and B's present age be 3x. • A's age 4 years ago = 5x - 4. • B's age 4 years hence = 3x + 4. • Equate them according to the 1 : 1 ratio: 5x - 4 = 3x + 4 • Solve for x: 2x = 8 \implies x = 4. • Calculate their exact present ages: A = 5(4) = 20 years. B = 3(4) = 12 years. • We need the ratio of A's age 4 years hence to B's age 4 years ago. • A's age in 4 years = 20 + 4 = 24. • B's age 4 years ago = 12 - 4 = 8. • Find the final ratio: 24 : 8 = 3 : 1. Exam Strategy & Shortcut: Logical Gap Analysis: The condition 5x - 4 = 3x + 4 shows that the difference between their current ages (2x) is exactly equal to the total time shift (8 years). This immediately gives x=4. Since you want A + 4 and B - 4, you are essentially pushing A one x-unit forward and pulling B one x-unit back. The new ratio is (5+1) : (3-1) = 6 : 2 = 3 : 1. Common Pitfall: Misreading the final question and finding the ratio of their *present* ages or accidentally calculating B's future age against A's past age again. Always double-check the final prompt direction. Final Answer: Therefore, the correct answer is 3 : 1.

Question 3

The ratio of the ages of a man and his wife is 4 : 3. After 4 years, this ratio will be 9 : 7. If at the time of their marriage, the ratio of their ages was 5 : 3, then how many years ago were they married?

Accepted Answer

Concept & Strategy: This is a three-stage timeline problem (Marriage \rightarrow Present \rightarrow Future). First, solve for the present ages using the present-to-future transition, then use those established ages to find the historical marriage timeline. \frac{Present_A + t}{Present_B + t} = Future Ratio Step-by-Step Solution: • Given: Present ratio = 4 : 3. Ratio in 4 years = 9 : 7. Marriage ratio = 5 : 3. • Calculation: Let the present ages of the man and wife be 4x and 3x. • Establish the future equation: (4x + 4/3x + 4) = (9/7) • Cross-multiply to solve for x: 7(4x + 4) = 9(3x + 4) • Expand: 28x + 28 = 27x + 36 • Isolate x: x = 8. • Calculate their exact present ages: Man = 4(8) = 32 years. Wife = 3(8) = 24 years. • Now, let y be the number of years ago they got married. • Set up the past ratio equation: (32 - y/24 - y) = (5/3) • Cross-multiply again: 3(32 - y) = 5(24 - y) • Expand: 96 - 3y = 120 - 5y • Solve for y: 2y = 24 \implies y = 12. Exam Strategy & Shortcut: Cross-Product Method: For the present-to-future step: Present = 4 : 3 Future (+4) = 9 : 7 Difference in cross products = (9 × 3) - (7 × 4) = 27 - 28 = 1 unit. Difference in time cross products = (9 × 4) - (7 × 4) = 36 - 28 = 8 years. So, 1 unit = 8 years. Present ages are 32 and 24. Check options for marriage y years ago. If y=12, ages were 20 and 12, which simplifies exactly to 5 : 3. Common Pitfall: Getting bogged down trying to create a massive single equation with both x and y simultaneously. Break the problem into two distinct, manageable phases: find present ages first, then find the marriage date. Final Answer: Therefore, the correct answer is 12 years.

Question 4

The difference between the ages of two men is 10 years.  15 years ago, the elder one was twice as old as the younger one.  The present age of the elder man is

Accepted Answer

Concept & Logic: The golden rule of age problems applies perfectly here: The difference in age between two individuals remains constant throughout their lives. If the difference is 10 years today, it was also exactly 10 years 15 years ago. Step-by-Step Solution: • Given: 15 years ago, the elder man was twice as old as the younger man. • Let the younger man's age 15 years ago be x. • Then, the elder man's age 15 years ago was 2x. • Calculation: We know the age difference is always 10 years. We can express this using their ages from 15 years ago: Elder - Younger = 10 2x - x = 10 \implies x = 10 • So, 15 years ago, the younger man was 10 years old, and the elder man was 2(10) = 20 years old. • Finding Present Age: The question asks for the present age of the elder man. • Present Age = (Age 15 years ago) + 15 • Elder man's present age = 20 + 15 = 35 years. Exam Strategy & Shortcut: This problem should take 5 seconds. Whenever one person is "twice as old" as another, the *difference* in their ages is exactly equal to the *younger person's age* (2x - x = x). Since the difference is 10, the younger person was 10 at that time. The elder was 10 × 2 = 20. Add 15 years to get the present age: 20 + 15 = 35. No written algebra required! Common Pitfall: A standard trap is solving for the age 15 years ago (20 years) and hastily marking it as the final answer if it were an option. Always re-read the final sentence of the question to confirm whether it asks for the *past*, *present*, or *future* age. Final Answer: Therefore, the correct answer is 35 years.

Question 5

The ages of Sulekha and Arunima are in the ratio 9 : 8 respectively.  After 5 years, the ratio of their ages will be 10 : 9.  What is the difference in their ages?

Accepted Answer

Concept & Formula: This is the most ideal scenario for the Direct Ratio Unit Method. When the difference in ratio units between two people is naturally balanced across time periods, the problem can be solved in a single step without equations. Step-by-Step Solution: • Given: The present age ratio of Sulekha to Arunima is 9 : 8. Let their ages be 9x and 8x. • Equation Setup: After 5 years, the ratio is 10 : 9. (9x + 5/8x + 5) = (10/9) • Calculation: 9(9x + 5) = 10(8x + 5) 81x + 45 = 80x + 50 x = 5 • Finding Target Value: The difference in their ages is 9x - 8x = x. • Since x = 5, the difference in their ages is 5 years. Exam Strategy & Shortcut: You can solve this instantly by visually inspecting the ratios. Present Ratio \implies 9 : 8 Future Ratio (+5 yrs) \implies 10 : 9 Notice that Sulekha's ratio unit increases from 9 to 10 (1 unit increase). Arunima's ratio unit increases from 8 to 9 (1 unit increase). Because the unit increase perfectly matches for both, we can directly equate it to the time passed: 1 ratio unit = 5 years The question asks for the difference in their ages. In the ratio (9:8), the difference is exactly 9 - 8 = 1 ratio unit. Since 1 unit = 5 years, the age difference is 5 years. No algebra needed. Common Pitfall: The only real pitfall here is overcomplicating it. Students often instinctively dive into cross-multiplication (81x + 45 = 80x + 50) instead of taking two seconds to realize the ratio units are already perfectly balanced. In a timed exam, recognizing this pattern saves crucial seconds. Final Answer: Therefore, the correct answer is 5 years.

Question 6

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. (a/2/4b) = (5/12) (a/8b) = (5/12) Cross-multiply to find a relationship: 12a = 40b \implies 3a = 10b \implies a = (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: (a + 12/2) = (b + 12) - 2 (a + 12/2) = b + 10 a + 12 = 2b + 20 a - 2b = 8 • Calculation (Part 3: Solving for B): Substitute a = (10b/3) into the new equation: (10b/3) - 2b = 8 (10b - 6b/3) = 8 (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: (a/8(6)) = (5/12) \implies (a/48) = (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.

Question 7

Directions : Each of the questions given below consists of a statement and / or a question and two statements numbered I and II given below it. You have to decide whether the data provided in the statement(s) is / are sufficient to answer the questi…

Accepted Answer

Concept & Logic: To find the age of a specific person (C) joining a group, you need to find the difference between the sum of the ages of the new group (A + B + C) and the sum of the ages of the old group (A + B) at the same point in time (present). Average formulas are the key to finding these sums. Step-by-Step Solution: • Evaluating Statement I alone: "Three years ago, the average of A and B was 18 years." Sum of A and B three years ago = 18 × 2 = 36 years. To find their present sum, we must add 3 years for A and 3 years for B. Present sum of A and B = 36 + 3 + 3 = 42 years. This tells us nothing about C. *Statement I alone is NOT sufficient.* • Evaluating Statement II alone: "With C joining them now, the average becomes 22 years." This means the present average of A, B, and C is 22. Present sum of A, B, and C = 22 × 3 = 66 years. Without knowing the sum of A and B, we cannot isolate C. *Statement II alone is NOT sufficient.* • Evaluating Statements I and II together: From I, we know the present sum of A + B = 42. From II, we know the present sum of A + B + C = 66. By subtracting the first sum from the second, we can easily find C's present age: C = (A + B + C) - (A + B) C = 66 - 42 = 24 years. Since combining both gives us the answer, both are necessary. Exam Strategy & Shortcut: When dealing with averages of groups, focus entirely on the Sum. The moment you realize Statement I gives you the present sum of (A+B) and Statement II gives you the present sum of (A+B+C), you instantly know subtraction will yield C. You do not need to calculate 66 - 42 = 24 to mark the correct Data Sufficiency option. Common Pitfall: The most dangerous pitfall is bringing past sums to the present incorrectly. Students often find the sum 3 years ago (18 × 2 = 36) and then just add 3 to get the present sum (36 + 3 = 39). You MUST add the time gap for *every* individual in the group (36 + 3 × 2 = 42). Final Answer: Therefore, the correct answer is The data in both Statements I and II together are necessary to answer the question.

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