Question 1

At present, Suresh's age is twice the age of his daughter. After 6 years from now, the ratio of the ages of Suresh and his daughter will be $23 : 13$. What is the present age of Suresh?

Accepted Answer

Concept & Formula: The logic revolves around defining present ages using a multiplier (twice), advancing those ages into the future mathematically, and equating them to a newly given ratio. Step-by-Step Solution: • Given: • Present: Suresh = 2 × Daughter • Future (+6 years): Suresh : Daughter = 23 : 13 • Calculation / Deduction: • Let the daughter's present age be x. • Then, Suresh's present age is 2x. • After 6 years, their ages will be: • Daughter = x + 6 • Suresh = 2x + 6 • The problem states the future ratio is 23 : 13. Set up the equation: • (2x + 6/x + 6) = (23/13) • Cross-multiply to solve for x: • 13(2x + 6) = 23(x + 6) • 26x + 78 = 23x + 138 • Group the variables and constants: • 26x - 23x = 138 - 78 • 3x = 60 • x = 20 • The daughter's present age is 20. • Suresh's present age = 2x = 2 × 20 = 40 years. Exam Strategy & Shortcut: Use Option Elimination with Divisibility Rules. Suresh's present age must be an even number (since he is twice his daughter's age). All given options are even. After 6 years, Suresh's age must be a multiple of 23 (from the 23:13 ratio). Let's test the options by adding 6: (a) 36 + 6 = 42 (Not divisible by 23) (b) 40 + 6 = 46 (Divisible by 23, since 23 × 2 = 46) (c) 46 + 6 = 52 (Not divisible by 23) Option (b) is the only mathematically viable answer. This takes 10 seconds and requires zero algebra. Common Pitfall: Students often incorrectly place the 6 years. They might multiply the present ratio by 6 or forget to add 6 to *both* the numerator and denominator when setting up the future fraction. Always remember that time passes equally for both individuals. Final Answer: Therefore, the correct answer is 40 years.

Question 2

The difference between the present ages of Arun and Deepak is 14 years. Seven years ago, the ratio of their ages was $5 : 7$ respectively. What is Deepak's present age?

Accepted Answer

Concept & Logic: The cardinal rule of age problems is that the age difference between two people remains absolutely constant throughout their lives. If the difference between Arun and Deepak is 14 years today, it was also exactly 14 years a decade ago, and will be 14 years a century from now. Step-by-Step Solution: • Given: • Present age difference = 14 years. • 7 years ago, Arun : Deepak = 5 : 7. • Calculation / Deduction: • Because the age difference never changes, the difference 7 years ago was also 14 years. • Let their ages 7 years ago be 5x (Arun) and 7x (Deepak). • Set up the equation for their difference: • 7x - 5x = 14 • 2x = 14 • x = 7 • Now, find Deepak's age 7 years ago: • Deepak (7 years ago) = 7x = 7 × 7 = 49 years. • To find Deepak's present age, add the 7 years back: • Deepak (Present) = 49 + 7 = 56 years. Exam Strategy & Shortcut: Look at the ratio gap. Seven years ago, the ratio was 5 : 7. The gap in ratio units is 2 units (7 - 5 = 2). We logically know this 2-unit gap represents the constant 14-year age difference. Therefore, 2 units = 14 years \Rightarrow 1 unit = 7 years. Deepak was 7 units (7 × 7 = 49 years old) in the past. Add 7 to reach the present (49 + 7 = 56). This completely bypasses algebraic equations. Common Pitfall: The most dangerous trap is solving for the past age (49) and selecting it from the options (Option C is specifically designed as a trap). Always perform a final check on the timeline the question is asking for ("What is Deepak's present age?"). Final Answer: Therefore, the correct answer is 56 years.

Question 3

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 × Son • 2 years hence (future): 2 × Man = 5 × 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 × 4 = 28. Present man = 28 + 10 = 38. Future man (+2 yrs) = 40. Future son (+2 yrs) = 16. Check condition: 2 × 40 = 80, and 5 × 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 4

The sum of the ages of a daughter and her mother is 56 years. After four years, the age of the mother will be three times that of the daughter. At present their ages are

Accepted Answer

Concept & Logic: The total sum of ages for a group increases proportionally with time. For two people, the total sum increases by 2 × (years passed). Calculate the future total sum first, then distribute it according to the future ratio condition to find the individual ages. Step-by-Step Solution: • Given: • Present sum (Daughter + Mother) = 56 years. • After 4 years: Mother = 3 × Daughter. • Calculation / Deduction: • In 4 years, both the mother and the daughter will age by 4 years. • Total sum after 4 years = 56 + 4 + 4 = 64 years. • Let the future daughter's age be x. The future mother's age is 3x. • x + 3x = 64 • 4x = 64 • x = 16 • Future daughter = 16 years. Future mother = 3(16) = 48 years. • To find their present ages, subtract the 4 years back: • Present Daughter = 16 - 4 = 12 years. • Present Mother = 48 - 4 = 44 years. Exam Strategy & Shortcut: Use Option Elimination by adding 4 to both numbers in the choices and checking if the mother is exactly thrice the daughter's age. (a) 10+4=14, 46+4=50 (Here, 50 
eq 3 × 14) (b) 12+4=16, 44+4=48 (Here, 48 = 3 × 16) - Immediate match! You don't need to test the rest. Common Pitfall: Students often set up the algebraic equation directly as x + 3x = 56, completely forgetting to advance the total sum by 8 years before applying the future ratio. Always synchronize your timeframes. Final Answer: Therefore, the correct answer is 12 years, 44 years.

Question 5

The present age of son is half of the present age of his mother. Ten years ago, his mother's age was thrice the age of her son. What is the present age of the son?

Accepted Answer

Concept & Formula: Define the present ages using a primary variable and a multiplier, then project those variables backwards in time by subtracting the time elapsed to establish a solvable algebraic equation. Step-by-Step Solution: • Given: • Present: Son = (1/2) × Mother (which means Mother = 2 × Son) • 10 years ago: Mother = 3 × Son • Calculation / Deduction: • Let the present age of the son be x. • Then, the present age of the mother is 2x. • Exactly 10 years ago, their ages were: • Son = x - 10 • Mother = 2x - 10 • According to the condition 10 years ago, the mother was thrice the son: • 2x - 10 = 3(x - 10) • 2x - 10 = 3x - 30 • 30 - 10 = 3x - 2x • x = 20 • Since x directly represents the present age of the son, he is 20 years old. Exam Strategy & Shortcut: Apply the Ratio Cross-Multiplication technique: Present Ratio (Mother : Son) = 2 : 1 Past Ratio (-10 years) = 3 : 1 Difference in cross-products = (3 × 1) - (2 × 1) = 1. Difference in past ratio parts = 3 - 1 = 2. Multiply the ratio parts difference by the time gap (10): 2 × 10 = 20. Value of 1 ratio unit = 20. The Son's present ratio part is 1, so his age is directly 20 years. Common Pitfall: A very common mistake is applying the past adjustment (-10) only to the mother's age while leaving the son as x in the equation setup. Both individuals age backwards together. Final Answer: Therefore, the correct answer is 20 years.

Question 6

Ratio between the ages of Subhash, Prasad and Amar is $3 : 6 : 7$. If the difference between the ages of Prasad and Amar is 10 years, then what is the difference between the ages of Subhash and Prasad?

Accepted Answer

Concept & Logic: Treat ratio numbers as distinct, uniform building blocks (units). The true numerical difference between any two individuals' ages corresponds perfectly to the difference in their respective ratio units. Step-by-Step Solution: • Given: • Subhash : Prasad : Amar = 3 : 6 : 7 • Age difference (Prasad and Amar) = 10 years • Calculation / Deduction: • Let their actual ages be 3x, 6x, and 7x respectively. • The difference between Prasad and Amar's ages algebraically is: • 7x - 6x = x • We are explicitly given that this difference is 10 years, making x = 10. • The question asks for the difference between Subhash and Prasad's ages: • Difference = 6x - 3x = 3x • Substitute the known value of x: • Difference = 3 × 10 = 30 years. Exam Strategy & Shortcut: Solve entirely in ratio units without deploying variables. Look at Prasad (6) and Amar (7). They have a gap of exactly 1 ratio unit. Since this 1 unit = 10 years, the scale is set. Now look at Subhash (3) and Prasad (6). They have a gap of exactly 3 ratio units. Therefore, 3 units × 10 years = 30 years. This takes under 5 seconds and requires zero algebra. Common Pitfall: Many students waste valuable exam time unnecessarily calculating the exact individual ages of all three people (30, 60, and 70 years old) before performing the final subtraction, rather than just comparing the ratio units directly. Final Answer: Therefore, the correct answer is 30 years.

Question 7

Rajan got married 8 years ago. His present age is $\frac{6}{5}$ times his age at the time of his marriage. Rajan's sister was 10 years younger to him at the time of his marriage. The age of Rajan's sister is

Accepted Answer

Concept & Strategy: There are two critical principles at play here: 1. Formulating an equation that connects present age and a past life event using a fractional multiplier. 2. The core rule of age problems: Age difference between two individuals is permanent and constant. If a sister was 10 years younger a decade ago, she is still exactly 10 years younger today. Step-by-Step Solution: • Given: • Married 8 years ago. • Present Age = (6/5) × (Age at Marriage) • Sister is 10 years younger. • Calculation / Deduction: • Let Rajan's present age be x. • His age at the time of his marriage (8 years ago) was x - 8. • Set up the fractional equation as given in the problem: • x = (6/5)(x - 8) • 5x = 6(x - 8) • 5x = 6x - 48 • x = 48 • Rajan's present age is 48 years. • His sister is 10 years younger than him. This gap remains constant throughout their entire lives. • Sister's present age = 48 - 10 = 38 years. Exam Strategy & Shortcut: Analyze the fraction (6/5) directly. This means Present Age is 6 ratio parts, and Marriage Age is 5 ratio parts. The difference between these states is exactly 1 ratio part. This 1 part represents the actual time passed since the marriage, which is 8 years. Therefore, 1 part = 8 years. Rajan's present age is 6 parts = 6 × 8 = 48 years. Sister's present age = 48 - 10 = 38 years. Common Pitfall: Students often overcomplicate finding the sister's age. They calculate Rajan's past marriage age (40), find the sister's past marriage age (30), and then add the 8 years back to get the present age (38). While technically correct, recognizing that the gap is permanent bypasses these extra steps entirely. Final Answer: Therefore, the correct answer is 38 years.

Question 8

A couple has a son and a daughter. The age of the father is four times that of the son and the age of the daughter is one-third of that of her mother. The wife is 6 years younger to her husband and the sister is 3 years older than her brother. The m…

Accepted Answer

Concept & Logic: This is a simultaneous equation problem involving four variables. The key is to express all family members' ages in terms of a single variable, typically the mother's or father's age, using substitution. M = F - 6 D = S + 3 Step-by-Step Solution: • Given: Father (F) = 4 × Son (S). Daughter (D) = Mother (M) / 3. Mother (M) = F - 6. Daughter (D) = S + 3. • Calculation: Substitute D and S in terms of the parents' ages into the sibling equation. • We know S = F / 4 and D = M / 3. • Substituting into D = S + 3, we get M / 3 = (F / 4) + 3. • Multiply the entire equation by 12 to clear fractions: 4M = 3F + 36. • Substitute F = M + 6 into the equation: 4M = 3(M + 6) + 36. • Expand and solve: 4M = 3M + 18 + 36 \implies 4M = 3M + 54. • Therefore, M = 54. Exam Strategy & Shortcut: Working Backwards from Options: The mother's age must be divisible by 3 (since the daughter's age is M/3 and usually an integer in these problems). Options (a) 42, (b) 48, (c) 54, (d) 63 are all divisible by 3. Test M = 54: D = 18. F = 54 + 6 = 60. S = 60 / 4 = 15. Is D = S + 3? 18 = 15 + 3. Yes! It matches perfectly. Common Pitfall: Misinterpreting "younger to" or "older than" relationships leads to reversed signs (e.g., writing M = F + 6 instead of F - 6). Always double-check who is the older person. Final Answer: Therefore, the correct answer is 54 years.

Question 9

The present ages of three persons are in the proportion 4 : 7 : 9. Eight years ago, the sum of their ages was 56 years. The present age of the eldest person is

Accepted Answer

Concept & Logic: When given a ratio for present ages and a sum for past ages, we must bring the sum to the present timeline. For every year that passes, the sum of n people's ages increases by n. Present Sum = Past Sum + (n × years) Step-by-Step Solution: • Given: Present ratio = 4:7:9. Sum 8 years ago = 56. Number of people = 3. • Calculation: Find the present sum of their ages. Since 8 years have passed for all 3 people, the total increase is 3 × 8 = 24 years. • Present sum = 56 + 24 = 80 years. • Let the present ages be 4x, 7x, and 9x. • 4x + 7x + 9x = 80 \implies 20x = 80 \implies x = 4. • The eldest person has the largest ratio share (9x). • Present age of the eldest = 9 × 4 = 36 years. Exam Strategy & Shortcut: Ratio Multiples: The present age of the eldest is represented by 9 units in the ratio. Therefore, the answer must be a multiple of 9. Among the options, 28 is not, 45 is (9 × 5), and 36 is (9 × 4). You only have to test two options, or just quickly find the sum multiplier. Common Pitfall: Forgetting to multiply the 8 years by the number of people. Many students simply add 8 to 56, getting 64, which leads to incorrect fractional ages and confusion. Final Answer: Therefore, the correct answer is 36 years.

Question 10

A man was asked to state his age in years. His reply was, "Take my age 3 years hence, multiply it by 3 and then subtract 3 times my age 3 years ago and you will know how old I am." What is the age of the man?

Accepted Answer

Concept & Logic: This is a direct word-to-algebra translation problem. Simply map the man's convoluted statement into a single linear equation representing his present age. x = 3(x + 3) - 3(x - 3) Step-by-Step Solution: • Given: Present age = 3 × (Age in 3 years) - 3 × (Age 3 years ago). • Calculation: Let the man's present age be x. • Age 3 years hence = x + 3. • Age 3 years ago = x - 3. • Construct the equation based on his statement: x = 3(x + 3) - 3(x - 3). • Expand both terms: x = (3x + 9) - (3x - 9). • Distribute the negative sign carefully: x = 3x + 9 - 3x + 9. • Combine like terms: The 3x and -3x cancel each other out. • x = 18. Exam Strategy & Shortcut: Pattern Recognition: Notice the structure a(x + y) - a(x - y). This expands to ax + ay - ax + ay, which always simplifies to 2ay. In this problem, a=3 and y=3. So the result is simply 2(3)(3) = 18. The x terms completely disappear, making it an instant calculation. Common Pitfall: Failing to distribute the negative sign properly on the second term. Writing (3x + 9) - 3x - 9 = 0 leads to a nonsensical result and panic during the exam. Final Answer: Therefore, the correct answer is 18 years.

Question 11

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 12

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 13

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 14

The ratio between the ages of Neelam and Shiny is 5 : 6 respectively. If the ratio between the one-third age of Neelam and half of Shiny's age is 5 : 9, then what is Shiny's age?

Accepted Answer

Concept & Logic: For a system of linear equations to have a unique numerical solution, the equations must be mathematically independent. If the second given condition is just a manipulated restatement of the first condition, you have infinite possible solutions. If Eq A \equiv Eq B, the exact value cannot be found. Step-by-Step Solution: • Given: Neelam : Shiny = 5 : 6. Also, ((1/3) Neelam) : ((1/2) Shiny) = 5 : 9. • Calculation: Let Neelam's age be N and Shiny's age be S. • The first condition states: (N/S) = (5/6) • Let's analyze the second condition mathematically: (\frac{N/3)}{(S/2)} = (5/9) • Simplify the complex fraction by multiplying by the reciprocal: (N/3) × (2/S) = (5/9) • This becomes: (2N/3S) = (5/9) • Now, isolate (N/S) by multiplying both sides by (3/2): (N/S) = (5/9) × (3/2) = (15/18) • Simplify the fraction (15/18) by dividing numerator and denominator by 3: (N/S) = (5/6) • The second condition simply yields (N/S) = (5/6), which is the exact same information provided in the first sentence. We have no absolute time shifts or fixed numerical anchors to determine their actual ages. Exam Strategy & Shortcut: Information Audit: Skim the problem for any real absolute numbers (like "5 years ago", "sum is 50", etc.). Both conditions provided here are purely relative ratios. Without at least one absolute numerical anchor tied to time or a sum, it is fundamentally impossible to extract a specific age. You can instantly mark "Cannot be determined". Common Pitfall: Blindly substituting 5x and 6x into the second equation and thinking you made a mistake when everything cancels out to 1 = 1 or 5 = 5. This cancellation is the mathematical proof that the data is insufficient. Final Answer: Therefore, the correct answer is Cannot be determined.

Question 15

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 16

6 years ago, the ratio of the ages of Kunal and Sagar was 6 : 5.  Four years hence, the ratio of their ages will be 11 : 10.  What is Sagar's age at present?

Accepted Answer

Concept & Strategy: For problems involving ratios of ages at two different points in time, the most efficient approach is the Ratio Unit Method. This method works because the time gap between the two scenarios is identical for both individuals, meaning the difference in their ratio units should ideally reflect that constant time gap. The total time gap between "6 years ago" and "4 years hence" is 6 + 4 = 10 years. Step-by-Step Solution: • Given: Let the ages of Kunal and Sagar 6 years ago be 6x and 5x. • Establish the Present Age: • Kunal's present age = 6x + 6 • Sagar's present age = 5x + 6 • Establish the Future Age: Four years hence (from the present), their ages will be: • Kunal = (6x + 6) + 4 = 6x + 10 • Sagar = (5x + 6) + 4 = 5x + 10 • Calculation: The future ratio is given as 11 : 10. (6x + 10/5x + 10) = (11/10) • Cross-multiply to solve for x: 10(6x + 10) = 11(5x + 10) 60x + 100 = 55x + 110 5x = 10 \implies x = 2 • Finding Target Age: We need Sagar's *present* age. • Sagar's age 6 years ago = 5x = 5(2) = 10 years. • Sagar's present age = 10 + 6 = 16 years. Exam Strategy & Shortcut: Use the Ratio Unit shift directly. Past Ratio (6 yrs ago) \implies K : S = 6 : 5 (Gap between ratios is 1 unit) Future Ratio (4 yrs hence) \implies K : S = 11 : 10 (Gap between ratios is 1 unit) Notice that for both Kunal and Sagar, the ratio increases from 6 	o 11 and 5 	o 10. That is exactly a 5-unit increase for both. This 5-unit increase corresponds exactly to the 10-year time gap (6 past + 4 future). 5 units = 10 years \implies 1 unit = 2 years Sagar's past age (5 units) = 5 × 2 = 10 years. Present age = 10 + 6 = 16 years. Done in 10 seconds. Common Pitfall: A very common mistake is finding the value of x (which is 2) and simply plugging it into the past ratio to get 10, then selecting an option (if 10 were present) without adding the 6 years to arrive back at the *present* age. Always verify the timeline the question asks for. Final Answer: Therefore, the correct answer is 16 years.

Question 17

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 18

Three years ago, the ratio of the ages of Amisha and Nimisha was 8 : 9 respectively.  3 years hence, the ratio of their ages will be 11 : 12 respectively.  What is the present age of Amisha?

Accepted Answer

Concept & Logic: This problem bridges two time periods without explicitly stating the present age ratio. The strategy is to find the total time gap between the two given ratios and use the ratio unit progression to solve it quickly. The total time gap between "3 years ago" and "3 years hence" is 3 + 3 = 6 years. Step-by-Step Solution: • Given: Let their ages 3 years ago be 8x and 9x. • Establish Future Ages: The gap between "3 years ago" and "3 years hence" is 6 years. So, we add 6 to their past ages. • Amisha's age 3 years hence = 8x + 6 • Nimisha's age 3 years hence = 9x + 6 • Calculation: The future ratio is given as 11 : 12. (8x + 6/9x + 6) = (11/12) • Cross-multiply to solve for x: 12(8x + 6) = 11(9x + 6) 96x + 72 = 99x + 66 99x - 96x = 72 - 66 3x = 6 \implies x = 2 • Finding Target Age: We need Amisha's *present* age. • Amisha's age 3 years ago = 8x = 8(2) = 16 years. • Amisha's present age = 16 + 3 = 19 years. Exam Strategy & Shortcut: Use visual ratio tracking. Past Ratio (3 yrs ago) \implies 8 : 9 Future Ratio (3 yrs hence) \implies 11 : 12 Look at the vertical jump: Amisha goes 8 	o 11 (3 units). Nimisha goes 9 	o 12 (3 units). The ratio jumps by exactly 3 units for both. This 3-unit jump covers the entire 6-year gap (3 past + 3 future). 3 units = 6 years \implies 1 unit = 2 years Amisha's age 3 years ago was 8 units \implies 8 × 2 = 16 years. Her present age is 16 + 3 = 19 years. Common Pitfall: The most frequent error is miscalculating the time gap. Students will often set up the equation as (8x + 3/9x + 3) = (11/12), mistakenly thinking the jump from "3 years ago" to "3 years hence" is only 3 years instead of 6 years. Always map the timeline explicitly. Final Answer: Therefore, the correct answer is 19 years.

Question 19

Eight year ago, Poorvi's age was equal to the sum of the present ages of her one son and one daughter.  Five years hence, the respective ratio between the ages of her daughter and her son that time will be 7 : 6.  If Poorvi's husband is 7 years elde…

Accepted Answer

Concept & Strategy: While this problem looks like a nightmare of four variables (P, H, S, D), it contains a massive Divisibility Loophole hidden in the future ratio. Using this loophole allows you to bypass the algebra entirely in a multiple-choice format. Step-by-Step Solution: • Standard Algebraic Method (For Understanding): • Let present ages be P (Poorvi), S (Son), D (Daughter), and H (Husband). • Equations: 1. P - 8 = S + D \implies P = S + D + 8 2. (D + 5/S + 5) = (7/6) \implies 6D + 30 = 7S + 35 \implies 7S = 6D - 5 3. H = P + 7 4. H = 3S • Substitution: Combine Eq 3 and 4: P + 7 = 3S \implies P = 3S - 7. • Equate the two expressions for P (Eq 1 and substituted Eq): S + D + 8 = 3S - 7 D + 15 = 2S • Multiply by 7 to make substituting Eq 2 easier: 7(D + 15) = 14S 7D + 105 = 2(7S) • Substitute 7S = 6D - 5 into the equation: 7D + 105 = 2(6D - 5) 7D + 105 = 12D - 10 5D = 115 \implies D = 23 Exam Strategy & Shortcut: The Divisibility Shortcut (MANDATORY): Look at the second condition: "Five years hence, the ratio between the ages of her daughter and son... will be 7 : 6." This means: \frac{Daughter + 5}{Son + 5} = (7/6) Therefore, (Daughter's Present Age + 5) MUST be a multiple of 7. Let's test the given options for the Daughter's present age: (a) 15 + 5 = 20 (Not divisible by 7) (b) 23 + 5 = 28 (Divisible by 7!) (c) 19 + 5 = 24 (Not divisible by 7) (d) 27 + 5 = 32 (Not divisible by 7) (e) 13 + 5 = 18 (Not divisible by 7) Only option (b) works. You can solve this 4-variable problem in literally 5 seconds by checking options. Common Pitfall: The biggest mistake students make is diving headfirst into complex substitution algebra without scanning the problem for ratio constraints. Spending 3-4 minutes solving a matrix of four variables destroys your exam pacing when a 5-second divisibility check was available. Final Answer: Therefore, the correct answer is 23 years.

Question 20

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.

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