Question 1

Directions: Each of these questions is followed by three statements. You have to study the question and all the three statements given to decide whether any information provided in the statement(s) is redundant and can be dispensed with while answer…

Accepted Answer

Concept & Logic: To find the ratio of their ages after 5 years, we must first find their current ages. If any two statements provide enough information to find both current ages, then the third statement is essentially giving us duplicate information (redundant). Step-by-Step Solution: Let Sam's age be S and Albert's age be A. Let's translate the statements: • I: S - A = 4 • II: A = 20 • III: (A/S) = (5/6) \Rightarrow 6A = 5S Let's test pairs to see if they can solve the system: • Using I and II: If A = 20, then from I, S = 24. We found both ages. (III is redundant) • Using II and III: If A = 20, then from III, 6(20) = 5S \Rightarrow S = 24. We found both ages. (I is redundant) • Using I and III: Substitute S = A + 4 into III: 6A = 5(A + 4) \Rightarrow 6A = 5A + 20 \Rightarrow A = 20. Then S = 24. We found both ages. (II is redundant) Since any pair of statements is sufficient to solve for both ages, any single statement can be discarded without losing the ability to answer the question. Exam Strategy & Shortcut: The Golden Triangle of Values: Statements I, II, and III give you the Difference, an Absolute Value, and a Ratio, respectively. Mathematically, having any two of these three elements allows you to find the exact numerical values for both variables. Recognizing this pattern means you can instantly conclude that any one statement is dispensable. Common Pitfall: A common mistake is selecting an option like "III only" because statements I and II seem like the "easiest" or most direct way to solve the problem. Redundancy means checking if *any* other combination works, not just finding the most convenient one. Final Answer: Therefore, the correct answer is I or II or III only.

Question 2

Directions: Each of these questions is followed by three statements. You have to study the question and all the three statements given to decide whether any information provided in the statement(s) is redundant and can be dispensed with while answer…

Accepted Answer

Concept & Logic: To find the difference between two variables (Ayush and Deepak), we need sufficient information to calculate both of their values. We must analyze whether the statements independently solve for a variable or link the variables together. Step-by-Step Solution: Let Ayush's present age be A and Deepak's present age be D. • Statement I: (A/A + 8) = (4/5) \Rightarrow 5A = 4A + 32 \Rightarrow A = 32. (This statement alone gives Ayush's exact age). • Statement II: (A/D) = (4/3) \Rightarrow 3A = 4D. (This is a linking relationship). • Statement III: (D/D - 4) = (6/5) \Rightarrow 5D = 6D - 24 \Rightarrow D = 24. (This statement alone gives Deepak's exact age). Let's evaluate combinations: • Using I and III: We know A = 32 and D = 24. Difference = 8. (II is redundant). • Using I and II: We know A = 32. Substituting into II: 3(32) = 4D \Rightarrow D = 24. Difference = 8. (III is redundant). • Using II and III: We know D = 24. Substituting into II: 3A = 4(24) \Rightarrow A = 32. Difference = 8. (I is redundant). Because we only ever need two statements to fully solve for both ages, any one of the three statements can be dropped. Exam Strategy & Shortcut: Single Variable Recognition: Notice that Statement I only talks about Ayush and Statement III only talks about Deepak. Equations with only one variable solve themselves instantly. Statement II connects them. Thus, picking the two independent statements works (I & III), or picking one independent statement plus the connector works (I & II, or II & III). Therefore, any single one is dispensable. Common Pitfall: Students often solve using the most obvious path (Statement I and Statement III) and incorrectly assume Statement II is the *only* redundant one. You must test all pairs to see if they yield a valid solution. Final Answer: Therefore, the correct answer is Any one of the three.

Question 3

If the ratio between the ages of $P$ and $Q$ is $2 : 3$ and that between the ages of $Q$ and $R$ is $4 : 5$, then the ratio between the ages of $P$ and $R$ is

Accepted Answer

Concept & Compounding Ratios: When given sequential ratios connecting three entities (like P to Q, then Q to R), the ratio of the first entity to the last can be found by multiplying the fractional forms of the ratios. (P/R) = (P/Q) × (Q/R) Step-by-Step Solution: 1. Write the given ratios as fractions: (P/Q) = (2/3) (Q/R) = (4/5) 2. To find the ratio of P to R, multiply the two fractions. The Q variable will cancel out: (P/R) = ((2/3)) × ((4/5)) 3. Multiply the numerators and the denominators: Numerator: 2 × 4 = 8 Denominator: 3 × 5 = 15 4. The resulting fraction is (8/15). 5. Express the fraction as a ratio: P : R = 8 : 15 Exam Strategy & Shortcut: Just like finding the ratio of a:c from a:b and b:c, simply multiply the numerical ratios straight across. (2/3) × (4/5) = (8/15). No need to find the combined ratio P:Q:R. Common Pitfall: A common error is to try and equalize the Q values (3 and 4) to form a combined ratio P:Q:R = 8:12:15 and then making an arithmetic mistake during the conversion. While correct, it introduces unnecessary steps. Final Answer: Therefore, the correct answer is 8 : 15.

Question 4

The present ages of a mother and her son are in the ratio 11 : 5. When the son becomes as old as his mother is now, then the ratio of his father's age to that of his mother is 19 : 17. When the son becomes as old as his father is now, then the sum o…

Accepted Answer

Concept & Logic: In age-related problems, the absolute difference in ages between any two individuals remains strictly constant over time, regardless of past or future scenarios. Step-by-Step Solution: 1. Let the present ages of the mother and son be 11x and 5x, respectively. 2. The difference in their ages is 11x - 5x = 6x. This difference is constant. 3. The first future scenario is "when the son becomes as old as his mother is now". This means the son becomes 11x. This happens after 6x years. 4. In this future scenario (+6x years): - Mother's age = 11x + 6x = 17x. - The ratio of Father : Mother is given as 19 : 17. - Since the Mother's age is exactly 17x, the Father's age at this time must be 19x. 5. Determine the Father's present age: - Father's present age = Father's future age - 6x = 19x - 6x = 13x. 6. The second future scenario is "when the son becomes as old as his father is now". This means the son becomes 13x. This happens after 8x years (since 13x - 5x = 8x). 7. In this second future scenario (+8x years): - Son's age = 13x. - Father's age = 13x + 8x = 21x. - The sum of their ages is 170: 13x + 21x = 170 34x = 170 \implies x = 5 8. Find the Father's present age: - Father's present age = 13x = 13 × 5 = 65 years. Exam Strategy & Shortcut: Track the age increments directly in terms of units (x). Son=5, Mother=11. Increment to reach Mother's age = 6 units. So future Mother = 17 units. Father is 19 units then. Present Father = 19 - 6 = 13 units. To reach Father's present age, Son needs 13 - 5 = 8 units. Future Father = 13 + 8 = 21 units. Sum = 13 + 21 = 34 units. 34 units = 170 \implies 1 unit = 5. Father = 13 × 5 = 65. No complex algebraic substitution required. Common Pitfall: A common mistake is adding the future years incorrectly (e.g., adding to the father's future age instead of present age) or losing track of which timeframe you are operating in. Final Answer: Therefore, the correct answer is 65 years.

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