Question 1

Mr. Joe's family consists of six people—himself, his wife and their four children. It is known that the average age of the family immediately after the birth of the first, second, third and fourth child was 16, 15, 16 and 15 years respectively. Find…

Accepted Answer

### Concept & Logic To solve this problem, we use the principle of **Total Age = Average Age $	imes$ Number of Members**. We must track the total age of the family at each given milestone (birth of each child). Note that the age of a newborn is exactly $0$ years. ### Step-by-step Solution * **At 1st child's birth:** Family size = $3$. Average age = $16$. Total age = $3 	imes 16 = 48$ years. (This is simply the sum of the parents' ages, as the newborn is $0$). * **At 2nd child's birth:** Let the gap between the 1st and 2nd child be $x$ years. During this time, the parents and the first child all age by $x$ years. Total age = $48 + 3x$. Given average = $15$, Family size = $4$. $48 + 3x = 4 	imes 15 = 60 \Rightarrow 3x = 12 \Rightarrow x = 4$ years. (The first child is now $4$ years old). * **At 3rd child's birth:** Let the gap between the 2nd and 3rd child be $y$ years. Total age = $60 + 4y$. Given average = $16$, Family size = $5$. $60 + 4y = 5 	imes 16 = 80 \Rightarrow 4y = 20 \Rightarrow y = 5$ years. * **At 4th child's birth:** Let the gap between the 3rd and 4th child be $z$ years. Total age = $80 + 5z$. Given average = $15$, Family size = $6$. $80 + 5z = 6 	imes 15 = 90 \Rightarrow 5z = 10 \Rightarrow z = 2$ years. At this point, the eldest son's age is the sum of the gaps: $x + y + z = 4 + 5 + 2 = 11$ years. * **Present Day:** Let $t$ years have passed since the 4th child's birth. Total present age = $90 + 6t$. Given present average = $16$, Family size = $6$. $90 + 6t = 6 	imes 16 = 96 \Rightarrow 6t = 6 \Rightarrow t = 1$ year. The present age of the eldest son is his age at the 4th birth plus $1$ year: $11 + 1 = 12$ years. ### Exam Strategy & Shortcut Instead of complex equations, just track the total age increment mentally. Start: Total age $48$ (3 people). To reach average $15$ for 4 people, total must be $60$. Difference is $12$, shared by 3 existing people $\Rightarrow 4$ years gap. To reach average $16$ for 5 people, total must be $80$. Difference is $20$, shared by 4 existing people $\Rightarrow 5$ years gap. To reach average $15$ for 6 people, total must be $90$. Difference is $10$, shared by 5 existing people $\Rightarrow 2$ years gap. Total gap = $4+5+2 = 11$ years. Present total is $96$, so gap is $96-90 = 6$, shared by 6 people $\Rightarrow 1$ year gap. Final Age $= 11+1 = 12$. This logic is lightning-fast on paper. ### Common Pitfall Students often forget that the parents *and all existing children* age during the gap years. They might divide the age difference incorrectly. Always divide the total age increase by the number of people who were alive *during* that time gap. ### Final Answer **Therefore, the correct answer is 12 years.**

Question 2

The average salary of all the workers in a workshop is ₹ 8000. The average salary of 7 technicians is ₹ 12000 and the average salary of the rest is ₹ 6000. The total number of workers in the workshop is

Accepted Answer

### Concept & Strategy This problem is a classic application of the **Weighted Average** or **Rule of Alligation**. It involves mixing two distinct groups (technicians and the rest) to form a combined whole (all workers). ### Step-by-Step Solution * **Given Data:** Overall average salary = ₹ $8000$ Average salary of technicians (Group 1) = ₹ $12000$ Number of technicians = $7$ Average salary of the rest (Group 2) = ₹ $6000$ * **Algebraic Approach:** Let the total number of workers be $x$. The total salary of all workers = $8000x$. The total salary of the 7 technicians = $7 	imes 12000 = 84000$. The number of remaining workers = $x - 7$. The total salary of the remaining workers = $6000 	imes (x - 7)$. Equating the total salaries: $$8000x = 84000 + 6000(x - 7)$$ $$8000x = 84000 + 6000x - 42000$$ $$8000x - 6000x = 42000$$ $$2000x = 42000$$ $$x = \frac{42000}{2000} = 21$$ ### Exam Strategy & Shortcut The fastest way to solve this is using the **Rule of Alligation**. Set up the averages of the two groups and the mean in the middle: Technicians (₹ $12000$) ----------- Rest (₹ $6000$) \ / Mean (₹ $8000$) / \ $(8000 - 6000)$ ----------- $(12000 - 8000)$ Ratio $ightarrow 2000 : 4000 = 1 : 2$ This means the ratio of Technicians to Rest is $1 : 2$. If $1$ part represents $7$ technicians, then the total parts ($1 + 2 = 3$ parts) represent the total workers. Total workers = $3 	imes 7 = 21$. ### Common Pitfall A common mistake is finding the number of "rest" workers ($14$) and forgetting to add back the $7$ technicians to find the *total* number of workers as the question asks. Always double-check what the final variable represents! ### Final Answer **Therefore, the correct answer is 21.**

Question 3

Direction: Each of the questions below consists of a question-statement and two statements I and II are given below it. You have to decide whether the data provided in the statements are sufficient to answer the question. Give answer.  Whose body we…

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### Concept & Logic To determine whose body weight is the second highest among five people, we must either find the exact weights of all five boys or establish a strict relative order among them. We use the formula for the sum of elements given their average: $$Sum = Average 	imes Number of Elements$$ ### Step-by-Step Solution **Evaluating Statement I alone:** * Given: Average weight of Arun (A), Suraj (S), and Vinay (V) = $68$ kg. * Sum of A, S, V = $3 	imes 68 = 204$ kg. * Given individual weights: S = $78$, R = $68$, V = $46$. * Substitute S and V to find A: A + $78$ + $46$ = $204$ kg $ightarrow$ A + $124$ = $204$ kg $ightarrow$ A = $80$ kg. * Given: Average weight of Raju (R) and Pratap (P) = $72$ kg. * Sum of R, P = $2 	imes 72 = 144$ kg. * Substitute R to find P: $68$ + P = $144$ kg $ightarrow$ P = $76$ kg. * Now we have all five weights: A = $80$, S = $78$, P = $76$, R = $68$, V = $46$. * Arranging in descending order: A, S, P, R, V. The second highest is Suraj (S). Statement I is sufficient. **Evaluating Statement II alone:** * Given: Average weight of A, S, V, R = $68$ kg. * Sum of A, S, V, R = $4 	imes 68 = 272$ kg. * Given weights: S = $78$, R = $68$, V = $46$. * Find A: A + $78$ + $68$ + $46$ = $272$ kg $ightarrow$ A + $192$ = $272$ kg $ightarrow$ A = $80$ kg. * We know four weights: A = $80$, S = $78$, R = $68$, V = $46$. * However, Statement II provides absolutely no information about Pratap (P). Without P's weight, we cannot guarantee who holds the second highest position. Statement II is not sufficient. ### Exam Strategy & Shortcut For data sufficiency questions involving averages, quickly check if the variables match up. In Statement I, you are given the average of 3 boys and the exact weights of 2 of them. This immediately tells you that you can find the 3rd. You are then given the average of the remaining 2 boys and the weight of 1 of them. This means you can find the final boy's weight. Since all 5 weights can be determined, Statement I is definitely sufficient. Statement II lacks Pratap entirely, so it is instantly insufficient. You do not even need to calculate the final values to mark the answer! ### Common Pitfall A major mistake is performing all the arithmetic calculations during a data sufficiency problem. While we solved it above for clarity, calculating $80$ and $76$ consumes precious time. Recognize that having $n$ independent variables and $n$ independent linear equations guarantees a solution. ### Final Answer **Therefore, the correct answer is The data in statement I alone was sufficient to answer the question while II alone are not sufficient to answer the question.**

Question 4

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 given question. Rea…

Accepted Answer

### Concept & Logic To find the value of a specific item ($R$) within a larger set, you can subtract the sum of all other items from the total sum of the entire set. $$Item = (Total \ Sum) - (Sum \ of \ Remaining \ Items)$$ ### Step-by-Step Solution **Given from the prompt:** * The average salary of 5 people ($P, Q, R, S, T$) is ₹ 45980. * Total salary of all 5 = $45980 	imes 5 = 229900$. * $$P + Q + R + S + T = 229900$$ **Evaluating Statement I:** * $P + T = 90670$. * This leaves $Q, R$, and $S$ unknown. * Statement I alone is not sufficient. **Evaluating Statement II:** * $Q + S = 76540$. * This leaves $P, R$, and $T$ unknown. * Statement II alone is not sufficient. **Evaluating Statements I and II Together:** * Substitute both known sums into the total equation: $$(P + T) + (Q + S) + R = 229900$$ $$90670 + 76540 + R = 229900$$ $$167210 + R = 229900$$ $$R = 229900 - 167210 = 62690$$ * By combining both statements, we can uniquely determine $R$'s salary. ### Exam Strategy & Shortcut You don't need to compute $45980 	imes 5$ or do the final subtraction during the exam! Just verify the variables. The prompt gives the sum for $P,Q,R,S,T$. Statement I knocks out $P$ and $T$. Statement II knocks out $Q$ and $S$. The only variable left standing is $R$. Since all other variables are accounted for, the data is sufficient. ### Common Pitfall Wasting time doing the heavy arithmetic. Data sufficiency tests your logical reasoning regarding whether a problem *can* be solved, not your ability to act as a human calculator. The moment you see that substituting the chunks isolates $R$, mark "both necessary" and move on. ### Final Answer **Therefore, the correct answer is If the data in both Statements I and II together are necessary to answer the question.**

Question 5

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 given question. Rea…

Accepted Answer

### Concept & Logic The sum of a specific subset of items (the best and worst scores) can be found by subtracting the sum of the remaining items from the total sum of all items. $$ 	ext{Sum of (Best + Worst)} = 	ext{Total Sum (8 tests)} - 	ext{Sum of Remaining (6 tests)} $$ ### Step-by-Step Solution * **From Statement I:** We are given the average of all 8 tests is 84%. Total sum of 8 tests = $8 	imes 84 = 672$. We know the grand total, but we do not know any individual scores to isolate the best and worst. Thus, Statement I alone is not sufficient. * **From Statement II:** We are given the average of the middle 6 tests is 86%. Total sum of 6 tests = $6 	imes 86 = 516$. We know the subset total, but we don't know the grand total to figure out what is missing. Thus, Statement II alone is not sufficient. * **Combining Statements I and II:** We now have the sum of all 8 tests (672) and the sum of the middle 6 tests (516). The difference between these two sums must equal the sum of the two dropped tests (the best and the worst). Sum of (Best + Worst) = $672 - 516 = 156$. The question specifically asks for the average of these two dropped scores. Average = $\frac{156}{2} = 78\%$. Since we can find a unique numerical answer by relying on both pieces of information, both statements together are required. ### Exam Strategy & Shortcut Visualize the underlying equation conceptually: $(	ext{Total Sum}) = (	ext{Sum of Middle 6}) + (	ext{Best} + 	ext{Worst})$. You need to find the average of (Best + Worst), which means you inherently need to find their sum. Statement I gives you the Total Sum piece. Statement II gives you the Sum of Middle 6 piece. It is instantly obvious you need both pieces to isolate the final unknown subset. You do not even need to execute the multiplication to confidently mark the answer. ### Common Pitfall Students sometimes overthink this problem and incorrectly believe they need to determine the precise individual values of the "best" score and the "worst" score to calculate their average. You do not need individual values to find an average, only their combined sum and the count of items (which is statically 2). ### Final Answer **Therefore, the correct answer is Data in both Statements I and II together are necessary.**

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