Direction :"You are presented with two quantities, labeled as Quantity I and Quantity II. Your task is to solve both quantities and determine the correct relationship between Quantity I and Quantity II. After solving, choose the appropriate option that describes the relationship correctly." Quantity-I: The monthly income of Tarun is ₹(z + 3300). He spent 'y'% on education and (1/3) of the remaining was spent on house rent. After that ₹652 was spent on travelling and (y + 14)% of the remaining was saved in bank and remaining was saved in the post office. If the amount spent on house rent by him is ₹12826 and (7y² – 476y + 8092 = 0), then find out the 2.5% of 'z'. Quantity-II: If 1.2P² = 2268750, then find out the value of 'P'.

### Concept & Quadratic Roots & Percentages Quantity I involves solving a quadratic equation to find a percentage variable, which is then used to reverse-calculate the total income based on sequential spending. Quantity II tests basic algebraic manipulation and emphasizes the dual roots (positive and negative) of a quadratic function $P^2 = k$. $$ \text{Remaining Amount} = \text{Initial} \times \left(1 - \frac{R}{100}\right) $$ ### Step-by-Step Solution **Solving Quantity I:** 1. Solve the quadratic equation for $y$: $7y^2 - 476y + 8092 = 0$ Divide the entire equation by 7: $y^2 - 68y + 1156 = 0$ This is a perfect square: $(y - 34)^2 = 0 \implies y = 34$. 2. Tarun spends $34\%$ on education. Remaining income = $(100 - 34)\% = 66\%$ of Total Income. 3. He spends $1/3$ of the remaining on house rent: House rent = $\frac{1}{3} \times 66\% = 22\%$ of Total Income. 4. Given that the amount spent on house rent is ₹12826: $22\%$ of Income = 12826 $\text{Income} = \frac{12826}{0.22} = \frac{1282600}{22} = 58300$. 5. Equate calculated income to given income formula to find $z$: $z + 3300 = 58300$ $z = 58300 - 3300 = 55000$. 6. Calculate $2.5\%$ of $z$: $2.5\% \text{ of } 55000 = \frac{2.5}{100} \times 55000 = 25 \times 55 = 1375$. **Quantity I = 1375** **Solving Quantity II:** 1. Solve for P: $1.2P^2 = 2268750$ $P^2 = \frac{2268750}{1.2} = \frac{22687500}{12} = 1890625$ 2. Take the square root of both sides. $P = \pm\sqrt{1890625}$ $P = 1375 \text{ or } -1375$. **Quantity II = +1375 or -1375** **Comparison:** Comparing Quantity I (1375) with Quantity II (1375, -1375): If $P = 1375$, Quantity I = Quantity II. If $P = -1375$, Quantity I > Quantity II. Therefore, combining both possibilities, Quantity I $\ge$ Quantity II. ### Exam Strategy & Shortcut When finding the square root of a large number ending in 25 like 1890625, you know the root ends in 5. By checking $1300^2 = 1690000$ and $1400^2 = 1960000$, you can easily deduce the root is exactly 1375 without complex long division methods. ### Common Pitfall The most common mistake in banking/aptitude exams involving quadratic variables ($P^2 = x$) is ignoring the negative root. Concluding $P = 1375$ only would lead to marking "Quantity-I = Quantity-II", which is incorrect because of the negative possibility. ### Final Answer Therefore, the correct answer is **Quantity-I ≥ Quantity-II**.