One-fourth of a herd of camels was seen in the forest. Twice the square root of the herd had gone to mountains and the remaining 15 camels were seen on the bank of a river. Find the total number of camels.

### Concept & Logic This word problem is solved by translating the given conditions into an algebraic equation. Since the problem involves the square root of the total number of camels, it will naturally form a quadratic equation when formalized. ### Step-by-Step Solution * **Given:** Let the total number of camels in the herd be $x$. * Number of camels in the forest = $\frac{x}{4}$ * Number of camels gone to the mountains = $2\sqrt{x}$ * Number of remaining camels on the bank = $15$ * The sum of these three groups must equal the total number of camels: * $\frac{x}{4} + 2\sqrt{x} + 15 = x$ * Rearrange the terms to group the $x$ variables on one side: * $x - \frac{x}{4} - 2\sqrt{x} = 15$ * $\frac{3x}{4} - 2\sqrt{x} = 15$ * Multiply the entire equation by 4 to clear the fraction: * $3x - 8\sqrt{x} = 60$ * Let $y = \sqrt{x}$. This implies $y^2 = x$. Substitute this into the equation to form a standard quadratic: * $3y^2 - 8y - 60 = 0$ * Factor the quadratic equation. We need two numbers that multiply to $(3 \times -60) = -180$ and add to $-8$. Those numbers are $-18$ and $10$. * $3y^2 - 18y + 10y - 60 = 0$ * $3y(y - 6) + 10(y - 6) = 0$ * $(3y + 10)(y - 6) = 0$ * This gives $y = -\frac{10}{3}$ or $y = 6$. * Since $y$ represents a square root of a physical count of camels, it must be positive. Thus, $y = 6$. * Substitute back to find $x$: $\sqrt{x} = 6$, so $x = 36$. ### Exam Strategy & Shortcut **Option Elimination is the fastest method here.** The problem states "Twice the square root of the herd...". This strongly implies that the total number of camels must be a perfect square integer (since you can't have a fractional number of live camels). Look at the options: (a) 32 - Not a perfect square. (b) 34 - Not a perfect square. (c) 35 - Not a perfect square. (d) 36 - Perfect square! ($\sqrt{36} = 6$) You can solve this question in exactly 2 seconds without writing a single equation. ### Common Pitfall The most common mistake is setting up the equation correctly but struggling to factor the resulting quadratic with surds. Students often panic when they see $3x - 8\sqrt{x} = 60$. Using substitution ($y = \sqrt{x}$) simplifies it, but relying on option elimination bypasses the risk of arithmetic errors entirely. ### Final Answer **Therefore, the correct answer is 36.**