Quadratic from a product — A number is multiplied by three-fourths of itself and the result is 10800. Find the number, showing how the quadratic simplifies cleanly.

Introduction / Context:
Phrases like “multiplied by three-fourths of itself” translate into simple algebraic equations. Interpreting the statement accurately as a product of the number and a fraction of itself leads to a quadratic equation that is straightforward to solve by isolating the square term.

Given Data / Assumptions:

• Let the unknown number be n.
• Equation: n * (3/4 * n) = 10800.
• We seek the positive real solution appropriate for a “number” in standard aptitude context.

Concept / Approach:
Combine factors: n * (3/4 * n) = (3/4) * n^2. Isolate n^2 by multiplying both sides by 4/3. Then take the principal square root to find n. Finally, verify by substitution to ensure no arithmetic errors occurred during scaling.

Step-by-Step Solution:
Write equation: (3/4) * n^2 = 10800.Multiply both sides by 4/3: n^2 = 10800 * (4/3) = 14400.Take square root: n = sqrt(14400) = 120 (positive branch).Therefore, the number is 120.

Verification / Alternative check:
Compute three-fourths of 120: (3/4) * 120 = 90. Then n * that = 120 * 90 = 10800, which matches the given result exactly.

Why Other Options Are Wrong:

• 210 / 180 / 160 / 240: Substituting any of these into (3/4) * n^2 does not yield 10800. Only n = 120 satisfies the equation precisely.

Common Pitfalls:
Misreading “three-fourths of itself” as “n + 3/4” times n; forgetting to multiply both sides by 4/3; taking the wrong square root magnitude.

Final Answer:
120