If a number is multiplied by two-thirds of itself the value so obtained is $864$. What is the number?

### Concept & Formula Unlike addition or subtraction problems that form linear equations, multiplying a number by a fraction of itself creates a quadratic equation (involving $x^2$). ### Step-by-step Solution **Given:** * Let the unknown number be $x$. * The product of the number and $\frac{2}{3}$ of itself is $864$. **Calculation:** Formulate the algebraic equation: $$ x \times \left(\frac{2}{3}x\right) = 864 $$ Multiply the variables: $$ \frac{2}{3}x^2 = 864 $$ Isolate $x^2$ by multiplying both sides by $\frac{3}{2}$: $$ x^2 = 864 \times \frac{3}{2} $$ $$ x^2 = 432 \times 3 $$ $$ x^2 = 1296 $$ Find the square root of both sides to solve for $x$: $$ x = \sqrt{1296} $$ $$ x = 36 $$ ### Exam Strategy & Shortcut **Option Elimination via Unit Digits:** You need a number $x$ such that $\frac{2}{3}x^2 = 864$, meaning $x^2 = 1296$. Instead of calculating the square root of $1296$, look at the options and their unit digits: (a) $34 \rightarrow 4^2$ ends in $6$ (b) $36 \rightarrow 6^2$ ends in $6$ (c) $38 \rightarrow 8^2$ ends in $4$ (Eliminate) (d) $44 \rightarrow 4^2$ ends in $6$ (e) $46 \rightarrow 6^2$ ends in $6$ Since $40^2 = 1600$, the answer must be significantly less than $40$. This immediately eliminates $44$ and $46$. We are left with $34$ and $36$. Since $35^2 = 1225$, and our target ($1296$) is larger, the answer must be $36$. ### Common Pitfall The most frequent error is treating the multiplication like addition, resulting in $\frac{2}{3}x = 864$ instead of $\frac{2}{3}x^2 = 864$. Always remember that $x \times x = x^2$. ### Final Answer **Therefore, the correct answer is 36.**