The difference between a number and its three-fifths is $50$. What is the number?

### Concept & Logic This is a classic linear equation problem. By translating the English statement directly into a mathematical equation involving an unknown variable $x$, we can easily isolate the variable. ### Step-by-step Solution **Given:** * Let the unknown number be $x$. * The difference between the number and $\frac{3}{5}$ of itself is $50$. **Calculation:** Set up the equation based on the given information: $$ x - \frac{3}{5}x = 50 $$ Factor out $x$: $$ x \left(1 - \frac{3}{5}\right) = 50 $$ Simplify the fraction: $$ x \left(\frac{2}{5}\right) = 50 $$ Isolate $x$ by multiplying both sides by $\frac{5}{2}$: $$ x = 50 \times \frac{5}{2} $$ $$ x = 25 \times 5 $$ $$ x = 125 $$ ### Exam Strategy & Shortcut **Ratio Method (Mental Math):** If a number is reduced by its $\frac{3}{5}$th part, the remaining part is simply $1 - \frac{3}{5} = \frac{2}{5}$th of the number. The question states this remaining part equals $50$. So, $2$ parts = $50$ $1$ part = $25$ Total number ($5$ parts) = $5 \times 25 = 125$. This can be done in your head in $3$ seconds. ### Common Pitfall A common error is writing the equation backward, such as $\frac{3}{5}x - x = 50$, which leads to a negative value. Always respect the phrasing "difference between a number [first term] and its three-fifths [second term]". ### Final Answer **Therefore, the correct answer is 125.**