A number whose fifth part increased by 4 is equal to its fourth part diminished by 10, is

### Concept & Formula This problem requires constructing an equation where two different modifications to fractions of the same unknown number are set equal to each other. $$ \frac{x}{5} + 4 = \frac{x}{4} - 10 $$ ### Step-by-Step Solution * Let the unknown number be $x$. * Its fifth part increased by 4 can be written as: $$ \frac{x}{5} + 4 $$ * Its fourth part diminished by 10 can be written as: $$ \frac{x}{4} - 10 $$ * Set these two expressions equal to each other: $$ \frac{x}{5} + 4 = \frac{x}{4} - 10 $$ * Bring all the numerical constants to one side and the variable terms to the other: $$ 4 + 10 = \frac{x}{4} - \frac{x}{5} $$ $$ 14 = \frac{x}{4} - \frac{x}{5} $$ * Find a common denominator (20) for the fractions on the right side: $$ 14 = \frac{5x - 4x}{20} $$ $$ 14 = \frac{x}{20} $$ * Multiply by 20 to find $x$: $$ x = 14 \times 20 = 280 $$ ### Exam Strategy & Shortcut Notice the fractions involved: $\frac{1}{4}$ and $\frac{1}{5}$. The difference between them is $\frac{1}{20}$. The problem tells us that a smaller fraction ($\frac{1}{5}$) plus 4 equals a larger fraction ($\frac{1}{4}$) minus 10. The total gap between the fractional parts is $4 + 10 = 14$. So, $\frac{1}{20}$ of the number is 14. The number must be $14 \times 20 = 280$. ### Common Pitfall A common mistake is mixing up the operations, for example, writing "diminished by 10" as $10 - \frac{x}{4}$ instead of $\frac{x}{4} - 10$. Always preserve the order of operations described in the text. ### Final Answer **Therefore, the correct answer is 280.**