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

### Concept & Logic This word problem requires translating independent operations on different fractional parts of a number into a single equality. "Diminished by" means subtracted. ### Step-by-step Solution **Given:** * Let the unknown number be $x$. * Fifth part increased by $4$: $\frac{x}{5} + 4$ * Fourth part diminished by $10$: $\frac{x}{4} - 10$ **Calculation:** Equate the two expressions: $$ \frac{x}{5} + 4 = \frac{x}{4} - 10 $$ Group the variable terms on one side and the constant terms on 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: $$ 14 = \frac{5x - 4x}{20} $$ $$ 14 = \frac{x}{20} $$ Solve for $x$ by multiplying both sides by $20$: $$ x = 14 \times 20 $$ $$ x = 280 $$ ### Exam Strategy & Shortcut **Option Elimination via Divisibility:** The problem talks about a "fifth part" ($\frac{x}{5}$) and a "fourth part" ($\frac{x}{4}$). This strongly implies the original number must be perfectly divisible by both $5$ and $4$ (so, divisible by $20$). Looking at the options: (a) $240$, (b) $260$, (c) $270$, (d) $280$. All except $270$ are multiples of $20$. So we can likely eliminate (c). Let's quick-test (d) $280$: Fifth part of $280 = 56$. Increased by $4 = 60$. Fourth part of $280 = 70$. Diminished by $10 = 60$. They match perfectly. Testing options is often faster than algebra here. ### Common Pitfall A standard error is misplacing the additions and subtractions, for instance, writing $\frac{x+4}{5}$ instead of $\frac{x}{5} + 4$. Read carefully: it is the *part* that is increased, not the whole number before division. ### Final Answer **Therefore, the correct answer is 280.**