If one-seventh of a number exceeds its eleventh part by 100, then the number is

### Concept & Formula The word "exceeds" indicates a difference between two quantities. We can establish an equation representing the subtraction of the smaller fractional part from the larger one. $$ \frac{x}{7} - \frac{x}{11} = 100 $$ ### Step-by-Step Solution * Let the original number be $x$. * Its one-seventh is $\frac{x}{7}$ and its eleventh part is $\frac{x}{11}$. * The problem states the one-seventh part is greater than the eleventh part by 100: $$ \frac{x}{7} - \frac{x}{11} = 100 $$ * Find a common denominator (77) for the left side: $$ \frac{11x - 7x}{77} = 100 $$ * Simplify the numerator: $$ \frac{4x}{77} = 100 $$ * Solve for $x$: $$ 4x = 7700 $$ $$ x = \frac{7700}{4} = 1925 $$ ### Exam Strategy & Shortcut Work with the fractions directly. The difference between $\frac{1}{7}$ and $\frac{1}{11}$ is $\frac{11 - 7}{77} = \frac{4}{77}$. The problem states that this $\frac{4}{77}$ gap is equal to 100. Therefore, $\frac{1}{77}$ of the number is $100 \div 4 = 25$. The total number is $25 \times 77$. Calculate $25 \times 77$ quickly: $(100 \div 4) \times 77 = 7700 \div 4 = 1925$. ### Common Pitfall A common calculation mistake occurs during cross-multiplication or finding the common denominator. Always verify that $\frac{11x}{77}$ and $\frac{7x}{77}$ correctly map back to the original fractions before subtracting. ### Final Answer **Therefore, the correct answer is 1925.**