One-third of a two-digit number exceeds one-fourth of its successive number by 1. The number is

### Concept & Logic This problem defines a relationship between a number and its successive (next consecutive) integer using fractional parts. "Exceeds" means the difference between the larger and smaller value is 1. $$ \frac{x}{3} - \frac{x + 1}{4} = 1 $$ ### Step-by-Step Solution * Let the required two-digit number be $x$. * Its successive number is $x + 1$. * One-third of the number is $\frac{x}{3}$. * One-fourth of its successive number is $\frac{x + 1}{4}$. * According to the problem, the first quantity exceeds the second by 1: $$ \frac{x}{3} - \frac{x + 1}{4} = 1 $$ * Find a common denominator (12) for the left side: $$ \frac{4x - 3(x + 1)}{12} = 1 $$ * Distribute the negative sign carefully: $$ \frac{4x - 3x - 3}{12} = 1 $$ $$ \frac{x - 3}{12} = 1 $$ * Solve for $x$: $$ x - 3 = 12 $$ $$ x = 15 $$ ### Exam Strategy & Shortcut Use **Option Elimination**, which is incredibly fast for small integers. You need a number that is easily divisible by 3, and whose successor is divisible by 4. * Check (a) 12: Successor is 13 (not cleanly divisible by 4). Skip. * Check (b) 15: One-third is $15 \div 3 = 5$. Successor is 16. One-fourth is $16 \div 4 = 4$. Difference is $5 - 4 = 1$. Matches perfectly! You can arrive at the answer in under 10 seconds without algebra. ### Common Pitfall A frequent algebraic error is forgetting to distribute the negative sign when finding the common denominator. Writing $4x - 3x + 3$ instead of $4x - 3x - 3$ will completely derail the final answer. ### Final Answer **Therefore, the correct answer is 15.**