The digit in the unit's place of a number is equal to the digit in the ten's place of half of that number and the digit in the ten's place of that number is less than the digit in unit's place of half of the number by 1. If the sum of the digits of the number is 7, then what is the number?

### Concept & Logic This problem introduces complex, interdependent positional relationships between the digits of a number $N$ and the digits of its quotient $N/2$. ### Step-by-Step Solution **Given:** * Let the original number be $N$. * Condition 1: Unit digit of $N$ = Ten's digit of $N/2$. * Condition 2: Ten's digit of $N$ = (Unit digit of $N/2$) - 1. * Condition 3: Sum of the digits of $N = 7$. **Calculation / Deduction:** * Creating algebraic formulations for extracting digits of $N/2$ creates complicated parity trees. We will deduce the answer by systematically testing the given valid options against the conditions. * Test Option (a) 34: $N = 34$, so $N/2 = 17$. The unit digit of $N$ is 4. The ten's digit of $N/2$ is 1. Since $4 \neq 1$, Condition 1 fails immediately. * Test Option (b) 52: $N = 52$, so $N/2 = 26$. Check Condition 1: Unit digit of $N$ is 2. Ten's digit of $N/2$ is 2. (Matches). Check Condition 2: Ten's digit of $N$ is 5. Unit digit of $N/2$ is 6. Is $5 = 6 - 1$? Yes, $5 = 5$. (Matches). Check Condition 3: Sum of digits of $N$ is $5 + 2 = 7$. (Matches). * Since option (b) flawlessly satisfies every distinct condition, it is the correct number. ### Exam Strategy & Shortcut **Pure Option Elimination:** Problems that mandate extracting specific digits from "half" of a variable number are traps designed to make you write messy algebraic bounds. Skip the algebra entirely. Grab the options, divide them by 2, and read the digits. You can verify the conditions for 52 ($N=52$, $N/2=26$) in roughly 15 seconds mentally. ### Common Pitfall Attempting to mathematically define the digits of $N/2$ as $\frac{10x+y}{2}$. If $y$ is an odd digit, this formula creates decimals, meaning you have to borrow from the tens place which completely breaks standard digit-extraction algebra. ### Final Answer **Therefore, the correct answer is 52.**