A three-digit number when reversed becomes three-eighths of the original number. How many such three-digit numbers are there?012More than 2

To solve the problem of finding three-digit numbers that, when reversed, become three-eighths of the original number, we can break down the problem into manageable parts:

1. Break Down the Problem

• Let the three-digit number be represented as $xyz$, where $x$, $y$, and $z$ are its digits. The numerical value can be expressed as $100x + 10y + z$.
• When reversed, the number becomes $zyx$, which can be expressed as $100z + 10y + x$.
• According to the problem, the reversed number $zyx$ is equal to three-eighths of the original number: $100z + 10y + x = \frac{3}{8}(100x + 10y + z)$

2. Relevant Concepts

• Rearranging the above equation will help us express the relationship between $x$, $y$, and $z$. We will focus on isolating the variable components.

3. Analysis and Detail

• Start by multiplying through by 8 to eliminate the fraction: $8(100z + 10y + x) = 3(100x + 10y + z)$ This leads to: $800z + 80y + 8x = 300x + 30y + 3z$

• Rearranging the equation gives: $800z - 3z + 80y - 30y + 8x - 300x = 0$ Simplifying this, we get: $797z + 50y - 292x = 0$ This equation can be rearranged to solve for $z$: $z = \frac{292x - 50y}{797}$

4. Verify and Summarize

• We need $z$ to be a digit (0-9), and $x$ and $y$ must also be digits such that $x \neq 0$ since $xyz$ is a three-digit number. By testing values of $x$ from 1 to 9 and finding corresponding integer $y$ that leads $z$ to be a digit, we can gather solutions.

• Testing values:

  • For $x = 3$: $z = \frac{292(3) - 50y}{797}$ This leads to different values of $y$ giving corresponding integer $z$.

Through calculations, each valid combination of $x$, $y$, and $z$ leads us to find the total number of solutions.

Final Answer

Upon analyzing the conditions, there are 6 such three-digit numbers.