Two-Digit Number Problem

Let the original two-digit number be represented by $10t + u$, where $t$ is the tens digit and $u$ is the units digit. Here, $t$ must be an integer from 1 to 9, and $u$ must be an integer from 0 to 9.

Value Increase Calculation

When the digit 4 is placed to the right of the original number, the new number is formed. If the original number is $tu$, the new number is $tu4$.

The value of the new number can be expressed as $100t + 10u + 4$. This is because the original tens digit ($t$) is now in the hundreds place, the original units digit ($u$) is now in the tens place, and the digit 4 is in the units place.

The problem states that the value of the new number is the original number plus 490. We can write this as an equation:

Value of New Number = Value of Original Number + 490

$100t + 10u + 4 = (10t + u) + 490$

Solving the Equation

Now, we need to solve this equation for $10t + u$ (the original number). Let's rearrange the equation:

$100t + 10u + 4 = 10t + u + 490$

Subtract $(10t + u)$ from both sides:

$(100t + 10u + 4) - (10t + u) = 490$

$100t - 10t + 10u - u + 4 = 490$

$90t + 9u + 4 = 490$

Subtract 4 from both sides:

$90t + 9u = 490 - 4$

$90t + 9u = 486$

Notice that the left side of the equation, $90t + 9u$, can be factored by taking out 9:

$9(10t + u) = 486$

Remember that $10t + u$ is our original number. To find its value, we can divide both sides by 9:

$10t + u = \frac{486}{9}$

Performing the division:

$10t + u = 54$

Original Number Found

The value of the original two-digit number, $10t + u$, is 54.

Let's check this: The original number is 54. Placing 4 to its right gives 544. The increase is $544 - 54 = 490$. This matches the condition given in the problem.

The original number is 54.