Finding the Two-Digit Number

This problem asks us to find a two-digit number based on two given conditions. We will explore two methods to solve this: an algebraic method and a method of checking the given options.

Understanding the Number Problem

Let the two-digit number be represented by its tens digit and its units digit. A two-digit number can be expressed mathematically as \(10 \times \text{tens digit} + \text{units digit}\).

• The first condition states that the sum of the digits is 9.
• The second condition states that if 45 is subtracted from the number, its digits are interchanged.

Algebraic Solution for the Number

Let's use variables to represent the digits and set up equations:

• Let \(t\) be the tens digit.
• Let \(u\) be the units digit.
• The original two-digit number can be written as \(10t + u\).
• When the digits are interchanged, the new number becomes \(10u + t\).

Step 1: Formulate Equations from Conditions

From the first condition, the sum of the digits is 9:

\[t + u = 9 \quad \text{(Equation 1)}\]

From the second condition, if 45 is subtracted from the number, its digits are interchanged:

\[(10t + u) - 45 = 10u + t \quad \text{(Equation 2)}\]

Step 2: Simplify Equation 2

Let's rearrange Equation 2 to make it simpler:

\[10t - t + u - 10u = 45\]

\[9t - 9u = 45\]

Divide the entire equation by 9:

\[t - u = 5 \quad \text{(Equation 3)}\]

Step 3: Solve the System of Equations

Now we have a system of two linear equations with two variables:

• Equation 1: \(t + u = 9\)
• Equation 3: \(t - u = 5\)

We can add Equation 1 and Equation 3 to eliminate \(u\):

\[(t + u) + (t - u) = 9 + 5\]

\[2t = 14\]

\[t = \frac{14}{2}\]

\[t = 7\]

Now substitute the value of \(t = 7\) back into Equation 1 to find \(u\):

\[7 + u = 9\]

\[u = 9 - 7\]

\[u = 2\]

Step 4: Determine the Number

With \(t = 7\) (tens digit) and \(u = 2\) (units digit), the original two-digit number is:

\[10t + u = 10(7) + 2 = 70 + 2 = 72\]

Verifying the Number with Options

We can also solve this problem by checking each of the given options against the two conditions.

As the table clearly shows, only the number 72 satisfies both conditions: its digits sum to 9, and when 45 is subtracted, its digits are interchanged.