A number consists of two digits. The sum of the digits is 9. If 45 is subtracted from the number, its digits are interchanged. What is the number?

The correct option is 72.

To find the two-digit number, we can translate the given conditions into algebraic equations and solve them step-by-step.

Let the tens digit of the number be represented by $x$ and the units digit be represented by $y$.
Any two-digit number can be written in its expanded form as:

$10x+y$

Step 1: Express the sum of the digits.
We are given that the sum of the digits is 9. We can write this as Equation (1):

$x+y=9$

Step 2: Express the condition when 45 is subtracted.
If we subtract 45 from the original number, its digits are interchanged. The number with interchanged digits is written as:
$10y+x$

According to the given condition, we can write the following equation:

$(10x+y)-45=10y+x$

Step 3: Simplify the second equation.
Let us rearrange the terms to group the variables on one side:

$10x-x+y-10y=45$

$9x-9y=45$

Divide the entire equation by 9 to simplify it further. This gives Equation (2):

$x-y=5$

Step 4: Solve the system of linear equations.
Now we have two simple equations:
Equation (1): $x+y=9$
Equation (2): $x-y=5$

Add Equation (1) and Equation (2) together:

$(x+y)+(x-y)=9+5$

$2x=14$

$x=7$

Substitute the value of $x=7$ back into Equation (1) to solve for $y$:

$7+y=9$

$y=2$

Step 5: Form the final number.
The tens digit $x$ is 7, and the units digit $y$ is 2. Therefore, the number is:

$10\left(7\right)+2=72$

Verification:
1. The sum of the digits is $7+2=9$, which is correct.
2. Subtracting 45 from 72 gives $72-45=27$. The digits of 72 are interchanged to form 27, which also matches the second condition.