Question

1. the sum of the digits of a two - digit counting number is 8. when the digits are reversed, the new number is 36 less than the original number. what are the two numbers?

Explanation:

Step 1: Define Variables

Let the tens digit be \( x \) and the units digit be \( y \). The original two - digit number can be expressed as \( 10x + y \), and the number with digits reversed is \( 10y+x \).

We know two conditions:

1. The sum of the digits is 8, so \( x + y=8 \).
2. When the digits are reversed, the new number is 36 less than the original number, so \( (10x + y)-(10y + x)=36 \).

Step 2: Simplify the Second Equation

Simplify \( (10x + y)-(10y + x)=36 \):
\[

$$\begin{align*} 10x + y-10y - x&=36\\ 9x-9y&=36\\ x - y&=4 \end{align*}$$

\]

Step 3: Solve the System of Equations

We have the system of equations:
\[

$$\begin{cases} x + y=8\\ x - y=4 \end{cases}$$

\]
Add the two equations together: \( (x + y)+(x - y)=8 + 4 \)
\[

$$\begin{align*} x + y+x - y&=12\\ 2x&=12\\ x&=6 \end{align*}$$

\]
Substitute \( x = 6 \) into \( x + y=8 \), we get \( 6+y = 8 \), then \( y=8 - 6=2 \).

Step 4: Find the Two Numbers

The original number is \( 10x + y=10\times6 + 2=62 \), and the reversed number is \( 10y + x=10\times2+6 = 26 \).

Answer:

The original number is 62 and the reversed number is 26.