Finding the Required Two-Digit Number

Video Explanation

Question

A two-digit number is such that the product of its digits is 20. If 9 is added to the number, the digits interchange their places. Find the number.

Solution

Step 1: Let the Variables

Let the tens digit = \(x\)

Let the units digit = \(y\)

Step 2: Form the Number

Original number = \(10x + y\)

Reversed number = \(10y + x\)

Step 3: Form the Equations

Product of digits:

\[ xy = 20 \quad (1) \]

If 9 is added, digits reverse:

\[ 10x + y + 9 = 10y + x \]

\[ 10x + y + 9 – 10y – x = 0 \]

\[ 9x – 9y + 9 = 0 \]

\[ x – y = -1 \quad (2) \]

Step 4: Solve the Equations

From equation (2):

\[ x = y – 1 \]

Substitute in equation (1):

\[ (y – 1)y = 20 \]

\[ y^2 – y – 20 = 0 \]

\[ (y – 5)(y + 4) = 0 \]

Since digit cannot be negative:

\[ y = 5 \]

Step 5: Find the Value of x

\[ x = 5 – 1 \]

\[ x = 4 \]

Conclusion

Original number:

\[ 10x + y = 10(4) + 5 \]

\[ = 45 \]

\[ \boxed{45} \]

Final Answer (For Exam)

The required number is 45.