Finding the Required Two-Digit Number

Video Explanation

Question

A two-digit number is 4 times the sum of its digits and twice the product of its digits. 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 = \(10x + y\)

Step 3: First Condition

Number is 4 times the sum of digits:

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

\[ 10x + y = 4x + 4y \]

\[ 6x – 3y = 0 \]

\[ 2x – y = 0 \quad (1) \]

Step 4: Second Condition

Number is twice the product of digits:

\[ 10x + y = 2xy \quad (2) \]

Step 5: Solve the Equations

From equation (1):

\[ y = 2x \]

Substitute in equation (2):

\[ 10x + 2x = 2x(2x) \]

\[ 12x = 4x^2 \]

\[ 4x^2 – 12x = 0 \]

\[ 4x(x – 3) = 0 \]

Since digit cannot be 0,

\[ x = 3 \]

Step 6: Find the Value of y

\[ y = 2(3) \]

\[ y = 6 \]

Conclusion

Original number:

\[ 10x + y = 10(3) + 6 \]

\[ = 36 \]

\[ \boxed{36} \]

Final Answer (For Exam)

The required number is 36.