Finding the Required Two-Digit Number

Video Explanation

Question

Seven times a two-digit number is equal to four times the number obtained by reversing the digits. If the difference between the digits is 3, find the number.

Solution

Step 1: Let the Variables

Let the tens digit = \(x\)

Let the units digit = \(y\)

Step 2: Form the Numbers

Original number = \(10x + y\)

Reversed number = \(10y + x\)

Step 3: Form the First Equation

\[ 7(10x + y) = 4(10y + x) \]

\[ 70x + 7y = 40y + 4x \]

\[ 70x – 4x = 40y – 7y \]

\[ 66x = 33y \]

\[ 2x = y \quad (1) \]

Step 4: Digits Differ by 3

Two possible cases:

Case 1: \[ x – y = 3 \quad (2) \]

Case 2: \[ y – x = 3 \quad (3) \]

Case 1: Solve (1) and (2)

From (1): \(y = 2x\) Substitute in (2):

\[ x – 2x = 3 \]

\[ -x = 3 \]

\[ x = -3 \quad \text{(Not possible for a digit)} \]

So, Case 1 is rejected. —

Case 2: Solve (1) and (3)

From (1): \(y = 2x\) Substitute in (3):

\[ 2x – x = 3 \]

\[ x = 3 \]

Then,

\[ y = 2(3) = 6 \]

Conclusion

Original number:

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

\[ = 36 \]

\[ \boxed{36} \]

Final Answer (For Exam)

The required number is 36.