Finding the Required Two-Digit Number

Video Explanation

Question

A two-digit number can be obtained either by:

• Multiplying the sum of its digits by 8 and subtracting 5
• Multiplying the difference of its digits by 16 and adding 3

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: Form the Equations

According to first condition:

\[ 10x + y = 8(x + y) – 5 \]

\[ 10x + y = 8x + 8y – 5 \]

\[ 2x – 7y = -5 \quad (1) \]

According to second condition:

\[ 10x + y = 16(x – y) + 3 \]

\[ 10x + y = 16x – 16y + 3 \]

\[ 6x – 17y = -3 \quad (2) \]

Step 4: Solve the Equations

Multiply equation (1) by 3:

\[ 6x – 21y = -15 \quad (3) \]

Subtract equation (2) from (3):

\[ (6x – 21y) – (6x – 17y) = -15 – (-3) \]

\[ -4y = -12 \]

\[ y = 3 \]

Step 5: Find the Value of x

Substitute \(y = 3\) in equation (1):

\[ 2x – 7(3) = -5 \]

\[ 2x – 21 = -5 \]

\[ 2x = 16 \]

\[ x = 8 \]

Conclusion

Original number:

\[ 10x + y = 10(8) + 3 \]

\[ = 83 \]

\[ \boxed{83} \]

Final Answer (For Exam)

The required number is 83.