Sum of a Two-Digit Number and Its Reverse | Linear Equations in Two Variables | Class 10 Maths
Problems involving two-digit numbers are among the most important applications of Linear Equations in Two Variables. In these questions, we represent the digits using variables, form equations based on the given conditions, and solve them using algebraic methods.
In this example, the sum of a two-digit number and the number obtained by reversing its digits is given. We are also told that the digits differ by 2. Our goal is to determine all such numbers.
Question
The sum of a two-digit number and the number obtained by reversing its digits is 66. If the digits of the number differ by 2, find the number. Also, determine how many such numbers are possible.
Step 1: Assume the Digits
Let
- x = Tens digit
- y = Units digit
Therefore,
Original number = 10x + y
Reversed number = 10y + x
Step 2: Form the First Equation
According to the question,
Original Number + Reversed Number = 66
(10x + y) + (10y + x) = 66
11x + 11y = 66
Divide both sides by 11:
x + y = 6 ...(1)
Step 3: Form the Second Equation
The digits differ by 2.
This gives two possible cases:
Case 1
x − y = 2 ...(2)
Case 2
y − x = 2 ...(3)
Case 1 Solution
Using Equations (1) and (2):
x + y = 6
x − y = 2
Adding both equations,
2x = 8
x = 4
Substitute into Equation (1):
4 + y = 6
y = 2
Required number:
10(4) + 2 = 42
Case 2 Solution
Using Equations (1) and (3):
x + y = 6
y − x = 2
Adding both equations,
2y = 8
y = 4
Substitute into Equation (1):
x + 4 = 6
x = 2
Required number:
10(2) + 4 = 24
Final Answer
The possible two-digit numbers are:
- 24
- 42
Hence,
Total Number of Such Numbers = 2
Verification
For 24
- Reverse = 42
- 24 + 42 = 66 ✓
- Difference of digits = 4 − 2 = 2 ✓
For 42
- Reverse = 24
- 42 + 24 = 66 ✓
- Difference of digits = 4 − 2 = 2 ✓
Concept Used
Whenever a two-digit number is involved:
| Representation | Formula |
|---|---|
| Two-digit number | 10x + y |
| Reversed number | 10y + x |
| Difference of digits | x − y or y − x |
| Sum of digits | x + y |
Important Formula to Remember
- If tens digit = x and units digit = y
- Original number = 10x + y
- Reversed number = 10y + x
- Difference of digits = |x − y|
Common Mistakes Students Make
- Writing the number as x + y instead of 10x + y.
- Forgetting that reversing digits changes the place values.
- Considering only one case for the difference of digits.
- Ignoring the second possible solution.
- Not verifying the final answers with the original conditions.
Quick Exam Trick
- Always represent the tens digit separately from the units digit.
- Whenever the statement says "digits differ by...", solve both cases:
- x − y = value
- y − x = value
- Check every answer by reversing the number.
- Verify all given conditions before writing the final answer.
Practice Questions
- The sum of a two-digit number and its reverse is 88. The digits differ by 4. Find the number.
- A two-digit number is 27 greater than the number formed by reversing its digits. Find the number.
- The sum of the digits of a two-digit number is 9. If reversing the digits decreases the number by 27, find the number.
- The difference between a two-digit number and its reverse is 45. Find all possible numbers.
Board Exam Tips
- Define the variables clearly before forming equations.
- Always write the original and reversed numbers correctly.
- Number every equation for easy presentation.
- Show complete algebraic steps.
- Don't forget to mention the total number of possible solutions.
Conclusion
This problem demonstrates how real-life situations involving numbers can be converted into a system of linear equations. By representing the digits using variables, forming equations from the given conditions, and solving both possible cases for the digit difference, we obtain the required numbers. Such questions are frequently asked in CBSE Class 10 Board Exams and help strengthen algebraic thinking and equation-solving skills.