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Question
The sum of the digits in a two-digits number is 9. The number obtained by interchanging the digits exceeds the original number by 27. Find the two-digit number.
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Solution
Step 1: Write the conditions as equations
- The sum of the digits is 9: x + y = 9.
- The number obtained by interchanging the digits exceeds the original number by 27: 10y + x = 10x + y + 27.
Step 2: Simplify the second equation
10y + x = 10x + y + 27.
10y − y = 10x − x + 27,
9y = 9x + 27.
y = x + 3
Step 3: Solve the system of equations
- x + y = 9,
- y = x + 3
Substitute y = x + 3 into x + y = 9:
x + (x + 3) = 9
2x + 3 = 9
2x = 6 ⇒ x = 3
Substitute x = 3 into y = x + 3:
y = 3 + 3 = 6
10x + y = 10(3) + 6 = 36.
The two-digit number is 36.
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Activity: Let the digit in the unit’s place be y and the digit in the ten’s place be x.
∴ The number = 10x + y
∴ The number obtained by interchanging the digits = `square`
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∴ 10x + y + 10y + x = `square`
∴ x + y = `square` ...(i)
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Digit in the ten’s place = digit in the unit’s place + 2
∴ x – y = 2 ...(ii)
Solving equations (i) and (ii)
∴ x = `square`, y = `square`
Ans: The original number = `square`
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