Question
The sum of the digits of a two-digit number is 9. If 27 is subtracted from the number, its digits are reversed. Find the number.
Solution
Step 1: Define the variables
Let the two-digit number be 10x+y10x + y, where:
- xx is the digit in the tens place.
- yy is the digit in the units place.
Step 2: Write the given conditions as equations
- The sum of the digits is 9: x+y=9.(1)x + y = 9 \tag{1}.
- If 27 is subtracted from the number, the digits are reversed: 10x+y−27=10y+x⇒9x−9y=27⇒x−y=3.(2)10x + y – 27 = 10y + x \quad \Rightarrow \quad 9x – 9y = 27 \quad \Rightarrow \quad x – y = 3 \tag{2}.
Step 3: Solve the equations
From equation (2): x−y=3⇒x=y+3.(3)x – y = 3 \quad \Rightarrow \quad x = y + 3 \tag{3}.
Substitute x=y+3x = y + 3 into equation (1): (y+3)+y=9.(y + 3) + y = 9. 2y+3=9⇒2y=6⇒y=3.2y + 3 = 9 \quad \Rightarrow \quad 2y = 6 \quad \Rightarrow \quad y = 3.
Substitute y=3y = 3 into equation (3): x=y+3=3+3=6.x = y + 3 = 3 + 3 = 6.
Step 4: Write the solution
The number is: 10x+y=10(6)+3=63.10x + y = 10(6) + 3 = 63.
Verification
- The sum of the digits is: 6+3=96 + 3 = 9 (correct).
- Subtracting 27 from 63 gives 63−27=3663 – 27 = 36, which is the reverse of 63 (correct).
This problem ensures the following are covered:
- Algebraic representation of word problems.
- Logical derivation and substitution.
- Verification of the result.
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