Digits and algebra
A number has two digits. When the digits are reversed, the new number is 27 more than the original number. The sum of the digits is 11.
What is the original number?
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Let the tens digit be x, and the units digit be y.
ReplyDeleteThen:
The original number = 10x + y
The reversed number = 10y + x
From the question:
1) Reversed number is 27 more than the original:
10y + x = 10x + y + 27
2) Sum of digits is 11:
x + y = 11
Step 1: Rearranging the first equation
10y + x = 10x + y + 27
→ 10y − y + x − 10x = 27
→ 9y − 9x = 27
→ y − x = 3 ← (Equation A)
From before:
x + y = 11 ← (Equation B)
Step 2: Solve the equations
From Equation A:
y = x + 3
Substitute into Equation B:
x + (x + 3) = 11
→ 2x + 3 = 11
→ 2x = 8
→ x = 4
Now find y:
y = x + 3 = 4 + 3 = 7
Final Answer:
Original number = 10x + y = 10×4 + 7 = 47