The sum of the digits of a two-digit number is 10. If 18 is subtracted from it, the digits in the resulting number will be equal. The number is:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 73
Let the number have tens digit \(a\) and units digit \(b\), so it is \(10a+b\), with \(a+b=10\). Taking away 18 means taking away 1 ten and 8 units, and the units digit decides whether a ten has to be borrowed.
The digits of the number minus 18:
If \(b\) were 8 or 9 (no borrowing):
The result would be \(10(a-1)+(b-8)\)
Equal digits need \(a-1=b-8\), that is, \(a=b-7\)
With \(a+b=10\), this gives \(b=8.5\), not a digit
So \(b\) is less than 8, and 1 ten is borrowed: the result is \(10(a-2)+(b+2)\)
Equal digits: \(a-2=b+2\), so \(a=b+4\) (1)
Sum of digits: \(a+b=10\) (2)
Solving (putting (1) into (2)):
\(2b+4=10\), so \(b=3\) and \(a=7\)
The number is 73, and \(73-18=55\)
Shortcut: Subtract 18 from each option and look for equal digits.
\(75-18=57\), \(73-18=55\), \(65-18=47\), \(64-18=46\)
Only 73 gives equal digits, and its digits add up to \(7+3=10\)