Question : By interchanging the digits of a two-digit number, we get a number which is four times the original number minus 24. If the digit at the unit's place of the original number exceeds its digit at ten's place by 7, then the original number is:
Option 1: 29
Option 2: 36
Option 3: 58
Option 4: 18
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Correct Answer: 29
Solution :
Let the digit in the ten's place and the digit in the unit's place be x and y respectively.
Then the original number = 10x + y and the new number = 10y + x.
According to the question,
10y + x = 4(10x + y) – 24
or, 10y + x = 40x + 4y– 24
or, 39x – 6y = 24
or, 13x – 2y = 8 -------------------------------(1)
Also, given that y – x = 7
so, y = x + 7 -----------------------------------(2)
Now, putting the value of y in (1)
13x – 2(x + 7) = 8
or, 13x – 2x – 14 = 8
or, 11x = 22
or, x = 2
Now, from (2) we get,
y = 2 + 7
or, y = 9
Putting the value of x and y in the original number, we get,
10x + y = (10 × 2) + 9 = 29
Hence, the correct answer is 29.
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